# Jamming pair of general run-and-tumble particles:
Exact results and universality classes

Léo Hahn  Arnaud Guillin  Manon Michel<EMAIL_ADDRESS>Laboratoire de
Mathématiques Blaise Pascal UMR 6620, CNRS, Université Clermont-Auvergne,
Aubière, France.

###### Abstract

We consider a general system of two run-and-tumble particles interacting by
hardcore jamming on the unidimensional torus. RTP are a paradigmatic active
matter model, typically modeling the evolution of bacteria. By directly
modeling the system at the continuous-space and -time level thanks to
piecewise deterministic Markov processes (PDMP), we derive the conservation
conditions which sets the invariant distribution and explicitly construct the
two universality classes for the steady state, the _detailed-jamming_ and the
_global-jamming_ classes. They respectively identify with the preservation or
not of a detailed symmetry at the level of the dynamical internal states
between probability flows entering and exiting jamming configurations. Thanks
to a spectral analysis of the tumble kernel, we give explicit expressions for
the invariant measure in the general case. The non-equilibrium features
exhibited by the steady state includes positive mass for the jammed
configurations and, for the global-jamming class, exponential decay and growth
terms, potentially modulated by polynomial terms. Interestingly, we find that
the invariant measure follows, away from jamming configurations, a catenary-
like constraint, which results from the interplay between probability
conservation and the dynamical skewness introduced by the jamming
interactions, seen now as a boundary constraint. This work shows the powerful
analytical approach PDMP provide for the study of the stationary behaviors of
RTP systems and motivates their future applications to larger systems, with
the goal to derive microscopic conditions for motility-induced phase
transitions.

Active matter systems such as bacterial colonies [1, 2, 3, 4] or robot swarms
[5, 6] are characterized by the breaking at the microscopic scale of energy
conservation and time reversibility [7, 8, 9]. Compared to their passive
counterparts, these systems exhibit rich behaviors including motility-induced
phase separation [10, 11], emergence of patterns [12] or collective motion
[13, 14]. Their study remains challenging because many tools from equilibrium
statistical mechanics do not apply. In particular there is no guarantee the
steady state follows some Boltzmann distribution, even up to the definition of
an effective potential. This has fueled interesting developments in both
physics and mathematics, but which typically rely on a coarse-grained approach
[15, 8, 16, 17, 18, 19]. While they successfully recover some out-of-
equilibrium macroscopic behaviors, they leave open the question of the
microscopic origin of a given phenomenon, a question strongly linked to the
fundamental one of universality.

It motivates the derivation of an exact theory of active-particle systems
directly from the microscopic scale of the individual particle, without any
coarse-graining apart the stochastic description of individual particle
motion. Recent works [20, 21] deal with the derivation of large-scale limit of
active lattice-gas models [22], typically aiming at deriving the steady state.
While being exact, such hydrodynamic limit requires strong conditions
(symmetric jumps, particular exclusion rule), in order to ensure a rigorous
limit derivation. Furthermore, any results at a large but finite size requires
a subtle computation of fluctuating terms, as done in Macroscopic Fluctuation
Theory [23, 24]. Whereas such terms impacts the existence of metastable
solutions, their derivation remains challenging [25].

Another approach focuses on the derivation of exact results for microscopic
models [26, 27], with a strong motivation to decipher the impact of the
reversibility breaking. They consider discrete models of two Run-and-Tumble
Particles (RTP) with a hardcore jamming interaction on a 1D-ring. The RTP is a
paradigmatic active matter model mimicking the moving pattern of bacteria such
as E. Coli [1]. RTPs perform a series of straight runs separated by stochastic
refreshments of the run direction called tumbles. Even a single RTP displays
interesting out-of-equilibrium features [28, 29, 30]. In [26, 27], the steady
state distribution is exactly determined from a Master equation on a lattice
by a generating function approach, respectively for the case of instantaneous
and finite tumbles. The continuous limit is also discussed. By doing so, they
identify the Dirac jamming contributions to the stationary distribution and
how a new lengthscale, linked to an exponential decay or growth, appears in
the case of the finite tumble. While these results are a first, questions
remain open regarding the rigorous continuum limit of such discrete lattice
models, the portability of such involved methods to more general systems and
dimensions and the actual capacity of lattice models to capture the particular
role that domain boundaries seem to play in such piecewise-ballistic systems.
More importantly, the definition of some general universality classes ruling
the steady state remains yet open.

In this work, we answer this precise question and show that a general periodic
two-RTP system with jamming interactions is entirely described by two explicit
universality classes that we name the _detailed-jamming_ and _global-jamming_
classes. They identify with the preservation or not of a detailed symmetry at
the level of the dynamical internal states between probability flows entering
and exiting jamming configurations. The key idea is to build a formalism of
the RTP dynamics incorporating all the present symmetries (particle
indistinguishability, space homogeneity and periodicity) and directly at the
continuous-space and -time level thanks to Piecewise deterministic Markov
processes (PDMP) [31, 32]. It then gives a direct access to the impact of any
dynamical skewness introduced by the jamming as a boundary constraint on the
bulk dynamical relaxation ruled by the tumbles. This approach is inspired by
developments in the sampling of equilibrium systems by non-reversible and
continuous-time Markov processes, known as _Event-chain Monte Carlo_ [33, 34]
and also formalized as PDMP [35]. Such stochastic process comes down in
particle systems to generating an artificial RTP dynamics. Time reversibility
is broken but the equilibrium Boltzmann distribution is still left invariant
by the preservation at the particle collisions, i.e. on the boundary of the
set of valid configurations, of some key invariances, be it pairwise [34],
translational [36] or rotational [37]. This motivated a similar investigation
of which symmetry is preserved on the boundaries, but in an active system, and
whether it is sufficient to fully determine the steady state.

Thus, we provide in Section I a full description of the two-RTP process as
PDMP, including their generator (an infinitesimal description of the
evolution). This is the basis to obtain in Section II the conservation
constraints on the exact steady state distribution. A careful analysis leads
in Section III to the explicit definition of the two universality classes and
their respective steady-state expressions for general continuous models, which
are consistent with the continuous-space limit of the particular discrete
cases studied in [26, 27]. Finally we discuss in Section IV the light now shed
on the interplay between reversibility breaking and jamming and the
perspectives it brings.

##  I Two-RTP system as PDMP

We consider a pair of interacting RTPs on a one-dimensional torus of length
$L$. A single-particle possible internal state is described by the variable
$v$, which typically codes for the velocity of the particle and commonly takes
values in $\\{-1,+1\\}$ (two-state RTP or instantaneous tumble) or in
$\\{-1,0,+1\\}$ (three-state RTP or finite tumble). Aiming at universality
class definition, we adopt a general setting and, therefore, we do not
restrict the value of $v$, which can be continuous. Both particles change
their internal state independently according to a Poisson jump process set by
the transition rate $\omega(v)>0$ and kernel $q$, which are general but
homogeneous in space and yielding ergodicity in the internal-state space. The
particles interact with each other through hardcore interactions leading to
jammed states where the two RTPs are colliding against each other until a
tumble allows the particles to escape, as illustrated in Fig. 1.

We now show how to formalize the system and all its symmetries in a
continuous-time and continuous-space setting by using PDMP. First, we note
$x_{t,1},x_{t,2}$ the positions of the particles, and $v_{t,1}$ and $v_{t,2}$
the particle internal states. Thanks to the homogeneity, periodicity and
particle indistinguishability, the system can entirely be described at time
$t$ by the periodic interdistance $0\leq r_{t}\leq L/2$ between particles,
defined as $\min(|x_{t,1}-x_{t,2}|,L-|x_{t,1}-x_{t,2}|)$. As shown in Fig. 1,
the interdistance $r_{t}$ then undergoes the following evolution, depending if
it is located in the _bulk_ (a), at the _periodic_ (b) or _jamming_ (c)
boundaries:

(a - Run) While in the _bulk_ ($0<r_{t}<L/2$), the interdistance $r_{t}$ is
updated through a deterministic evolution in three possible regimes:
increasing ($(v_{t,1}-v_{t,2})(x_{t,1}-x_{t,2})\gtrless 0$ if
$|x_{t,1}-x_{t,2}|\lessgtr L/2$), decreasing
($(v_{t,1}-v_{t,2})(x_{t,1}-x_{t,2})\lessgtr 0$ if $|x_{t,1}-x_{t,2}|\lessgtr
L/2$) and stalling ($v_{t,1}=v_{t,2}$).

(a - Tumble) The transitions inside and between regimes stem from a particle
tumble and are ruled by the superposed Poisson process of rate
$\omega(v_{1})+\omega(v_{2})$. No direct transition between two states of the
stalling regime can occur, as it would require a simultaneous tumble from both
particles, which is of null probability.

(b) When reaching the _periodic_ boundary ($r_{t}=L/2$) in an increasing
regime, the interdistance automatically switches, by periodicity, from an
increasing to a decreasing regime.

(c) Finally, and most interestingly, when the _jamming_ boundary ($r_{t}=0$)
is reached in a decreasing regime, the interdistance is stalling in $0$ until
a tumble occurs allowing the particles to separate from each other and making
$r_{t}$ re-enter the _bulk_ in an increasing regime.

Figure 1: Trajectory of two RTPs undergoing instantaneous tumble (left), and
of their corresponding periodic interdistance (right) ($\omega=0.2$, $L=10$).

We now characterize the stochastic dynamics of $r_{t}$ by a PDMP [31, 32],
which is entirely encoded:

(a) by its infinitesimal generator in the bulk, $0<r_{t}<L/2$,

$\mathcal{A}_{\mathcal{B}}\\!=\\!\underbrace{\nu\phi(s)\partial_{r}}_{\text{Run
phase}}+\underbrace{\tilde{\omega}(s)(Q((\nu,s),\cdot)-\text{Id})}_{\text{Tumble
phase}},$ (1)

where we introduce the two auxiliary dynamical variables
$(\nu_{t},s_{t})\in\mathcal{V}$ (so that the complete bulk set is
$\mathcal{B}=]0,L/2[\times\mathcal{V}$): The variable $\nu_{t}\in\\{-1,0,1\\}$
sets the nature of the regime as defined in (a) and $\mathcal{V}$ can be
decomposed as $\mathcal{V}_{-1}\cup\mathcal{V}_{0}\cup\mathcal{V}_{1}$. The
variable $s_{t}=\\{v_{t,1},v_{t,2}\\}$, encoding the particle
indistinguishability, sets the amplitude of the evolution by
$\phi(s_{t})=|v_{t,1}-v_{t,2}|$ and the tumble by its rate
$\tilde{\omega}(s_{t})=\omega(v_{t,1})+\omega(v_{t,2})$ and its transition
Markov kernel $Q((\nu,s),\cdot)$,

$\begin{split}Q((\nu,\\{v,\tilde{v}\\}),(\nu^{\prime},\\{v^{\prime},\tilde{v}\\}))\\!=\\!q(v,v^{\prime})\big{(}\tfrac{1}{2}\mathbb{1}_{\\{0\\}}(\nu)\mathbb{1}_{\\{-1,1\\}}(\nu^{\prime})\\\
\
+\tfrac{\omega(v)}{\tilde{\omega}(\\{v,\tilde{v}\\})}\mathbb{1}_{\\{-1,1\\}}(\nu)\mathbb{1}_{\\{\text{sign}(\nu\nu^{\prime})\\}}\big{(}\text{sign}\big{(}\tfrac{v^{\prime}-\tilde{v}}{v-\tilde{v}}\big{)}\big{)}\big{)},\end{split}$

all other transitions being of probability 0, as they would involve tumble of
both particles. Thus, the particle indistinguishability leads to an isotropic
symmetry for $Q$,

$Q((\nu,s),(\nu^{\prime},s^{\prime}))=Q((-\nu,s),(-\nu^{\prime},s^{\prime})),$
(2)

as illustrated on Fig. 2 for the instantaneous and finite tumbles. The
symmetry (2) will have an important impact on the possible invariant
distributions and justifies the introduction of the representation $(\nu,s)$.

Figure 2: Transitions between the internal states at the single particle
(left) and interdistance level (right) in the case of instantaneous (top, and
finite tumble (left), in a reduced representation for $s$ thanks to the
single-particle tumble isotropy (instantaneous: $s$ can be omitted, finite:
$s$ codes for the number of moving particles).

(b) by its behavior at the periodic boundary: at
$(r,\nu,s)\in\\{L/2\\}\times\mathcal{V}_{1}=\Gamma^{*}_{P}$, the periodic
boundary kernel
$Q_{P}((r,\nu,s),(r^{\prime},\nu^{\prime},s^{\prime}))=\mathbb{1}_{\\{(r,-\nu,s)\\}}(r^{\prime},\nu^{\prime},s^{\prime})$
codes for a switch from an increasing ($\nu=1$) to a decreasing
($\nu^{\prime}=-1$) regime.

(c) and finally by its behavior at the jamming boundary: for
$(r,\nu,s)\in\\{0\\}\times\mathcal{V}_{-1}=\Gamma^{*}_{J}$, the jamming
boundary kernel,
$Q_{J}((r,\nu,s),(r^{*},\nu^{*},s^{*}))=\mathbb{1}_{\\{(r,\nu,s)\\}}(r^{*},\nu^{*},s^{*})$
makes the system jumps into a jammed configuration
$(r^{*},\nu^{*},s^{*})\in\mathcal{J}=\\{0\\}\times\mathcal{V}_{-1}\cup\mathcal{V}_{0}$,
which is ruled by the following infinitesimal generator,

$\mathcal{A}_{\mathcal{J}}\\!=\\!\tilde{\omega}(s^{*})(Q((\nu^{*},s^{*}),\cdot)-\text{Id}),$
(3)

which translates the persistence of the jamming state until the tumbles lead
to an unjamming state ($\nu=+1$), which, rigorously combined with an unjamming
boundary kernel analogous to $Q_{\mathcal{J}}$, generates an increasing regime
in the bulk.

