\begin{equation}\label{CUSTRJMSONGJMSD-287729ou35t4jrg24qwerty44} \frac{ \kappa }\rho\Delta u=g,\end{equation}

which is a form of the so-called Poisson's equation (compare with (<ref>)).

In the absence of gravity, (<ref>) reduces to
the so-called Laplace's equation (compare with (<ref>))

\begin{equation*}
\Delta u(x)=0.\end{equation*}

This gives that (at least in the linear approximation) elastic membranes constrained at their boundaries
are described by the graph of harmonic functions.
Once again, this fact brings us a useful hint, suggesting a smoothing effect
for harmonic functions, somewhat inherited from the fact that elastic membranes
do not develop spikes (the elasticity would indeed reduce the peak
to find a balance): this intuition will be mathematically formalized, developed
and consolidated in Chapter <ref>.

§.§ Elasticity theory and torsion of bars

A topical argument in material sciences deals with the determination of the shape of objects
subject to external forces and to the understanding of the relations between the forces
applied and the resulting deformations. Providing a comprehensive account of this elasticity theory goes well beyond
the scopes of these notes (see e.g. <cit.> and the references therein for further readings):
here we just introduce, in a rather simplified form, some basic concepts which will lead us to the study
of an interesting[Actually, the final equation that we will obtain in (<ref>)
is precisely the same as the one arising from fluid dynamics in (<ref>). This coincidence shows
once again the extraordinary unifying power of mathematics.]
elliptic equation (which actually will be studied in detail in Sections <ref> and <ref>).
For concreteness, we focus on the three-dimensional case and we consider an infinitesimal portion of the material under consideration,
modeled as a very tiny cubes with faces oriented along the coordinate axes, see Figure <ref>.
If the cube has the form $(0,\delta)^3$, for some small $\delta>0$, we denote by $S_1$, $S_2$ and $S_3$
three facets of the cubes in direction $e_1$, $e_2$ and $e_3$ respectively, i.e.

$$ S_1:=\{\delta\}\times(0,\delta)\times(0,\delta),\qquad

For each $i$, $j\in\{1,2,3\}$ we denote by $\sigma_{ij}$ the force per unit of area applied in direction $e_i$ to the surface $S_j$.
In jargon, the matrix $\{\sigma_{ij}\}_{i,j\in\{1,2,3\}}$ is sometimes called “stress tensor”stress tensor.

The stress tensor.

By Newton's Third Law, at equilibrium the forces acting on the above ideal infinitesimal cube
must balance, therefore the forces acting on the facets of the cubes opposite to $S_1$ (or $S_2$, or $S_3$) are the ones discussed above, but with a sign changed.
In particular, given $i\in\{1,2,3\}$, at equilibrium the total forces per unit of area applied in direction $e_i$ must balance out
and therefore (noticing that if $x\in S_j$ then $x-\delta e_j$ belongs to the facet opposite to $S_j$)

$$ 0=\sum_{j=1}^3\Big(\sigma_{ij}(x)-\sigma_{ij}(x-\delta e_j)\Big)=
\delta \sum_{j=1}^3\partial_j\sigma_{ij}(x)+o(\delta),$$

leading to the following balance prescription for the stress tensor:

\begin{equation}\label{356um1ijkialivj}
\sum_{j=1}^3\partial_j\sigma_{ij}=0\qquad{\mbox{for all }}\;i\in\{1,2,3\}.
\end{equation}

Additionally, the angular momentum contribution $\vartheta_1$ coming from $S_1$ is proportional to the vector product
between the vectorial force per unit of area on $S_1$, corresponding to $\sigma_{11}e_1+\sigma_{21}e_2+\sigma_{31}e_3$,
and the vector joining normally the center of the cube to $S_1$, namely $e_1$. Hence, up to normalizing constants,

$$ \vartheta_1=\det\left(\begin{matrix}
e_1 & e_2 & e_3\\
\sigma_{11} & \sigma_{21} & \sigma_{31}\\
1 & 0 &0
\end{matrix}\right)=\sigma_{31}e_2-\sigma_{21}e_3.

the angular momentum contribution $\vartheta_2$ coming from $S_2$ is proportional to the vector product
between the vectorial force per unit of area on $S_2$, corresponding to $\sigma_{12}e_1+\sigma_{22}e_2+\sigma_{32}e_3$,
and the vector joining normally the center of the cube to $S_2$, namely $e_2$, whence

$$ \vartheta_2=\det\left(\begin{matrix}
e_1 & e_2 & e_3\\
\sigma_{12} & \sigma_{22} & \sigma_{32}\\
0 & 1 &0
\end{matrix}\right)=-\sigma_{32}e_1+\sigma_{12}e_3.

And also the angular momentum contribution $\vartheta_3$ coming from $S_3$ is proportional to the vector product
between the vectorial force per unit of area on $S_3$, corresponding to $\sigma_{13}e_1+\sigma_{23}e_2+\sigma_{33}e_3$,
and the vector joining normally the center of the cube to $S_2$, namely $e_3$, whence

$$ \vartheta_3=\det\left(\begin{matrix}
e_1 & e_2 & e_3\\
\sigma_{13} & \sigma_{23} & \sigma_{33}\\
0 & 0 &1
\end{matrix}\right)=\sigma_{23}e_1-\sigma_{13}e_2.

On this account, the
conservation of the total angular momentum entails that, at equilibrium,

$$ 0=\vartheta_1+\vartheta_2+\vartheta_3=(\sigma_{23}-\sigma_{32})e_1

This establishes that the stress tensor is symmetric, namely

\begin{equation}\label{SYMME}
\sigma_{ij}=\sigma_{ji}\qquad{\mbox{for all }}\,i,j\in\{1,2,3\}.
\end{equation}

Now, so as to model the distortion of the material caused by these forces, we consider a displacement vector $v$
describing at any point the change in the configuration of the given body.
The displacement vector itself
may not be the most interesting object to take into account in an elasticity theory, since
translations and rotations do produce significant displacements without affecting the shape of the body under consideration.
It is therefore common to try to detect the deformation effects that displacements produce
by accounting for relative displacements. Namely, if a point $x$ is moved to $S(x):=x+v(x)$, given $\omega\in\partial B_1$,
the point $y:=x+\delta\omega$ is moved to ${\mathcal{P}}(y)=y+v(y)=x+\delta\omega+v(x+\delta\omega)$. To compute the corresponding
relative displacement, one checks how $y-x$ is affected by this transformation, namely we point out that

\begin{eqnarray*}
{\mathcal{P}}(y)-{\mathcal{P}}(x)={\mathcal{P}}(x+\delta \omega)-{\mathcal{P}}(x)=\delta\,D{\mathcal{P}}(x)\omega+O(\delta^2)=D{\mathcal{P}}(x)(y-x)+O(\delta^2).\end{eqnarray*}

Namely, the “deformation gradient” $D{\mathcal{P}}$ is a linear approximation measuring relative displacements.

If we want to measure the change of square length, we have that

\begin{equation}\label{UJSvherEDIMAPOMKE}\begin{split}& |{\mathcal{P}}(y)-{\mathcal{P}}(x)|^2-|y-x|^2=|D{\mathcal{P}}(x)(y-x)+O(\delta^2)|^2-|y-x|^2\\&\qquad=
\big(D{\mathcal{P}}(x)(y-x)+O(\delta^2)\big)\cdot\big(D{\mathcal{P}}(x)(y-x)+O(\delta^2)\big)-|y-x|^2\\&\qquad=
\big(D{\mathcal{P}}(x)(y-x)\big)\cdot\big(D{\mathcal{P}}(x)(y-x)\big)-|y-x|^2+O(\delta^3)\\&\qquad=
\big(D{\mathcal{P}}(x)\big)^\top D{\mathcal{P}}(x)(y-x)\cdot(y-x)-|y-x|^2+O(\delta^3)\\&\qquad=\Big(
\big(D{\mathcal{P}}(x)\big)^\top D{\mathcal{P}}(x)-{\rm Id}\Big)(y-x)\cdot(y-x)+O(\delta^3),
\end{split}\end{equation}

where ${\rm Id}$ is the identity matrix and the superscript “$\top$” denotes matrix transposition.

It is also useful to observe that

\begin{eqnarray*}&&
D{\mathcal{P}}^\top D{\mathcal{P}}-{\rm Id}=\big({\rm Id}+Dv\big)^\top\big({\rm Id}+Dv)-{\rm Id}=
Dv+Dv^\top+Dv^\top Dv.
\end{eqnarray*}

In particular, if one assumes that the displacements are small, the quadratic term $Dv^\top Dv$
is negligible with respect to the “symmetric gradient” $Dv+Dv^\top$ which is of linear type.
Therefore, it is customary to consider the approximation $
D{\mathcal{P}}^\top D{\mathcal{P}}-{\rm Id}\simeq Dv+Dv^\top$ and rewrite (<ref>) as

\begin{equation*}
|{\mathcal{P}}(y)-{\mathcal{P}}(x)|^2-|y-x|^2\simeq \Big(Dv(x)+\big(Dv(x)\big)^\top\Big)
\end{equation*}

or equivalently

\begin{equation}\label{STRTENDE}
|{\mathcal{P}}(y)-{\mathcal{P}}(x)|^2-|y-x|^2\simeq \sum_{i,j=1}^3 \e_{ij}(x)
\end{equation}

\begin{equation}\label{STRTENDE-2}
\e_{ij}(x):=\Big(Dv(x)+\big(Dv(x)\big)^\top\Big)_{ij}
=\partial_i v_j(x)+\partial_j v_i(x).
\end{equation}

In jargon, $\e_{ij}$ is called the “strain tensor”strain tensor (and notice the symmetry feature that $\e_{ij}=\e_{ji}$)
and (<ref>) expresses in a quantitative way (though after some simplifying approximation)
the relative change in the position of points under the deformation.

We now relate strain and stress.
To this end, once again,
the simplest possible ansatz, leading to a linear theory of elasticity, is now that
deformations (or, more precisely, strains, to avoid translations and rotations) are directly proportional
to the forces excerpt (this is indeed a possible rephrasing of Hooke's Law for linear elastic materials).
That is, neglecting proportionality constants, we will suppose that

\begin{equation}\label{STRESTAP}
\e_{ij}=\sigma_{ij}.\end{equation}

It is interesting to remark that this identity is also algebraically compatible with the fact that
both the tensors above are symmetric.

We now apply the general framework of the linear theory of elasticity showcased so far
to the special situations of straight bars subject to torsion. This will be indeed one of the founding motivations
in Sections <ref> and <ref>. To model a straight bar, we consider
a body of the form

\begin{equation}\label{HANXTHECRPR4RO4O}
\Omega\times[0,+\infty),\end{equation}
with $\Omega\subset\R^{2}$
smooth and contractible, see Figure <ref>.

A bar subject to torsion.

We assume that such a bar is constrained at the level $\{x_3=0\}$ but it gets “twisted” from the top.
The effect of the twist would be to make the cross section rotate by some angle, depending on the level that we take into account.
Assuming again a linear behavior, we suppose that the rotation performed at level $\{x_3=\zeta\}$
is proportional to $\zeta$ (i.e., the angle of rotation increases linearly with the height,
which is not so unreasonable, at least for small heights, since the bar is constrained at ground zero).
Accordingly, we model the twist at level $\{x_3=\zeta\}$ by a rotation of angle $\Theta_\zeta= c_0\zeta$, for some $c_0>0$.
The corresponding displacement sends therefore the point $x=(x_1,x_2,\zeta)$
to a point $z:=x+v(x)$, with the horizontal component of $v$ being prescribed by the above rotation by the angle $\Theta_\zeta$,
namely, for small heights $\zeta$,

\begin{eqnarray*} \left( \begin{matrix} z_1\\z_2\end{matrix}\right)&=&
\left( \begin{matrix} \cos\Theta_\zeta&\sin\Theta_\zeta\\-\sin\Theta_\zeta&\cos\Theta_\zeta\end{matrix}\right)
\left( \begin{matrix} x_1\\x_2\end{matrix}\right)\\&=&
\left( \begin{matrix} \cos(c_0\zeta)&\sin(c_0\zeta)\\-\sin(c_0\zeta)&\cos(c_0\zeta)\end{matrix}\right)
\left( \begin{matrix} x_1\\x_2\end{matrix}\right)\\&=&
\left( \begin{matrix} 1+O(\zeta^2)& c_0\zeta+O(\zeta^2)\\-c_0\zeta+O(\zeta^2)&1+O(\zeta^2)\end{matrix}\right)
\left( \begin{matrix} x_1\\x_2\end{matrix}\right).
\end{eqnarray*}

This gives that

\begin{equation*}
\left( \begin{matrix} v_1(x)\\v_2(x)\end{matrix}\right)=
\left( \begin{matrix} z_1\\z_2\end{matrix}\right)-
\left( \begin{matrix} x_1\\x_2\end{matrix}\right)
\left( \begin{matrix} O(\zeta^2)& c_0\zeta+O(\zeta^2)\\-c_0\zeta+O(\zeta^2)&O(\zeta^2)\end{matrix}\right)
\left( \begin{matrix} x_1\\x_2\end{matrix}\right)=
\left( \begin{matrix} c_0\zeta x_2\\-c_0\zeta x_1\end{matrix}\right)+O(\zeta^2).
\end{equation*}

Hence, by sloppily[This just because here we are playing around with motivations.
From next chapter on, this kind of sloppiness will no longer be tolerated!] neglecting the higher order terms, and using the fact that $x=(x_1,x_2,x_3)\in\{x_3=\zeta\}$,
we will simply write that

\begin{equation}\label{STRESTAP-3}
v_1(x)= c_0 x_2 x_3\qquad{\mbox{and}}\qquad v_2(x)=-c_0 x_1 x_3.
\end{equation}

As for the vertical component of the displacement, a common ansatz is to consider it independent of the height:
roughly speaking, the torsion of the bar may exhibit a distortion of the cross section out of its own plane
(known in jargon as “warping deformation”warping deformation), but usually this effect is less visible than other types of deformations
and we are assuming here that each layer of the bar (i.e., each horizontal cross section) “equally pushes
the next one”, regardless of its height (once again, this is not a completely unrealistic assumption
especially if we are restricting our attention to the layers close to ground zero). To model that this warping effect
does not depend on the height, we assume that the vertical component of the displacement vector is independent of $x_3$, that is

\begin{equation}\label{STRESTAP-4} v_3(x)=w(x_1,x_2),\end{equation}

for some function $w$.

Now, from (<ref>), (<ref>), (<ref>) and (<ref>) we arrive at

$$ \sigma_{13}=\e_{13}=\partial_1 v_3+\partial_3 v_1 =\partial_1 w+c_0x_2.$$

$$ \sigma_{23}=\e_{23}=\partial_2 v_3+\partial_3 v_2 =\partial_2 w-c_0x_1$$

$$ \sigma_{33}=\e_{33}=2\partial_3 v_3=0.$$

Interestingly, the above expressions for $\sigma_{i3}$ with $i\in\{1,2,3\}$ are all independent of the height $x_3$
(which was possibly not obvious from the beginning). Furthermore, recalling the
balance prescription in (<ref>) and the symmetry property in (<ref>),

\begin{eqnarray*}&& 0=\sum_{j=1}^3\partial_j\sigma_{3j}=\sum_{j=1}^3\partial_j\sigma_{j3}=\partial_1\sigma_{13}+\partial_2\sigma_{23}.
\end{eqnarray*}

That is, the differential form $\sigma_{23}\,dx_1-\sigma_{13}\,dx_2$ is exact, and therefore closed, thanks to the
Poincaré Lemma (see e.g. <cit.>). This yields that there exists a function $\phi=\phi(x_1,x_2)$,
sometimes refereed to with the name of “warping potential”warping potential, such that $d\phi=\sigma_{23}\,dx_1-\sigma_{13}\,dx_2$.

This gives that $\partial_1\phi=\sigma_{23}$ and $\partial_2\phi=-\sigma_{13}$. Consequently,

\begin{equation}\begin{split}\label{TRUNPDOmrfgEPPJO}&
\Delta\phi=\partial_1\sigma_{23}-\partial_2\sigma_{13}=
\partial_1(\partial_2 w-c_0x_1)-\partial_2(\partial_1 w+c_0x_2)\\&\qquad\qquad\qquad=
\partial_{12} w-c_0-\partial_{12} w-c_0=-2c_0.\end{split}
\end{equation}

Another notion of interest for engineering purposes is the “traction”traction ${\mathcal{T}}$ along the boundary of the bar,
which is defined as the normal component of the vertical stress. More explicitly, if we define $\sigma_3:=(\sigma_{31},\sigma_{32},\sigma_{33})$ and we consider the outer unit normal $\nu^\star$ of the bar, we set

\begin{equation}\label{CA2LT} {\mathcal{T}}:=\sigma_3\cdot\nu^\star.\end{equation}

As a matter of fact, since the straight bar has the form given in (<ref>),
we can identify $\nu^\star\in\R^3$ with the vector $(\nu,0)$, being $\nu\in\R^2$ the
outer unit normal at $\partial\Omega$ and consequently

$$ {\mathcal{T}}=\sigma_{31} \nu_1+\sigma_{32}\nu_2=\sigma_{13} \nu_1+\sigma_{23}\nu_2=

where $\tau:=(-\nu_1,\nu_2)$ can be taken as the unit tangent vector along $\partial\Omega$ (say, clockwise oriented).
In this spirit, we find that the traction is the tangential derivative of the warping potential.

