# The law of entropy increase and the Meissner effect

A. V. Nikulov<EMAIL_ADDRESS>Institute of Microelectronics Technology and
High Purity Materials, Russian Academy of Sciences, 142432 Chernogolovka,
Moscow District, RUSSIA

###### Abstract

The law of entropy increase postulates the existence of irreversible processes
in physics: the total entropy of an isolated system can increase but cannot
decrease. The annihilation of an electric current in normal metal with the
generation of Joule heat because of a non-zero resistance is well-known
example of irreversible process. The persistent current, an undamped electric
current observed in a superconductor, annihilates after the transition into
the normal state. Therefore this transition was considered as irreversible
thermodynamic process before 1933. But if this transition is irreversible then
the Meissner effect discovered in 1933 is experimental evidence of a process
reverse to the irreversible process. Belief in the law of entropy increase
forced physicists to change their understanding of the superconducting
transition, which is considered a phase transition after 1933. This change has
resulted to the internal inconsistency of the conventional theory of
superconductivity, which is created within the framework of reversible
thermodynamics but predicts Joule heating. The persistent current annihilates
after the transition into the normal state with the generation of Joule heat
and reappears during the return to the superconducting state according to this
theory and contrary to the law of entropy increase. The success of the
conventional theory of superconductivity forces to consider the validity of
belief in the law of entropy increase.

##  I Introduction

The theory of superconductivity GL1950 ; BCS1957  is an outstanding
achievement of twentieth-century physics. The famous Ginzburg-Landau (GL)
theory GL1950  published in 1950 describes superconductivity as a macroscopic
quantum phenomenon with the help of the GL wave function. The GL theory GL1950
is based on the assumption that numerous electrons can be on the same level.
It is impossible according to the basis of quantum mechanics since electrons
are fermions rather than bosons. The main efforts of the theorists dealing
with superconductivity in the forties and fifties were aimed at explaining how
electrons can turn into bosons. According to the majority of physicists, this
was done in 1957 by J. Bardeen, L.N. Cooper, and J.R. Schrieffer (BCS) BCS1957
. According to the BCS theory, electrons become bosons due to the formation of
Cooper pairs of electrons BCS1957 .

The conventional theory of superconductivity GL1950 ; BCS1957  successfully
describes numerous quantum phenomena observed in superconductors. The Meissner
effect, discovered experimentally in 1933 Meissner1933 , is described as a
special case of the quantization of the magnetic flux, discovered
experimentally in 1961 FluxQuant1961 , and the the quantization of the
persistent current, discovered experimentally in 1962 LP1962 . One of the most
outstanding achievements of the theory is the description of the Abrikosov
state Abrikosov  observed in type II superconductors Huebener . The
achievements of the conventional theory of superconductivity GL1950 ; BCS1957
are so great that until recently no one questioned this theory.

But recently Jorge Hirsch drew attention to the internal inconsistency of this
theory: on the one hand, this theory was created within the framework of
equilibrium reversible thermodynamics, and on the other hand, it predicts
Joule heating Hirsch2020EPL ; Hirsch2020Physica ; Hirsch2020ModPhys . The
inconsistency is obvious since Joule heating is irreversible thermodynamic
process which cannot be described in the framework of equilibrium reversible
thermodynamics. Therefore, it is strange that no one noticed this
inconsistency earlier. The inconsistency is considered in the paper
Physica2021  as a consequence of the history of superconductivity
investigation and the belief of the most scientists in the law of entropy
increase.

The transition of a bulk superconductor into the normal state in a magnetic
field $H$ was considered as an irreversible process before the discovery the
Meissner effect Meissner1933 . Expert in superconductivity D. Shoenberg wrote
in the book Shoenberg1952  published in 1952: ”At that time [before 1933], it
was assumed that the transition in a magnetic field is substantially
irreversible, since the superconductor was considered as a perfect conductor
(in the sense which was discussed in Chapter II), in which, when the
superconductivity is destroyed, the surface currents associated with the field
are damped with the generation of Joule heat”.