##  II Conservation constraints in the steady state and jamming boundary role

The steady-state distribution of the periodic interdistance $r$ of two passive
hardcore particles is the uniform density over $[0,L/2]$, with in particular a
null probability to find $r$ in the exact state $0$. The addition of non-
equilibrium features (activity, jamming) may introduce deviations from this
equilibrium behavior, which we now study. The measure $\pi$ left invariant by
the considered PDMP should satisfy the following condition,

$\int_{\mathcal{B}}\mathcal{A}_{B}f(r,\nu,s)\textrm{d}\pi(r,\nu,s)+\int_{\mathcal{J}}\mathcal{A}_{J}f(0,\nu,s)\textrm{d}\pi(0,\nu,s)=0,$
(4)

with $f$ a suitably smooth test function. As the considered process is
trivially irreducible thanks to the ergodicity of the tumbles, the
distribution $\pi$ is unique. Now, as $f$ can be any test function, we obtain
by integration by part and boundary conditions, as detailed in Appendix A, the
equivalent condition system,

$\left\\{\begin{array}[]{l}(\textrm{C}_{\mathcal{B}})\
\int_{\mathcal{V}}\pi(r,\textrm{d}\nu^{\prime}\textrm{d}s^{\prime})\tilde{\omega}(s^{\prime})Q((\nu^{\prime},s^{\prime}),(\nu,s))\\\
\ \ \ \ =\tilde{\omega}(s)\pi(r,\nu,s)+\nu\phi(s)\partial_{r}\pi(r,\nu,s)\\\
(\textrm{C}_{P})\ \pi(L/2,1,s)=\pi(L/2,-1,s)\\\ (\textrm{C}_{\mathcal{J}})\
\int_{\mathcal{J}}\textrm{d}\pi(0,\nu^{\prime},s^{\prime})\tilde{\omega}(s^{\prime})Q((\nu^{\prime},s^{\prime}),(\nu,s))\\\
\ \ \ \
=\mathbb{1}_{\\{-1,0\\}}(\nu)\tilde{\omega}(s)\pi(0,\nu,s)+\nu\phi(s)\pi(0^{+},\nu,s)\\\
\end{array}\right.$ (5)

This condition system completely constrains the invariant measure $\pi$. It
can be understood as a set of conservation equations of probability flows in
the bulk $\mathcal{B}$, and on the jamming and periodic boundaries. It leads
to the following conclusions:

(i) $(\textrm{C}_{P})$ is a direct translation of the periodicity of the
system and is not impacting the physical meaning of the form of $\pi$, in
particular potential deviation from the passive equilibrium form.

(ii) In the bulk, $(\textrm{C}_{\mathcal{B}})$ is setting a balance between
the Run and Tumble contributions. Any product measure of the form
$\tfrac{2}{L}\mathbb{1}_{]0,L/2[}(r)\mu_{\mathcal{B}}(\cdot)$, with
$\tilde{\omega}(\cdot)\mu_{\mathcal{B}}(\cdot)$ the unique invariant measure
of the tumble kernel $Q$, is satisfying $(\textrm{C}_{B})$ with null Run term
$\phi(\cdot)\partial_{r}\pi(r,\pm 1,\cdot)$. Furthermore, the isotropy of $Q$
(2) leads to $\mu_{\mathcal{B}}(\nu,s)=\mu_{\mathcal{B}}(-\nu,s)$, satisfying
$(\textrm{C}_{P})$. Reciprocally, any distribution such that
$\pi(r,\nu,s)=\pi(r,-\nu,s)$ is obeying $(\textrm{C}_{B})$ only if
$\partial_{r}\pi=0$ for all $r>0$, i.e. only if
$\tilde{\omega}(\cdot)\pi(\cdot)$ is invariant by $Q$ and then identifies with
$\tilde{\omega}(\cdot)\mu_{\mathcal{B}}(\cdot)$.

(iii) The jamming condition $(\textrm{C}_{J})$ creates Dirac components in
$r=0$ in $\pi$ for finite $\omega$, which are not present in the passive
counterpart.

(iv) More interestingly, it appears clearly now that the jamming interaction
can impact the bulk behavior by the term $\phi(s)\pi(0^{+},\pm 1,s)$ in
$(\textrm{C}_{\mathcal{J}})$. It can be understood as a boundary source term
for the bulk equation set by $(\textrm{C}_{\mathcal{B}})$, which can force the
system away from the uniform distribution in $r$ in the bulk. This term comes
from the integration by part, which stems from the Run term in
$\mathcal{A}_{B}$. It is indeed the activity of the particles that pushes the
exploration up to the boundary of the state space.

(v) In spite of the non-equilibrium features, we can derive some _active_
_global_ balance condition. Indeed, $(\textrm{C}_{\mathcal{J}})$ yields the
global balance of the probability flows in and out of jamming,
$\int\textrm{d}s\phi(s)\pi(0^{+},-1,s)=\int\textrm{d}s\phi(s)\pi(0^{+},1,s)$
and $(\textrm{C}_{B})$ imposes the global balance at any $r>0$
$\int\textrm{d}s\phi(s)\partial_{r}\pi(r,-1,s)=\int\textrm{d}s\phi(s)\partial_{r}\pi(r,1,s)$.
Both combined, we obtain the following _active_ _global_ balance of the
probability flows at any $r>0$,

$\int\textrm{d}s\phi(s)\pi(r,-1,s)=\int\textrm{d}s\phi(s)\pi(r,1,s).$ (6)

From (ii), any distribution $\pi$ obeying the stricter _detailed_ counterpart
of (6), i.e., for any $r,s$,

$\pi(r,-1,s)=\pi(r,1,s),$ (7)

must identify with
$\tfrac{2}{L}\mathbb{1}_{]0,L/2[}(r)\mu_{\mathcal{B}}(\cdot)$ in the bulk.
Actually, by symmetry argument, any distribution $\pi$ obeying (7) in at least
a single point $r$ satisfies (7) in all $r$.

Thus, analyzing the conservation conditions in (5) unravels the potential
impact of the non-equilibrium features on the invariant measure: First, the
presence of Dirac terms in $r=0$ for finite $\omega$; Second, the deviation
from a product measure uniform in $r$ in the bulk for unjamming scenarios
which sets boundary conditions $\pi(0^{+},\cdot)$ different from the tumble
stationary distribution $\mu_{\mathcal{B}}(\cdot)$.

##  III Universality classes and expression of the steady state

Therefore, we identify two universality classes, based on the deviation or not
of $\pi(0^{+},\pm 1,s)$ from $\mu_{\mathcal{B}}(\pm 1,s)$, which can always be
explicitly tested in $(C_{\mathcal{J}})$. It is equivalent to the satisfaction
or breaking of the detailed symmetry upon entering and exiting jamming for any
$s$,

$\pi(0^{+},1,s)=\pi(0^{+},-1,s).$ (8)

Detailed-jamming class. The detailed symmetry (8) is satisfied. As a
consequence, the global-flow balance (6) condition is obeyed through the
detailed-flow one (7) and the non-equilibrium nature is only apparent through
the presence of Dirac term at the jamming point in the invariant distribution
$\pi_{\textrm{det}}(r,\nu,s)$, which is of the form,

$w_{\mathcal{J}}\mathbb{1}_{\\{0\\}}(r)\mu_{\mathcal{J}}(\nu,s)+w_{\mathcal{B}}\tfrac{2}{L}\mathbb{1}_{]0,L/2[}(r)\mu_{\mathcal{B}}(\nu,s),$
(9)

with $\mu_{\mathcal{J}}$ the invariant jamming distribution and
$w_{\mathcal{J}},w_{\mathcal{B}}$ the average time ratio respectively spent in
jamming and the bulk, details on their derivation can be found in Appendix B.
Very close to an equilibrium counterpart, it follows in the bulk a product
form, which, once marginalized over the dynamical variables, identifies with
the passive uniform steady-state. In particular, this is the case of the
instantaneous tumble [26] but also of any systems with only two internal
states $\pm v$.

Global-jamming class. The detailed symmetry (8) is not satisfied. This is for
instance the case when the particles unjam at the smallest possible velocity
when they jam together at different ones, implying some energy dissipation.
This class then gathers systems further away from equilibrium, as we now
explain. First, while the global-flow balance (6) still holds, there is no
detailed-flow balance (7) at any point $r>0$ but at the periodic boundary
$r=L/2$. Sign of a stronger non-equilibrium nature, the stationary
distribution $\pi_{\text{glob}}(r,\nu,s)$ cannot longer be put under the form
of Dirac contributions in $r=0$ and a product form in the bulk $r>0$, and
writes itself,

$w_{\mathcal{J}}\mathbb{1}_{\\{0\\}}(r)\mu_{\mathcal{J}}(\nu,s)+w_{\mathcal{B}}\tfrac{2}{L}\mathbb{1}_{]0,L/2[}(r)\mu(r,\nu,s),$
(10)

where
$\mu(r,\nu,s)\\!=\\!w_{\text{eq}}\mu_{\mathcal{B}}(\nu,s)\\!+\\!w_{\text{rel}}\gamma(r,\nu,s)$
and $\gamma(\cdot)$ stands for the relaxation in the bulk from the non-
detailed constraint on its jamming boundary to the detailed constraint on its
periodic one. Supposing that the internal states are of discrete values and as
detailed in Appendix C.1, the bulk condition $(C_{\mathcal{B}})$ gives that
$(\gamma(r,0,s))_{s}$ is determined by the sum of
$\bm{\gamma}_{\pm}(r)=(\gamma(r,\pm 1,s))_{s}$, which obeys some relaxation
equation,

$\partial_{r}(\bm{\gamma}_{+}(r),\bm{\gamma_{-}}(r))=B(\bm{\gamma}_{+}(r),\bm{\gamma_{-}}(r)),$
(11)

where, by the symmetry (2), the bulk matrix $B$ is of the form
$B=\left(\begin{array}[]{cc}B^{+}&B^{-}\\\ -B^{-}&-B^{+}\end{array}\right)$,
leading to the uncoupled second-order system,

$\left\\{\begin{array}[]{l}\partial^{2}_{r}(\bm{\gamma}_{+}\\!+\\!\bm{\gamma}_{-})=(B^{+}\\!-\\!B^{-})(B^{+}\\!+\\!B^{-})(\bm{\gamma}_{+}\\!+\\!\bm{\gamma}_{-})\\\
\partial^{2}_{r}(\bm{\gamma}_{+}\\!-\\!\bm{\gamma}_{-})=(B^{+}\\!+\\!B^{-})(B^{+}\\!-\\!B^{-})(\bm{\gamma}_{+}\\!-\\!\bm{\gamma}_{-})\end{array}\right.$
(12)

The symmetry (2) also constrains the Jordan form of $B$, as it admits as
eigenvalues symmetrical finite $(\pm\lambda_{k})_{k}$ of respective
eigenvectors $(\bm{\mu}^{\pm}_{k,1},\bm{\mu}^{\mp}_{k,1})$ and necessarily $0$
of eigenvector $(\bm{\mu}_{\mathcal{B}},\bm{\mu}_{\mathcal{B}})$, see Appendix
C.2.

The general solution of (11), whose precise expression is in Appendix C.3,
derives from the spectral properties of $B$ and decomposes over a sum of
exponentials of rate $(\pm\lambda_{k})_{k}$, modulated by polynomial terms
stemming from possible degeneracies of the $\lambda_{k}$ and the boundary
conditions. As $B$ admits $0$ as an eigenvalue, the general solution also
admits a pure polynomial term.

An interesting case is when $B$ does not admit Jordan blocks of size bigger
than $3$, as, from $(C_{P})$, any polynomial decay disappears and we recover
an exact symmetry between decaying and increasing exponentials and such for
any jamming scenario, leading to,

$\begin{array}[]{r}\bm{\gamma}^{\pm}(r)\\!=\\!\sum_{k}a_{k}\Big{[}\cosh\left(\lambda_{k}\left(\frac{L}{2}\\!-\\!r\right)\right)\par(\bm{\mu}_{k,1}^{+}\\!+\\!\bm{\mu}_{k,1}^{-})\\\
\\!\mp\sinh\left(\lambda_{k}\left(\frac{L}{2}\\!-\\!r\right)\right)(\bm{\mu}_{k,1}^{+}\\!-\\!\bm{\mu}_{k,1}^{-})\Big{]},\end{array}$
(13)

with $(a_{k})_{k}$ set by $(C_{\mathcal{J}})$. It yields a catenary-like
relaxation once marginalized over $(\nu,s)$, as hinted by (12). A particular
case is any system with particle internal states symmetrical enough so that
$B$ only admits a single non-null $\pm\lambda_{k}$, so that
$\lambda_{k}^{2}\propto\text{Tr}((B^{+}+B^{-})(B^{+}-B^{-}))$, as for the
finite tumble (see Appendix C.4), which simplifies the formula in [27].

##  IV Discussion

The approach presented here gives a direct continuous-time and continuous-
space setting, which encodes the fundamental system symmetries (periodicity,
particle indistinguishability). By adequately capturing the jamming behavior
as some boundary effect on the bulk, we are able to define the two
universality classes of the steady-state behavior for systems of two RTPs with
homogeneous tumbles. Interestingly, they are entirely defined from the
satisfaction or not of a detailed symmetry (8) at the jamming point, which
translates in the bulk into the satisfaction of the necessary active global
balance (6) in a stricter –and closer to equilibrium– detailed manner (7) or
not. Then, the further away from the detailed symmetry (8) the jamming
scenario constrains the system, the more involved structure and relaxation
behavior set by (11) are eventually exhibited in the bulk, see Appendix C.3.3.
Indeed, the system is analogous to a catenary problem, where the active global
balance acts as the fixed-length constraint on a cable or equivalently the RTP
activity acts as some tension, while the tumble process acts as a weight
force.

Thus, the impact of the time-reversibility breaking by the particle activity
simply translates itself into the capacity of the system to explore its
boundaries. There, the now _seen_ jamming interactions may further impact the
steady state, depending on some symmetry preservation. It is indeed
reminiscent of the situation in non-reversible sampling by ECMC, where
particle collisions can also be understood as infinite-rate jamming preserving
some key symmetries.