In the special situation in which the bar is subject to no traction, it follows that $\nabla\phi\cdot\tau=0$ on $\partial\Omega$. This gives that $\phi$ is constant along $\partial\Omega$, because if $\partial\Omega$ is locally parameterized by a curve $\gamma:(-1,1)\to\R^2$ with $\dot\gamma$ proportional to $\tau$, we have that

$$ \frac{d}{dt}\phi(\gamma(t))=\nabla\phi(\gamma(t))\cdot\dot\gamma(t)=0.$$

Thus, if the traction of the bar vanishes, we can write that

\begin{equation}\label{TRUNPDOmrfgEPPJO2}
{\mbox{$\phi=c_2$ on~$\partial\Omega$,}}\end{equation}

for some $c_2\in\R$.

Recalling (<ref>), we call the magnitude of the traction on the surface of the bar $\partial\Omega$ the scalar $|\sigma_3|$.
We remark that

$$ |\nabla\phi|=\sqrt{ (\partial_1\phi)^2+(\partial_2\phi)^2}=\sqrt{\sigma_{23} ^2+\sigma_{13}^2}=
\sqrt{\sigma_{23} ^2+\sigma_{13}^2+\sigma_{33}^2}=|\sigma_3|$$

therefore the magnitude of the traction coincides also with the norm of the gradient of the warping potential.

Thus, a consequence of (<ref>) is that $|\partial_\nu\phi|=|\nabla\phi|$,
hence in this situation the magnitude of the traction coincides also (possibly up to a sign) with the normal
derivative of the warping potential along the surface of the bar.

As a consequence, in the special situation in which the bar is subject to no traction and
the magnitude of the traction is constant, we have that

\begin{equation}\label{TRUNPDOmrfgEPPJO2b}
{\mbox{$\partial_\nu\phi=c_3$ on~$\partial\Omega$,}}\end{equation}

for some $c_3\in\R$.

Thus, if we define

$$ u:=\frac{c_2-\phi}{2c_0},$$

we deduce from (<ref>),
(<ref>) and (<ref>)
that for a straight bar
subject to torsion and no traction, if the magnitude of the traction along the surface of the bar is constant then there exists a solution of

\begin{equation}\label{EUMSD-OS-32456i7DNSSE234R-BNMBAR}
\begin{dcases}
\Delta u=1&{\mbox{ in }}\Omega,\\
u=0&{\mbox{ on }}\partial\Omega,\\
\partial_\nu u=c&{\mbox{ on }}\partial\Omega.
\end{dcases}\end{equation}

Notice the perfect coincidence[As noticed in footnote <ref>
on page OJSLLAFOr5O3OSMEr3r,
equation (<ref>) possesses a solution when $\Omega$
is a disk, that is when the bar has a circular cross section.
We will discuss in Sections <ref> and <ref>
whether bars of other shapes maintain the same property of presenting no traction and constant
traction magnitude along their surface.]
of this set of prescriptions with that in (<ref>),
which also reveals a possibly unexpected connection between the dynamics of viscous fluids
and the elastic reactions of bars subject to torsion.

Bending steel beams with bare hands by the power of $\e$.

§.§ Bending beams and plates

Now we deal with deflection of beams, which is a classical topic in all superhero comics, see Figure <ref>.
Besides, the question has obvious applications in engineering, see e.g. Figure <ref>.
The beam equation beam equation

\begin{equation}\label{BARREbi96}
\end{equation}

describes[The linear model of elasticity accounting for the load-carrying and deflection of beams
is often called “Euler-Bernoulli beam theory”.
No confusion should arise between (<ref>) and the model presented
in Section <ref> where instead the torsion theory
of bars, and not the deformation under a load, was taken into account.]
the small deflection $u$ of a homogeneous beam at equilibrium in terms of the applied load $q$.

In (<ref>), $u$ and $q$ are functions of one variable $x\in\R$ (the beam is assumed to be infinitely long).
In a nutshell, the derivation of (<ref>) from prime principles relies on balance of forces and moments.
Roughly speaking, the bending of the beam produces a compression force on one side
of the beam and a tensile stress on the other side: these deformation
stresses will be modeled via an elastic Hooke's Law and produce
turning forces which, at equilibrium, together with the corresponding moments,
need to be balanced with the external load on the beam.

Narrows Bridge, Perth (image by Speddie23 from
licensed under the Creative Commons Attribution-Share Alike 4.0 International license).

The details go as follows. We assume that the beam is displayed along the horizontal axis and slightly deformed
into a graph of the form $z=u(x)$ (the beam may well be three-dimensional, in which case we assume
that its
transversal sections possess some given shape that remains essentially invariant under small bending, and the three-dimensional
coordinates are denoted, as usual, by $(x,y,z)$).

More precisely, the graph $z=u(x)$ describes the “neutral fiber” of the beam,
that is a fiber which maintains infinitesimally the original length that it possessed at rest, see Figure <ref>.
The transversal sections of the beam that were perpendicular to the neutral fiber before the beam deforms are assumed to remain perpendicular to the neutral fiber after the bending.
We stress that this neutral fiber may not be located at the mid-height of the beam. In any case, we assume that on one side
of this neutral fiber compression takes place and on the other side tension occurs
(specifically, in Figure <ref>, the upper fibers are in compression and the lower fibers are under tension).

To efficiently describe the elastic forces produced by this interplay of compression and tension, it is profitable to consider
the osculating circle at a given point of the neutral fiber, see Figure <ref>.
Considering an infinitesimal quantity $\e>0$ and an infinitesimal element of the beam between $x$ and $x+\e$,
we thereby replace the graph of $u$ with a small portion of the osculating circle of the graph of $u$ tangent to $(x,u(x))$.
We denote by $\varrho(x)$ the radius of this osculating circle and by $\vartheta$ the corresponding polar angle infinitesimally
joining $(x,u(x))$ to $(x+\e,u(x+\e))$.

A deflected beam in the $(x,z)$-plane with an osculating circle.

We consider an upper fiber located at distance $z$ above the neutral fiber and we
try to quantify the force produced by compression.
To this end, we consider the ratio $\upsilon(z)$
between the variation of infinitesimal
length of the fiber and its original length before bending
and we suppose that the corresponding compression force is linear with respect to $\upsilon(z)$
(linearity with respect to length variations is indeed the main ingredient of
elasticity according to Hooke's Law, and note that the bigger the distance $z$ from the neutral fiber the bigger
the compression). We also observe that the infinitesimal length of this fiber after bending (in the osculating circle approximation)
is given by $(\varrho(x)-z)\vartheta$ while its original length before bending was equal to that of the neutral fiber,
which is $\varrho(x)\vartheta$.

and accordingly the compression force related to an upper fiber at distance $z$
from the neutral fiber is proportional to $\frac{z}{\varrho(x)}$.

Similarly, the lower fiber located at distance $z$ below the neutral
one would produce by tension a force which is
proportional to $\frac{z}{\varrho(x)}$
(notice that the forces produced by tension are opposed to the ones produced by compression).
Thus, if we suppose that the elastic modulus of the material is the same for compression and tension
(which is another classical ansatz in the elasticity theory according to Hooke's Law) and we use the convention that a positive $z$
corresponds to the upper fibers and a negative $z$
corresponds to the lower fibers, the resulting elastic force $F$ is approximately oriented along the tangential axis and equals to $\frac{Ez}{\varrho(x)}$, where $E>0$ denotes the elastic modulus of the material (that we suppose to be constant).

We stress that the fact that the force $F$ changes sign with $z$ (accordingly, that compression and tension produce forces
in opposite directions above and below the neutral fiber) produces a total torque $M(x)$.
Indeed, to compute this torque at the point $(x,u(x))$, for each $z$ we take into consideration
the vector product between the position vector and the corresponding force, namely
if $(y,z)$ belongs to the cross section $A$ of the beam corresponding to $(x,u(x))$,
the magnitude of the torque at $(y,z)$ is approximately given by $zF=\frac{Ez^2}{\varrho(x)}$.

Loads and moments of a beam.

The magnitude of the total torque $M(x)$ is thereby the corresponding integral for $(y,z)\in A$, that is

\begin{equation}\label{uJNSEMM0olSdfg2} M(x)=\iint_A \frac{Ez^2}{\varrho(x)}\,dy\,dz.\end{equation}

It is also customary[Of course, the axial second moment of area depends on the shape of the cross section of the beam.
For instance, for a circular cross section we have that $A=\{(y,z)\in\R^2$ s.t. $y^2+z^2<r^2\}$, for some $r>0$, and therefore
in this case

$$I=\iint_{\{ y^2+z^2<r^2\}} z^2\,dy\,dz=2\int_{-r}^{r}z^2\sqrt{r^2-z^2} \,dz=\frac{\pi r^4}{4}.

Instead, if the cross section is a rectangle $(-a,a)\times(-b,b)$ for some $a$, $b>0$,

$$I=\iint_{(-a,a)\times(-b,b) } z^2\,dy\,dz=\frac{4ab^3}{3}.
to define the axial second moment of area axial second moment of area as

$$ I:=\iint_A z^2\,dy\,dz.$$

With this notation, we can write (<ref>) as

\begin{equation}\label{uJNSEMM0olSdfg223} M(x)=\frac{EI}{\varrho(x)}.\end{equation}

We also recall that the osculating radius $\varrho(x)$ can be written as $\frac{(1+(u'(x))^2)^{3/2}}{u''(x)}$,
see e.g. <cit.>.
Hence, for small deformations, the osculating radius $\varrho(x)$ can be approximated by $\frac{1}{u''(x)}$
and then, with this approximation, equation (<ref>) reduces to

\begin{equation}\label{uJNSEMM0olSdfg224} M(x)= EI\,u''(x).\end{equation}

To carry on with the derivation of (<ref>),
we now take into consideration the balance of moments, see Figure <ref>.
For this, we assume that the beam is subject to a vertical distributed load of magnitude $q=q(x)$
and we compute its torque at the reference point corresponding to $x+\e$. For this,
given $\ell\in(0,\e)$, we calculate the magnitude of the vector product between the distributed load at the point $x+\ell$,
which is $(0,-q(x+\ell))$ and the vector joining the reference point to the point corresponding to $x+\ell$, which is $\big(x+\ell,u(x+\ell)\big)-\big(x+\e,u(x+\e)\big)=\big( \ell-\e, u(x+\ell)-u(x+\e)\big)$, thus finding the quantity $ q(x+\ell)(\ell-\e)$.
Correspondingly, the magnitude of the total torque produced by the external load on the beam between $x$ and $x+\e$ corresponds to

\begin{equation}\label{BAMO-MAHNdfo-21} \int_0^\e q(x+\ell)(\ell-\e)\,d\ell=
\int_0^\e \big(q(x)+O(\e)\big)(\ell-\e)\,d\ell=

\begin{equation}\label{BAMO-MAHNdfo-22} M(x+\e)-M(x)=\e \partial_xM (x)+\frac{\e^2}2 \partial_x^2 M(x)+O(\e^3).\end{equation}

The balance of moments at the second order in $\e$ between (<ref>) and (<ref>)
leads[The first order in $\e$ in (<ref>) reveals the existence of a shear force
acting on the faces of the beam: namely, the first order in $\e$ moments balance would produce that this shear force equates
the term $\partial_xM$ (possibly up to a sign convention).]

$$ \partial_x^2 M(x)= q(x).$$

By combining this with (<ref>) we obtain

$$ q(x)=\partial_x^2( EI\,u''(x))=EI\,u''''(x),$$

which corresponds to the beam equation in (<ref>) (up to normalizing constants).
See e.g. <cit.> for further details on the beam equation and on related topics.

Augustus Edward Hough Love (Public Domain image from

It is interesting to generalize the model discussed so far to comprise the case of[The mathematical model commonly used to describe stresses and deformations in thin plates
is sometimes called “Kirchhoff-Love theory”,
after Gustav Robert Kirchhoff and Augustus Edward Hough Love.
We have already met Kirchhoff on page SIMKIRAoGREENFIDItangeFI.
Besides his work in elasticity theory, Love dedicated himself to the study of
the spin-orbit locking, that is the phenomenon occurring when an astronomical body
always has the same face toward the object it is orbiting
(for example, up to some minor variability,
the same side of the Moon always faces the Earth).
See Figure <ref> for a wood engraved print representing Love taken from the British Newspaper The Graphic.]
bending plates. For instance, one could model a thin plate as a graph $x_{n+1}=u(x_1,\dots,x_{n})$, with $x=(x_1,\dots,x_n)\in\R^n$
(of course, bearing in mind that concretely $n=2$ and the corresponding plate is two-dimensional).
In this setting, the beam equation in (<ref>) leads to the plate equation plate equation

\begin{equation}\label{BARREbi96-PLA}
\Delta^2 u=q,
\end{equation}

where $\Delta^2$ is the Laplace operator applied twice.

To deduce the plate equation in (<ref>) from the beam equation in (<ref>) one can argue as follows.
Let us suppose that the vertical bending force of a horizontal plate is described, for small deformations, by a linear differential operator ${\mathcal{L}}$ of the form

\begin{equation}\label{BARREbi96-PLA-DAD2} {\mathcal{L}}:=\sum_{{\alpha\in\N^n}\atop{|\alpha|\le m}}c_\alpha\partial^\alpha,\end{equation}

for some $m\in\N$ and $c_\alpha\in\R$. Under this assumption, the force balance
would lead to the equation ${\mathcal{L}}u=q$, being $q$ the external load.

Then, to check (<ref>), one needs to show that

\begin{equation}\label{BARREbi96-PLA-DAD}
{\mathcal{L}}=\Delta^2 .
\end{equation}

For this, we assume that, consistently with the model of beam deformation in (<ref>),
the operator ${\mathcal{L}}$ acts as a fourth derivative on any one-dimensional function.
Namely, let us suppose that if $u_\omega(x):=u_0(\omega\cdot x)$ for some $u_0:\R\to\R$ and $\omega\in\partial B_1$

\begin{equation}\label{BARREbi96-PLA-DAD3} {\mathcal{L}}u_\omega(x)=u_0''''(\omega\cdot x).\end{equation}

We observe that, for all $j\in\{1,\dots,n\}$,

$$ \partial_j u_\omega(x)=\omega_j \,u_0'(\omega\cdot x)$$

and therefore, if $\alpha=(\alpha_1,\dots,\alpha_n)\in\N^n$,

$$ \partial^\alpha u_\omega(x)=\partial_1^{\alpha_1}\dots\partial_n^{\alpha_n} u_\omega(x)
=\omega_1^{\alpha_1}\dots\omega_n^{\alpha_n} \,u_0^{(|\alpha|)}(\omega\cdot x)=\omega^\alpha\,u_0^{(|\alpha|)}(\omega\cdot x).

Hence, by (<ref>) and (<ref>),

$$ u_0''''(\omega\cdot x)={\mathcal{L}}u_\omega(x)=\sum_{{\alpha\in\N^n}\atop{|\alpha|\le m}}c_\alpha\, \omega^{\alpha} \,u_0^{(|\alpha|)}(\omega\cdot x)

Since $u_0$ is arbitrary, we infer that $c_\alpha=0$ unless $|\alpha|=4$ and

$$ 1=\sum_{{\alpha\in\N^n}\atop{|\alpha|=4}}c_\alpha\, \omega^{\alpha}

Now, given $X\in\R^n\setminus\{0\}$ we let $\rho:=|X|$ and $\omega:=\frac{X}{|X|}$ and we deduce that

\begin{eqnarray*}&& \sum_{i,j=1}^n X_i^2 X_j^2=\left(\sum_{i=1}^n X_i^2\right)^2
=\big(|X|^2\big)^2=|X|^4=|X|^4\sum_{{\alpha\in\N^n}\atop{|\alpha|=4}}c_\alpha\, \omega^{\alpha}\\&&\qquad=
\sum_{{\alpha\in\N^n}\atop{|\alpha|=4}}c_\alpha\, \rho^{|\alpha|}\omega^{\alpha}=\sum_{{\alpha\in\N^n}\atop{|\alpha|=4}}c_\alpha\,\big( \rho \omega\big)^{\alpha}=\sum_{{\alpha\in\N^n}\atop{|\alpha|=4}}c_\alpha\,X^{\alpha}

Consequently, by the Identity Principle for polynomials,

$$ c_\alpha=\begin{dcases}
1 & {\mbox{ if $\alpha=2e_i+2e_j$ for some }}i,j\in\{1,\dots,n\},\\
0&{\mbox{ otherwise.}}
\end{dcases}$$

From this and (<ref>) we arrive at

$$ {\mathcal{L}}=\sum_{i,j=1}^n \partial^2_i\partial_j^2=\sum_{i=1}^n \partial^2_i\Delta=\Delta^2$$

and this establishes (<ref>), as desired.