But if this transition is an irreversible thermodynamic process during which
the total entropy increases on $\Delta S=E_{k}/T$ due to the dissipation of
the kinetic energy $E_{k}$ of the electric current into Joule heat then the
Meissner effect is experimental evidence of a process reverse to the
irreversible thermodynamic process during which Joule heat is converted back
into the kinetic energy and the total entropy decreases on $\Delta
S=-E_{k}/T$. Belief in the law of entropy increase forced physicists to change
their understanding of the superconducting transition, which began to be
considered as a phase transition after 1933. Therefore the conventional theory
of superconductivity GL1950 ; BCS1957  was created within the framework of
equilibrium reversible thermodynamics.

This theory explains the Meissner effect Meissner1933 , the quantization of
the magnetic flux FluxQuant1961  and of the persistent current LP1962  as a
consequence of the appearance of the persistent current after the transition
into the superconducting state because of the quantization of the angular
momentum of Cooper pairs. The persistent current annihilates after the
transition into the normal state. No generation of Joule heat should be at
this transition in order it can be considered as the phase transition.
Therefore the famous physicist W. H. Keesom wrote in 1934 that ”it is
essential that the persistent currents have been annihilated before the
material gets resistance, so that no Joule-heat is developed” Keesom1934 .

But the conventional theory of superconductivity GL1950 ; BCS1957  does not
explain how the persistent currents can be annihilated before the material
gets resistance. This theory considers the process of disappearance of the
persistent current during the transition into the normal state in the same way
as this process was considered before 1933: the persistent currents are damped
with the generation of Joule heat. This internal inconsistency, indicated for
the first time in Hirsch2020EPL ; Hirsch2020Physica ; Hirsch2020ModPhys ,
means that the conventional theory of superconductivity GL1950 ; BCS1957
contradicts to the law of entropy increase since the appearance of the
persistent current after the transition into the superconducting state is the
process reverse to the irreversible thermodynamic process according to this
theory.

The success of the conventional theory of superconductivity GL1950 ; BCS1957
in the description of numerous phenomena observed in superconductors prompts
us to consider the validity of the almost universal belief in the law of
entropy increase. The history and the basis of this belief will be considered
in the next section. In the third section, reader’s attention will be drawn to
the change of the sense of the law of entropy increase after the victory of
the atomistic-kinetic worldview over the thermodynamic-energy worldview. The
assumption of molecular disorder needed for the validity of the law of entropy
increase and examples of its violation in quantum systems will be considered
in the fourth section.

Jorge Hirsch states that the conventional BCS theory of superconductivity
BCS1957  is wrong because of the prediction of Joule heating Hirsch2020EPL ;
Hirsch2020Physica ; Hirsch2020ModPhys . He ignores numerous experimental
evidences of Joule heating observed in superconductors which corroborate this
prediction of GL1950 ; BCS1957 . Hirsch also ignores the obvious mistakes that
physicists had to make after the discovery of the Meissner effect in order to
maintain their faith in the law of entropy increase. These mistakes will be
considered in the five section. The urgency of an unbiased and open discussion
of the possibility of violating the law of entropy increase is justified in
the Conclusion.

##  II The history of the law of entropy increase

The concept of entropy as a physical property was established for
thermodynamic system by the second law of thermodynamics. This concept came
from German scientist Rudolf Clausius who formulated in the 1850s the Carnot
principle Carnot  with the help of this concept. Therefore as Smoluchowski
wrote in 1914: ”We call the Carnot principle as the second law of
thermodynamics since Clausius’s time” Smoluchowski . Sadi Carnot postulated in
1824 that the efficiency of conversion of heat into work with the help a heat
engine has the upper limit

$\eta_{max}=1-\frac{T_{co}}{T_{he}}$ $None$

which is determined by the ratio $T_{co}/T_{he}$ of the cooler temperature
$T_{co}$ to the heater temperature $T_{he}$. In particular, the efficiency of
any heat engine is zero $\eta=0$ without a temperature difference
$T_{he}-T_{co}$, according to the Carnot equation (1). Therefore the
temperature difference $T_{he}-T_{co}$ must be created and maintained in order
the heat energy $\Delta Q$ can be converted in an useful work $W$.