The presented conclusions straightforwardly generalize into more general
jamming scenario $\mathcal{A}_{J}$, set by some tumbling rates and kernel
different from the bulk, as it only impacts the exact form of
$w_{\mathcal{J}}$ and $\mu_{\mathcal{J}}(\cdot)$. Some future work lies in the
study of the impact of non-homogeneity, with the tumbling depending on the RTP
positions, as in presence of an external potential or soft interactions. The
question of the relaxation of the system to its steady state should also find
in the PDMP formalism a powerful approach that should be explored. More
importantly, a compelling research perspective lies in the use of PDMPs and
their faculty to isolate the impacts of boundary effects on the bulk to model
more than two RTPs with hardcore jamming interactions. An explicit formula or
a good quantitative understanding of the invariant measure would provide a new
perspective on clustering phenomena such as MIPS, with a direct understanding
of the impact of the microscopic details.

###### Acknowledgements.

A.G. is supported by the French ANR under the grant ANR-17-CE40-0030 (project
_EFI_) and the Institut Universtaire de France. M.M. acknowledges the support
of the French ANR under the grant ANR-20-CE46-0007 (_SuSa_ project).

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Supplementary Materials
Jamming pair of general run-and-tumble particles:
Exact results and universality classes

##  Appendix A Derivation of the system of conservation conditions

Piecewise deterministic Markov processes (PDMP) have been introduced in [31,
32]. A PDMP refers to a Markov process which evolves ballistically according
to a deterministic differential flow, until a jump following some Poisson
process occurs or the process reaches the domain boundary. We consider the
PDMP defined by the infinitesimal generators $\mathcal{A}_{\mathcal{B}}$ and
$\mathcal{A}_{\mathcal{J}}$ and by the periodic and jamming boundaries, as
defined in (a), (b), (c).

The generator is an efficient tool to study the invariance of a measure $\pi$
(e.g. [32, Prop. 34.7]), as done in [35] for PDMP generated in particle
systems by ECMC methods. The measure $\pi$ is indeed left invariant by the
considered PDMP if the following condition is satisfied,

$\int_{\mathcal{B}}\mathcal{A}_{B}f(r,\nu,s)\textrm{d}\pi(r,\nu,s)+\int_{\mathcal{J}}\mathcal{A}_{J}f(0,\nu,s)\textrm{d}\pi(0,\nu,s)=0,$
(14)

with $f$ a suitably smooth test function (here we do not detail the
description of the extended domain of the generator and a suitable core, as
required by [32, Th. 5.5, ]). Furthermore, stemming from the periodic boundary
kernel $Q_{P}$, $f$ shall verify the condition on the periodic boundary,

$f(L/2,\nu,s)=f(L/2,-\nu,s).$ (15)

Similar conditions at the jamming boundary are required, i.e.

$f(0^{+},\nu,s)=f(0,\nu,s),$ (16)

where we omit out of simplicity the superscript ∗ characterizing the jamming
state. Now, expliciting the bulk term in 14,

$\begin{split}\int_{\mathcal{V}_{-1}\cup\mathcal{V}_{0}}\mathcal{A}_{J}f(0,\nu,s)\pi(0,d\nu,ds)+\int_{]0,L/2[\times\mathcal{V}}\textrm{d}\pi(r,\nu,s)\nu\phi(s)\partial_{r}f(r,\nu,s)\\\
+\int_{]0,L/2[\times\mathcal{V}}\textrm{d}\pi(r,\nu,s)\tilde{\omega}(s)\int_{\mathcal{V}}Q((\nu,s),(\textrm{d}\nu^{\prime}\textrm{d}s^{\prime}))\Big{(}f(r,\nu^{\prime},s^{\prime})-f(r,\nu,s)\Big{)}=0,\end{split}$
(17)

we obtain by integration by parts in the second term,

$\begin{split}\int_{\mathcal{V}_{-1}\cup\mathcal{V}_{0}}\mathcal{A}_{J}f(0,\nu,s)\pi(0,\textrm{d}\nu,\textrm{d}s)-\int_{\mathcal{V}_{-1}\cup\mathcal{V}_{1}}\nu\phi(s)\pi(0^{+},\textrm{d}\nu,\textrm{d}s)f(0,\nu,s)+\int_{\mathcal{V}_{-1}\cup\mathcal{V}_{1}}\nu\phi(s)\pi(L/2^{-},\textrm{d}\nu,\textrm{d}s)f(L/2,\nu,s)\\\
+\int_{]0,L/2[\times\mathcal{V}}\textrm{d}r\textrm{d}\nu\textrm{d}sf(r,\nu,s)\Big{[}\int_{\mathcal{V}}\pi(r,\textrm{d}\nu^{\prime},\textrm{d}s^{\prime})\tilde{\omega}(s^{\prime})Q((\nu^{\prime},s^{\prime}),(\nu,s))-\pi(r,\nu,s)\tilde{\omega}(s)-\nu\phi(s)\partial_{r}\pi(r,\nu,s)\Big{]}=0.\end{split}$
(18)

As, $f(L/2,\nu,s)=f(L/2,-\nu,s)$ by the condition on the periodic boundary,
$\pi(L/2,-\nu,s)=\pi(L/2,\nu,s)$ by definition of the periodic interdistance,
we have the cancellation of the term in $L/2$ by periodicity,

$\int_{\mathcal{V}_{1}}\nu\phi(s)\pi(L/2,\textrm{d}\nu,\textrm{d}s)f(L/2,\nu,s)+\int_{\mathcal{V}_{-1}}-(-\nu)\phi(s)\pi(L/2,-\textrm{d}\nu,\textrm{d}s)f(L/2,-\nu,s)=0.$
(19)

Thus we have, expliciting the jamming generator,

$\begin{split}\int_{\mathcal{V}}\textrm{d}\nu\textrm{d}sf(0,\nu,s)\Big{[}\int_{\mathcal{V}_{-1}\cup\mathcal{V}_{0}}\pi(0,\textrm{d}\nu^{\prime},\textrm{d}s^{\prime})\tilde{\omega}(s^{\prime})Q((\nu^{\prime},s^{\prime}),(\nu,s))-\mathbb{1}_{\\{-1,0\\}}(\nu)\pi(0,\nu,s)\tilde{\omega}(s)-\nu\phi(s)\pi(0^{+},\nu,s)\Big{]}\\\
+\int_{]0,L/2[\times\mathcal{V}}\textrm{d}r\textrm{d}\nu\textrm{d}sf(r,\nu,s)\Big{[}\int_{\mathcal{V}}\pi(r,\textrm{d}\nu^{\prime},\textrm{d}s^{\prime})\tilde{\omega}(s^{\prime})Q((\nu^{\prime},s^{\prime}),(\nu,s))-\pi(r,\nu,s)\tilde{\omega}(s)-\nu\phi(s)\partial_{r}\pi(r,\nu,s))\Big{]}\\\
=0.\end{split}$ (20)

As $f$ can be any test function and recalling the periodic condition, we
obtain the following equivalent condition system,

$\left\\{\begin{array}[]{ll}(\textrm{C}_{B})&\int_{\mathcal{V}}\pi(r,\textrm{d}\nu,\textrm{d}s^{\prime})\tilde{\omega}(s^{\prime})Q((\nu^{\prime},s^{\prime}),(\nu,s))=\pi(r,\nu,s)\tilde{\omega}(s)+\nu\phi(s)\partial_{r}\pi(r,\nu,s)\text{
for any }(r,\nu,s)\in\mathcal{B},\\\
(\textrm{C}_{P})&\pi(L/2,1,s)=\pi(L/2,-1,s)\text{ for any }(\pm
1,s)\in\mathcal{V}_{\pm 1},\\\
(\textrm{C}_{J})&\int_{\mathcal{V}_{-1}\cup\mathcal{V}_{0}}\pi(0,\textrm{d}\nu^{\prime}\textrm{d}s^{\prime})\tilde{\omega}(s^{\prime})Q((\nu^{\prime},s^{\prime}),(\nu,s))=\mathbb{1}_{\\{-1,0\\}}(\nu)\pi(0,\nu,s)\tilde{\omega}(s)+\nu\phi(s)\pi(0^{+},\nu,s)\text{
for any }(\nu,s)\in\mathcal{V}\end{array}\right.,$ (21)

which identifies with the system (5), whose consequences are further discussed
in Section II.

##  Appendix B Derivation of the detailed-jamming steady state

We now determine in more details the steady-state distribution in the
detailed-jamming class as introduced in (9) in Section III and as recalled
here,

$\pi_{\textrm{det}}(r,\nu,s)=w_{\mathcal{J}}\mathbb{1}_{\\{0\\}}(r)\mu_{\mathcal{J}}(\nu,s)+w_{\mathcal{B}}\tfrac{2}{L}\mathbb{1}_{]0,L/2[}(r)\mu_{\mathcal{B}}(\nu,s).$

Plugging the detailed-jamming symmetry (8) into $(\textrm{C}_{J})$ (5), we
obtain,

$\int_{\mathcal{V}_{-1}\cup\mathcal{V}_{0}}\textrm{d}\mu_{\mathcal{J}}(\nu^{\prime},s^{\prime})\tilde{\omega}(s^{\prime})Q((\nu^{\prime},s^{\prime}),(1,s))=\phi(s)\pi(0^{+},1,s)=-(-1)\phi(s)\pi(0^{+},-1,s),$

and derive an explicit condition on the jamming equivalent to (8) ,

$(\textrm{C}_{\textrm{det}})\left\\{\begin{array}[]{l}\int_{\mathcal{V}_{-1}\cup\mathcal{V}_{0}}\textrm{d}\mu_{\mathcal{J}}(\nu^{\prime},s^{\prime})\tilde{\omega}(s^{\prime})Q((\nu^{\prime},s^{\prime}),(0,s))=\tilde{\omega}(s)\mu_{\mathcal{J}}(0,s)\\\
\int_{\mathcal{V}_{-1}\cup\mathcal{V}_{0}}\textrm{d}\mu_{\mathcal{J}}(\nu^{\prime},s^{\prime})\tilde{\omega}(s^{\prime})(Q((\nu^{\prime},s^{\prime}),(-1,s))+Q((\nu^{\prime},s^{\prime}),(1,s)))=\tilde{\omega}(s)\mu_{\mathcal{J}}(-1,s)\\\
\int_{\mathcal{V}_{-1}\cup\mathcal{V}_{0}}w_{\mathcal{J}}\textrm{d}\mu_{\mathcal{J}}(\nu^{\prime},s^{\prime})\tilde{\omega}(s^{\prime})Q((\nu^{\prime},s^{\prime}),(1,s))=\phi(s)w_{\mathcal{B}}\tfrac{2}{L}\mu_{\mathcal{B}}(1,s)\
\text{for any}\ (1,s)\in\mathcal{V}_{1}\end{array}\right.,$ (22)

so that the jamming invariant measure $\mu_{\mathcal{J}}$ is completely
characterized by the first two lines and only depends on the jamming scenario.
Then, the ratio $w_{\mathcal{J}}/w_{\mathcal{B}}$ of time spent respectively
while jamming and in the bulk is set by the last line. The ratio
$w_{\mathcal{J}}/w_{\mathcal{B}}$ does not depend on $s$ as the jamming exit
probability in state $(1,s)$ (left term of the last line) is indeed
proportional to the invariant bulk probability $\mu_{\mathcal{B}}(\pm 1,s)$ of
this state, which is known from $(C_{\mathcal{B}})$, as we recall,

$\int_{\mathcal{V}}\mu_{\mathcal{B}}(\textrm{d}\nu^{\prime}\textrm{d}s^{\prime})\tilde{\omega}(s^{\prime})Q((\nu^{\prime},s^{\prime}),\textrm{d}\nu\textrm{d}s)=\mu_{\mathcal{B}}(\textrm{d}\nu,\textrm{d}s)\tilde{\omega}(s).$

An interesting particular case is systems with two internal states
$v_{1},v_{2}$, which includes the case of the instantaneous tumble
($v_{1},v_{2}=-1,+1$, $\omega(v_{1})=\omega(v_{2})$), as illustrated in Figure
2. These systems are always detailed-jamming ones, as there is only one entry
state into jamming and one exit state out of jamming, realizing the detailed-
jamming symmetry (8). For these systems, we obtain,

$(\textrm{C}_{\textrm{det}})\left\\{\begin{array}[]{l}\mu_{\mathcal{J}}(-1,\\{v_{1},v_{2}\\})\omega(v_{1})=2\omega(v_{2})\mu_{\mathcal{J}}(0,\\{v_{2},v_{2}\\})\\\
\mu_{\mathcal{J}}(-1,\\{v_{1},v_{2}\\})\omega(v_{2})=2\omega(v_{1})\mu_{\mathcal{J}}(0,\\{v_{1},v_{1}\\})\\\
w_{\mathcal{J}}\omega(v_{1})\mu_{\mathcal{J}}(0,\\{v_{1},v_{1}\\})+w_{\mathcal{J}}\omega(v_{2})\mu_{\mathcal{J}}(0,\\{v_{2},v_{2}\\})=w_{\mathcal{B}}\tfrac{2|v_{1}-v_{2}|}{L}\mu_{\mathcal{B}}(1,\\{v_{1},v_{2}\\})\\\
\iff\tfrac{1}{2}w_{\mathcal{J}}(\omega(v_{1})+\omega(v_{2}))\mu_{\mathcal{J}}(-1,\\{v_{1},v_{2}\\})=w_{\mathcal{B}}\tfrac{2|v_{1}-v_{2}|}{L}\mu_{\mathcal{B}}(1,\\{v_{1},v_{2}\\})\end{array}\right.,$
(23)

and,

$\left\\{\begin{array}[]{l}\omega(v_{2})\mu_{\mathcal{B}}(\pm
1,\\{v_{1},v_{2}\\})=\omega(v_{1})\mu_{\mathcal{B}}(0,\\{v_{1},v_{1}\\})\\\
\omega(v_{1})\mu_{\mathcal{B}}(\pm
1,\\{v_{1},v_{2}\\})=\omega(v_{2})\mu_{\mathcal{B}}(0,\\{v_{2},v_{2}\\})\\\
\end{array}\right..$