§.§ Gravitation and electrostatics

One of the reasons for which the Laplace operator became very popular, especially in the eighteenth andgravitation
nineteenth centuries, is its pivotal role in the description of the gravity and electromagneticelectrostatics potentials.
The physical intuition of these phenomena was actually very helpful for the beautiful minds of those ages
in understanding a number of very deep concepts which paved the way to the modern theory of
partial differential equations and which are now widely used in many branches of mathematics, physics
and engineering.

We give here a couple of motivations[Also, a different but related
perspective on electrostatics and magnetism will be given in the forthcoming
Section <ref>.] relating the gravity field and harmonic functions
(similar arguments would link the electrostatic field and harmonic functions as well,
just changing the notion of mass with that of electric charge, and possibly allowing
a sign change, given the fact that unlike charges attract each other, but
like charges repel).

One observation is that the gravity force (in $\R^3$) is inversely proportional to the square of the distance,
namely a point mass located at $x$ is attracted by a Permanent Center of Gravity located at the origin
via a force $f(x)$ equal (up to dimensional constant) to $-\frac{x}{|x|^3}$, with the minus
sign to stress that this force is attractive, hence tends to reduce the distance of
the mass located in $x$ to the origin.

It is readily seen that $f$ is originated by a gravity potential $u$: more precisely, if we set $u(x):=\frac{1}{|x|}$,
we have that $\nabla u(x)=-\frac{x}{|x|^3}=f(x)$. Remarkably, the gravity potential is harmonic away[We will learn in equation (<ref>)
a more handy way to perform the calculation in (<ref>).]
the origin (here it is important to work in dimension $3$) since, if $x=(x_1,x_2,x_3)\in\R^3\setminus\{0\}$,

\begin{equation}\label{WRBSY0perIKSMaismoBBeJPREB}
\Delta u(x)=-\sum_{i=1}^3 \partial_i \frac{x_i}{|x|^3}
=-\sum_{i=1}^3 \left(\frac{1}{|x|^3}
-\frac{3 x_i^2}{|x|^5}\right)=-\frac{3}{|x|^3}
+\frac{3 |x|^2}{|x|^5}=0.
\end{equation}

These considerations can be recast in a possibly more general, and more “geometric”,
framework, also allowing a higher dimensional presentation.
For this, we first employ the fact that the
work done by the gravitational force depends only on initial and final positions, and not on the path between them: this gives that the gravitational vector field $f$ is conservative and thus can be written as the gradient of some function $u$, which is usually called the gravitational potential.

Furthermore, Gauß's Flux Law states that the flux of the gravity
field through an arbitrary closed surface enclosing no masses (or no electric) charges is necessarily zero (roughly speaking,
the field lines going into the region enclosed by the surface balance exactly the ones coming out).
Thus, from the Divergence Theorem, for every bounded region $\Omega\subset\R^n$ (say, with smooth boundary)

$$ 0=\int_{\partial\Omega} f(x)\cdot\nu(x)\,d{\mathcal{H}}_x^{n-1}=
\int_{\Omega} \div f(x)\,dx=\int_{\Omega} \div (\nabla u(x))\,dx=\int_{\Omega} \Delta u(x)\,dx.$$

Since $\Omega$ is arbitrary we thus conclude that the gravity potential (as well as the electrostatic potential) is harmonic away from the masses that are present in the environment.

The study of functions that are harmonic away from a point singularity will be
the main topic of the forthcoming Section <ref>. There we will also appreciate
how different dimensions affect the precise expression of the above potentials.

§.§ Classical electromagnetism

The theory of classical electromagnetismelectromagnetism has its roots in the so-called Maxwell's equationsMaxwell equations
that describe the behavior and mutual interaction of electric and magnetic fields.
These equations are somewhat a unified version of several specific information arising from
Gauß's Flux Law, Faraday's Law and
Ampère's Law.
In the vacuum (hence, in a region of $\R^3$ with no electric charges and no electric currents)
Maxwell's equations take the form

\begin{equation}\label{KMSJUNDNSMDEJDN} \div E =0,\qquad \curl E=-{\frac {\partial B }{\partial t}},\qquad
\div B=0
\qquad{\mbox{and}}\qquad\curl B=\frac1{c^2}{\frac {\partial E }{\partial t}}.\end{equation}

Here above, $E$ is the electric field,
$B$ is the magnetic field and $c$ is a physical constant (corresponding to the speed of light in vacuum).

We also observe that, for all smooth vector fields $v:\R^3\to\R^3$,

\begin{equation}\label{KMSJUNDNSMDEJDN2}
\nabla (\div v)-\curl(\curl v)=\Delta v.
\end{equation}

To check this classical identity in an elementary way, given $v=(v_1,v_2,v_3)$,
we write $v=v^{(1)}+v^{(2)}+v^{(3)}$,
where $v^{(1)}:=(v_1,0,0)$, $v^{(2)}:=(0,v_2,0)$ and $v^{(3)}:=(0,0,v_3)$,
and we remark that, thanks to the linear structure of (<ref>),
it suffices to check it for the vector fields $v^{(1)}$, $v^{(2)}$ and $v^{(3)}$.
We can focus our attention on $v^{(1)}$, up to reordering coordinates.
In this situation, since, for every smooth vector field $V:\R^3\to\R^3$,

$$ \curl V=\left(
{\frac {\partial V_3}{\partial x_2}}-{\frac {\partial V_2}{\partial x_3}} ,\;
{\frac {\partial V_1}{\partial x_3}}-{\frac {\partial V_3}{\partial x_1}},\;
{\frac {\partial V_2}{\partial x_1}}-{\frac {\partial V_1}{\partial x_2}}\right),$$

we have that

$$ W:=\curl v^{(1)}=\big(0 ,\partial_3 v_1, -\partial_2 v_1\big)$$

and subsequently

\begin{eqnarray*}&&
\nabla (\div v^{(1)})-\curl(\curl v^{(1)})\\&=&
\nabla(\partial_1 v_1)
-\curl W\\
&=&\left(\partial_{11} v_1,\,
\partial_{12} v_1,\,
\partial_{13} v_1
\right)
\left(
{\frac {\partial W_3}{\partial x_2}}-{\frac {\partial W_2}{\partial x_3}} ,\;-
{\frac {\partial W_3}{\partial x_1}},\;
{\frac {\partial W_2}{\partial x_1}}\right)\\
&=&\left(\partial_{11} v_1,\,
\partial_{12} v_1,\,
\partial_{13} v_1
\right)-
\left(
-\partial_{22} v_1-\partial_{33} v_1 ,\,
\partial_{12} v_1,\,\partial_{13} v_1\right)\\&=&
\left(\partial_{11} v_1+
\partial_{22} v_1+\partial_{33} v_1 ,\,0,\,0\right)\\&=&\Delta v^{(1)},
\end{eqnarray*}

thus establishing (<ref>).

Now, in the light of (<ref>) and (<ref>),

\begin{equation}\label{EFEDLFIemfdL}
\partial_{tt}E=c^2\partial_t(\curl B)=c^2\curl(\partial_t B)=-c^2\curl(\curl E)
=c^2\big(\Delta E-\nabla (\div E)\big)=c^2\Delta E
\end{equation}

and similarly

\begin{equation*}
\partial_{tt}B=-\partial_t(\curl E)=-\curl(\partial_t E)=-c^2\curl(\curl B)=
c^2\big(\Delta B-\nabla (\div B)\big)=c^2\Delta B.
\end{equation*}

These observations show that
both the electric and the magnetic fields propagate as waves do, since they satisfy
the wave equation in (<ref>).

§.§ Quantum-mechanical systems

A popular equation governing the wave function of a quantum-mechanical system
was proposed by  Erwin Schrödinger
(this equation formed the basis for the work that resulted in
Schrödinger's 1933 Nobel Prize).

In a nutshell, the Schrödinger equation reads

\begin{equation}\label{SCHREQOR}
-i\hbar\partial_t\psi=\frac{\hbar^2 \Delta \psi}{2m}-V\psi,
\end{equation}

where $i=\sqrt{-1}$, $\psi=\psi(x,t)$ is the wavefunction[Roughly speaking, a wavefunction is
a function that assigns a complex number to each point $x\in\R^3$ at each time $t\in\R$.
The magnitude squared of this function
represents the probability density of measuring the particle as being at the point $x$ at a given time $t$.]
of a particle of mass $m$ subject to a potential $V$
and $\hbar$ is a suitable positive physical constant (called the “reduced Planck constant”reduced Planck constant).

Particularly interesting solutions of (<ref>)
are the so-called standing waves, namely solutions which oscillate in time but whose amplitude profile does not change in space.
From the mathematical point of view, these solutions are of the form

$$ \psi(x,t)=u(x)\,e^{i\phi t},$$

where $u$ is a real valued function and $\phi>0$ is the time frequency of oscillation.

Since for standing waves we have that

$$ \partial_t\psi(x,t)=i\phi\,u(x)\,e^{i\phi t}
\qquad{\mbox{and}}\qquad
\Delta \psi(x,t)=\Delta u(x)\,e^{i\phi t},$$

these special solutions of (<ref>) are actually solutions[According to the classification
presented in footnote <ref>
on page CLASSIFICATIONFOOTN, equation (<ref>)
is of elliptic type. When $V=V(x)$ equation (<ref>)
is linear in $u$. More complex situations arise when $V$ also depends on the magnitude of the wavefunction,
say $V=V(x,u(x))$ or even $V=V(u(x))$, since in this situation equation (<ref>)
is not anymore linear in $u$ (though it is linear with respect to the Hessian of $u$). A study of these “semilinear” equations semilinear equation
will be started on page Pohozaev Identity.
Further comments on semilinear equations will be given in footnote <ref>
on page TRIS Pohozaev Identity.]

\begin{equation}\label{CLASSIFICATIONFOOTNSV}
\hbar \phi u
=\frac{\hbar^2 \Delta u }{2m}-V u.
\end{equation}

Going back to the initial discussion of this section,
one may wonder why and how Schrödinger introduced the equation in (<ref>) above.
This would be a rather long and complicated story and a short answer
was given in <cit.>:
“Where did we get that from? Nowhere. It’s not possible to derive it from anything you know. It came out of the mind of Schrödinger, invented in his struggle to find an understanding of the experimental observations of the real world”.
However, several quite convincing motivations of the Schrödinger equationSchrödinger equation are possible (see e.g. <cit.> and the references therein)
and we present one of them here below, based on the notion of canonical quantizationcanonical quantization.

To this end, we recall that the energy ${\mathcal{E}}$ of a photon is proportional to its temporal frequency $\phi$, according to the so-called Planck-Einstein energy-frequency relation

$$ {\mathcal{E}}=\hbar\phi.$$

Also, Einstein's relativistic energy formula reads

\begin{equation}\label{ENEJHND8iujMNSID-77} {\mathcal{E}}=\sqrt{ (|p|\, c)^2 + (mc^2)^2},\end{equation}

being $p$ the relativistic momentum, $m$ the mass at rest and $c$ the speed of light. For massless particles, such as photons, this reduces to
\begin{equation}\label{ENEJHND8iujMNSID} {\mathcal{E}}= |p|\,c,\end{equation}
from which we obtain

\begin{equation}\label{PPho} |p|=\frac{\mathcal{E}}c=\frac{\hbar\phi}{c}.\end{equation}

Now, we relate a photon to its electric field $E$ traveling through space and manifesting itself as a solution of the wave equation (recall (<ref>)). That is, we consider a direction of propagation $\varpi\in\partial B_1$, a spatial frequency $\kappa$ and a temporal frequency $\phi$ and we suppose that,
for large distances from the photon's source, the field is modeled by a simple plane wave, say

$$ E=E_0 e^{i(\kappa\varpi\cdot x-\phi t)},$$
where $E_0\in(0,+\infty)$.
More precisely, by (<ref>),

$$ -\phi^2 E_0 e^{i(\kappa\varpi\cdot x-\phi t)}=
\partial_{tt} E=c^2\Delta E=-c^2 \kappa^2 E_0 e^{i(\kappa\varpi\cdot x-\phi t)}$$

\begin{equation}\label{PPho22536-0i2rkjfmMS}
\phi=c \kappa,\end{equation}
thus relating temporal and spatial frequencies of the photon.

We thus obtain from (<ref>)

\begin{equation}\label{PPho22536}
and therefore

\begin{equation}\label{PPho2}|p|E=\hbar\kappa E=-i\hbar\nabla E\cdot\varpi.\end{equation}

Taking the direction of the momentum to coincide with the spatial direction $\varpi$ of the wave, i.e. taking $p=|p|\varpi$,
we rewrite (<ref>) in the form

\begin{equation}\label{PPho3}pE=-i\hbar\nabla E.\end{equation}

We can also reconsider (<ref>) in view of (<ref>) and (<ref>) and write that

$${\mathcal{E}}= |p|\,c=\hbar c\kappa=\hbar\phi$$
and consequently

\begin{equation}\label{ENEJHND8iujMNSID-2krwjeg}
{\mathcal{E}}E=\hbar \phi E=i\hbar\partial_t E.
\end{equation}

With this, without the aim of being exhaustive, we can recall some of the main ideas leading to canonical quantization.
This procedure, originally introduced by Paul Dirac in his 1926 PhD thesis, wishes to recast a classical theory into a quantum
one by preserving (as much as possible) its symmetries
and formal structures. Since the Hamiltonian formalism is one of the key tools to understand symmetries of classical mechanical systems, a natural idea in this framework is to try to (at least partially) preserve the Hamiltonian structure in the quantum description of nature.

To this end, a simple observation is that the Hamiltonian formalism relies on conjugated variables $q$ and $p$, with the physical meaning of position and momentum. The first goal of the canonical quantization is therefore to rephrase position and momentum in a way that is compatible with the quantum description of the observables.

In particular, in quantum mechanics,
all significant features
of a particle are contained in a certain state $\psi$.
The observables are represented by operators acting on states. For instance, the eigenvalues of an operator represent the values
of the measurements of the corresponding fundamental states of a particle (namely, the eigenfunctions, or eigenstates);
since any state is represented as a linear combination of eigenstates, the application of an operator to the state $\psi$
corresponds to the determination of a measurable parameter and the physical act of measuring corresponds to make the
values of the state “collapse” from a superposition of eigenstates to a single eigenstate due to the
interaction with the external world.

With these basic principia in mind, we therefore aim at replacing the classical position and momentum variables $q$ and $p$ with two operators, say $Q$ and $P$, by preserving the original structure as much as possible.

The most natural setting is therefore to consider the position operator $Q$ as the operator that corresponds to the position evaluation of a state (simply, applying $Q$ to a state $\psi$ being the evaluation of the function $\psi$ at a given point in space).

As for the corresponding momentum operator, in light of (<ref>), a natural choice is to take $P$ as the differential operator $-i \hbar \nabla$, thus obtaining a quantum analogue of (<ref>) via the relation

\begin{equation}\label{ENEJHND8iujMNSID-2krwjeg-21-ktg22} P\psi=-i\hbar\nabla\psi.\end{equation}

Pushing this analogy a bit further, we can also obtain a quantum counterpart ${\mathcal{H}}$ of
the total energy ${\mathcal{E}}$ (say, the Hamiltonian) in (<ref>).
In this setting, the operator analogue of (<ref>) would then be

\begin{equation}\label{ENEJHND8iujMNSID-2krwjeg-21-ktg}
{\mathcal{H}}\psi=-i\hbar\partial_t \psi.
\end{equation}

While this canonical formalism for quantum mechanics was obtained by analyzing the special case of the wave produced by a photon,
we can believe that the same protocol governs the evolution of the wave function of a particle (not necessarily a photon)
subject to a given potential $V$. For instance, for a particle
of mass $m>0$ one can push the previous construction by considering the quantum analogue of the mechanical energy
(perhaps neglecting for the moment some relativistic effects,
which we will briefly discuss in the forthcoming footnote <ref>).
Indeed, in classical mechanics the total energy would be given by the sum of the kinetic energy
of a particle (equal to $\frac{m|v|^2}{2}$, being $v$ its velocity) and the potential energy $V$. That is,
recalling the classical momentum definition $p=mv$,

$$ {\mathcal{E}}=\frac{m|v|^2}{2}+V=\frac{|p|^2}{2m}+V.$$

By formally applying the quantization in (<ref>) and (<ref>), we thus obtain its quantum counterpart as

$$ i\hbar\partial_t={\mathcal{H}}=\frac{|P|^2}{2m}+V=
\frac{P\cdot P}{2m}+V=-
\frac{\hbar^2 \nabla\cdot\nabla}{2m}+V=-\frac{\hbar^2 \Delta}{2m}+V,

that is the Schrödinger equation[It is interesting to point out that variations
in the previous arguments lead to other equations of interest: for instance,
thinking over the full relativistic energy formula in (<ref>) and formally inserting the quantum mechanical operator in (<ref>), the quantum analogue of the kinetic energy becomes

$$ \sqrt{ -\hbar^2\, c^2\Delta + (mc^2)^2},$$

that is the “square root of the Laplacian”square root of the Laplacian (see e.g. <cit.>
for a basic introduction to this very interesting object). Thus,
in the presence of an external potential $V$, the conservation of the full energy and the quantization
in (<ref>) lead to the balance

$$-i\hbar\partial_t =\sqrt{ -\hbar^2\, c^2\Delta + (mc^2)^2}-V,$$

which produces the equation

$$ -i\hbar\partial_t \psi=\sqrt{ -\hbar^2\, c^2\Delta \psi+ (mc^2)^2}-V\psi.$$

This equation and variants of it are usually
called “relativistic Schrödinger equations”relativistic Schrödinger equation, see e.g. equation (1.4) in <cit.>.