The efficiency factor of any heat engine is equal by definition to the part
$\eta=W/Q_{he}$ of the thermal energy of the heater $Q_{he}$ that is converted
into useful work $W$. This part has a maximum value when all other part of the
thermal energy created in the heater is spent only on increasing the termal
energy of the cooler, i.e. $W=Q_{he}-Q_{co}$. Therefore the maximum efficiency
should be equal $\eta_{max}=1-Q_{co}/Q_{he}$. This equation corresponds to the
Carnot equation (1) when the value $Q_{co}/T_{co}-Q_{he}/T_{he}$ cannot be
less than zero. The inequality $Q_{co}/T_{co}-Q_{he}/T_{he}\geq 0$ expresses
the essence the first rigorous definition of the second law of thermodynamics
proposed by Clausius in the 1850s: heat can never pass from a colder to a
warmer body without some other change, connected therewith, occurring at the
same time. This definition was based on the concept of entropy as the ratio
$S=Q/T$ of the thermal energy $Q$ of a thermodynamic body to its thermodynamic
temperature $T$. The entropy of a closed system cannot decrease, i.e. the law
of entropy increase is another formulation of the second law of
thermodynamics.

###  II.1 The irreversibility is needed for the impossibility of a perpetuum
mobile

Sadi Carnot substantiated his principle by the centuries-old belief in the
impossibility of perpetuum mobile. He wrote in his brilliant work of 1824
Carnot  that ”it would be not only a perpetual motion, but also an unlimited
creation of the driving force without the cost of phlogiston or any other
agents” if the efficiency of a heat engine could exceed the maximum value (1).
The genius of Carnot is that using the notion of phlogiston, he determined the
relationship between maximum efficiency of any heat engine and the
impossibility of a perpetuum mobile, which is valid in different ideas about
heat.

It became clear that Carnot postulated irreversibility in physics when the
heat began to consider as a form of energy. The law of entropy increase
postulates most clear the irreversibility of some thermodynamic processes. The
car stops when it runs out of fuel, because the kinetic energy $E_{k}$ of its
directional movement is converted into thermal energy $Q=E_{k}$. The total
entropy increases on the value $Q/T$ in this process since the kinetic energy
of a directed motion of any macroscopic body does not contribute to the
entropy. This process is irreversible according to the law of entropy
increase. The heat energy $Q$ cannot be convert back complitely into the
kinetic energy $E_{k}$ of the car and an useful work cannot be obtained from
thermal energy without a temperature difference $T_{he}-T_{co}$ according to
this law. Therefore we have to burn fuel in the heat engines to create the
temperature difference $T_{he}-T_{co}$.

We would not have to burn fuels if the law of entropy increase could be
violated. Therefore Elliott Lieb and Jakob Yngvason PhysRep1999  wrote that
”The world’s energy problems would be solved at one stroke” at any
reproducible violation of the second law of thermodynamics. The heat energy
$Q$ could be convert back in the kinetic energy $E_{k}$ of the car if all the
laws of physics were reversible. But a perpetuum mobile would be inevitable
according to the law of energy conservation without the irreversibility
postulated by the law of entropy increase.

Therefore most scientists believe in this law. Arthur Eddington wrote: ”The
second law of thermodynamics holds, I think, the supreme position among the
laws of Nature. If someone points out to you that your pet theory of the
universe is in disagreement with Maxwell’s equations - then so much the worse
for Maxwell’s equations. If it is found to be contradicted by observation,
well, these experimenters do bungle things sometimes. But if your theory is
found to be against the second law of thermodynamics I can give you no hope;
there is nothing for it but collapse in deepest humiliation” Eddington .

###  II.2 The centuries-old belief in the impossibility of a perpetuum mobile

The supreme position of the second law of thermodynamics is definitely
connected with the centuries-old belief of scientists in the impossibility of
a perpetuum mobile. The perpetuum mobile is one of the oldest problems of
science. First attempts to invent a perpetuum mobile were known beginning with
13 century but already Leonardo da Vinci (1452-1519) stigmatized its inventors
equally with alchemists. Stivin (1548-1620) postulated first the principle of
impossibility of any perpetuum mobile. Because of such history most scientists
interpret perpetuum mobile as a problem of the ignorant Middle Ages on a level
with alchemy.