We then get,

$\begin{array}[]{lllll}\left\\{\begin{array}[]{l}\mu_{\mathcal{J}}(0,\\{v_{1},v_{1}\\})=\tfrac{\omega(v_{2})^{2}}{\mathcal{N}_{\mathcal{J}}}\\\
\mu_{\mathcal{J}}(0,\\{v_{2},v_{2}\\})=\tfrac{\omega(v_{1})^{2}}{\mathcal{N}_{\mathcal{J}}}\\\
\mu_{\mathcal{J}}(-1,\\{v_{1},v_{2}\\})=2\tfrac{\omega(v_{1})\omega(v_{2})}{\mathcal{N}_{\mathcal{J}}}\\\
\mathcal{N}_{\mathcal{J}}=\big{(}\omega(v_{1})+\omega(v_{2})\big{)}^{2}\end{array}\right.,&&\left\\{\begin{array}[]{l}\mu_{\mathcal{B}}(0,\\{v_{1},v_{1}\\})=\tfrac{\omega(v_{2})^{2}}{\mathcal{N}_{\mathcal{B}}}\\\
\mu_{\mathcal{B}}(0,\\{v_{2},v_{2}\\})=\tfrac{\omega(v_{1})^{2}}{\mathcal{N}_{\mathcal{B}}}\\\
\mu_{\mathcal{B}}(\pm
1,\\{v_{1},v_{2}\\})=\tfrac{\omega(v_{1})\omega(v_{2})}{\mathcal{N}_{\mathcal{B}}}\\\
\mathcal{N}_{\mathcal{B}}=\big{(}\omega(v_{1})+\omega(v_{2})\big{)}^{2}\end{array}\right.&\text{
and
}&\left\\{\begin{array}[]{l}w_{\mathcal{B}}=\frac{1}{1+\tfrac{2|v_{1}-v_{2}|}{L(\omega(v_{1})+\omega(v_{2}))}}\\\
w_{\mathcal{J}}/w_{\mathcal{B}}=\tfrac{2|v_{1}-v_{2}|}{L(\omega(v_{1})+\omega(v_{2}))}\end{array}\right..\end{array}$
(24)

where the ratio $w_{\mathcal{J}}/w_{\mathcal{B}}$ of time spent in jamming
versus in the bulk increases with the strength of the ballistic nature of the
considered RTP dynamics (decrease of the time needed to cover the length $L$
of the torus, decrease of the total tumble rate). Computations for systems
with more than two internal states are similar. We finally also stress that
the conditions determined here also applied for a continuous distribution of
internal states.

##  Appendix C Derivation of the global-jamming steady state

We now determine in more details for discrete distributions of internal states
the steady-state distribution in the global-jamming class as introduced in
(10) in Section III and as recalled here,

$\pi_{\textrm{glob}}(r,\nu,s)=w_{\mathcal{J}}\mathbb{1}_{\\{0\\}}(r)\mu_{\mathcal{J}}(\nu,s)+w_{\mathcal{B}}\tfrac{2}{L}\mathbb{1}_{]0,L/2[}(r)(w_{\text{eq}}\mu_{\mathcal{B}}(\nu,s)+w_{\text{rel}}\gamma(r,\nu,s)).$

We do so by first deriving in Section C.1 the bulk relaxation equation, which
is ruled by the bulk relaxation matrix $B$. We then study in Section C.2 the
spectral properties of $B$, which are strongly constrained by the symmetry (2)
of $Q$. We then obtain a general explicit expression for $\gamma(\cdot)$ in
Section C.3, which we completely determine in particular cases including the
finite tumble in Section C.4.

###  C.1 Bulk evolution equation

We obtain, by plugging $\pi_{\textrm{glob}}$ into the bulk condition
$(C_{\mathcal{B}})$ and exploiting the symmetry (2) of $Q$,

$\left\\{\begin{array}[]{rl}\gamma(r,0,s)\tilde{\omega}(s)=&\int\textrm{d}s^{\prime}\mathbb{1}_{\mathcal{V}_{1}}((1,s^{\prime}))(\gamma(r,-1,s^{\prime})+\gamma(r,1,s^{\prime}))\tilde{\omega}(s^{\prime})Q((1,s^{\prime}),(0,s))\\\
\pm\partial_{r}\gamma(r,\pm 1,s)=&-\phi(s)^{-1}\gamma(r,\pm
1,s)\tilde{\omega}(s)\\\
&+\phi(s)^{-1}\int\textrm{d}s^{\prime}\mathbb{1}_{\mathcal{V}_{1}}((1,s^{\prime}))\Big{[}\gamma(r,1,s^{\prime})Q((1,s^{\prime}),(\pm
1,s))+\gamma(r,-1,s^{\prime})Q((-1,s^{\prime}),(\pm 1,s))\\\
&+(\gamma(r,1,s^{\prime})+\gamma(r,-1,s^{\prime}))\int\textrm{d}s^{\prime\prime}\mathbb{1}_{\mathcal{V}_{0}}((0,s^{\prime\prime}))Q((1,s^{\prime}),(0,s^{\prime\prime}))Q((0,s^{\prime\prime}),(1,s))\Big{]}\tilde{\omega}(s^{\prime})\end{array}\right.$
(25)

We note,

$\bm{\gamma}(r)=\left(\begin{array}[]{l}\bm{\gamma}^{+}(r)=(\gamma(1,s))_{(1,s)\in\mathcal{V}_{1}}\\\
\bm{\gamma}^{-}(r)=(\gamma(\\!-\\!1,s))_{(-1,s)\in\mathcal{V}_{-1}}\end{array}\right).$

and summarize (25) into,

$\partial_{r}\bm{\gamma}(r)=B\bm{\gamma}$ (26)

with $B$ the bulk relaxation matrix,

$B=\left(\begin{array}[]{cc}B^{+}&B^{-}\\\ -B^{-}&-B^{+}\end{array}\right)\in
M_{2|\mathcal{V}_{1}|\times 2|\mathcal{V}_{1}|}(\mathbb{R})$ (27)

and,

$B^{+}=D^{-1}_{\phi}(Q^{+}-I)D_{\omega},\ \
B^{-}=D^{-1}_{\phi}Q^{-}D_{\omega}\in
M_{|\mathcal{V}_{1}|\times\mathcal{V}_{1}|}(\mathbb{R})$ (28)
$D_{\phi,ss^{\prime}}=\phi(s)\delta_{ss^{\prime}}\ \text{and}\
D_{\omega,ss^{\prime}}=\omega(s)\delta_{ss^{\prime}},$ (29)

and,

$Q^{\pm}_{ss^{\prime}}=Q((\pm
1,s^{\prime}),(1,\textrm{d}s))+\sum_{s^{\prime\prime}}Q((\pm
1,s^{\prime}),(0,\textrm{d}s^{\prime\prime}))Q((0,s^{\prime\prime}),(1,\textrm{d}s)).$
(30)

As $Q$ is a Markov kernel and as there is no tumble between two states with
$\nu=0$ since it will require a tumble of each particle at the same time, we
have for any $(1,s^{\prime})\in\mathcal{V}_{1}$ the conservation identity,

$\sum_{s}(Q^{+}_{ss^{\prime}}+Q^{-}_{ss^{\prime}}-\delta_{ss^{\prime}})=0$
(31)

i.e.,

$\sum_{s}\phi(s)\big{(}B^{+}+B^{-}\big{)}_{ss^{\prime}}=0,$ (32)

which is consistent with the active global balance condition (6), as clearly
appearing when rewritten as,

$\sum_{s}\phi(s)(\partial_{r}\bm{\gamma}_{s}^{+}(r)-\partial_{r}\bm{\gamma}_{s}^{-}(r))=0=\sum_{s}\phi(s)\big{(}B^{+}+B^{-}\big{)}_{ss^{\prime}}(\bm{\gamma}_{s^{\prime}}^{+}(r)+\bm{\gamma}_{s^{\prime}}^{-}(r)).$
(33)

###  C.2 Spectral analysis of bulk matrix $B$

####  C.2.1 Jordan reduction of $B$ and symmetry constraints

We now consider the Jordan reduction of $B=P^{-1}JP$, with $J$ a Jordan
matrix. For the Jordan block $k$ of size $p_{k}$, we note the corresponding
eigenvalue $\lambda_{k}$ and the Jordan chain of linearly-independent
generalized eigenvectors
$\bm{\mu_{k,n}}=(\bm{\mu}^{+}_{k,n},\bm{\mu}^{-}_{k,n})$, with $1\leq n\leq
p_{k}$, $\bm{\mu_{k,1}}$ being an eigenvector and
$B\bm{\mu}_{k,n}=\lambda_{k}\bm{\mu}_{k,n}+\bm{\mu}_{k,n-1}$ for $n>1$.

By the symmetry of $B$, any block $k$ of eigenvalue $\lambda_{k}>0$ is
mirrored by a block $k^{*}$ of eigenvalue $\lambda_{k^{*}}=-\lambda_{k}$ and
generalized eigenvectors
$\bm{\mu}_{k^{*},n}=(-1)^{n+1}(\bm{\mu}^{-}_{k,n},\bm{\mu}^{+}_{k,n})$, as
$B(\bm{\mu}^{-}_{k,1},\bm{\mu}^{+}_{k,1})=-\lambda_{k}(\bm{\mu}^{-}_{k,1},\bm{\mu}^{+}_{k,1})$
and, for $n>1$,

$(-1)^{n+1}B(\bm{\mu}^{-}_{k,n},\bm{\mu}^{+}_{k,n})=-\lambda_{k}(-1)^{n+1}(\bm{\mu}^{-}_{k,n},\bm{\mu}^{+}_{k,n})+(-1)^{n}(\bm{\mu}^{-}_{k,n-1},\bm{\mu}^{+}_{k,n-1}).$

Furthermore, as $\hat{D}^{\pm}_{\phi}B\hat{D}^{-1}_{\omega}$, with
$\hat{D}^{\pm}_{\phi}=\left(\begin{smallmatrix}D_{\phi}&0\\\ 0&-D_{\phi}\\\
\end{smallmatrix}\right)$ and
$\hat{D}^{-1}_{\omega}=\left(\begin{smallmatrix}D^{-1}_{\omega}&0\\\
0&D^{-1}_{\omega}\\\ \end{smallmatrix}\right)$ , identifies with the tumble
Markov kernel collapsed in $\nu=0$ of the considered process, it is
diagonalizable with real eigenvalues ranging from $0$ to $-1$ and a unique
eigenvector $\bm{\mu}$ so that
$\hat{D}^{\pm}_{\phi}B\hat{D}^{-1}_{\omega}\bm{\mu}=0$. The vector $\bm{\mu}$
identifies with the unique stationary bulk solution as
$\bm{\mu}=\hat{D}_{\omega}(\bm{\mu}_{\mathcal{B}},\bm{\mu}_{\mathcal{B}})$. As
$\hat{D}_{\phi}$ is invertible,
$(\bm{\mu}_{\mathcal{B}},\bm{\mu}_{\mathcal{B}})$ is the unique eigenvector of
eigenvalue $0$ of $B$. Then, for the unique block $k$ with $\lambda_{k}=0$,
the generalized eigenvectors are of the following form
$\bm{\mu_{k,n}}=(\bm{\mu}^{+}_{k,n},(-1)^{n+1}\bm{\mu}^{+}_{k,n})$, as,

$\left\\{\begin{array}[]{l}\bm{\mu}_{k,1}=(\bm{\mu}_{\mathcal{B}},\bm{\mu}_{\mathcal{B}})\\\
B\bm{\mu}_{k,n}=\bm{\mu}_{k,n-1},\ n>1\end{array}\right.\ \ \text{gives}\ \
\left\\{\begin{array}[]{ll}(B^{+}+(-1)^{n+1}B^{-})\bm{\mu}^{+}_{k,n}&=\bm{\mu}^{+}_{k,n-1}\\\
-((-1)^{n+1}B^{+}+B^{-})\bm{\mu}^{+}_{k,n}&=(-1)^{n}\bm{\mu}^{+}_{k,n-1}\\\
\end{array}\right.$

It is finally noteworthy that
$(\bm{\mu}_{\mathcal{B}},\bm{\mu}_{\mathcal{B}})$ is the only symmetrical
eigenvector of $B$, as any symmetrical eigenvector is of eigenvalue $0$, as
can be seen directly from $B$ or from an equivalent argument of linear
independence for $\lambda_{k}\neq 0$ of the eigenvectors
$(\bm{\mu}^{+}_{k,n},\bm{\mu}^{-}_{k,n})$ (e.v. $\lambda_{k}$) and
$(\bm{\mu}^{-}_{k,n},\bm{\mu}^{+}_{k,n})$ (e.v. $-\lambda_{k}$). This is
consistent with the fact that a symmetrical eigenvector is by definition a
stationary solution of the detailed-jamming class which is unique and of
eigenvalue $0$. Hence the complete characterization only by the detailed-
symmetry condition ($\pi(r,\nu,s)=-\pi(r,-\nu,s)$) of the detailed-jamming
class, which is equivalent to $\pi(r,\nu,s)=\mu_{Q}(\nu,s)$.

Thus, the number of Jordan blocks of $J$ is odd and is noted $2d+1$. We rank
the blocks from $-d$ to $d$ in a symmetrical manner: The $k$-th block of
eigenvalue $\lambda_{k}$ and generalized eigenvectors
$(\bm{\mu}^{+}_{k,n},\bm{\mu}^{-}_{k,n})$ is mirrored by the $(-k)$-th block
of eigenvalue $-\lambda_{k}$, generalized eigenvectors
$((-1)^{n+1}\bm{\mu}^{-}_{k,n},(-1)^{n+1}\bm{\mu}^{+}_{k,n})$ . We then set
the position of the block linked to the eigenvector
$(\bm{\mu}_{0,1},\bm{\mu}_{0,1})=(\bm{\mu}_{\mathcal{B}},\bm{\mu}_{\mathcal{B}})$
to the $0$-th rank. It is composed of generalized eigenvectors of the form
$(\bm{\mu}_{0,n},(-1)^{n+1}\bm{\mu}_{0,n})$.