Another approach for taking into account relativistic effects consists in taking the square of the
relativistic energy formula in (<ref>), thus writing that

$$ {\mathcal{E}}^2= (|p|\, c)^2 + (mc^2)^2.$$

In the absence of external potentials (i.e., if this represents the square of the total energy of the system), one can think about exploiting the quantization procedure in (<ref>)
and (<ref>), formally finding that

$$ -\hbar^2\partial_{tt}= -\hbar^2 c^2\Delta+ (mc^2)^2.$$

This leads to the equation

$$ \partial_{tt}\psi=  c^2\Delta\psi-\frac{ m^2c^4\psi}{\hbar^2},$$

which is often referred to with the name of “Klein-Gordon equation”Klein-Gordon equation,
see e.g. equation (34) in <cit.>.] in (<ref>).

§.§ Who wants to be a millionaire?

One way to become a millionaire is to solve a Millennium Prize Problem (see pages 106106
and 106106-bisPE), but this is probably by far the most difficult way to get rich.
Also, one should take into consideration the possibility that mathematics may not make us rich,
but it makes us happy, which is inestimable.

James Harris Simons in 2007
(image from
licensed under the
Creative Commons Attribution-Share Alike 2.0 Germany license).

There are however truly outstanding mathematicians, such as Jim Simons (see Figure <ref>)
who, after having revolutionized the theory of minimal surfaces and the topological quantum field theory,
and also after having broken codes for the NSA
during the Cold War,
decided to improve his portfolio:
allegedly, Simons' net worth is estimated to be $25.2\times 10^9$ US $,
making him the 66th-richest person in the world, and likely the richest[Simons has not forgotten
his love for mathematics. Not only because he owns a motor yacht named Archimedes,
but also because he
co-founded the Simons Foundation,
supporting projects related to scientific research, education and health, and
he sponsored the Simons Institute for the Theory of Computing
and the Mathematical Sciences Research Institute at Berkeley.

Here is an interesting story by the way. While Simons was working for the Institute for Defense Analyses by cracking Russian codes, his boss General Maxwell Taylor wrote an article in the New York Times Magazine strongly supporting the Vietnam War. Then, Simons published a counter-editorial in the Times, claiming that not everyone who worked for Taylor subscribed to his views. This resulted in an interview of Simons from Newsweek. When Simons told his superiors about this interview, they fired him right away.

Allegedly, Simons said “Getting fired once can be a good experience. You just don't want to make a habit of it”.]
among the mathematicians.
His financial success is related to the foundation of
a quantitative hedge fund (see below for the notion of hedging)
which trades using quantitative mathematical models, so mathematics can be sometimes useful after all.

One of the chief uses of partial differential equations
in mathematical finance is related to the Black-Scholes equationBlack-Scholes equation

\begin{equation}\label{B:LA:CK}
\frac{\partial V}{\partial t}+{\frac {1}{2}}\sigma ^{2}S^{2}
\frac{\partial ^{2}V}{\partial S^2}=rV-rS\frac {\partial V}{\partial S},\end{equation}

named after Fischer Black and Myron Scholes.

The meaning of this equation is the following.
Suppose that we have a stockstock (that is, a security that represents the ownership of a fraction of a corporation)
whose price is denoted by $S$.
The price of a stock fluctuates based on supply and demand
(if more people want to buy than sell it, the price will rise) hence we take $S=S(t)$ to be a function of time.

To model the fluctuations of $S$ in the simplest possible way, we can make two assumptions.

On the one hand, we can assume that the stock refers to some company
with revenues, earnings, dividends, etc., and we can estimate the growth rate
of the company and of the corresponding stocks: for this, we assume that the variation of $S$
in the unit of time is proportional, by some factor $\mu$, to the value of $S$
(the more stocks one possesses, and the higher the coefficient $\mu$, the more one receives from the growth of the company's revenues; actually positive values of $\mu$ correspond to a growth
and negative ones to a degrowth).

On the other hand, the market presents a number of uncertainties which are almost
impossible to fully take into account, such as economic crises, political developments,
real estate bubbles, technological innovations, pandemics, landing of aliens from outer space, etc.
If we have no specific information on the matter, we can just assume that this volatility
is modeled by a Brownian motion (pretty much as the random walk presented on page RANDOW).
This is accounted for by a process $W(t)$ that “randomly wiggles up and down” and
that quantifies the source of uncertainty in the price. This term is modulated by a volatility coefficient $\sigma\in[0,+\infty)$ (the higher this coefficient, the bigger the impact of the uncertainties in the price oscillation).

Assuming that these two effects contribute somewhat consistently to the variation in time of the stock,
we therefore write that[Equation (<ref>)
is sometimes called the “geometric Brownian motion”geometric Brownian motion
equation. It is a very popular tool in quantitative finance since it relies
on the simple, but often reasonable, assumption that the relative variation $\frac{dS}S$
of a value follows a diffusive process.]

\begin{equation}\label{3456yuioiuhgfdr5678iokmnbvfdr5678ijhgfde45678ijhgfdety3hfe2}
dS=\mu S\,dt+\sigma S\, dW.\end{equation}

Strictly speaking, this is not really a differential equation,
since we cannot “divide both terms by $dt$”, since Brownian motions
are “not differentiable”: see e.g. <cit.>
for a precise statement and for an elegant introduction to stochastic calculus.
Indeed, (<ref>)
is a stochastic differential equationstochastic differential equation,
which is a type of equations that we do not treat here,
dealing with it just at an intuitive level. Basically, the only bit of information that
we need from stochastic calculus is the so called[Kiyosi Itô
was the pioneer of the stochastic calculus that brings his name.
See Figure <ref>
for a picture of Kiyosi Itô
at age 22 (Kiyosi is the second from the left; the first from the left is his brother
Seizô, who also became a mathematician).]
Itô's Chain RuleItô's Chain Rule (see e.g. Sections 1.3,  4.3, 4.4 and in particular the theorem on page 80 of <cit.>) according to which if $dX=F\,dt+G\,dW$,
then the random variable $H(X(t),t)$ satisfies

\begin{equation} \label{ITOPIOTNNRRRESIDEKKA}
dH(X,t)=\left( \partial_t H +F \partial_X H+\frac{G^2}{2} \partial_X^2 H\right)\,dt+G\partial_X H\,dW.\end{equation}

Let us now come back to our financial setting.
Suppose now that we perceive the buying of a given stock as too risky
and we prefer instead to buy only “the possibility of buying the stock
in the future”, if this turns out to be convenient.
This type of contract is called “option”option
and conveys its owner
the right, but not the obligation, to buy the stock at a specified strike price either
prior to or on a specified date.
For simplicity, let us consider the case in which the option allows, but not obliges, the holder
to buy the stock for a specified price $K$ (called “strike price”strike price
in jargon)
at a specific time $T$, that is on the date of option maturity:
these options are called in jargon “European options”European option;
the options which instead
can be exercised any time up to and including the date of expiration
are called “American options”American option
(perhaps the name originated
by the alleged fact that one type of option was commonly
traded in London and the other in New York).

The Itô family (Public Domain image from

We denote by $V$ the price of this option
and we suppose that $V$ depends on time and on the value $S$ of the underlying stock, that is $V=V(t,S)$, for $t$ less than the maturity time $T$.
For instance, at the maturity date the value of this option is either $S(T)-K$,
namely the difference between the actual value of the stock $S$ at time $T$
and the strike price $K$ that was agreed at the beginning, if $S(T)-K$ is positive
(in which case the option turned out to be a convenient investment),
or null, if $S(T)-K$ is negative (in this case the value of the stock is lower than the strike price,
so no need to use the option to pay more, if one is interested in the stock can just forget
about the option and buy the stock at the market prince). More specifically, these observations yield that[For American options, one also has that

$$ V(t, S(t))\ge\max\{S(t)-K,0\} $$

for all $ t\in[0,T]$,
the inequality coming from the fact that one can, but is not obliged to, utilize the option at time $t$
and this provides an “obstacle problem”obstacle problem
for equation (<ref>), since it forces solutions of (<ref>)
to stay above a constraint
(or to drop the equation when they meet the constraint;
this situation is significantly more complicated than that of a single equation, and this is the reason for which
we limited ourselves here to the case of European options).

With respect to (<ref>), we may consider it as a “terminal condition”terminal condition
for equation (<ref>).

Interestingly, with respect to the classification discussed in footnote <ref> on page CLASSIFICATIONFOOTN,
we have that equation (<ref>) is of parabolic type, but the sign in front of the time derivative
is opposite to the case of the heat equation (compare with (<ref>);
a closer relation
between the Black-Scholes equation and the heat equation will be discussed on page BSDAGB-ADkrVoiweLL4re2346ytmngrrUj72).
In this sense, equation (<ref>) shares some similarity with heat type equations,
but going backwards in time. This makes sense, since while the heat equation
smooths out differences as time flows, the value of an option becomes quite rigid close to its maturity

\begin{equation}\label{TERM01-2pwf} V(T,S)=\max\{S-K,0\}.\end{equation}

Our objective is thus to understand the time evolution of the value $V$ of the option:
this is important, since if one possesses an option, may decide to sell it at some time $t$,
knowing just the strike price $K$ agreed at the beginning and the present value of the stock $S(t)$,
hence it is relevant to establish a fair price $V(t,S(t))$ for the option
under this information (since one would like to answer this question for all possible values
of the stocks, this reduces to understand $V(t,S)$ just knowing the strike price $K$,
with this one would substitute $S=S(t)$ to obtain the fair price of the option).

To model the time evolution of $V$, and thus provide a convincing derivation of
the Black-Scholes equation in (<ref>), we
use a method called in finance “delta hedging”delta hedging,
namely we adopt a trading strategy that reduces, or “hedges”, the risk related to the volatility
of the process.
For this, we suppose to buy a quantity $\alpha=\alpha(t,S(t))$ of options and some quantity $\beta=\beta(t,S(t))$ of the stocks.
The goal is to choose $\alpha$ and $\beta $ to fully cover the risk
(or, at least, to cover the risk to the best of our possibilities). More precisely, we consider a portfolio $P:=\alpha V+\beta S$
and we compare this investment with another one that carries zero risk,
that is some operation which ensure a certain (or, more realistically, almost certain)
future return and (virtually) no possibility of loss: for instance a treasury bill, or treasury bond,
namely a bond in which the face value is repaid at the time of maturity with some interest.
We denote by $r\ge0$ the risk-free interest rate (this will be precisely the parameter $r$
appearing in (<ref>)). The value of this zero risk portfolio $P_0$ is thus
described by ${dP_0}=r P_0\,dt$.

Since, via a suitable choice of $\alpha$ and $\beta$, the portfolio $P$ is set up to have zero risk, it must be equivalent
to the portfolio $P_0$, otherwise there would be a better choice to gain money at zero risk! This says that
also $ dP=rP\,dt$ and therefore

\begin{equation}\label{93456yuioiuhgfdr5678iokmnbvfdr5678ijhgfde45678ijhgfdety3hfe3CP8}
\big(r\alpha V+r\beta S\big)\,dt=
rP\,dt =d\Big(\alpha V+\beta S \Big).\end{equation}

We stress that the portfolio $P$ is assumed to be self-financingself-financing portfolio,
namely there are no inflows or outflows of money. More specifically, at every instant of time,
the purchase of a new option or stock must be financed by the sale of an old one:
hence, since there is no external infusion or withdrawal of money,
the variation of the value of the portfolio corresponds precisely to the variation of the values
of the options and stocks possessed, that is

\begin{equation}\label{93456yuioiuhgfdr5678iokmnbvfdr5678ijhgfde45678ijhgfdety3hfe3CP89} dP=\alpha\, dV+\beta \,dS.\end{equation}

We stress that this does not mean necessarily that $\alpha$ and $\beta$ are constants,
just that the market value of the portfolio $P$ at a given time
equals the purchase value of the new portfolio.
Namely, if in the infinitesimal interval of time $dt$ the value of the options has increased respectively by $dV$
and the value of the stocks by $dS$, then the gain obtained would correspond to the number of options
possessed times the options' increment value (that is $\alpha$ times $dV$) plus the number of stocks
possessed times the stocks' increment value (that is $\beta$ times $dS$): this gives
that the total gain in the infinitesimal interval of time $dt$ is $\alpha\, dV+\beta \,dS$, which is precisely
the right hand side in (<ref>).
Accordingly, the balance prescribed by (<ref>)
states that this gain is used precisely to reinvest to enlarge the portfolio itself
(this, in the optimistic scenario that there is a gain; if there is a loss then the right hand side of (<ref>) is negative and
the portfolio needs to be correspondingly shrunk).

Now we let $\varpi$ be the ratio[In most of the literature,
this ratio is denoted by either $\Delta$ or $\delta$, whence the name
of delta hedging. Here, for typographical convenience, and to avoid confusion with
the Laplace operator and with increments and derivatives, we preferred to call it $\varpi$.] between $\beta$ and $\alpha$, namely we take $\varpi=\varpi(t,S(t))$ such that $\beta=\alpha\varpi$.
Then (<ref>)
can be stated in the form

\begin{equation*}
\big(r\alpha V+r\alpha\varpi S\big)\,dt=
\big(r\alpha V+r\beta S\big)\,dt=dP
\end{equation*}

and therefore (assuming for simplicity $\alpha\ne0$)

\begin{equation}\label{93456yuioiuhgfdr5678iokmnbvfdr5678ijhgfde45678ijhgfdety3hfe38}
\big(r V+r\varpi S\big)\,dt
\end{equation}

We also point out that

\begin{equation*}
dV=\left( \partial_t V +\mu S \partial_S V+\frac{\sigma^2 S^2}{2} \partial_S^2 V\right)\,dt+\sigma S \partial_S V\,dW

thanks to (<ref>).

From this, (<ref>) and (<ref>) we infer that

\begin{equation}\label{0olLur2345609iuygfadTbSan2sDMrote}
\begin{split}
\big(rV+r\varpi S\big)\,dt\\
\left( \partial_t V +\mu S\partial_S V+\frac{\sigma^2 S^2}{2} \partial_S^2 V\right)\,dt+\sigma S\partial_S V\,dW
\big(rV+r\varpi S\big)\,dt\\&=
\left( \partial_t V +\mu S\partial_S V+\frac{\sigma^2 S^2}{2} \partial_S^2 V
-rV-r\varpi S
\right)\,dt+\sigma S\partial_S V\,dW
+\varpi\,\big(\mu S\,dt+\sigma S\, dW\big)
\\&=
\left( \partial_t V +\mu S\big(\partial_S V+\varpi\big)+\frac{\sigma^2 S^2}{2} \partial_S^2 V
-rV-r\varpi S
\right)\,dt+ \sigma S\big(\partial_S V+\varpi \big)\,dW.\end{split}
\end{equation}

Consequently, since $\varpi$ is chosen to make the portfolio free from the risk caused by the stochastic term
this gives that one must choose $\varpi$ such that $\partial_SV+\varpi=0$
(note that this cancels in (<ref>)
the term involving $W$ and the linear term in $\sigma$). That is, we take

\begin{equation*}
\varpi=-{\partial_SV}
\end{equation*}

and we thus plug this information into (<ref>),
concluding that

\begin{equation*}
0=\left( \partial_t V +\frac{\sigma^2 S^2}{2} \partial_S^2 V

which leads to the the Black-Scholes equation in (<ref>), as desired.

It is also interesting to revisit the Black-Scholes equation in (<ref>)
in view of the financial model that it describes. We note in particular that the
right hand side of (<ref>) is described only in terms
of the risk-free interest rate $r$: this is therefore the risk-free part of the investment
and it consists of the superposition of a long term strategy dictated by the option $V$ (which will become effective only at its maturity time) and a short term one embodied by the stock $S$ (which fluctuates
in a market subject to randomness).

On the left hand side of (<ref>) instead we see two terms. The first is the time derivative
of the option. The second term on the left hand side of (<ref>)
is more “geometric” since it reflects the convexity properties of the dependence of the option
on the underlying stock.