The Paris Academy of Sciences has decided in 1775 year to do not consider any
project of perpetuum mobile. Some scientists still refer to this decision when
it comes to a violation of the second law of thermodynamics. Such references
can hardly have a scientific basis, since the Parisian academicians could not
have known in 1775 not only, for example, quantum mechanics, but even
thermodynamics. Nevertheless the centuries-old belief in the impossibility of
perpetuum mobile has played an important role in the a dramatic history of the
second law of thermodynamics. The belief in impossibility of a perpetuum
mobile had been persisted despite several fundamental changes in our ideas
about Nature over the past centuries. One of the most fundamental changes in
the ideas of most scientists about thermodynamics processes occurred in the
beginning of the 20th century.

##  III The struggle between thermodynamic-energy and atomistic-kinetic
worldviews

Smoluchowski wrote in his article ”Limits of Validity of the Second Law of
Thermodynamics” published in 1914: ”I begin my presentation of the above topic
with a brief historical overview. Anyone who has been involved in the struggle
between thermodynamic-energy and atomistic-kinetic worldviews for the past
forty years knows why I do this. It is no longer easy for us to imagine the
way of thinking that prevailed at the end of the last century. After all, at
that time, scientists in Germany and France were convinced that the kinetic
theory of atoms had already played a role. The principle of Carnot,
intuitively understood by him, we call since the time of Clausius the second
law of thermodynamics. Because of the confidence in the great achievements of
thermodynamics, this principle has been elevated to the rank of the absolute,
exact dogma without exclusion. And since at that time molecular kinetics in
the interpretation of this principle faced certain difficulties associated
with the irreversibility of processes, it, together with atomistics, was
immediately condemned as untenable. Although Boltzmann tried to prove that if
there are contradictions, they still cannot practically become tangible”
Smoluchowski .

Smoluchowski made very important remark that the way of thinking in particular
about the second law of thermodynamics in 1914 differed fundamentally from the
one prevailed at the end of the 19th century. Most scientists in the late 19th
and even early 20th century negatively related to the Maxwell-Boltzmann
statistical theory because of its contradiction with the second law of
thermodynamics. Many scientists, supporters of the thermodynamic-energy
worldview, for example Wilhelm Ostwald, Nobel Prize Winner 1909, denied even
the existence of atoms and their perpetual thermal motion. Smoluchowski
pointed out one of the main objections to the atomistic-kinetic theory: the
laws of mechanics describing the thermal motion of atoms are reversible, while
the second law of thermodynamics postulates irreversibility as a necessary
condition for the impossibility of a perpetuum mobile.

###  III.1 The difference in the understanding of irreversibility between
scientists of the 19th and 20th centuries

Scientists of both the 19th and 20th centuries understood that the law of
entropy increase postulates irreversibility in Nature. But the understanding
of irreversibility by the most scientists of the 19th century differed
fundamentally from the one in the 20th century. Smoluchowski wrote in 1914
about how irreversibility was understood in the 19th century: ”In the
phenomena of fluctuation experimentally observed in recent years, it seems
extremely strange to a supporter of classical thermodynamics that he sees with
his own eyes the reverse course of processes that are generally regarded as
irreversible. Because according to the classical theory, the second principle
of thermodynamics should disappear if at least one process, regarded as
irreversible, admits reversibility” Smoluchowski .

Supporters of classical thermodynamics denied the possibility of any motion at
thermodynamic equilibrium, in particular the perpetual thermal motion of
atoms. The investigations of such fluctuation phenomena as the Brownian motion
by Einstein, Smoluchowski and others have convinced even supporters of the
thermodynamic - energy worldview in the existence of perpetual thermal motion
of atoms, molecules, electrons, ions, and other particles. None of the modern
scientists doubt that the random motion of particles suspended in a medium, a
liquid or a gas, first described by the botanist Robert Brown as far back as
in 1827, is a result of the perpetual thermal motion of atoms or molecules of
the liquid or the gas.

The great scientists, Albert Einstein was one of the first who modeled in 1905
the motion of the pollen particles as a result of the thermal motion of
individual water molecules Einstein1905Bm . This explanation of Brownian
motion was further verified experimentally by Jean Perrin and other
experimenters. Perrin was awarded the Nobel Prize in Physics in 1926 ”for his
work on the discontinuous structure of matter”. The Brownian motion is the
result of the many-body interactions. No model accounting for every involved
molecule can describe this motion. Therefore Einstein, Smoluchowski and others
used the statistical theory proposed in the 19th century by Maxwell,
Boltzmann, Gibbs, and others for the description of the Brownian motion. The
success of this description was one of the main reasons for the victory of the
atomistic-kinetic worldview over the thermodynamic-energy worldview in the
20th century.