It is noteworthy that, as $B$ is of even size, the Jordan block associated to
$\lambda_{0}=0$ must be of an even size $p_{0}>0$, so that $B$ is never
diagonalizable. This is a direct consequence of the symmetry imposed on $Q$ by
the particle indistinguishability and space homogeneity.

####  C.2.2 Spectral relationship with submatrices $B_{+}$ and $B_{-}$ and
catenary equation

The family
$((\bm{\mu}^{+}_{k,n},\bm{\mu}^{-}_{k,n})_{k,n},(\bm{\mu}_{0,n},(-1)^{n+1}\bm{\mu}_{0,n})_{n},(\bm{\mu}^{-}_{k,n},\bm{\mu}^{+}_{k,n})_{k,n})$
naturally forms a basis of $\mathbb{R}^{2|\mathcal{V}_{1}|}$. Furthermore, the
vectors $(\bm{\mu}^{+}_{k,n}+\bm{\mu}^{-}_{k,n})_{k,n}$ (resp.
$(\bm{\mu}^{+}_{k,n}-\bm{\mu}^{-}_{k,n})_{k,n}$) are linearly independent.
Indeed, if it exists $\\{b_{k,n}\\}$ so that
$\sum_{k,n}b_{k,n}\bm{\mu}^{+}_{k,n}=-\sum_{k,n}b_{k,n}\bm{\mu}^{-}_{k,n}$
(resp. $=\sum_{k,n}b_{k,n}\bm{\mu}^{-}_{k,n}$), then we have
$\sum_{k,n}b_{k,n}\bm{\mu}_{k,n}=-\sum_{k,n}b_{k,n}(-1)^{n+1}\bm{\mu}_{-k,n}$
(resp. $=\sum_{k,n}b_{k,n}\bm{\mu}_{k,n}$) , which is not possible.
Eventually, by considering the size of the sets
$(\bm{\mu}^{+}_{k,n}\pm\bm{\mu}^{-}_{k,n})$, they form two basis of
$\mathbb{R}^{|\mathcal{V}_{1}|}$.

In addition, we have the relationships, noting $M^{+}=(B^{+}+B^{-})$ and
$M^{-}=(B^{+}-B^{-})$,

$\begin{aligned} \text{For $k>0$,
}&\left\\{\begin{array}[]{l}M^{+}(\bm{\mu}^{+}_{k,1}+\bm{\mu}^{-}_{k,1})=\lambda_{k}(\bm{\mu}^{+}_{k,1}-\bm{\mu}^{-}_{k,1})\\\
M^{-}(\bm{\mu}^{+}_{k,1}-\bm{\mu}^{-}_{k,1})=\lambda_{k}(\bm{\mu}^{+}_{k,1}+\bm{\mu}^{-}_{k,1})\end{array}\right.&\text{and}&\left\\{\begin{array}[]{l}M^{+}\bm{\mu}_{0,1}=0\\\
M^{-}\bm{\mu}_{0,2}=\bm{\mu}_{0,1}\\\ \end{array}\right.\end{aligned},$ (34)

so that, the vectors $(\bm{\mu}^{+}_{k,1}\pm\bm{\mu}^{-}_{k,1})_{k>1}$ are
respectively the eigenvectors of $M^{\mp}M^{\pm}$ of eigenvalues
$\lambda_{k}^{2}$ and $\bm{\mu}_{0,1}$ (resp. $\bm{\mu}_{0,2}$) is the
eigenvector of $M^{-}M^{+}$ (resp. $M^{+}M^{-}$) of eigenvalue $0$.

And furthermore, for $n>1$, we have,

$\begin{aligned} \text{For $k>0$,
}&\left\\{\begin{array}[]{l}M^{+}(\bm{\mu}^{+}_{k,n}+\bm{\mu}^{-}_{k,n})=\lambda_{k}(\bm{\mu}^{+}_{k,n}-\bm{\mu}^{-}_{k,n})+(\bm{\mu}^{+}_{k,n-1}-\bm{\mu}^{-}_{k,n-1})\\\
M^{-}(\bm{\mu}^{+}_{k,n}-\bm{\mu}^{-}_{k,n})=\lambda_{k}(\bm{\mu}^{+}_{k,n}+\bm{\mu}^{-}_{k,n})+(\bm{\mu}^{+}_{k,n-1}+\bm{\mu}^{-}_{k,n-1})\end{array}\right.&\text{and}&\left\\{\begin{array}[]{l}M^{+}\bm{\mu}_{0,2n+1}=\bm{\mu}_{0,2n}\\\
M^{-}\bm{\mu}_{0,2n}=\bm{\mu}_{0,2n-1}\\\ \end{array}\right.\end{aligned}.$
(35)

Giving, for $p<n$,

For $k>0$,
$\displaystyle\left\\{\begin{array}[]{l}(M^{-}M^{+}-\lambda_{k}^{2})^{p}(\bm{\mu}^{+}_{k,n}+\bm{\mu}^{-}_{k,n})\neq\bm{0}\\\
(M^{+}M^{-}-\lambda_{k}^{2})^{p}(\bm{\mu}^{+}_{k,n}-\bm{\mu}^{-}_{k,n})\neq\bm{0}\end{array}\right.$
and
$\displaystyle\left\\{\begin{array}[]{l}(M^{-}M^{+})^{p}\bm{\mu}_{0,2n+1}\neq\bm{0}\\\
(M^{+}M^{-})^{p}\bm{\mu}_{0,2n}\neq\bm{0}\end{array}\right.$ (36) For $k>0$,
$\displaystyle\left\\{\begin{array}[]{l}(M^{-}M^{+}-\lambda_{k}^{2})^{n-1}(\bm{\mu}^{+}_{k,n}+\bm{\mu}^{-}_{k,n})=\bm{\mu}^{+}_{k,1}+\bm{\mu}^{-}_{k,1}\\\
(M^{+}M^{-}-\lambda_{k}^{2})^{n-1}(\bm{\mu}^{+}_{k,n}-\bm{\mu}^{-}_{k,n})=\bm{\mu}^{+}_{k,1}-\bm{\mu}^{-}_{k,1}\end{array}\right.$
and
$\displaystyle\left\\{\begin{array}[]{l}(M^{-}M^{+})^{n-1}\bm{\mu}_{0,2n-1}=\bm{\mu}_{0,1}\\\
(M^{+}M^{-})^{n-1}\bm{\mu}_{0,2n}=\bm{\mu}_{0,2},\end{array}\right.$

so that, $((\bm{\mu}^{+}_{k,n}+\bm{\mu}^{-}_{k,n})_{k,1<n\leq
p_{k}},(\bm{\mu}_{0,2n-1})_{1<n\leq p_{0}/2})$ (resp.
$((\bm{\mu}^{+}_{k,n}-\bm{\mu}^{-}_{k,n})_{k,1<n\leq
p_{k}},(\bm{\mu}_{0,2n})_{1<n\leq p_{0}/2})$) are generalized eigenvectors of
$M^{-}M^{+}$ (resp. of $M^{+}M^{-}$).

More generally, the spectral properties of $B$ and $M^{+}M^{-}$, $M^{-}M^{+}$
are strongly related. Indeed, the characteristic polynomial of $B$ writes
itself in terms of $B^{2}$, as,

$\displaystyle
B^{p_{0}}\prod_{k=1}^{d}(B-\lambda_{k}I)^{p_{k}}(B+\lambda_{k}I)^{p_{k}}=(B^{2})^{p_{0}/2}\prod_{k=1}^{d}(B^{2}-\lambda_{k}^{2}I)^{p_{k}}$
$\displaystyle=0$ (37) $\displaystyle
B^{2(p_{0}/2)}\sum_{m=0}^{d}B^{2m}\sum_{\\{i_{k}\\}_{k=1}^{d-m}}\prod_{k=1}^{d-m}\lambda_{i_{k}}^{2}$
$\displaystyle=0.$

And, as,

$B^{2m}=\frac{1}{2}\left(\begin{array}[]{cc}(M^{-}M^{+})^{m}+(M^{+}M^{-})^{m}&(M^{-}M^{+})^{m}-(M^{+}M^{-})^{m}\\\
(M^{-}M^{+})^{m}-(M^{+}M^{-})^{m}&(M^{-}M^{+})^{m}+(M^{+}M^{-})^{m}\end{array}\right)$

it is equivalent to,

$\left\\{\begin{array}[]{rl}(M^{+}M^{-})^{p_{0}/2}\sum_{m=0}^{d}(M^{+}M^{-})^{m}\sum_{\\{i_{k}\\}_{k=1}^{d-m}}\prod_{k=1}^{d-m}\lambda_{i_{k}}^{2}&=0\\\
(M^{-}M^{+})^{p_{0}/2}\sum_{m=0}^{d}(M^{-}M^{+})^{m}\sum_{\\{i_{k}\\}_{k=1}^{d-m}}\prod_{k=1}^{d-m}\lambda_{i_{k}}^{2}&=0.\\\
\end{array}\right.$ (38)

So that, the characteristic polynomials of $M^{+}M^{-}$ and $M^{-}M^{+}$ are,

$(M^{\pm}M^{\mp})^{p_{0}/2}\prod_{k=1}^{d}(M^{\pm}M^{\mp}-\lambda_{k}^{2}I)^{p_{k}}=0.\\\
$ (39)

A similar derivation can be carried out regarding the minimal polynomial and,
thus, the fact that $B$ admits no generalized eigenvectors apart the necessary
$\bm{\mu}_{0,2}$ (i.e. $p_{0}=2$, $p_{k\neq 0}=1$) is equivalent to
$M^{+}M^{-}$ and $M^{-}M^{+}$ both diagonalizable.

Therefore, studying the squared global bulk matrix $B^{2}$ or equivalently the
submatrices $M^{+}M^{-}$ and $M^{-}M^{+}$, seems the natural way to deal with
the bulk evolution. Indeed, it underlines the fact that, instead of
considering the global bulk evolution $\partial_{r}\bm{\gamma}=B\bm{\gamma}$,
one can consider the system composed of the symmetrical bulk evolution,

$\partial_{r}(\bm{\gamma}_{+}+\bm{\gamma}_{-})=M^{-}(\bm{\gamma}_{+}-\bm{\gamma}_{-}),$
(40)

and of the antisymmetrical bulk evolution,

$\partial_{r}(\bm{\gamma}_{+}-\bm{\gamma}_{-})=M^{+}(\bm{\gamma}_{+}+\bm{\gamma}_{-})$
(41)

which then gives the uncoupled second-order system,

$\left\\{\begin{array}[]{l}\partial^{2}_{r}(\bm{\gamma}_{+}+\bm{\gamma}_{-})=M^{-}M^{+}(\bm{\gamma}_{+}+\bm{\gamma}_{-})\\\
\partial^{2}_{r}(\bm{\gamma}_{+}-\bm{\gamma}_{-})=M^{+}M^{-}(\bm{\gamma}_{+}-\bm{\gamma}_{-})\end{array}\right.,$
(42)

which leads to catenary-like solutions in common cases, as the finite tumble.

###  C.3 General global-jamming solution

####  C.3.1 Bulk general form

Exploiting the Jordan form of $B$, the solution to Eq. 26 is,

$\begin{array}[]{rl}\left(\begin{array}[]{c}\bm{\gamma}^{+}\\\
\bm{\gamma}^{-}\\\
\end{array}\right)(r)&=\sum_{n=1}^{p_{0}}\sum_{l=n}^{p_{0}}a_{0,l}\frac{r^{l-n}}{(l-n)!}\left(\begin{array}[]{c}\bm{\mu}_{0,n}\\\
(-1)^{n+1}\bm{\mu}_{0,n}\\\ \end{array}\right)\\\
&+\sum_{k=1}^{d}\Big{[}\exp\left(-\lambda_{k}\left(\frac{L}{2}-r\right)\right)\sum_{n=1}^{p_{k}}\sum_{l=n}^{p_{k}}a_{k,l}\frac{r^{l-n}}{(l-n)!}\left(\begin{array}[]{c}\bm{\mu}_{k,n}^{+}\\\
\bm{\mu}_{k,n}^{-}\\\ \end{array}\right)\\\ &\ \ \ \ \ \ \ \ \ \
+\exp\left(\lambda_{k}\left(\frac{L}{2}-r\right)\right)\sum_{n=1}^{p_{k}}\sum_{l=n}^{p_{k}}a_{-k,l}\frac{r^{l-n}}{(l-n)!}(-1)^{n+1}\left(\begin{array}[]{c}\bm{\mu}_{k,n}^{-}\\\
\bm{\mu}_{k,n}^{+}\\\ \end{array}\right)\Big{]},\end{array}$ (43)

with $(a_{\pm k,l})_{k,l}$ real constants and $a_{0,1}=0$ (no uniform
component along $(\mu_{\mathcal{B}},\mu_{\mathcal{B}})$). It is
straightforward to check that if $\bm{\gamma}^{+}(r)=\bm{\gamma}^{-}(r)$ for
all $r$, then $\bm{\gamma}^{+}(r)=\bm{\gamma}^{-}(r)=0$ and we recover the
detailed-jamming case. We can also check for the active global balance,

$\displaystyle\sum_{s}\phi(s)(\bm{\gamma}^{+}(r)-\bm{\gamma}^{-}(r))_{s}$
$\displaystyle=0$ (44)
$\displaystyle\sum_{k=1}^{d}\sum_{n=1}^{p_{k}}\sum_{l=n}^{p_{k}}\frac{r^{l-n}}{(l-n)!}\big{(}a_{k,l}e^{-\lambda_{k}(\frac{L}{2}-r)}-(-1)^{n+1}a_{-k,l}e^{\lambda_{k}(\frac{L}{2}-r)}\big{)}\sum_{s}\phi(s)(\bm{\mu}_{k,n}^{+}(s)-\bm{\mu}_{k,n}^{-}(s))$
$\displaystyle+\sum_{n=1}^{p_{0}/2}\sum_{l=2n}^{p_{0}}\frac{r^{l-2n}}{(l-2n)!}2a_{0,l}\sum_{s}\phi(s)\bm{\mu}_{0,2n}(s)=0.$