It is also useful to relate the Black-Scholes equation in (<ref>) to
the classical heat equation. As already observed on page BSDAGB-ADkrVoiweLL4re2346ytmngrrUj7,
to make this connection one needs to revert the arrow of time, hence it is convenient to look at a new time $\tau:=T-t$ which transforms the terminal condition (<ref>) into an initial condition.
Furthermore, it comes in handy to introduce the nonlinear transformation $x=\ln \left({\frac {S}{K}}\right)+\left(r-{\frac {\sigma^{2}}{2}}\right)\tau$ in order to simplify the right hand side of (<ref>). More specifically, one sets

$$ u(x,\tau):=e^{r\tau} \,V\left(T-\tau,Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)$$

and observes that if $V$ solves the Black-Scholes equation in (<ref>) then,
using the notation $S:=Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}$ and $t:=T-\tau$,

\begin{eqnarray*}&&
\frac{\sigma^2}2 \partial_{xx} u(x,\tau)-\partial_\tau u(x,\tau)\right]\\&=&
\frac{\sigma^2}2 \partial_{x}
\left[ Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\partial_S
-r  V\left(T-\tau,Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)\\&&\qquad
+Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau} \left(r-\frac{\sigma^2}2\right)\partial_S
+\partial_t V\left(T-\tau,Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)
\\&=&
\frac{\sigma^2}2
\left[
\left(Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)^2\partial_{SS}
\right]\\&&\qquad
-r  V\left(T-\tau,Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)
+Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau} \left(r-\frac{\sigma^2}2\right)\partial_S
+\partial_t V\left(T-\tau,Ke^{x-\left(r-\frac{\sigma^2}2\right)\tau}\right)\\
\left[ S\partial_S
\right]-r  V(t,S)+S \left(r-\frac{\sigma^2}2\right)\partial_S
V(t,S)+\partial_t V(t,S)\\
&=&\frac{\sigma^2 S^2}2\partial_{SS}V(t,S)-r  V(t,S)+rS\partial_S
V(t,S)+\partial_t V(t,S)\\&=&0.
\end{eqnarray*}

$$ u(x,0)=V\left(T,Ke^{x}\right).$$

Consequently, the Black-Scholes equation in (<ref>) and the terminal condition (<ref>)
produce the following heat diffusion problem for time $\tau>0$ with initial datum:

\begin{equation}\label{CALOCA}
\begin{dcases}
&\partial_\tau u(x,\tau)=\displaystyle\frac{\sigma^2}2 \partial_{xx} u(x,\tau),\\
\end{dcases}
\end{equation}

where, as usual, we used the notation for the characteristic function of a set, namely

$$ \chi_A(x):=\begin{dcases}1 & {\mbox{ if }}x\in A,\\
0 &{\mbox{ otherwise.}}\end{dcases}$$

Plot of a solution of (<ref>)
(with $K:=1$ and $\sigma:=\sqrt2$).

The plot of a solution of (<ref>) is sketched in Figure <ref>.
The advantage of this formulation is that one can focus on the solution of the classical heat equation in (<ref>)
and deduce useful information on the solution of the Black-Scholes equation in (<ref>)
by transforming back to the original variables.

See e.g. <cit.> for additional readings on the Black-Scholes equation[Of course,
as every equation, the Black-Scholes equation
must be taken with a pinch of salt.
For instance, according to Ian Stewart's article appeared in the British newspaper
The Observer
the Black-Scholes equation was “one ingredient in a rich stew of financial irresponsibility, political ineptitude,
perverse incentives and lax regulation” that ultimately led to
the 2007 financial crisis.
“The equation itself wasn't the real problem. It was useful, it was precise, and its limitations were clearly stated [...]. The trouble was its potential for abuse”. In particular, we stress that the equation relies on the knowledge of the market volatility $\sigma$.
“This is a measure of how erratically its market value changes. The equation assumes that the asset's volatility remains the same for the lifetime of the option, which needs not be correct. Volatility can be estimated by statistical analysis of price movements but it can't be measured in a precise, foolproof way, and estimates may not match reality [...].
The equation also assumes that there are no transaction costs, no limits on short-selling and that money can always be lent and borrowed at a known, fixed, risk-free interest rate. Again, reality is often very different”.] and related models.

§.§ Diffusion of transition probabilities

In a stochastic model, one is often interested in the so-called transition probability,transition probability
namely the likelihood of transitioning from one state to another.

In particular, Markov processesMarkov process are stochastic models “without memory”, i.e.
in which predictions about future outcomes can be done based solely on the knowledge of its present state.
In this setting, if $P(x_0,t_0| x,t)$ denotes the probability of going from the point $x_0$ at time $t_0$
to the point $x$ at time $ t$ (say, with $t>t_0$),
we have that such a probability can be expressed as the superposition of all the
probabilities of going first to a point $y$ at some time $s\in(t_0,t)$
and from there reaching $x$ at time $t$: more explicitly, given $s\in (t_0,t)$, it holds that

\begin{equation}\label{CHAKOL}
P(x_0,t_0|x,t)=\int_{\R^n} P(x_0,t_0|y,s)\,
\end{equation}

This relation is often called
Chapman-Kolmogorov equationChapman-Kolmogorov equation,
after[Chapman was a mathematician and geophysicist. At age 16 he entered the University of Manchester with a scholarship (he was the last student selected) and graduated with an engineering degree, but his passion for mathematics drove him to study for one further year to take a mathematics degree.

Besides his eminent contributions to stochastic processes, Chapman contributed to the theory of geomagnetism, investigating the beautiful phenomenon of aurorae (see Figure <ref>) in relation to the Earth's magnetic field and the solar wind,
contributing to the understanding of the photochemical mechanisms that produce the ozone layer,
and predicting the existence of the magnetosphere (confirmed experimentally 30 years later).

Kolmogorov's scientific contributions are paramount
and comprise basically all fields of mathematics, including probability, topology, logic, mathematical physics, harmonic analysis, mathematical biology and numerical analysis. His mathematical talent appeared quite early in his life: at the age of five, he wrote his first mathematical paper, published in the journal of his school. The content of the paper was the formula

$$ \sum_{k=0}^{N-1} (2k+1)=N^2,$$

which is easy peasy for all of us, but not quite a piece of cake for a five-year-old boy (by the way, the little kid also became editor of the mathematical section of the school journal).

His former student and prominent mathematician Vladimir Igorevich Arnold used to place Kolmogorov in his top 5 mathematicians list (with Poincaré, Gauß, Euler and Newton, see Figure <ref>
for a picture of Kolmogorov preparing his talk during a conference in Tallin in 1973).

A mark on Kolmogorov's reputation is however produced by his involvement in the so-called "Luzin Affair" during the Great Purge in 1936
(see also footnote <ref> on page PURFO: on this occasion, Nikolai Nikolaevich Luzin, prominent mathematical analyst and point-set topologist and Kolmogorov's former doctoral advisor (and author of the classical Luzin's Theorem, see e.g. <cit.>) became a target of Stalin's regime. Luzin was accused of plagiarism, nepotism, of being an enemy of the Soviet people and a servant to “fascistoid science” (whatever it means), as confirmed by the fact that Luzin published some of his results in foreign journals.

In view of this allegation, Luzin lost his academic position (not too bad after all,
his advisor Dimitri Fyodorovich Egorov, author of Egorov's Theorem, see <cit.>, was arrested during a previous purge and died after a hunger strike initiated in prison).

Regrettably, Kolmogorov took part in the hearing at the Commission of the Academy of Sciences of the USSR, where the allegations against Luzin were formalized. The question of whether Kolmogorov was coerced by the police into testifying against Luzin, possibly
using as a threat an alledged long-lasting homosexual relationship,
remains a topic of speculation, see e.g. <cit.>. In any case, a sad story of envy, violence, indifference, discrimination and brutality.
Which is good to know, since
those who fail to learn from the mistakes done in the past
are doomed to repeat them.]
Sydney Chapman and Andrey Nikolaevich Kolmogorov.

Collection of pictures of aurorae from around the world (photos by
Mila Zinkova, Samuel Blanc, Joshua Strang, Varjisakka and Jerry Magnum Porsbjer;
images from
licensed under the Creative Commons Attribution-Share Alike 1.0 Generic license).

It is clearly desirable, given an initial point $x_0$ at an initial time $t_0$,
to know more about the likelihood of being somewhere else at some future time, that is to have more information on
the function $p(x,t):=P(x_0,t_0|x,t)$.

Getting ready for a big presentation (photo by Terrence L. Fine; image from
licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).

This is obtained
through a partial differential equation of the type[Equation (<ref>)
is usually called Fokker-Planck equationFokker-Planck equation.
The name comes after Adriaan Daniël Fokker and Max Karl Ernst Ludwig Planck.

Fokker was a physicist and a musician, inventor of
a 31-tone equal-tempered organ
(we recall that
the 12-tone equal temperament
is the most common musical system today, recall the musical digression on page JS-PGAGPA).
A picture of Fokker playing his organ is available at https://upload.wikimedia.org/wikipedia/en/a/ae/AdriaanFokker.jpg

Planck is the famous originator of quantum theory: fortunately, he did not follow the advice of
his professor Philipp von Jolly at the
Ludwig-Maximilians-Universität München, according to whom it was not worth to go into physics
since “in this field, almost everything is already discovered, and all that remains is to fill a few holes”.

The Planck family lived for many years in a villa in Berlin-Grunewald, located in Wangenheimstraße 21,
who became a gathering place for other professors, such as Albert Einstein and theologian Adolf von Harnack
(brother of the mathematician portrayed on Figure <ref>
on  page HARAXELHARLROAGIJ7soloDItangeFIPAGI).

Planck family suffered hard times due to the deplorable political situation around.
At the onset of World War I, Planck was one of the signatories of the so-called
Manifesto of the Ninety-Three
stating an unequivocal support of German military actions; among the other ninety-two prominent signatories,
Fritz Haber, Adolf von Harnack, Philipp Lenard (Nobel Prize for cathode rays research,
future supporter of the Nazi ideology and future crusader against Einstein's relativity theory)
and Felix Klein (the outstanding mathematician who devise the “the Klein bottle”). During World War I,
Planck's oldest son Karl was killed in action at the Battle of Verdun
and his second son Erwin was taken prisoner by the French.

When the Nazis came to power, Planck was 74 and, when asked
to gather distinguished professors to issue a public proclamation against the expulsion of Jewish academics,
Planck replied, “If you are able to gather today 30 such gentlemen, then tomorrow 150 others will come and speak against it, because they are eager to take over the positions of the others”.
However, Planck tried to avoid the expulsion of Fritz Haber
(who had pioneered chemical warfare
by introducing the use of poisonous gases during World War I and
personally overseeing the first use of chlorine gas during the Second Battle of Ypres,
and who had received the Nobel Prize in Chemistry in 1918 for the invention of a process to synthesize ammonia from nitrogen and hydrogen gases, with broad application in the synthesis of fertilizers and explosives).
Planck's support in this circumstance was unsuccessful
and Haber died in exile the following year.

Planck's son Erwin was later arrested by the Gestapo following the attempted assassination of Hitler on 20 July 1944,
sentenced to death and executed by hanging.

On a positive note, see Figure <ref>
for a social dinner
among Nobel Laureates:
from left to right: Walther Nernst (1920 Nobel Prize for Chemistry), Albert Einstein
(1921 Nobel Prize for Physics), Max Planck (1918 Nobel Prize for Physics),
Robert A. Millikan (1923 Nobel Prize for Physics)
and Max von Laue (1914 Nobel Prize for Physics).]

\begin{equation}\label{FOKPLA}
\partial_t p(x,t)\,dx=-\div\big(\alpha(x,t)\,p(x,t)\big)+
\sum_{i,j=1}^n \partial_{ij}\big(\beta_{ij}(x,t)\,p(x,t)\big)

where $\alpha$ and $\beta_{ij}$ are called drift vector and diffusion tensor, respectively.

Nobel Laureates only
(Public Domain image from

We now give a motivation for equation (<ref>).
Let $\varphi$ be a smooth and compactly supported function in $\R^n$.
Using the Chapman-Kolmogorov equation in (<ref>), we know that, given $s\in (t_0,t)$,

\begin{eqnarray*}&&
\int_{\R^n}\varphi(x)\,p(x,t)\,dx=
\int_{\R^n}\varphi(x)\,P(x_0,t_0|x,t)\,dx\\&&\qquad=\iint_{\R^n\times\R^n} P(x_0,t_0|y,s)\,
\end{eqnarray*}

and, as a result, differentiating in $t$,

\begin{equation}\label{frasuetartialarphi}
\int_{\R^n}\varphi(x)\,\partial_t p(x,t)\,dx=\iint_{\R^n\times\R^n } P(x_0,t_0|y,s)\,
\partial_t P(y,s|x,t)\,\varphi(x)\,dx\,dy

It is also useful to recall that the total probability is normalized to $1$, namely

\begin{equation*} \int_{\R^n} P(y,s|x,t)\,dx=1,\end{equation*}

whence, taking a derivative in $t$,

\begin{equation}\label{BASDBAefTTrotlk4453} \int_{\R^n} \partial_t P(y,s|x,t)\,dx=0.\end{equation}

Given $y\in\R^n$, we also use a Taylor expansion to write that, when $|x-y|$ is small,

$$ \varphi(x)=\varphi(y)+\nabla\varphi(y)\cdot(x-y)+\frac12 D^2\varphi(y)(x-y)\cdot(x-y)+O(|x-y|^3).$$

This and (<ref>) give that

\begin{equation}\label{poe0586uyrhqwertyuiolkjhgfdszxcvbnm0987654}
\begin{split}
&\int_{\R^n} \partial_t P(y,s|x,t)\,\varphi(x)\,dx\\
=&\int_{\R^n} \partial_t P(y,s|x,t)\,\left(\varphi(y)+\nabla\varphi(y)\cdot(x-y)+\frac12 D^2\varphi(y)(x-y)\cdot(x-y)+O(|x-y|^3)
\right)\,dx\\
=&\int_{\R^n} \partial_t P(y,s|x,t)\,\left(\nabla\varphi(y)\cdot(x-y)+\frac12 D^2\varphi(y)(x-y)\cdot(x-y)+O(|x-y|^3)
\right)\,dx\\
=&\,\alpha(y,s,t)\cdot\nabla\varphi(y)+\frac12 \sum_{i,j=1}^n\beta_{ij}(y,t)\partial_{ij}\varphi(y)+
\int_{\R^n} \partial_t P(y,s|x,t)\,O(|x-y|^3)\,dx,
\end{split}\end{equation}

where[To be precise, we note that
the latter integral in (<ref>)

occurs only in the support of $\varphi$, which is a bounded set.]

\begin{eqnarray*}
&& \alpha(y,s,t):=
\int_{\R^n} \partial_t P(y,s|x,t)\,(x-y)\,dx\\
{\mbox{and }}&&
\beta_{ij}(y,s,t):=\frac12\int_{\R^n} \partial_t P(y,s|x,t)\,(x_i-y_i)(x_j-y_j)\,dx.
\end{eqnarray*}

Combining this and (<ref>), we find that

\begin{equation}\label{NEHDF-HSJDJI2}
\begin{split}&
\int_{\R^n}\varphi(x)\,\partial_t p(x,t)\,dx\\
\Bigg[
\alpha(y,s,t)\cdot\nabla\varphi(y)+ \sum_{i,j=1}^n\beta_{ij}(y,s,t)\partial_{ij}\varphi(y)\\&\qquad\qquad+
\int_{\R^n} \partial_t P(y,s|x,t)\,O(|x-y|^3)\,dx\Bigg]\,p(y,s)\,dy \\
\int_{\R^n} \alpha(y,s,t)\cdot\nabla\varphi(y)\,p(y,s)\,dy\\&\qquad\qquad+\int_{\R^n}
\sum_{i,j=1}^n\beta_{ij}(y,s,t)\partial_{ij}\varphi(y)\,p(y,s)\,dy +{\mathcal{R}}(s,t),\end{split}\end{equation}

\begin{eqnarray*}{\mathcal{R}}(s,t):=
\iint_{\R^n\times\R^n} \partial_t P(y,s|x,t)\,O(|x-y|^3)\,dx\,dy.
\end{eqnarray*}

We suppose additionally that the stochastic process is homogeneous in time, namely
the transition probability depends only on the points $x_0$ and $x$ and on the elapsed time $t-t_0$, that is

$$ P(x_0,t_0+\tau|x,t+\tau)=P(x_0,t_0|x,t)$$

for all $\tau\in\R$.