###  III.2 The law of chaos increase

The interpretation of the law of entropy increase changed fundamentally after
this victory: this law began to be understood as the law of chaos increase.
Modern scientists follow Maxwell, Boltzmann and Gibbs in this interpretation.
J.C. Maxwell wrote in 1878 Maxwell1878  that ”the second law is drawn from our
experience of bodies consisting of an immense number of molecules”. And Joel
Lebowitz wrote approximately the same after more than century: ”It is not
every microscopic state of a macroscopic system that will evolve in accordance
with the second law, but only the ’majority’ of cases - a majority which
however becomes so overwhelming when the number of atoms in the system becomes
very large that irreversible behavior becomes a near certainty” Lebowitz1999 .

Smoluchowski wrote in 1914: ”Thus, the issue is considered very different
today than it was twenty years ago. Atomistics is recognized as the basis of
modern physics in general; the second law of thermodynamics has once and for
all lost its significance as an unshakable dogma, as one of the basic
principles of physics” Smoluchowski . But the dogma has changed rather than
lost its significance. The victory of the atomistic-kinetic worldview has
created the illusion that the law of entropy increase and thus, the
irreversibility in Nature belong to our a priory rather than empirical
knowledge. The gas of molecules or atoms are distributed evenly in a closed
box after the partition separating the two parts of the box with different gas
pressure is removed. It is difficult to doubt in this case that the Boltzmann
entropy

$S=-k_{B}\sum_{i}p_{i}\ln p_{i}$ $None$

has increased and its decrease to the initial value is extremely unlikely.
Here $p_{i}$ is is the probability that the system is in a $i$ state; $k_{B}$
is the Boltzmann constant.

The law of chaos increase is based on our a priori rather than empirical
knowledge. Thus, the kinetic theory of atoms, rejected by many physicists in
the 19th century because of the contradiction with the irreversibility
postulated by the second law of thermodynamics, gave an a priori argument to
justify the belief in the impossibility of a perpetuum mobile. Most scientists
do not doubt in this argument in spite of the reversibility paradox,
formulated by Johann Loschmidt and William Thomson, and the recurrence
paradox, formulated by Henri Poincare and Ernst Zermelo.

##  IV The assumption of molecular disorder belong to our empirical rather
than a priori knowledge

The reversibility paradox, i.e. the contradiction between the reversibility of
mechanics and the irreversibility of thermodynamics, was considered as
important objections against Boltzmann’s H-theorem which is a mathematical
substantiation of the law of chaos increase. But most scientists are sure that
the Boltzmann H - theorem overcame this contradiction. C.G. Weaver argues in a
recent publication Weaver2021  that the reversibility paradox can be resolved
with the help of the hypothesis of molecular chaos. But he does not discuss
the validity of the hypothesis itself.

This hypothesis is crucial to the modern belief in the law of entropy
increase. Hendrik Lorentz recognized in 1887 Lorentz1887  that this hypothesis
is necessary for the proof of Boltzmann’s H-theorem. Max Planck questioned the
H - theorem because of the groundlessness of this hypothesis, or rather the
assumption. He noted in his Scientific Autobiography that ”Boltzmann omitted
in his deduction every mention of the indispensable presupposition of the
validity of his theorem namely, the assumption of molecular disorder. He must
have simply taken it for granted” Planck . Rather most scientists of the 20th
century than Boltzmann taken the assumption of molecular disorder for granted.
Boltzmann understood the need for the assumption of molecular disorder. He
wrote in 1896: ”We shall now explicitly make the assumption that the motion is
molar- and molecular - disordered, and also remains so during all subsequent
time” Boltzmann1896 .

But Boltzmann did not substantiate the universality of the assumption of
molecular disorder in any way. Therefore, Planck was right to doubt this
assumption. The assumption of molecular disorder needed for the modern
substantiation of the law of entropy increase implies that no ordered thermal
motion of atoms, molecules, electrons, ions, Brownian particles and other
particles is possible. We cannot be sure a priori that such a motion is
impossible. Therefore the assumption of molecular disorder cannot belong to
our a priori knowledge. Empirically, the assumption of the impossibility of
ordered thermal motion can only be refuted, but not proven.