This is verified as, we have for all $k\geq 1$ ($\lambda_{k}\neq 0$) by
summing along $\phi(s)$ in Eq. 35 and reminiscing Eq. 32,

$\displaystyle\sum_{s}\phi(s)(\bm{\mu}_{k,n}^{+}(s)-\bm{\mu}_{k,n}^{-}(s))$
$\displaystyle=-\frac{1}{\lambda_{k}}\sum_{s}\phi(s)(\bm{\mu}_{k,n-1}^{+}(s)-\bm{\mu}_{k,n-1}^{-}(s))$
$\displaystyle=\frac{(-1)^{n-1}}{\lambda_{k}^{n-1}}\sum_{s}\phi(s)(\bm{\mu}_{k,1}^{+}(s)-\bm{\mu}_{k,1}^{-}(s))$
$\displaystyle=\frac{(-1)^{n-1}}{\lambda_{k}^{n}}\sum_{s^{\prime}}\underbrace{\left[\sum_{s}\phi(s)(B^{+}+B^{-})_{ss^{\prime}}\right]}_{=0\
\text{see
Eq.~{}\ref{eq:tr_0}}}(\bm{\mu}_{k,1}^{+}(s^{\prime})+\bm{\mu}_{k,1}^{-}(s^{\prime}))=0,$

And for $k=0$,

$\displaystyle\sum_{s}\phi(s)\bm{\mu}_{0,2n}(s)$
$\displaystyle=\sum_{s^{\prime}}\underbrace{\left[\sum_{s}\phi(s)(B^{+}+B^{-})_{ss^{\prime}}\right]}_{=0\
\text{see Eq.~{}\ref{eq:tr_0}}}\bm{\mu}_{0,2n+1}(s^{\prime}).$

####  C.3.2 Periodic boundary constraint

We now constrain the constants $(a_{\pm k,n})_{k,n}$ by the boundary
conditions. First, rewriting the general solution (43) into,

$\begin{array}[]{rl}\left(\begin{array}[]{c}\bm{\gamma}^{+}\\\
\bm{\gamma}^{-}\\\
\end{array}\right)(r)&=\sum_{n=1}^{p_{0}}\sum_{l=n}^{p_{0}}\Big{[}\sum_{m=l}^{p_{0}}a_{0,m}\frac{\big{(}\tfrac{L}{2}\big{)}^{m-l}}{(m-l)!}\Big{]}\frac{\big{(}r-\tfrac{L}{2}\big{)}^{l-n}}{(l-n)!}\left(\begin{array}[]{c}\bm{\mu}_{0,n}\\\
(-1)^{n+1}\bm{\mu}_{0,n}\\\ \end{array}\right)\\\
&+\sum_{k=1}^{d}\Big{[}\exp\left(-\lambda_{k}\left(\frac{L}{2}-r\right)\right)\sum_{n=1}^{p_{k}}\sum_{l=n}^{p_{k}}\Big{[}\sum_{m=l}^{p_{k}}a_{k,m}\frac{\big{(}\tfrac{L}{2}\big{)}^{m-l}}{(m-l)!}\Big{]}\frac{\big{(}r-\tfrac{L}{2}\big{)}^{l-n}}{(l-n)!}\left(\begin{array}[]{c}\bm{\mu}_{k,n}^{+}\\\
\bm{\mu}_{k,n}^{-}\\\ \end{array}\right)\\\ &\ \ \ \ \ \ \ \ \ \
+\exp\left(\lambda_{k}\left(\frac{L}{2}-r\right)\right)\sum_{n=1}^{p_{k}}\sum_{l=n}^{p_{k}}\Big{[}\sum_{m=l}^{p_{k}}(-1)^{n+1}a_{-k,m}\frac{\big{(}\tfrac{L}{2}\big{)}^{m-l}}{(m-l)!}\Big{]}\frac{\big{(}r-\tfrac{L}{2}\big{)}^{l-n}}{(l-n)!}\left(\begin{array}[]{c}\bm{\mu}_{k,n}^{-}\\\
\bm{\mu}_{k,n}^{+}\\\ \end{array}\right)\Big{]},\end{array}$ (45)

the periodic boundary condition
$\bm{\gamma}^{+}(L/2^{-})=\bm{\gamma}^{-}(L/2^{-})$ yields,

$\sum_{n=1}^{p_{0}/2}2\Big{[}\sum_{m=2n}^{p_{0}}a_{0,m}\frac{\big{(}\tfrac{L}{2}\big{)}^{m-2n}}{(m-2n)!}\Big{]}\bm{\mu}_{0,2n}+\sum_{k=1}^{d}\sum_{n=1}^{p_{k}}\Big{[}\sum_{m=n}^{p_{k}}(a_{k,m}-(-1)^{n+1}a_{k,m})\frac{\big{(}\tfrac{L}{2}\big{)}^{m-n}}{(m-n)!}\Big{]}(\bm{\mu}_{k,n}^{+}-\bm{\mu}_{k,n}^{-})=0.$
(46)

As the vectors
$(\bm{\mu}_{k,n}^{+}-\bm{\mu}_{k,n}^{-})_{k,n},(\bm{\mu}_{0,2n})_{n}$ are
linearly independent, we obtain the condition,

$\left\\{\begin{array}[]{l}\sum_{m=2n}^{p_{0}}a_{0,m}\frac{(L/2)^{m-2n}}{(m-2n)!}=0,\text{for
all}\ n\leq p_{0}/2\\\
\sum_{m=n}^{p_{k}}(a_{k,m}-a_{-k,m}(-1)^{n+1})\frac{(L/2)^{m-n}}{(m-n)!}=0,\text{for
all}\ n\leq p_{k}\\\ \end{array}\right.,$ (47)

constraining $\bm{\gamma}$ into the following symmetrical decomposition in
respect to $\bm{\gamma}^{+}(r)$ and $\bm{\gamma}^{-}(L-r)$,

$\begin{array}[]{rl}\left(\begin{array}[]{c}\bm{\gamma}^{+}(r)\\\
\bm{\gamma}^{-}(r)\\\
\end{array}\right)&=\sum_{l=0}^{p_{0}/2-1}b_{0,2l+1}\frac{\big{(}\tfrac{L}{2}-r\big{)}^{2l}}{(2l)!}\left(\begin{array}[]{c}\bm{\mu}_{\mathcal{B}}\\\
\bm{\mu}_{\mathcal{B}}\\\ \end{array}\right)\\\
&+\sum_{n=1}^{p_{0}/2-1}\sum_{l=n}^{p_{0}/2-1}b_{0,2l+1}\frac{\big{(}\tfrac{L}{2}-r\big{)}^{2(l-n)}}{(2(l-n))!}\Big{[}\left(\begin{array}[]{c}\bm{\mu}_{0,2n+1}\\\
\bm{\mu}_{0,2n+1}\\\
\end{array}\right)-\frac{\big{(}\tfrac{L}{2}-r\big{)}}{2(l-n)+1}\left(\begin{array}[]{c}\bm{\mu}_{0,2n}\\\
-\bm{\mu}_{0,2n}\\\ \end{array}\right)\Big{]}\\\
&+\sum_{k=1}^{d}\sum_{n=1}^{p_{k}}\sum_{l=n}^{p_{k}}b_{k,l}\frac{\big{(}\tfrac{L}{2}-r\big{)}^{l-n}}{(l-n)!}\Big{[}(-1)^{l-n}e^{-\lambda_{k}\left(\frac{L}{2}-r\right)}\left(\begin{array}[]{c}\bm{\mu}_{k,n}^{+}\\\
\bm{\mu}_{k,n}^{-}\\\
\end{array}\right)+e^{\lambda_{k}\left(\frac{L}{2}-r\right)}\left(\begin{array}[]{c}\bm{\mu}_{k,n}^{-}\\\
\bm{\mu}_{k,n}^{+}\\\ \end{array}\right)\Big{]},\end{array}$ (48)

with,

$\left\\{\begin{array}[]{l}b_{0,2l+1}=\sum_{m=2l+1}^{p_{0}-1}a_{0,m}\frac{\big{(}L/2\big{)}^{m-2l-1}}{(m-2l-1)!},\
\text{for}\ 1\leq 2l+1\leq p_{0}-1\\\ a_{0,1}=0\ \text{and}\
a_{0,2l}=-\sum_{m=2l+1}^{p_{0}}a_{0,m}\frac{\big{(}L/2\big{)}^{m-2l}}{(m-2l)!},\
\text{for}\ 2\leq 2l\leq p_{0}-2\\\
b_{k,l}=\sum_{m=l}^{p_{k}}a_{k,m}\frac{\big{(}L/2\big{)}^{m-l}}{(m-l)!}=(-1)^{l+1}\sum_{m=l}^{p_{k}}a_{-k,m}\frac{\big{(}L/2\big{)}^{m-l}}{(m-l)!},\
\text{for}\ 1\leq l\leq p_{k}\end{array}\right..$ (49)

####  C.3.3 Jamming boundary constraint

We now consider the constraint imposed by the jamming interaction. The
condition in $0^{+}$ set by $(C_{\mathcal{J}})$ leads to,

$\left\\{\begin{array}[]{l}w_{\mathcal{J}}B^{+}\bm{\mu}_{\mathcal{J}}^{-}=-\bm{\pi}_{\text{glob}}(0^{+},-1)\\\
w_{\mathcal{J}}B^{-}\bm{\mu}_{\mathcal{J}}^{-}=\bm{\pi}_{\text{glob}}(0^{+},1)\end{array}\right.,$
(50)

with $\bm{\mu}_{\mathcal{J}}^{-}=({\mu}_{\mathcal{J}}(-1,s))_{s}$,
$\bm{\pi}_{\text{glob}}(0^{+},-1)=(\pi_{\text{glob}}(0^{+},-1,s))_{s}$,
$\bm{\pi}_{\text{glob}}(0^{+},1)=(\pi_{\text{glob}}(0^{+},1,s))_{s}$. It
decomposes into,

$\left\\{\begin{array}[]{l}\tfrac{w_{\mathcal{J}}}{w_{\mathcal{B}}}(B^{+}+B^{-})\bm{\mu}_{\mathcal{J}}^{-}=w_{\text{rel}}(\bm{\gamma}^{+}(0^{+})-\bm{\gamma}^{-}(0^{+}))\\\
-\tfrac{w_{\mathcal{J}}}{w_{\mathcal{B}}}(B^{+}-B^{-})\bm{\mu}_{\mathcal{J}}^{-}=(2w_{\text{eq}}\bm{\mu}_{\mathcal{B}}+w_{\text{rel}}(\bm{\gamma}^{+}(0^{+})+\bm{\gamma}^{-}(0^{+})))\end{array}\right..$
(51)

It clearly appears that recovering a detailed-jamming solution (i.e.
$w_{\text{rel}}=0$) imposes $(B^{+}+B^{-})\bm{\mu}_{\mathcal{J}}^{-}=0$, i.e.
$\bm{\mu}_{\mathcal{J}}^{-}\propto\bm{\mu}_{\mathcal{B}}$, which, combined
with the second condition, then makes the detailed-jamming condition
equivalent to $\bm{\mu}_{\mathcal{B}}$ being the unique eigenvector shared by
$B^{+}$ and $B^{-}$ with opposite eigenvalues. It also imposes that
$(B^{+}-B^{-})\bm{\mu}_{\mathcal{J}}^{-}\propto\bm{\mu}_{\mathcal{B}}$, i.e.
$\bm{\mu}_{\mathcal{J}}^{-}\propto\bm{\mu}_{0,2}$, which leads to the
condition $\bm{\mu}_{0,2}\propto\bm{\mu}_{\mathcal{B}}$.

Now assuming $w_{\text{rel}}\neq 0$, i.e.
$\bm{\mu}_{\mathcal{J}}^{-}\not\propto\bm{\mu}_{\mathcal{B}}$ nor
$\bm{\mu}_{\mathcal{J}}^{-}\not\propto\bm{\mu}_{0,2}$, we decompose
$\bm{\mu}_{\mathcal{J}}^{-}$ along the two basis of general eigenvectors,

$\begin{array}[]{rl}\bm{\mu}^{-}_{\mathcal{J}}&=\sum_{n=1}^{p_{0}/2}c_{0,2n}\par\bm{\mu}_{0,2n}+\sum_{k=1}^{d}\sum_{n=1}^{p_{k}}c^{-}_{k,n}\left(\bm{\mu}_{k,n}^{+}-\bm{\mu}_{k,n}^{-}\right)\\\
&=\sum_{n=0}^{p_{0}/2-1}c_{0,2n+1}\par\bm{\mu}_{0,2n+1}+\sum_{k=1}^{d}\sum_{n=1}^{p_{k}}c^{+}_{k,n}\left(\bm{\mu}_{k,n}^{+}+\bm{\mu}_{k,n}^{-}\right)\end{array},$
(52)

so that,

$\begin{array}[]{rl}-(B^{+}-B^{-})\bm{\mu}^{-}_{\mathcal{J}}&=-\sum_{n=1}^{p_{0}/2}c_{0,2n}\par\bm{\mu}_{0,2n-1}-\sum_{k=1}^{d}\Big{[}\lambda_{k}c^{-}_{k,p_{k}}\left(\bm{\mu}_{k,p_{k}}^{+}+\bm{\mu}_{k,p_{k}}^{-}\right)\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \
+\sum_{n=1}^{p_{k}-1}(\lambda_{k}c^{-}_{k,n}+c^{-}_{k,n+1})\left(\bm{\mu}_{k,n}^{+}+\bm{\mu}_{k,n}^{-}\right)\Big{]}\\\
(B^{+}+B^{-})\bm{\mu}^{-}_{\mathcal{J}}&=\sum_{n=1}^{p_{0}/2-1}c_{0,2n+1}\par\bm{\mu}_{0,2n}+\sum_{k=1}^{d}\Big{[}\lambda_{k}c^{+}_{k,p_{k}}\left(\bm{\mu}_{k,p_{k}}^{+}-\bm{\mu}_{k,p_{k}}^{-}\right)\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \
+\sum_{n=1}^{p_{k}-1}(\lambda_{k}c^{+}_{k,n}+c^{+}_{k,n+1})\left(\bm{\mu}_{k,n}^{+}-\bm{\mu}_{k,n}^{-}\right)\Big{]}\end{array}.$
(53)