Differentiating in $\tau$, this gives that

$$ \partial_{t_0} P(x_0,t_0|x,t)+\partial_{t} P(x_0,t_0|x,t)=0.$$

As a consequence, observing that $
when $x\ne y$ (unless one is provided with the “gift of ubiquity”),

\begin{eqnarray*}
\alpha(y,t)&:=&\lim_{\e\searrow0}
\frac1\e\int_{t-\e}^t\alpha(y,s,t)\,ds\\&=&\lim_{\e\searrow0}
\frac1\e
\iint_{\R^n\times(t-\e,t)} \partial_t P(y,s|x,t)\,(x-y)\,dx\,ds\\&
\frac1\e\iint_{\R^n\times(t-\e,t)} \partial_s P(y,s|x,t)\,(x-y)\,dx\,ds\\&
\frac1\e\int_{\R^n}\Big( P(y,t-\e|x,t)-P(y,t|x,t)\Big)
\,(x-y)\,dx\\&
\frac1\e\int_{\R^n}P(y,t-\e|x,t)\,(x-y)\,dx
\\&=&\lim_{\e\searrow0}
\int_{\R^n}\frac{P(y,0|x,\e)\,(x-y)}\e\,dx.
\end{eqnarray*}

\[ \beta_{ij}(y,t):=\lim_{\e\searrow0}
\frac1\e\int_{t-\e}^t\beta_{ij}(y,s,t)\,ds=\lim_{\e\searrow0}
\frac12\int_{\R^n} \frac{ P(y,0|x,\e)}{\e}\,(x_i-y_i)(x_j-y_j)\,dx\]

\begin{equation}\label{NEHDF-HSJDJI}
\frac1\e\int_{t-\e}^t{\mathcal{R}}(s,t)\,ds=\lim_{\e\searrow0}
\iint_{\R^n\times\R^n} \frac{ P(y,0|x,\e)}{\e}\,O(|x-y|^3)\,dx\,dy.
\end{equation}

Formally, we also have that

\begin{eqnarray*}&&
\lim_{\e\searrow0}\frac1\e
\iint_{\R^n\times(t-\e,t)} \alpha(y,s,t)\cdot\nabla\varphi(y)\,p(y,s)\,dy\,ds\\&=&
\lim_{\e\searrow0}\frac1\e\left(
\iint_{\R^n\times(t-\e,t)} \alpha(y,s,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy\,ds\right.\\&&\qquad\quad\left.+
\iint_{\R^n\times(t-\e,t)} \alpha(y,s,t)\cdot\nabla\varphi(y)\,\big(p(y,s)-p(y,t)\big)\,dy\,ds\right)\\&=&
\int_{\R^n} \alpha(y,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy+
\lim_{\e\searrow0}\frac1\e
\iint_{\R^n\times(t-\e,t)} \alpha(y,s,t)\cdot\nabla\varphi(y)\,\big(p(y,s)-p(y,t)\big)\,dy\,ds
\\&=&
\int_{\R^n} \alpha(y,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy+
\lim_{\e\searrow0}\frac1\e
\iint_{\R^n\times(t-\e,t)} |\nabla\varphi(y)|\,O(\e)\,dy\,ds\\&=&
\int_{\R^n} \alpha(y,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy
\end{eqnarray*}

and, in a similar way,

\begin{eqnarray*}
\lim_{\e\searrow0}\frac1\e
\iint_{\R^n\times(t-\e,t)} \beta_{ij}(y,s,t)\partial_{ij}\varphi(y)\,p(y,s)\,dy
=\int_{\R^n} \beta_{ij}(y,t)\partial_{ij}\varphi(y)\,p(y,t)\,dy.\end{eqnarray*}

Therefore, integrating (<ref>) in $s\in(t-\e,\e)$, dividing by $\e$ and sending $\e\searrow0$,
we see that

\begin{equation}\label{NEHDF-HSJDJI2COSetyt6}
\begin{split}&
\int_{\R^n}\varphi(x)\,\partial_t p(x,t)\,dx
\int_{\R^n} \alpha(y,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy\\&\qquad\qquad\qquad+\int_{\R^n}
\sum_{i,j=1}^n\beta_{ij}(y,t)\partial_{ij}\varphi(y)\,p(y,t)\,dy +{\mathcal{R}}(t).\end{split}\end{equation}

Leonard Ornstein and George Uhlenbeck (Public Domain images from

Now, for a sufficiently small time $\e$, if the stochastic process is continuous
we expect the probability of going from $y$ to $x$ in the elapsed time $\e$
to be “rather small” except when $|x-y|$ is small. It is therefore customary to consider the quantity ${\mathcal{R}}(t)$ in (<ref>) as a “negligible term”
(one may also want to recall the observation in footnote <ref>
to get rid of the contribution in ${\mathcal{R}}(t)$ coming from infinity).
Hence, taking the liberty of formally dropping it from the equation, we reduce (<ref>) to

\begin{equation*}\begin{split}&
\int_{\R^n}\varphi(x)\,\partial_t p(x,t)\,dx=\int_{\R^n} \alpha(y,t)\cdot\nabla\varphi(y)\,p(y,t)\,dy\\&\qquad\qquad+ \int_{\R^n}
\sum_{i,j=1}^n\beta_{ij}(y,t)\partial_{ij}\varphi(y)\,p(y,t)\,dy.\end{split}\end{equation*}

Integrating by parts (and renaming $y$ into $x$ in the variable of integration) we obtain

\begin{equation*}\begin{split}&
\int_{\R^n}\varphi(x)\,\partial_t p(x,t)\,dx\\&\qquad=-\int_{\R^n} \varphi(x)\,\div\big(\alpha(x,t)\,p(x,t)\big)\,\,dx+\int_{\R^n}
\sum_{i,j=1}^n \varphi(x)\,\partial_{ij}\big(\beta_{ij}(x,t)\,p(x,t)\big)\,dx,\end{split}\end{equation*}

which, by the arbitrariness of $\varphi$, produces the
Fokker-Planck equation in (<ref>), as desired.

Leonard Ornstein mural (image by Hansmuller from
licensed under the Creative Commons Attribution-Share Alike 4.0 International license).

Another possible derivation of the Fokker-Planck equation in (<ref>) comes from
stochastic differential equations (we present this argument in dimension $1$ for the sake of simplicity, but the same
ideas would carry over in higher dimensions too). Namely, if $dX=F\,dt+G\,dW$
we can use Itô's Chain Rule in (<ref>)
to every random variable $H(X(t),t)$, integrate and find that

\begin{equation*}
\int_0^T \,dH=
\int_0^T \left[
\left( \partial_t H +F \partial_X H+\frac{G^2}{2} \partial_X^2 H\right)\,dt+G\partial_X H \,dW\right].\end{equation*}

In particular, if $H=H(X(t))$,

\begin{equation}\label{WIENE-01}
\int_0^T \left[
\left(F \partial_X H+\frac{G^2}{2} \partial_X^2 H\right)\,dt+G\partial_X H \,dW\right].\end{equation}

Now we take the expected value ${\mathbb{E}}$ of this identity (say, at a given time $T$). For this, it is useful to recall that,
for a “reasonable” $\phi$, we have that

\begin{equation}\label{WIENE-02} {\mathbb{E}}\left(\int_0^T \phi\,dW\right)=0,\end{equation}

see e.g. <cit.>
(the bottom line of this formula being that $W$ presents independent increments hence the expectation of a function $\phi$
that is adapted to the stochastic process will vanish).

Thus, it follows from (<ref>) and (<ref>) that

\begin{equation}\label{WIENE-05} {\mathbb{E}}\Big(
\int_0^T
\left( F \partial_X H+\frac{G^2}{2} \partial_X^2 H\right)\,dt\right).\end{equation}

Now we recall the definition of expected value of a random variable $Y$ (see e.g. <cit.>), according to which

$$ {\mathbb{E}}(Y)=\int_{\R^n} Y(x) \,p(x,T)\,dx.$$

In this way, equation (<ref>) becomes

\begin{equation*}\begin{split}& \int_{\R^n} \Big(
\iint_{\R^n\times(0,T)}
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,p(x,T)\,dx\,dt.\end{split}\end{equation*}

We now write this identity for $T+\e$, subtract the one for $T$, divide by $\e$
and formally send $\e\searrow0$:
namely, recalling (<ref>),

\begin{eqnarray*}&&
\int_{\R^n} H(x)\,\partial_t p(x,T)\,dx\\&=&
\int_{\R^n} \Big(
H(x)-H(X(0))\Big)\,\partial_t p(x,T)\,dx\\
\int_{\R^n} \Big(
\lim_{\e\searrow0}\frac1\e
\left[\int_{\R^n} \Big(
\int_{\R^n} \Big(
\Bigg[\iint_{\R^n\times(0,T+\e)}
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,p(x,T+\e)\,dx\,dt\\&&\qquad
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,p(x,T)\,dx\,dt\Bigg]\\&=&\lim_{\e\searrow0}
\Bigg[\iint_{\R^n\times(0,T+\e)}
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,\frac{p(x,T+\e)-p(x,T)}\e\,dx\,dt\\&&\qquad
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,p(x,T)\,dx\,dt\Bigg]\\&=&
\left( F(x,T) \partial_x H(x)+\frac{G^2(x,T)}{2} \partial_x^2 H(x)\right)\,p(x,T)\,dx,
\end{eqnarray*}

$$ {\mathcal{S}}(T):=\iint_{\R^n\times(0,T)}
\left( F(x,t) \partial_x H(x)+\frac{G^2(x,t)}{2} \partial_x^2 H(x)\right)\,\partial_t p(x,T)\,dx\,dt.$$

The Ornstein-Uhlenbeck process, detail from the Leonard Ornstein mural (image by Hansmuller from
licensed under the Creative Commons Attribution-Share Alike 4.0 International license).

Since the term ${\mathcal{S}}(T)$ is small when $T$ is small, if we neglect it from the previous
computation we reduce the problem to

\[ \int_{\R^n} H(x)\,\partial_t p(x,T)\,dx=
\int_{\R^n}
\left( F(x,T) \partial_x H(x)+\frac{G^2(x,T)}{2} \partial_x^2 H(x)\right)\,p(x,T)\,dx.\]

Integrating by parts in $x$, we thus conclude that

\[ \int_{\R^n} H(x)\,\partial_t p(x,T)\,dx=
\int_{\R^n}
\left( -\partial_x\Big(F(x,T) \,p(x,T)\Big)+\frac12\partial_x^2\Big(
{G^2(x,T)}\, p(x,T)\Big)\right)\,H(x)\,dx.\]

From the arbitrariness of $H$
we thus obtain

\begin{equation}\label{KMS76ygiTSg12rZasdG} \partial_t p=-\partial_x \big(F p\big)+\frac12\partial_x^2\big({G^2} p\big),\end{equation}

which is the Fokker-Planck equation (<ref>) in this setting:
in the expression (<ref>) one can directly related the drift coefficient of the Fokker-Planck equation
with the “deterministic” term of the stochastic equation and the diffusion coefficient of the Fokker-Planck equation
with the “random” term of the stochastic equation.

A classical application for the Fokker-Planck equation in (<ref>) is given by the
so-called[The process in (<ref>) is named after
Leonard Salomon Ornstein and George Eugene Uhlenbeck, see Figure <ref>.

Compare also the photo of Ornstein in his lab with
Figure <ref>, showing a very elegant
mural in Utrecht, The Netherlands, depicting Ornstein,
the random walk (embodied by a drunkard, check the bottle of
booze in his left hand) and the Ornstein–Uhlenbeck
process (note the formula next to Ornstein's head,
as enlarged in Figure <ref>).

The mural has been painted by the Dutch painting collective
De Strakke Hand, see Figure <ref>.
See also http://www.destrakkehand.nl/ for other beautiful pieces of art in the urban landscape.

Sometimes, models similar to (<ref>) are also referred to with the name
of Langevin equation, Langevin equation
after Paul Langevin.] Ornstein-UhlenbeckOrnstein-Uhlenbeck process stochastic differential equation.
In this case, one considers in the background a stochastic differential equation of the form

\begin{equation}\label{ORST-ULHL-S-04} dX=-\vartheta \,X\,dt+\sigma \,dW,
\end{equation}

for suitable positive constant parameters $\vartheta$ and $\sigma$.

De Strakke Hand Team (image from De Strakke Hand website

To develop an intuition of the process in (<ref>), we can imagine an
elastic spring in the presence of a large dumping and thermal fluctuations. In a nutshell, Hooke's Law would prescribe
an equation of motion of the type $\ddot X=-\kappa X$, with $\kappa>0$ being the elastic constant
of the spring. In presence of friction, the previous equation becomes $\ddot X=-\kappa X-\gamma \dot X$,
being $\gamma$ the friction coefficient. Rewriting this equation in the form $\frac{\ddot X}{\gamma}=-\vartheta X-\dot X$,
with $\vartheta:=\kappa/\gamma$, for large values of elastic and dumping coefficients we can reduce ourselves
to $0=-\vartheta X-\dot X$, that we can formally write as $dX=-\vartheta X\,dt$. In this sense,
the Ornstein-Uhlenbeck equation in (<ref>) consists simply in adding to this model a random fluctuation
(e.g. due to thermal deviations)
and therefore it can be considered as describing a noisy relaxation process of an oscillator.

Notice that the setting in (<ref>) corresponds to that in (<ref>)
with the choices $F:=-\vartheta \,X$ and $G:=\sigma$, whence the corresponding
Fokker-Planck equation for the Ornstein-Uhlenbeck process takes the form

\begin{equation}\label{ORST-ULHL-S-05} \partial_t p=\vartheta\partial_x \big(x p\big)+\frac{\sigma^2}2\partial_x^2 p.\end{equation}

It is interesting to observe that the drift term modulated by $\vartheta$ in (<ref>)
corresponds to an attraction towards the center $x=0$ for the transition probability $p$.
To convince ourselves of this fact one can just compare (<ref>)
with (<ref>), which was describing the evolution of a biological population following a chemical attractant:
notice that $w$ in (<ref>) would correspond, up to constants, to $-\vartheta x^2$ in (<ref>),
hence $p$ is “attracted” towards the higher values of $-\vartheta x^2$ (namely, $x=0$) in the same way
as a biological population is attracted towards the higher density regions of the chemotactic factor.

Plotting the evolution of the transition probability for the Ornstein-Uhlenbeck process
according to the Fokker-Planck equation in (<ref>).

A plot of the solution $p$ of (<ref>), for a suitable choice of the parameters $\vartheta$ and $\sigma$, is sketched in Figure <ref>:
the initial datum at $t=0$ corresponds to the probability of finding the random particle located at $x=1$
(and for this reason the picture exhibits a blowup at $(x,t)=(1,0)$) and we can appreciate that, as time flows,
the solution has the tendency to move its mass towards $x=0$.

Evolution of the transition probability for the Ornstein-Uhlenbeck process
at subsequent times.

See also Figure <ref> for
frames of this solution at times $t=0.1$, $1$, $2$, $10$.

The situation in which the general tendency of reaching an objective (such as the origin
in Figures <ref>
and <ref>)
is weighted against the possibility of diffusing around and missing it
is actually quite common in our everyday experience, hence
the plots in Figures <ref>
and <ref>
should be somewhat close to our intuition. One can think for instance
to the case of rogaining, in which, in principle compass bearings point you straight to
the next control, but in reality one has to account for some fluctuation around the goal
(due to some miscalculations of angles and distances for which the teammate has to be blamed, see e.g. Figure <ref>,
to the roughness of the terrain, to possible modifications of the landscape, to possible inaccuracies of the map, etc.): accordingly, the evolution of an initial position is realistically not given
by a single point (that is, a Dirac Delta Function located at a moving point)
but rather by a probability distribution describing how likely one is exactly at that point.
In this scenario, the origin in Figures <ref>
and <ref> represents the ideal destination corresponding to the maximal value of the probability distribution, but the tails of this distribution represent the possibility of having missed the control during navigation
due to whatever fluctuation (at that point, the rogainer will have to seek the control patiently
and accurately; knowing the probability density would be perhaps a good indication on how far one should search).

For additional information about Brownian motion,
random disturbances and the Fokker-Planck equation,
see <cit.> and the references therein.

The difference between planning routes on a map and getting there.

§.§ Phase coexistence models

A topical problem in material sciences is the understanding of the separation
patterns between different phases of a given substance.
The two phases could be related to molecule orientation, magnetization,
multi-component alloy systems, state of the matter (including superconductive
or superfluid states). In this context, the description of a physical system often relies on the understanding of suitable “order parameters”, that are suitable quantities capable of distinguishing different phases of a given system.

For instance, the magnetization vector $M$ expresses the density of magnetic dipole moment in a magnetic material. Roughly speaking, at each point $x$ of a given magnetic material, the direction of $M(x)$ reproduces the direction of the magnetic field near $x$ and the magnitude $|M(x)|$ represents the amount of magnetization.

Another classical example of order parameter arises in nematic crystals. nematic crystalThese materials consist of long and thin molecules that prefer to align with one another. In this situation, the two ends of the elongated molecules are essentially indistinguishable, therefore an efficient order parameter for these crystals is given by a vector $N(x)$ modulo a sign (that is, an element of the projective plane, or equivalently a vector on a hemisphere with opposing points along the equator identified). That is, while magnetic materials tend to displace the magnetic vectors in a parallel fashion, the nematic vectors can be equally displayed either in a parallel or antiparallel form (the technological advantage is thus that these crystals can be aligned by an external magnetic or electric field, as it happens for instance in the liquid crystal displays for video games).