###  IV.1 The assumption of molecular disorder is needed for the impossibility
of an useful perpetuum mobile

Noting that, with the victory of the atomistic-kinetic worldview, the second
law of thermodynamics lost its importance as an unwavering dogma, Smolukhovsky
argued: ”On the contrary, from the point of view of molecular statistics, the
position of thermodynamics about the impossibility of perpetuum mobile of the
second kind is correct, if this expression is given a more precise meaning,
namely: ’an automatic machine continuously making work by consuming the heat
of another body with a lower temperature’” Smoluchowski .

Smoluchowski stated that ”the widespread opinion that molecular fluctuations
could be used directly to construct a simple perpetuum mobile is completely
wrong” Smoluchowski  and was proving his statement with the help of the
following example: ”First, let’s imagine a gummigut particle, which, despite
its gravity, will remain suspended above the bottom of the vessel due to
Brownian motion. We will attach to the bottom of the vessel a vertical rail
with sawtooth teeth or, better, a rail in which a zigzag groove is cut; at a
certain height in each groove we will adapt a sleeve that is pressed by an
elastic spring on one side, so that due to the latching of this device, the
particle in the normal state can move only from the bottom up, but not in the
opposite direction” Smoluchowski .

Smoluchowski understood that the impossibility of directed thermal motion is
needed for the impossibility of an useful perpetuum mobily. But he, in
contrast to the great scientists Max Planck, did not understand that the
assumption of molecular disorder cannot have a scientific substantiation.
Smoluchowski proved that an useful perpetuum mobily cannot be created on the
base of the mechanical machine considered in his example since all part of
this mechanical machine move chaotically due to thermal fluctuations.

The famous physicists Richard Feynman understood also that the Carnot
principle can be valid only if no directed thermal motion is possible. He
repeated in Chapter 46 ”Ratchet and pawl” of his lectures on physics
Feynman1963  the Smoluchowski proof that no useful work can be obtained from
heat energy without a temperature difference $T_{he}-T_{co}$ with the help of
a mechanical machine. This ’proof’ rather testifies to the belief of
Smoluchowski, Feynman and most physicists in the impossibility of a perpetuum
mobily than is a real universal proof of the assumption of molecular disorder.

###  IV.2 The chaotic of the Nyquist current as a consequence of the
assumption of molecular disorder

The centuries-old belief in the impossibility of perpetuum mobile turned out
to be so persistent that even paradoxical quantum phenomena could not question
the assumption of molecular disorder. Many scientists are ready to abandon
realism, even macroscopic Leggett1985 ; Mooij2010 ; LGineq2016 , because of
the paradoxical nature of some quantum phenomena. But only a few scientists
Entropy2004  are willing to question the law of entropy increase because of
these phenomena. Although the latter has more grounds already due to
quantization, which is observed not only in atoms.

An useful work cannot be obtained from the Brownian motion because of its
disorder. According to the equation

$m\frac{d{\bf v}}{dt}={\bf F_{dis}}+{\bf\chi}(t)$ $None$

proposed by Paul Langevin in 1908 Langevin1908  a Brownian particle, with a
small mass $m$, moves under the influence of a random force
$\bf{\chi}=(\chi_{x},\chi_{y},\chi_{z})$, representing the effect of the
collisions with the molecules of the fluid, and is decelerate by a viscous
force (friction force or dissipation force) ${\bf F_{dis}}=-\lambda{\bf v}$
proportional to the velocity of the particle ${\bf v}=(v_{x},v_{y},v_{z})$.

The random force $\bf{\chi}$ has a Gaussian probability distribution with
correlation function

$<\chi_{i}(t)\chi_{j}(t^{\prime})>=2\lambda
k_{B}T\delta_{i,j}\delta(t-t^{\prime})$ $None$

The delta-function form of the correlations $\delta_{i,j}$,
$\delta(t-t^{\prime})$ in (4) is the mathematical expression for the
assumption of molecular disorder: the i-th component of the vector $\bf{\chi}$
are completely uncorrelated between themselves, and the Langevin force at a
time $t$ is completely uncorrelated with the force at any other time
$t^{\prime}$. The dependence of the correlation function of the random
Langevin force (4) from the damping coefficient $\lambda$ is known as the
Einstein relation.