From the expression of $\bm{\gamma}^{\pm}$ (48), we get that,

$\begin{array}[]{rl}&\bm{\gamma}^{+}(0)+\bm{\gamma}^{-}(0)=\sum_{n=0}^{p_{0}/2-1}\sum_{l=n}^{p_{0}/2-1}2b_{0,2l+1}\frac{\big{(}\tfrac{L}{2}\big{)}^{2(l-n)}}{(2(l-n))!}\par\bm{\mu}_{0,2n+1}\\\
&+\sum_{k=1}^{d}\sum_{n=1}^{p_{k}}\sum_{l=n}^{p_{k}}b_{k,l}\frac{\big{(}\tfrac{L}{2}\big{)}^{l-n}}{(l-n)!}\Big{[}\exp\left(\lambda_{k}\frac{L}{2}\right)+(-1)^{l-n}\exp\left(-\lambda_{k}\frac{L}{2}\right)\Big{]}\left(\bm{\mu}_{k,n}^{+}+\bm{\mu}_{k,n}^{-}\right),\end{array}$
(54)

and,

$\begin{array}[]{rl}&\bm{\gamma}^{+}(0)-\bm{\gamma}^{-}(0)\par=-\sum_{n=1}^{p_{0}/2-1}\sum_{l=n}^{p_{0}/2-1}2b_{0,2l+1}\frac{\big{(}\tfrac{L}{2}\big{)}^{2(l-n)+1}}{(2(l-n)+1)!}\bm{\mu}_{0,2n}\\\
&+\sum_{k=1}^{d}\sum_{n=1}^{p_{k}}\sum_{l=n}^{p_{k}}b_{k,l}\frac{\big{(}\tfrac{L}{2}\big{)}^{l-n}}{(l-n)!}\Big{[}-\exp\left(\lambda_{k}\frac{L}{2}\right)+(-1)^{l-n}\exp\left(-\lambda_{k}\frac{L}{2}\right)\Big{]}\left(\bm{\mu}_{k,n}^{+}-\bm{\mu}_{k,n}^{-}\right).\end{array}$
(55)

Plugging those expression in (51), we obtain,

$\left\\{\begin{array}[]{l}w_{\mathcal{J}}c_{0,2}=-2w_{\mathcal{B}}\big{(}w_{\text{eq}}+w_{\text{rel}}\sum_{l=0}^{p_{0}/2-1}b_{0,2l+1}\frac{(L/2)^{2l}}{(2l)!}\big{)}\\\
w_{\mathcal{J}}c_{0,2(n+1)}=-2w_{\mathcal{B}}w_{\text{rel}}\sum_{l=n}^{p_{0}/2-1}b_{0,2l+1}\frac{(L/2)^{2(l-n)}}{(2(l-n))!}\
\text{for}\ n\geq 1\\\
w_{\mathcal{J}}c_{0,2n+1}=-2w_{\mathcal{B}}w_{\text{rel}}\sum_{l=n}^{p_{0}/2-1}b_{0,2l+1}\frac{(L/2)^{2(l-n)+1}}{(2(l-n)+1)!}\
\text{for}\ n\geq 1\\\
w_{\mathcal{J}}(\lambda_{k}c^{-}_{k,n}+c^{-}_{k,n+1})=-w_{\mathcal{B}}w_{\text{rel}}\sum_{l=n}^{p_{k}}b_{k,l}\frac{(L/2)^{l-n}}{(l-n)!}\Big{[}e^{\lambda_{k}\frac{L}{2}}+(-1)^{l-n}e^{-\lambda_{k}\frac{L}{2}}\Big{]}\
\text{for}\ n<p_{k}\\\
w_{\mathcal{J}}(\lambda_{k}c^{+}_{k,n}+c^{+}_{k,n+1})=-w_{\mathcal{B}}w_{\text{rel}}\sum_{l=n}^{p_{k}}b_{k,l}\frac{(L/2)^{l-n}}{(l-n)!}\Big{[}e^{\lambda_{k}\frac{L}{2}}-(-1)^{l-n}e^{-\lambda_{k}\frac{L}{2}}\Big{]}\
\text{for}\ n<p_{k}\\\
w_{\mathcal{J}}\lambda_{k}c^{-}_{k,p_{k}}=-w_{\mathcal{B}}w_{\text{rel}}b_{k,p_{k}}\Big{[}e^{\lambda_{k}\frac{L}{2}}+e^{-\lambda_{k}\frac{L}{2}}\Big{]}=w_{\mathcal{J}}\lambda_{k}c^{+}_{k,p_{k}}\text{cotanh}\left(\lambda_{k}\frac{L}{2}\right)\\\
\\\ \end{array}\right.$ (56)

Recovering a catenary-like solution (i.e. no polynomial modulation) then
requires

$\left\\{\begin{array}[]{l}w_{\mathcal{J}}c_{0,2}=-2w_{\mathcal{B}}w_{\text{eq}}\\\
c_{0,2(n+1)}=c_{0,2n+1}=0\ \text{for}\ n\geq 1\\\
w_{\mathcal{J}}\lambda_{k}c^{-}_{k,1}=-w_{\mathcal{B}}w_{\text{rel}}b_{k,1}\Big{[}\exp\left(\lambda_{k}\frac{L}{2}\right)+\exp\left(-\lambda_{k}\frac{L}{2}\right)\Big{]}=w_{\mathcal{J}}\lambda_{k}c^{+}_{k,1}\text{cotanh}\left(\lambda_{k}\frac{L}{2}\right)\\\
c^{-}_{k,n}=c^{+}_{k,n}=b_{k,n}=0\ \text{for}\ n>1\end{array}\right.,$ (57)

which, following (52), is possible only if the generalized eigenvector
$\mu_{0,2}$ can be decomposed over
$\big{(}\mu_{\mathcal{B}},(\mu_{k,1}^{+}+\mu^{-}_{k,1})_{k\in N}\big{)}$, $N$
some subset of $\llbracket 1,d\rrbracket$, as well as the corresponding
vectors $(\mu_{k,1}^{+}-\mu^{-}_{k,1})_{k\in N}$ so that the third relation in
(57) is obeyed.

More generally, the more vectors of
$\big{(}\mu_{\mathcal{B}},(\mu_{k,n}^{+}+\mu^{-}_{k,n})_{k,n}\big{)}$ are
needed to decompose $\mu_{0,2}$ over, the richer and less close to its
equilibrium counterpart the bulk relaxation behavior will be. A special case
is the detailed-jamming situation where $\mu_{0,2}$ is proportional to
$\mu_{\mathcal{B}}$ and $N=\emptyset$ or the catenary-like relaxation where
only the eigenvectors are involved in the decomposition.

That analysis of the impact of the jamming boundary constraint on the bulk
relaxation can be generalized to a tumble Markov kernel $Q^{\prime}$ different
from $Q$, leading to different matrices $B^{\prime+},B^{\prime-}$ in (51) and
the consideration of their eigenvectors and generalized ones.

###  C.4 Case of a single non-null eigenvalue

As in the case of the finite tumble represented in Figure 2, an interesting
situation is when the particles have an internal state space and tumbling
mechanism presenting enough symmetries so that it can be summarized into a
four-state space. In this collapsed representation, the corresponding matrix
$B_{\text{coll}}$ is of size $4$, and, as the Jordan block associated to $0$
is at least of size $2$, can only admit two Jordan blocks of size $1$ linked
to the eigenvalues $\pm\lambda\neq 0$. From the analysis carried on in
Appendix C.2.2, $\lambda^{2}$ is the only non-null eigenvalue of
$M_{\text{coll}}^{+}M_{\text{coll}}^{-}$ (or
$M_{\text{coll}}^{-}M_{\text{coll}}^{+}$), so that
$\lambda^{2}=\text{Tr}(M_{\text{coll}}^{+}M_{\text{coll}}^{-})$.

Recovering the finite-tumble case, we now detail the case of such a
symmetrical three-state RTP system. We note the three states
$\\{v_{1},v_{2},v_{3}\\}$. The symmetry leading to a possible collapse to a
four-state space is obeyed if two pair states, e.g. $\\{v_{1},v_{2}\\}$ and
$\\{v_{2},v_{3}\\}$ are equivalent, i.e. present the same run and tumble
mechanisms, leading to $\pi(r,\pm 1,\\{v_{2},v_{3}\\})=\pi(r,\pm
1,\\{v_{1},v_{2}\\})$, and more precisely framed by the conditions,

$\left\\{\begin{array}[]{l}q(v_{3},v_{1})=q(v_{1},v_{3})\\\
q(v_{2},v_{1})=q(v_{2},v_{3})=\tfrac{1}{2}\\\ q(v_{1},v_{2})=q(v_{3},v_{2})\\\
|v_{3}-v_{2}|=|v_{2}-v_{1}|\\\ \omega(v_{3})=\omega(v_{1})\\\
\end{array}\right..$ (58)

In the finite-tumble case, the central state $v_{2}$ is the state $0$. This
leads to the following matrices,

$\displaystyle B^{+}+B^{-}$
$\displaystyle=\frac{(1+q(v_{1},v_{3}))}{|v_{1}-v_{2}|}\left(\begin{array}[]{lll}-\omega(v_{1})(1-q(v_{1},v_{3}))&\tfrac{1}{4}\omega(v_{2})&\tfrac{1}{4}\omega(v_{2})\\\
\omega(v_{1})(1-q(v_{1},v_{3}))&-\tfrac{1}{2}(\omega(v_{1})+\omega(v_{2}))&\tfrac{1}{2}\omega(v_{1})\\\
\omega(v_{1})(1-q(v_{1},v_{3}))&\tfrac{1}{2}\omega(v_{1})&-\tfrac{1}{2}(\omega(v_{1})+\omega(v_{2}))\\\
\end{array}\right)$ (59) $\displaystyle B^{+}-B^{-}$
$\displaystyle=\frac{1}{|v_{1}-v_{2}|}\left(\begin{array}[]{lll}-\omega(v_{1})&\tfrac{1}{4}\omega(v_{2})&\tfrac{1}{4}\omega(v_{2})\\\
\omega(v_{1})(1-q(v_{1},v_{3}))&-(\omega(v_{1})+\omega(v_{2}))&-\omega(v_{1})q(v_{1},v_{3})\\\
\omega(v_{1})(1-q(v_{1},v_{3}))&-\omega(v_{1})q(v_{1},v_{3})&-(\omega(v_{1})+\omega(v_{2}))\\\
\end{array}\right),$

so that the eigenvalues of $M^{+}M^{-}$ and $M^{-}M^{+}$ are $0$ and

$\lambda^{2}=(\omega(v_{1})(1-q(v_{1},v_{3}))+\omega(v_{2}))(\omega(v_{1})+\tfrac{1}{2}\omega(v_{2}))\frac{1+q(v_{1},v_{3})}{(v_{1}-v_{2})^{2}}.$
(60)

In particular, in the case of the finite tumble, i.e.
$v_{1}=-1,v_{2}=0,v_{3}=1$ and $\omega(v_{1})=\omega(v_{3})=\alpha$,
$\omega(v_{2})=\beta$ and $q(v_{1},v_{3})=q(v_{3},v_{1})=0$, the typical
inverse lengthscale of the catenary relaxation is,

$\lambda=\sqrt{(\alpha+\beta)(\alpha+\tfrac{1}{2}\beta)}.$ (61)

More generally, if the persistent run dominates over the tumbles
($|v_{1}-v_{2}|>>\omega(v_{1})+\omega(v_{2})$), then $\lambda\to 0$ and the
system does not decorrelate from the jamming constraint. In the opposite
situation where the tumbles dominates over the run
($|v_{1}-v_{2}|<<\omega(v_{1})+\omega(v_{2})$), then $\lambda\to\infty$ and
the density immediately relaxes to the bulk uniform solution set by
$\mu_{\mathcal{B}}$.


# Formation of narrow atomic lines of Rb in the UV region using a magnetic
field

Ara Tonoyana Armen Sargsyana Rodolphe Momiera,b Claude Leroyb David Sarkisyana
,∗ a Institute for Physical Research, National Academy of Sciences - Ashtarak
2, 0203, Armenia bLaboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS
6303, Université de Bourgogne, 21000 Dijon, France<EMAIL_ADDRESS>
###### Abstract

Magnetically induced (MI) transitions (Fg = 1 $\rightarrow$ Fe= 3) of 87Rb D2
line are among the most promising atomic transitions for applications in laser
physics. They reach their maximum intensity in the 0.2–2 kG magnetic field
range and are more intense than many conventional atomic transitions. An
important feature of MI transitions is their large frequency shift with
respect to the unperturbed hyperfine transitions which reaches $\sim$12 GHz in
magnetic fields of $\sim$ 3 kG, while they are formed on the high-frequency
wing of the spectrum and do not overlap with other transitions. Some important
peculiarities have been demonstrated for the MI 5S1/2$\rightarrow$ 5P3/2
transitions ($\lambda$=780 nm). Particularly, it was shown that using a
nanocell with thickness $L=100$ nm it is possible to realize 1 $\mu$m-spatial
resolution which is important when determining magnetic fields with strong
spatial gradient (of $>$ 3G /$\mu$m). Earlier, our studies have been performed
for 5S1/2 $\rightarrow$ $n$P3/2 transition with $n=5$, while it is also
theoretically shown to be promising for the transitions with $n=6,7,8$ and
$9$, corresponding to the transition wavelengths of 420.2 nm, 358.7 nm, 334.9
nm and 322.8 nm, respectively.