While the above mentioned $M$ and $N$ are vectorial order parameters,
there are also ideal cases in which a scalar order parameter is sufficiently efficient to
describe some features of a system. For example, nematic crystals
may exhibit at some sufficiently large scale a “preferred direction” along which most
of its elements are aligned, which is called in jargon “local nematic director”. local nematic director In this context, scalar state parameters which are often adopted for practical purposes
are the functions of the angle $\vartheta$ between the liquid-crystal molecular axis and the local nematic director.

The simplest of this function may be constructed as follows. We take, up to a rotation, the
local nematic director to be the first vector $e_1$ of the Euclidean basis
and we consider a state parameter $S(\vartheta)$ such that:

(i). $S$ is $2\pi$-periodic (consistently with the fact that $\vartheta$ is an angle),

(ii). $S(\vartheta)=S(\vartheta+\pi)$ (consistently with the fact that
the two endings of the nematic crystals are indistinguishable),

(iii). $S(\vartheta)=S(-\vartheta)$ (consistently with the fact that this
order parameter should account for the “angular distance” of a molecule to the local nematic director),

(iv). $S(\vartheta)\le 1=S(0)$ (normalizing the maximal of the state parameter with $1$,
corresponding to the full alignment case),

(v). $\displaystyle\fint_{\partial B_1} S\big(\arccos(\omega\cdot e_1)\big)\,d{\mathcal{H}}^{n-1}_\omega=0$
(normalizing at $0$ the case of random directions, using the notation $\omega\cdot e_1=\cos\vartheta$),

see Figure <ref>.

A scalar order parameter using the local nematic director.

By (i), one can seek $S$ in the form of a Fourier Series, say

$$ S(\vartheta)=\frac{a_0}2+\sum_{j=1}^{+\infty} \left(a_j\cos(j\vartheta)+b_j\sin(j\vartheta)\right).$$

By (iii), we have that $b_j=0$ for all $j\in\{1,2,\dots\}$. Hence, using (ii),

$$ 0=S(\vartheta+\pi)-S(\vartheta)
=\sum_{j=1}^{+\infty} a_j\left( \cos(j(\vartheta+\pi))-\cos(j\vartheta)\right)
=-2\sum_{{j\ge1}\atop{j{\tiny\mbox{ odd}}}} a_j \cos(j\vartheta)

and therefore $a_j=0$ for all $j$ odd.

Thus, neglecting higher orders, one can take

$$ S(\vartheta)=\frac{a_0}2+a_2\cos(2\vartheta).$$

By (iv), we have that $\frac{a_0}2=1-a_2$, whence, in view of (v),

\begin{eqnarray*}&&
0=\fint_{\partial B_1} S(\arccos(\omega\cdot e_1))\,d{\mathcal{H}}^{n-1}_\omega
=1-a_2+a_2\fint_{\partial B_1} \cos(2\arccos\omega_1)\,d{\mathcal{H}}^{n-1}_\omega\\&&\qquad\qquad=
1-a_2+a_2\fint_{\partial B_1} \big(2\cos^2(\arccos\omega_1)-1\big)\,d{\mathcal{H}}^{n-1}_\omega\\&&\qquad\qquad
=1-2a_2+2a_2\fint_{\partial B_1} \omega_1^2\,d{\mathcal{H}}^{n-1}_\omega\\&&\qquad\qquad
\end{eqnarray*}

which, in the concrete case $n=3$, yields that $a_2=3/4$.

As a result,

$$ S(\vartheta)=\frac14+\frac34\cos(2\vartheta)=\frac12\big(3\cos^2\vartheta-1\big).$$

Interestingly, we will discover that this coincides with the
second Legendre polynomialLegendre polynomial
(see footnote <ref> on page LISTACON).

Coming back to the study of phase transitions,
Landau Landau theory
theory[This theory is named after Lev Davidovich Landau,
1962 Nobel Prize in Physics for his mathematical theory of superfluidity.
In 1938, Landau wrote a leaflet, joint with Mosey Korets, condemning
Stalin and the People's Commissariat for Internal Affairs (the forerunner of
MVD and KGB) on the occasion of the Great Purge in the Soviet Union
(see also footnote <ref> on page PURFO2).
According to the leaflet,
Stalin and his acolytes
“can only beat defenseless prisoners, catch unsuspecting innocent people, plunder national wealth and invent ridiculous trials against nonexistent conspiracies”.
The authors of the leaflet were sure that
“Proletariat [...] will throw off the fascist dictator and his clique”.

Things didn't work out as expected and Landau was arrested and held in the Lubyanka prison for more
than one year, see Figure <ref>.

There were however other features which possibly compensated Landau's dissident leaflet
in the eyes of Stalin. Landau led a team of mathematicians supporting Soviet atomic and hydrogen bomb development, thus receiving for his work the Stalin Prize in 1949 and 1953
and the title Hero of Socialist Labor in 1954. We should be always suspicious about ourselves
when we receive too many prizes and awards, because,
as all Spider-Man fans know, “with great power comes great responsibility”.

By the way, according to Wikipedia,
“Landau believed in free love rather than monogamy and encouraged his wife
and his students to practise free love. However, his wife was not enthusiastic”.]
attempts to understand the phase transitions related to continuous order parameters. Its main ansatz is that the equilibria of the system should come from a “free energy” produced by the order parameter under consideration. This energy may also be sensitive to the temperature of the system. For instance, one can consider an order parameter $\eta$ for a given system at temperature $T$ and describe the energy $E$ of the system as a function of $\eta$ and $T$ (for the sake of simplicity,
we take here $\eta$ to be a scalar order parameter, but the case
of vector valued order parameters can be treated in a similar manner).

Up to a normalization, one can suppose that the energy corresponding to $\eta=0$ is zero. Moreover, in many concrete cases, the energy of the physical system is invariant if we exchange $\eta$ with $-\eta$ (this is the case, for instance, of magnetic materials, since exchanging the North with the South pole should not alter the energy state of the system, and it is also the case of nematic crystals, since the ending of their molecules is symmetric, recall e.g. point (iii) here above).

These considerations lead to an energy $E(\eta,T)$ which is even in $\eta$ and such that $E(0,T)=0$.
The corresponding Taylor expansion
must therefore contain only even terms and takes the form

$$ E(\eta,T)=a(T)\eta^2+b(T)\eta^4+\dots$$

and we will indeed neglect higher order terms, thus reducing simply to the case

\begin{equation}\label{ENEFRE} E(\eta,T)=a(T)\eta^2+b(T)\eta^4.\end{equation}

The energy coefficients $a(T)$ and $b(T)$ depend on the temperature $T$
and their signs play a decisive role in the formation and separation of phases. More specifically,
one usually considers stable solutions in correspondence to minimal energy levels:
therefore, to avoid minimizers corresponding to an energy equal to $-\infty$, a natural structural assumption
is to suppose that $b(T)>0$ for every temperature $T$. Instead, to allow for
possible phase changes, one can assume that $a(T)$ changes sign above and below some critical temperature $T_c$,
for instance taking $a(T)>0$ when $T<T_c$ and $a(T)<0$ when $T>T_c$.

Plot of the function $\eta\mapsto\frac{(\eta^2-1)^2-1}4$.

In this setting, below the critical temperature the free energy exhibits the null value of the order parameter $\eta$
as its only minimizer, but above the critical temperature the stable
phase corresponds to $\pm\eta_0$ where $\eta_0$ is the minimizer for $E$ given by

$$ \eta_0:=\sqrt{-\frac{a}{2b}},$$

see Figure <ref>.

In particular, if given $T>T_c$, one wishes to normalize the stable phases at the levels $\pm1$
(as done for instance in point (iv) here above), up to normalizing factors one can choose $a(T)=-\frac12$ and $b(T)=\frac14$,
so that (<ref>) reduces to

\begin{equation}\label{DWHADEKLTSTMONAKN2}E(\eta)=-\frac12\eta^2+\frac14\eta^4=\frac{(\eta^2-1)^2-1}4\end{equation}

and the stable phases are the zeros of

\begin{equation}\label{DWHADEKLTSTMONAKN}

It is interesting to remark that this analysis of the free energy is useful to detect
the stable phases, namely $\pm1$ in (<ref>),
but it does not give any information on the separation between the two possible phases:
indeed, all configurations only attaining the stable phases $\pm1$ would indeed be
zeros of (<ref>), as well as minimizers of (<ref>).
However, in many concrete situations, one expects the separation between phases to
be somewhat minimal as well: one often experiences situations in which the two phases
occupy two separate “bulks” of the material which are separated by an interface, thus showing
a distinctive coarsening and phase separation.
This phenomenon is possibly the outcome of a “ferromagnetic”
effect, which tends to align the direction of the molecules of the material,
hence avoiding oscillations of the order parameter.

To include this phenomenon into the phase separation model, one
can modify the energy in (<ref>) by adding
a small penalization term which charges the formation of interfaces.
The simplest version of this procedure is typically to add to (<ref>)
a small “gradient term” (a gradient indeed detects the “local oscillation” of phases), i.e.
replace (<ref>) by

\begin{equation}\label{JohnWCahnandSam AllenE} {\mathcal{G}}(\eta)=\frac{\e}2\int_\Omega|\nabla\eta(x)|^2\,dx+\int_\Omega
\frac{(\eta^2(x)-1)^2-1}4\,dx,\end{equation}

for a small parameter $\e>0$ (where $\Omega$ is the region occupied by the material).

Looking for the corresponding minimal points of the full energy ${\mathcal{G}}$ leads to the equation

\begin{equation}\label{JohnWCahnandSam Allen}\e
\Delta\eta=\eta^3-\eta,
\end{equation}

which is often referred[Equation (<ref>)
is named after John Cahn and Sam Allen, who presented
a theoretical treatment of metal alloys <cit.>.

With respect to the classification
presented in footnote <ref> on page CLASSIFICATIONFOOTN,
we see that equation (<ref>) is of elliptic type.

We think that it is interesting to relate the “democratic” features of the Laplace operator,
according to the discussions on pages DEEFFNVST
and DEEFFNVSTBIS, to the ferromagnetic tendency of some materials which avoids
oscillations in spins and molecule alignments.

Furthermore, equation (<ref>) provides one of the chief examples
of “semilinear equations”, within a structure that we will investigate further  semilinear equation
from page Pohozaev Identity on. See also footnote <ref>
on page BIS Pohozaev Identity.

One can also appreciate the link between minimizers of the energy functional in (<ref>)
and interfaces of minimal area, by using
the Cauchy-Schwarz Inequality and the Coarea Formula (see e.g. <cit.>), namely setting $W(\eta):=
\frac{(\eta^2-1)^2}4$ and
noticing that

\begin{eqnarray*}&&{\mathcal{G}}(\eta)+\frac{|\Omega|}4=
\frac{\e}2\int_\Omega|\nabla\eta(x)|^2\,dx+\int_\Omega W(\eta(x))\,dx
\ge2\int_\Omega \sqrt{ \frac{\e}2|\nabla\eta(x)|^2W(\eta(x))}\,dx\\&&\qquad=
\sqrt{ 2{\e}}\int_\Omega |\nabla\eta(x)|\sqrt{W(\eta(x))}\,dx=\sqrt{2{\e}}
\int_\R \left[\int_{\{\eta=\tau\}} \sqrt{W(\eta(x))}\,d{\mathcal{H}}^{n-1}_x
\right]\,d\tau.
\end{eqnarray*}

Hence, heuristically, it should be convenient for a minimizer $\eta_\e$ to sit “as much as possible”
into the minima of $W$ (which correspond to the stable phases $\pm1$): if we assume that the level sets of $\eta_\e$
are surfaces “more or less parallel” to an interface ${\mathcal{S}}$
and $\eta_\e$ approaches “quite fast” the stable phases $\pm1$, the previous
computation suggests that the minimization of ${\mathcal{G}}$ reduces, as a first approximation,
to the minimization of the area of the interface ${\mathcal{S}}$.

Of course, it is not easy to transform the above heuristic argument into a rigorous proof, since the setting in (<ref>) is that of a singular perturbation singular perturbation,
namely the small parameter $\e$ affects precisely the most significant term in equation (<ref>).
As a matter of fact, the coherent development of arguments of this sort required the introduction
of a novel notion of asymptotics called $\Gamma$-convergence, $\Gamma$-convergence
see <cit.>.]
to with the name of Allen-Cahn equation. Allen-Cahn equation

See e.g. <cit.> and the references therein for additional details
on phase separation models.

Lev Landau in prison, photo by the People's Commissariat for Internal Affairs (Public Domain image from

§.§ Growth of interfaces

A number of scientific problems of interest are associated with the growth of the profile of suitable surfaces which describe, for instance, tumors, flame fronts, clusters, etc.

Opening ceremony at the Italian Parliament (October 10, 2021;
attribution: Presidenza della Repubblica https://www.quirinale.it/elementi/60152#&gid=1&pid=10).

A typical model describing this phenomenon is given  by the so-called[Equation (<ref>) is named after Mehran Kardar, Giorgio Parisi and Yi-Cheng Zhang, who introduced this model in <cit.>.
Parisi is also known for his contributions in
quantum fluctuations, spin glasses and whirling flocks of birds. See Figure <ref>
in which the Italian President Sergio Mattarella (on the right) congratulates Parisi (on the left)
for having received the 2021 Nobel Prize in Physics (they are wearing face masks due to
COVID regulations).]
Kardar-Parisi-Zhang equationKardar-Parisi-Zhang equation

\begin{equation}\label{KPZEQUA}
\partial_t h=\mu \Delta h+\lambda |\nabla h|^2+f,
\end{equation}

where $\mu>0$ is a diffusion parameter, $\lambda$ is a parameter related to the
surface growth and $f$ is a forcing term (which could be either a deterministic function
or a stochastic term, such as a white noise). Of course, when $\lambda=0$ the setting in (<ref>)
essentially boils down to that of the heat equation in (<ref>), but when $\lambda\ne0$
the equation presents a nonlinear term of geometric importance.

Indeed, the motivation underpinning (<ref>) goes as follows. Suppose
that we have some aggregate with an active zone of growth on its surface
which is described, at a given time $t$, by the graph of a certain function $y=g(x,t)$.
We consider an “infinitesimal” time step $\tau$ in which the surface growth takes
place and suppose that such a growth is regulated by three ingredients:
first, in time $\left(t,t+\frac\tau3\right)$, some particle is added to the surface,
then, in time $\left(t+\frac\tau3,t+\frac{2\tau}3\right)$ a forcing term kicks in,
finally, in time $\left(t+\frac{2\tau}3,t+\tau\right)$ some diffusion takes place (the order
in which these phenomena occurs is not really important, but we fix this order just to keep a concrete case in mind).

To describe the first step, we assume that spherical particles of diameter $\tau$
are added uniformly along the surface of the cluster, as described in Figure <ref>.
In this setting, if $\vartheta$ denotes the angle between the normal
of the spherical particle and the vertical direction, we have that the infinitesimal vertical surface growth
corresponds to $\frac\tau{\cos\vartheta}$. Hence we write that

\begin{equation}\label{KPZ-Eq-1} g\left(x,t+\frac\tau3\right)=g(x,t)+\frac\tau{\cos\vartheta}.\end{equation}

Actually, the normal of the spherical particle agrees with the normal of the original surface $\frac{\left(-\nabla g(x,t),1\right)}{\sqrt{1+|\nabla g(x,t)|^2}}$ and accordingly

$$\cos\vartheta=\frac{\left(-\nabla g(x,t),1\right)}{\sqrt{1+|\nabla g(x,t)|^2}}\cdot e_n=
\frac{1}{\sqrt{1+|\nabla g(x,t)|^2}}.$$

We substitute this information into (<ref>) and we find that

\begin{equation}\label{KPZ-Eq-2} g\left(x,t+\frac\tau3\right)=g(x,t)+ \tau \sqrt{1+|\nabla g(x,t)|^2}.\end{equation}

Now we describe the second step, namely the action of a forcing term: this effect is encoded in the equation

\begin{equation}\label{KPZ-Eq-3} g\left(x,t+\frac{2\tau}3\right)= g\left(x,t+\frac\tau3\right)+ \tau f\left(x,t+\frac\tau3\right).\end{equation}

As for the third step, we assume that diffusion takes place in the form

\begin{equation}\label{KPZ-Eq-4} g(x,t+\tau)= g\left(x,t+\frac{2\tau}3\right)+ \tau \Delta g\left(x,t+\frac{2\tau}3\right)

Thus, by collecting the observations in (<ref>), (<ref>) and (<ref>),

\begin{eqnarray*}
&&\frac{g(x,t+\tau)-g(x,t)}\tau\\&=&\frac{\displaystyle g(x,t+\tau)-g\left(x,t+\frac{2\tau}3\right)}\tau
+\frac{ \displaystyle g\left(x,t+\frac{2\tau}3\right)-g\left(x,t+\frac{\tau}3\right)}\tau+
\frac{\displaystyle  g\left(x,t+\frac{\tau}3\right)-g(x,t)}\tau\\&=& \Delta g\left(x,t+\frac{2\tau}3\right)
+ f\left(x,t+\frac\tau3\right)+\sqrt{1+|\nabla g(x,t)|^2},
\end{eqnarray*}

whence, formally taking the limit as $\tau\searrow0$,

\begin{equation}\label{KPZ-Eq-5} \partial_t g(x,t)= \Delta g(x,t)
+ f(x,t)+\sqrt{1+|\nabla g(x,t)|^2}

When the slope of the growing interface is small, one can consider $|\nabla g|$ as a small perturbation
and employ the approximation $\sqrt{1+|\nabla g|^2}\simeq 1+\frac{|\nabla g|^2}2$.
In this framework, one reduces (<ref>) to

\begin{equation*} \partial_t g= \Delta g
+ f+1+\frac{1}{2}|\nabla g|^2.\end{equation*}

Considering the velocity shift $h(x,t):=g(x,t)-t$, we thus conclude that

\begin{equation*} \partial_t h=
\partial_t g-1=
\Delta g
+ f+\frac{1}{2}|\nabla g|^2
=\Delta h
+ f+\frac{1}{2}|\nabla h|^2,
\end{equation*}

providing a motivation for the
Kardar-Parisi-Zhang equation in (<ref>).

Time evolution of a growing interface.

We mention that, in spite of its apparent simplicity, equations such as the one in (<ref>)
do capture complex phenomena and lead to the study of important
universality classes sharing the same characteristic exponents and scale invariant limits,
see e.g. <cit.>.

Sketch of Harry Bateman (Public Domain image from

Furthermore, equation (<ref>) is also directly related to other classical equations.
In particular, defining $v:=\nabla h$, we have that

\begin{eqnarray*}&&
\partial_t v+v\cdot\nabla v-\mu\Delta v-\nabla f=\nabla\Big(
\partial_t h-\mu\Delta h-f\Big)+\sum_{j=1}^n \partial_jh\, \partial_j \nabla h\\&&\qquad=\nabla\Big(
\partial_t h-\mu\Delta h-f\Big)+\frac12\nabla \left(\sum_{j=1}^n |\partial_jh |^2\right)\\&&\qquad=\nabla\left(
\partial_t h-\mu\Delta h-f+\frac12|\nabla h|^2\right),
\end{eqnarray*}

from which it follows that the
Kardar-Parisi-Zhang equation in (<ref>) with $\lambda:=\frac12$
is equivalent to the vectorial equation

\begin{equation}\label{VERBURG}
\partial_t v+v\cdot\nabla v=\mu\Delta v+\nabla f,
\end{equation}

which is a version of the Navier-Stokes equation in (<ref>) in the absence of gravity
(in this setting, $v$ corresponds to the speed of a given fluid, $\mu$ is its viscosity coefficient
and $f$ is minus the pressure).

For constant pressure, in dimension $n=1$, equation (<ref>) reduces to the scalar equation

\begin{equation}\label{VERBURG2}
\partial_t v+v\partial_x v=\mu\partial_{xx} v,
\end{equation}

which is known in jargon[Equation (<ref>)
is named after Harry Bateman and Johannes Martinus Burgers.
See Figure <ref> for a nice drawing of Harry Bateman
from the 1931 yearbook of the California Institute of Technology.] as the Bateman-Burgers equation Bateman-Burgers equation,
which models the speed of a fluid in a thin ideal pipe.

In this context, it is interesting to point out that in the absence of viscosity equation (<ref>) boils down to

\begin{equation}\label{VERBURG3}
\partial_t v+v\partial_x v=0,
\end{equation}

known as the inviscid Bateman-Burgers equation. A feature of the solutions of (<ref>)
is that they may develop “shock waves”, shock wave
i.e. singularities at a breaking time. For instance,
it is readily seen that the relation $v(x,t)=\arctan\big( tv(x,t)-x\big)$
provides, via the Implicit Function Theorem, a solution of (<ref>) for small times, with $v(x,0)=-\arctan x$.
This solution exhibits a shock wave since otherwise, for $t>1$,

$$ |\partial_x v(0,t)|=\left|\frac{t\partial_x v(0,t)}{\cos^2\big(tv(0,t)\big)}\right|\ge | t\partial_x v(0,t)|>|\partial_x v(0,t)|,

which is a contradiction. See Figure <ref>
for a sketch of the formation of such a shock wave.

Implicit plot of $v=\arctan( tv-x)$ for $t\in\{0.5,\,0.75,\,0.99,\,2\}$.

See e.g. <cit.> for more information on shock waves
in the context of the Bateman-Burgers equation.
See also Figure <ref> for a rather dramatic view of
expanding spherical atmospheric shock waves from a gun firing on the surface of the water.

The battleship USS Iowa of the	United States Navy
during a training exercise in Puerto Rico (Public Domain image from

§.§ Definitions come later on

Given $a$, $b$, $c\ge0$, the equation

\begin{equation}\label{HEATELEEQ0}
\frac{\partial ^{2} u}{\partial x^{2}}-a\frac{\partial ^{2} u}{\ \partial t^{2}}=
b \frac {\partial u }{\partial t} +c u
\end{equation}

is called[The inventor of equation (<ref>)
was Oliver Heaviside, see Figure <ref>.

Heaviside's uncle was Sir Charles Wheatstone, co-inventor of the first commercially successful telegraph. Following his uncle's advice, Heaviside became a telegraph operator and an electrician, continuing to study and do science while working.

His contributions to mathematics, physics and engineering were deep and remained as classical contributions: they include the use of complex numbers to solve circuit analysis differential equations, the calculus of the deformations of an electromagnetic field surrounding a moving charge and the derivation of the magnetic force on a moving charged particle (the first contribution towards the understanding of the Lorentz Force) and the prediction of the existence of a reflective layer of the ionosphere (which allowed radio waves radiated into the sky to return to Earth beyond the horizon).

Several works by Heaviside
were of great practical use, including the possible exploitation of loading coils in telephone and telegraph lines to increase their self-induction and correct the distortion which they suffered.

Several years later, some American telecommunications companies hired their own scientists to extend Heaviside's work and adapt the use of coils previously introduced by Heaviside. Some corporations later offered Heaviside money in exchange for his rights, but he declined to accept any money unless the company were to give him full recognition for his discoveries and inventions (it turns out that Heaviside remained for all his life chronically poor).

Heaviside was perhaps a bit eccentric too. He was a firm opponent of Einstein's theory of relativity, possibly beyond reasonable scientific arguments, and at the end of his life he developed a very strong aversion to meeting people and became a recluse. He died at age 74 after falling from a ladder.

Heaviside said: “Mathematics is an experimental science, and definitions do not come first, but later on. They make themselves, when the nature of the subject has developed itself”.]
the telegrapher's equation. telegrapher's equation

In equation (<ref>), we have that $u=u(x,t)$ with $x\in\R$, $t\in[0,+\infty)$.
When $a:=0$, $c:=0$ and $b>0$, equation (<ref>) boils down to
the heat equation in (<ref>).
When $b:=0$, $c:=0$ and $a>0$, it reduces[With respect to the terminology
in footnote <ref> on page CLASSIFICATIONFOOTN,
we note that equation (<ref>) is hyperbolic when $a>0$, parabolic when $a:=0<b$
and elliptic when $a:=0$ and $b:=0$.]
to the wave equation
in (<ref>).

When $a$, $b$, $c>0$ however the behavior of the solutions of (<ref>) are interestingly different from
those of the wave equation,
since solutions of the telegrapher's equation present damping and dispersion effects
that are not present in the wave equation.
See e.g. Figure <ref>
in which the evolution of the solution of the telegrapher's equation
exhibits a visible dispersion, in which the velocity of travel depends on the frequency thus
enlarging the support of the solution, and losses of intensity, causing the peaks of the traveling front to reduce over time.

The solution $u$ in equation (<ref>) describes the electric current intensity (or, in a similar manner,
the voltage) along a transmission line.
This transmission line can be a telegraph wire, a overhead electrical conductor, a telephone line, etc.,
see Figure <ref>. See also Figure <ref>
for a peculiar use of transmission lines.

To get to the bottom of the telegrapher's equation (<ref>), we consider
an electric line made of two parallel electrical wires.
There is a voltage difference between the wires (e.g., produced by an electric generator
“at the end of the wires”: since the wires in our idealized model are straight lines, the generator is essentially
“at infinity”).

Oliver Heaviside (Public Domain image from

Thinking at the two wires just as parallel straight lines would be however reductive,

* each straight line is actually made by a conductor which presents some
distributed electric resistance (say, modeled by a series resistor) and also some
distributed inductance (e.g., due to the magnetic field around the wires and to self-inductance, modeled by a series inductor),

* furthermore the dielectric material separating the two conductors is also not completely neutral from
the electric point of view, since it can carry
capacitance and conductance (modeled by
a shunt capacitor and resistor located between each “infinitesimal” portion $dx$ of the conductors).

To describe these phenomena,
as customary in the physical description of electric phenomena, we
reserve the name $I$ for the current, $V$ for the voltage, $R$ for the
resistance, $L$ for the inductance and $C$ for the capacity. We also use the letter $G$ to
denote the conductance, which is the reciprocal of a resistance.
The parameters $R$, $L$, $C$ and $G$ will be treated as structural constants
and we will be interested in the description of the functions $I=I(x,t)$
and $V=V(x,t)$ with respect to the position $x\in\R$ on the transmission line
and the time $t\ge0$.

More specifically, for concreteness we focus on the current in the upper conductor, see Figure <ref>: in this setting,
the distributed resistance in the infinitesimal elemental length $dx$ of the conductor
is denoted by $R\,dx$
and the distributed inductance is denoted by $L\,dx$.

The conductance of the dielectric material separating the two conductors
(accounting for bulk conductivity of the dielectric and dielectric loss) is denoted by $G\,dx$
and the capacity by $C\,dx$ (notice that, in “real life” there is no wire connecting the top and the bottom
cable, but Figure <ref> translates the behavior of the dielectric
between the two conductors into the language of electric circuits).

Now, to describe the current flowing through the upper wire, we denote by $I(x,t)$ the current intensity on the left end
of the elementary upper conductor in Figure <ref>
and by $I(x+dx,t)$ the one on the right end (say, with the convention that
the current is traveling from left to right).
Similarly, we denote by $V(x,t)$ the voltage on the left end
of the elementary upper conductor and by $V(x+dx,t)$ the one on the right end
(actually, $V$ would stand for the difference of voltage from the upper and the lower conductors;
for simplicity[Actually, telegraphs originally used two wires, utilizing a forward and a return paths
in a closed circuit to move energy along the transmission line.
But it was then noticed that Earth itself can be used as the return path. In this setting, the bottom wire is just the ground, which is normalized to be approximatively at constant zero volt (from practical purposes however, Earth is an adequate, but certainly not optimal, return path). Thus, in the situation in which the transmission line is modeled by one forward cable using Earth as a return path, the poles are also considered as dielectric and provide some capacitance and conductance, see Figure <ref>. In this case,
the vertical elements related to $G\,dx$ and $C\,dx$ in Figure <ref>
can be thought as “concrete objects”, such as the telegraph poles, located at an “infinitesimal” distance $dx$
at a large scale (in which the transmitter and the receiver are located “at infinity”).

See e.g. https://youtu.be/ySuUZEjARPY for a very thorough video about electric transmission.] one can think that the conductor at the bottom is at zero voltage).
Let also $\iota_1$ be the current between the nodes of the upper conductor (oriented left to right),
and $\iota_2$ and $\iota_3$ be the current through the
shunt capacitor $C\,dx$ and resistor $G\,dx$ respectively (oriented downwards).

With this notation, by Kirchhoff's Junction Rule (or simply by conservation of charge), we have that

$$ I(x,t)=\iota_1+\iota_3\qquad{\mbox{and}}\qquad
\iota_1=I(x+dx,t)+\iota_2.$$

Also, by the Laws of Ohm and Faraday,

$$ V(x,t)=\frac{\iota_3}{G\,dx}\qquad{\mbox{and}}\qquad
V(x+dx,t)-V(x,t)=-RI(x,t)\,dx-L\partial_t I(x,t)\,dx.$$

Additionally, by the definition of capacity,

Comparison between the evolution in time of the solution of the wave equation and of the
telegrapher's equation (author
Jacopo Bertolotti, https://twitter.com/j_bertolotti/status/1172517281374572551,
images from
Wikipedia, licensed under the Creative Commons CC0 1.0 Universal Public Domain Dedication).

These considerations lead to

\begin{eqnarray*}
\partial_x V(x,t)\,dx\simeq V(x+dx,t)-V(x,t)=-RI(x,t)\,dx-L\partial_t I(x,t)\,dx
\end{eqnarray*}

\begin{eqnarray*}&&
\partial_x I(x,t)\,dx\simeq
=-\iota_2-\iota_3=-C\partial_t V(x,t)\,dx-GV(x,t)\,dx.
\end{eqnarray*}

\begin{equation}\label{ihkbdDAvbsdewI089JHGSFDCokwfIUyb-X1}
\begin{dcases}
\partial_x V=-L \partial_t I-R I,\\
\partial_x I=-C\partial_t V-GV.
\end{dcases}
\end{equation}

We can take the derivative with respect to $x$ of the first equation
and compare with the derivative with respect to $t$ of the second equation
in (<ref>), finding that

\begin{equation}\label{ihkbdDAvbsdewI089JHGSFDCokwfIUyb-X2}
\partial_x^2 V-LC \partial_t^{2} V=(RC+GL) \partial_t V+GRV.
\end{equation}

Left: utility pole for a telephone line; center: electrical wires;
right: overhead lines in Queensland
(images from
Wikipedia, Public Domain for the first, photo by Novoklimov, licensed under the Creative Commons Attribution-Share Alike 4.0 International, licensed under the Unported license
for the second, photo by Pytomelon87 licensed under the Creative Commons Attribution-Share Alike 4.0 International license for the third).

Similarly, we can take the derivative with respect to $t$ of the first equation
and compare with the derivative with respect to $x$ of the second equation
in (<ref>), which yields that

\begin{equation}\label{ihkbdDAvbsdewI089JHGSFDCokwfIUyb-X3}
\partial_x^{2} I-LC\partial_t^{2}I=(RC+GL) \partial_t I+GRI.
\end{equation}

We stress that, up to replacing voltage and current, the structure of (<ref>)
and (<ref>) is the same.
Also, (<ref>) and (<ref>)
give that both voltage and current are solutions of the telegrapher's equation in (<ref>).

White storks nesting on an utility pole
(photo by Myrabella; image from
Wikipedia, licensed under the
Creative Commons Attribution-Share Alike 3.0 Unported license).

It is interesting to note that, in this situation, the structural parameters
in the telegrapher's equation in (<ref>)
correspond to $a:=LC$, $b:=RC+GL$ and $c:=GR$.
Thus, a lossless transmission in which the roles of resistances can be neglected corresponds to $R:=0$ and $G:=0$
(this is a perfect conductor offering no resistance and a perfect electrical insulator as a dielectric allowing for no
conductance), which in turn produce $b:=0$ and $c:=0$ in (<ref>), thus
reducing to the wave equation.
With this observation in mind, one can also have a second look at Figure <ref> to appreciate how the damping and dispersion effects
are the outcome of the imperfect conductors and dielectric in terms of resistance and electrical insulation.

Schematic for an elemental length $dx$ of transmission line (image by Omegatron from
Wikipedia, licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).

Transmission line for a telegraph, using Earth as return path.

§.§ Nerd sniping

Nerd sniping is a new sport invented in https://xkcd.com/356/
in which the value of mathematicians is frightfully appreciated,
see Figure <ref>. Here, we discuss the solution[The infinite resistor problem
happened to be studied quite in detail, see in particular <cit.>,

$\phantom{123456789}$ on-this-infinite-grid-of-resistors-whats-the-equivalent-resistance

and the references therein.

Strictly speaking, without additional information, the problem is possibly not uniquely posed, but in these pages we confine ourselves
to the case in which the solution is assumed to decay “fast enough” at infinity to validate the formal computations
showcased here, though we gloss over any technical aspect related to uniqueness, decay, convergence
and infinite cancellations (but let us mention, for instance, that if $\Gamma(k)$ is a solution of (<ref>), then
so is $\Gamma(k)+k_1$ and the equivalent resistance $R$ in (<ref>)  would thus
be affected by an additional term $-4$, showing a uniqueness issue unless we add some decay assumption on $\Gamma$).

Also, the webcomic xkcd was created by Randall Patrick Munroe, see Figure <ref>.]
of the infinite resistor problem presented in the fourth cartoon of Figure <ref>
and its strong relation with elliptic partial differential equations (we mean, not the relation between nerds and
elliptic partial differential equations, but the one between the infinite resistor problem
and a discrete variant of elliptic partial differential equations). Actually, the answer to the problem is

\begin{equation}\label{NERD:ANS}
\frac4\pi-\frac12,\end{equation}

but you'll see, it's about the journey, not the destination.