††journal: oe

##  1 INTRODUCTION

A high interest has recently been focused on atomic transitions between ground
and excited levels of the hyperfine structures satisfying the conditions
$F_{e}-F_{g}=\Delta F=\pm 2$ (the probability of these transitions is zero,
when the external magnetic field is zero), while the probabilities in a
magnetic field increase significantly; for this reason, we refer to these
transitions as Magnetically Induced (MI) transitions [1, 2]. For the case of
$\sigma^{+}$ radiation, the MI transitions $F_{g}=1\rightarrow F_{e}=3$ of
87Rb D2 line are among the most promising. The following rule has been found
for the intensities of MI transitions: the intensities of magnetically induced
transitions with $F_{e}-F_{g}=\Delta F=+2$ are maximal (the number of such MI
transitions is also maximal) in case of $\sigma^{+}$ polarized radiation,
whereas the intensities of MI transitions with $\Delta$F= -2 are maximal (the
number of such transitions is also maximal) in the case of $\sigma^{-}$
polarized radiation. An attractive feature of $F_{g}=1\rightarrow F_{e}=3$ MI
transitions is their large frequency shift with respect to the unperturbed
hyperfine transitions in a magnetic field, which reaches ${\sim}$12 GHz in a
magnetic field $B$ ${\sim}$ 3 kG. It is important to note that they are formed
on the high-frequency wing of the spectrum and do not overlap with others,
which is of practical interest for the study of new frequency ranges.

Previous experiments have been performed for 5S1/2 $\rightarrow$ nP3/2
transitions for $n=5$ ($\lambda$=780 nm), while theoretically it is shown in
Section 4 to be promising also transitions with $n=6,7,8$ and 9, the
corresponding wavelengths being 420.2 nm, 358.7nm, 334.9 nm and 322.8 nm.

The theoretical model used to calculate the behavior of the Zeeman transitions
based on building the magnetic Hamiltonian including the whole hyperfine
manifold in the |F, mF${>}$ basis is described in detail in [3-6].

Figure 1: Sketch of the experimental setup. ECDL – diode laser, $\lambda$ =
780nm, FI - Faraday isolator, 1 \- Rb nanocell,2\- quarter-wave plate , 3 are
the permanent magnets, 4 \- photodetectors, 5-SA unit for frequency reference,
6 – digital storage oscilloscope, E–electric field of laser radiation.
Configurations of B, E and k are presented in the insets,
$\textbf{B}\parallel\textbf{k}$, where k is wave vector of the laser field.

##  2 EXPERIMENTAL SETUP

The nanocell (NC) construction, which consists of 2.4mm-thick windows made of
technical sapphire (Al2O3) and a vertical side arm (a sapphire tube filled
with Rb) is presented in [7]. The NC is filled with a natural mixture of 85Rb
(72.2 %) and 87Rb (27.8%). The region of the NC L ${\approx}$ $\lambda$/2 is
used, where $\lambda$= 780 nm is the transition wavelength of Rb D2 line. The
NC operated with a specially designed oven with two ports for laser beam
transmission. The temperature of the NC reservoir (which contains the Rb
atoms) was ${\sim}$130 ∘C, but the windows were maintained at a temperature
that was 20 ∘C higher (to avoid condensation of the atomic vapor on the
windows ). The experimental setup is depicted on Fig 1. The Rb nanocell (2) is
located between the strong magnets (3). To avoid feedback (reflection from the
different optical elements), a Faraday insulator (FI) was used. A part of the
laser radiation was directed to the unit (5) for frequency reference using the
saturated absorption technique. The transmission signal was detected by a
photodiode (4) and was recorded with a Tektronix TDS 2014B four-channel
storage oscilloscope (6). In order to produce strong magnetic fields two
strong permanent magnets were used. Magnetic field B is directed along the
laser radiation propagation k ($B$$\mid$$\mid$k). Configuration of $B$ and
vertical linear polarization (which correspond to circular polarizations
$\sigma$+ and $\sigma$- when a longitudinal magnetic field is applied) is
shown in the inset of Fig.1. The$B$-field was measured with a Teslameter HT201
magnetometer.

Figure 2: (a) The left and right sides show an experimental SD absorption
spectrum of atomic transitions Fg = 1$\rightarrow$ Fe=2,3 of the 87Rb D2 line
for linearly polarized laser radiation (only spectra for $\sigma$+ are shown)
for a $B$ = 1.15 kG and 3 kG, respectively. The inset shows diagram of the Fg
= 1$\rightarrow$ Fe=2, 3 transitions. (b) Calculated frequency shift of Fg =
1$\rightarrow$ Fe=3 versus $B$-field.

The nanocell (NC) thickness in the direction of laser beam propagation was
chosen to be equal to half the resonant wavelength of Rb D2 line (L=$\lambda$
/2 = 390 nm). The technique allowing to measure the thickness of an atomic
vapor column in the NC is described in [8]. It has been demonstrated earlier
that in this case (so called $\lambda$/2 method), narrowing of atomic
transitions (lines) in the absorption spectra A($\omega$) of the NC occurs
[8,9]. This effect is called Coherent Dicke Narrowing. To obtain further
narrowing of the atomic lines, we performed second derivative (SD) of the
absorption spectrum A′′ ($\omega$) [5, 10,11]. In Fig. 2a, the left and right
sides show experimental SD absorption spectra of atomic transitions Fg =
1$\rightarrow$ Fe=2, 3 of the D2 line of 87Rb for linearly polarized laser
radiation (only spectra for $\sigma$+ are shown) for $B$ = 1.15 kG and 3 kG,
respectively. In the case of $B$ = 3 kG the frequency shift of the strongest
MI transition Fg=1, mF=-1$\rightarrow$Fe=3, mF=0 reaches ${\sim}$9 GHz, with
respect of the initial position shown in Fig.2b, while for $B$=4 kG (these MI
transitions still have high intensity) it reaches ${\sim}$ 17 GHz. The inset
shows diagram of the Fg = 1$\rightarrow$ Fe=2, 3 transitions.

Figure 3: Theoretical SD absorption spectra: the red curve has been calculated
for $B$1 = 3000 G, and the blue curve has been calculated for $B$2 = 3003.3 G.
When moving the NC by 1 $\mu$m (fields $B$1 and $B$2), the frequency shift
become 13 MHz which can be measured well. The inset shows the frequency slope
of 1$\rightarrow$3′ MI transitions. Figure 4: The curve labelled
“transmission” shows an EIT-resonance using a NC with L=100 nm, using two
diode lasers, the coupling and probe laser power are 30 and 0.1 mW
respectively. The temperature of the NC is 180oC. Left inset shows the
$\Lambda$-system formed by the probe radiation (the MI transition 1-3′) and
the coupling radiation 2-3′. Right Inset shows the second derivative of the
EIT-resonance, which is fitted by the Gaussian curve, the line-width is of
around 30 MHz.

As shown in the works [12,13], an important advantage of the nanocell use is
the possibility to measure strong magnetic fields gradients with a high
spatial resolution. The paper [14] provides a description of a Stern-Gerlach
type deflecting magnet, intended to deflect beams of paramagnetic
nanoclusters, molecules and atoms using a magnetic field with a gradient of
3.3 G/$\mu$m. Figure 3 shows theoretical SD absorption spectra (red and blue
solid lines with FWHM of 10 MHz and fitted with a Voigt profile) that can be
used with the help of the MI transition 3 (see Fig. 2b) of 87Rb to measure
such a gradient. If a NC with a thickness L = 100 nm is placed on a micrometer
stage (magnetic field $B$1 = 3 kG) and shifted by 1$\mu$m with respect to the
initial position, $B$ becomes $B$2 = 3003.3G. The red curve has been
calculated for B1 = 3000 G, and the blue curve has been calculated for $B$2 =
3003.3 G. As can be seen from Fig.3, when moving the NC by 1 $\mu$m (fields B1
and B2), the transition was shifted of 13 MHz, which can be measured well. The
inset of Fig. 3 shows the frequency slope (MHz/G) of 1$\rightarrow$3′ MI
transitions, which is 3.7 MHz/G for B${\sim}$3 kG and tend to 4 MHz/G at
higher $B$-field.

In the work [2] it has been experimentally demonstrated that 87Rb MI
transitions 1-3′ have been used successfully to form EIT-resonances in
$\Lambda$-systems in magnetic field up to 3 kG. For this purpose the NC was
used with L${\sim}$800 nm. In Fig.4 the curve labeled "transmission" shows a
transmission spectrum with an EIT-resonance obtained experimentally using a
nanocell with L=100 nm in the transmission spectrum, using two independent
diode lasers with 1 MHz linewidth. The $\Lambda$-system which is formed by the
probe radiation (the MI transition 1-3′) and the coupling radiation 2-3′ is
shown in the left inset. The curve labeled SD is the second derivative of the
transmission spectrum, which is well fitted by a Gaussian curve with FWHM 30
MHz, as presented in the right inset. In order to achieve 1 $\mu$m-spatial
resolution shown in Fig. 4 it is needed to reduce the EIT-resonance to
${\sim}$10 MHz or less. To implement this, we turn to the works [15-17] in
which coherently coupled probe and coupling radiations derived from a single
narrow-band laser beam were used. Using a vapor thin cell with L=100 $\mu$m
they obtained 10-kHz narrow EIT-resonance. This means that using coherently
coupled probe and coupling radiations and reducing the thickness L by a factor
of 1000 (this will increase the collisional linewidth by the same factor),
therefore it will be possible to obtain an EIT resonance with a width of at
least 10 MHz for L=100 nm and to achieve 1 $\mu$m-spatial resolution, as
explained in Fig 3a and above.

##  3 5S1/2\- nP3/2 TRANSITION FOR PRINCIPAL QUANTUM NUMBER n = 5,6,7,8 AND 9

Figure 5: (a) Intensities of 87Rb , 5S${}_{1/2\ }$\- nP3/2, MI Fg=1, mF${}_{\
}$= -1$\rightarrow$Fe=3, mF${}_{\ }$= 0 transition intensity versus $B$-field
for the $\sigma$${}^{+\ }$radiation, for principal quantum number n =
5,6,7,8,9. Since the intensity of the MI transition decreases with increasing
in n, the figure indicates by what value the intensity of the corresponding
transition is multiplied. (b) Spectra of 5S${}_{1/2\ }$\- nP${}_{3/2\ }$MI
Fg=1$\rightarrow$Fe=3, transitions versus $B$-field for the $\sigmaup$${}^{+\
}$radiation, longitudinal magnetic field of $B$ = 3kG is applied. The MI are
formed on the high-frequency wing of the spectrum and do not overlap with
others.

All previous studies have been done for 5S1/2\- nP${}_{3/2\ }$transition for n
= 5 ($\lambda$=780 nm), while theoretically is shown to be promising also
transitions with n=6,7,8 and 9, the corresponding laser radiation wavelengths
are 420.2 nm 358.7nm 334.9 nm and 322.8 nm. Numerical simulations of the
strongest MI transition for atomic transitions of 5S1/2\- nP3/2 of 87Rb D2
line, where n = 6,7,8,9 while the corresponding laser radiation wavelengths
420.2 nm 358.7nm 334.9 nm and 322.8 nm versus magnetic field are presented in
Fig.5a. With an increase in principal quantum number n, the magnetic field at
which the maximum intensity of the MI transition is reached decreases, which
is important for practical applications. Calculations show that for n $\geq$
17 ($\lambda$ $\leq$ 301.5nm) the earth’s magnetic field is sufficient for the
“forbidden” MI transition to become allowed. It’s important to note that the
usage of a nanocell filled with the Rb vapor with a half-wavelength thickness
[2,9] could make it possible to form narrow atomic transitions Fg = 1
$\rightarrow$ Fe= 3 in the UV region. Since the probabilities of all
5S${}_{1/2\ }$\- nP${}_{3/2\ }$transitions decreases with increasing of the
principal quantum number n, the MI transition remains among the strongest
atomic transitions in their group.

In Fig.5b, spectra of the Fg=1$\rightarrow$Fe=3 MI transitions of the
5S${}_{1/2\ }$\- nP3/2 (n = 5,6,7,8,9)${}_{\ }$lines of${}_{\ }$87Rb versus
magnetic field for the $\sigma^{+}$${}^{\ }$polarized radiation, for principal
quantum number are presented. Longitudinal magnetic field of B = 3kG is
applied. The MI are formed on the high-frequency wing of the spectrum and do
not overlap with others, which is of practical interest in laser spectroscopy.

The frequency shift of MI transitions in strong magnetic fields can reach
several tens of gigahertz with respect to the unperturbed hyperfine
transitions, which is significantly interesting for potential application of
using new frequency ranges, e.g. for laser frequency stabilization at
frequencies strongly shifted from the initial transition frequencies of
unperturbed atoms [18,19]. Note that in the case of sensitive photo receiver
use the formation of the narrow atomic lines could be obtained in a NC at a
room temperature [20].

##  4 CONCLUSIONS

All previous experimental and theoretical studies of the MI transitions of
87Rb, Fg=1$\rightarrow$Fe=3 have been done for 5S${}_{1/2\ }$\- nP${}_{3/2\
}$transition for n = 5 ($\lambda$ = 780 nm). In the manuscript an experimental
second derivative absorption spectra of atomic transitions Fg = 1$\rightarrow$
Fe=2,3 for $\sigma$+ circularly polarized laser radiation for a longitudinal
$B$-field of 1.15 kG and 3 kG, respectively are presented which show good
spectral resolution, when using a Rb nanocell with L${\sim}$ 400 nm A
description is given of a technique based on the EIT process, a nanocell with
L=100 nm, and a magnetically induced transition, which makes it possible to
realize a 1-$\mu$m spatial resolution of a magnetic field with a big gradient.

Here is theoretically shown the use of the transitions with n = 6,7,8 and 9,
with the corresponding laser radiation wavelengths are 420.2 nm, 358.7nm,
334.9 nm and 322.8 nm. The technique based on the nanocell use described above
could be also successfully adopted for the UV spectra.

## Funding

The work was supported by the NATO Science for Peace and Security Project
under grant G5794. The work was also supported by the Science Committee of RA,
in the frames of project N 1-6/23-I/IPR.

## Disclosures

The authors declare no conflicts of interest.

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