The Langevin equation (3) describes enough well the observed motion of
Brownian particles, which is random. The average velocity and the average
Langevin force of such motion equal zero. But can we be sure a priori that the
average velocity of any type of Brownian motion should always be zero? We can
be a priori sure that the average velocity can be nonzero only in a circular
motion, since Brownian motion is observed at thermodynamic equilibrium, in
which no macroscopic transfer of mass or energy from one region of space to
another can be. The Nyquist Nyquist  (or Johnson Johnson ) noise current

$\overline{I_{Nyq}^{2}}=\frac{k_{B}T\Delta\omega}{R}$ $None$

observed in a metal ring with a non-zero resistance $R>0$ is the example of
the circular Brownian motion Feynman1963 .

The Nyquist noise cannot be used to obtain useful work at thermodynamic
equilibrium, since all elements of any electrical circuit have the same
distribution of the power $P_{Nyq}=k_{B}T\Delta\omega$ in frequency $\omega$.
This distribution is a consequence of the randomness of this type of Brownian
motion. The power $P_{Nyq}=0$ at the zero frequency $\omega=0$ and the average
value of the Nyquist current $\overline{I_{Nyq}}=0$ equal zero also because of
the randomness.

###  IV.3 Quantization and the persistent current

The absence of the directed component in the average value of the Nyquist
current $\overline{I_{Nyq}}=0$ can be explain a priori by the equal
probability of the current with the opposite direction,
$\overline{I_{Nyq}}=\int_{0}^{Imax}dI(P(+I)I-P(-I)I)=0$ at $P(+I)=P(-I)$. But
this explanation is valid only according classical physics when all states in
the ring is permitted. According to quantum mechanics, some states of a
particle with a mass $m$ and a charge $q$ can be forbidden in the ring with a
radius $r$ because of the quantization of the canonical momentum $p=mv+qA$

$\oint_{l}dlp=\oint_{l}dl(mv+qA)=n2\pi\hbar$ $None$

The velocity of such particle

$\oint_{l}dlv=\frac{2\pi\hbar}{m}(n-\frac{\Phi}{\Phi_{0}})$ $None$

cannot be zero when the magnetic flux inside the ring $\Phi=\oint_{l}dlA$ is
not divisible $\Phi\neq n\Phi_{0}$ by the flux quantum $\Phi_{0}=2\pi\hbar/q$
and the permitted velocity $v=(\hbar/mr)(n-\Phi/\Phi_{0})$ in a one direction
corresponds the forbidden velocity in the opposite direction at $\Phi\neq
n\Phi_{0}$ and $\Phi\neq(n+0.5)\Phi_{0}$. This asymmetry is the cause of such
well known quantum phenomenon as the persistent current observed in
superconductor JETP2007 ; Moler2007 ; Letters2007  and normal metal
PC2009Science ; Moler2009  rings. The persistent current $I_{p}$ is observed
at the thermodynamic equilibrium, like the Nyquist noise current (5), and does
not decay in spite of a non-zero resistance of both superconductor ring
Moler2007 ; Letters2007  and normal metal ring PC2009Science ; Moler2009 . But
the directed component of the persistent current $\overline{I_{p}}$, i.e. the
current at the zero frequency $\omega=0$, is not zero
$\overline{I_{p}}=I_{p}(\omega=0)\neq 0$ JETP2007 ; Moler2007 ; Letters2007 ;
PC2009Science ; Moler2009  in contrast to the Nyquist noise current (5).

Obviously, this fundamental difference the persistent current from the Nyquist
current (5) became possible due to the violation of symmetry between the
opposite directions due to quantization of the canonical momentum (6). This is
confirmed by observations: $\overline{I_{p}}\neq 0$ at $\Phi\neq n\Phi_{0}$
and $\Phi\neq(n+0.5)\Phi_{0}$ Moler2007 ; PC2009Science ; Moler2009  when
there is the asymmetry between the opposite directions (7). The quantization
(7) can be visible unless the energy difference between the permitted states
is much less than the energy of thermal fluctuations $k_{B}T$. The energy
difference between the permitted states of a quantum particle

$\Delta
E_{n+1,n}=\frac{mv_{n+1}^{2}}{2}-\frac{mv_{n}^{2}}{2}\approx(2n+1)\frac{\hbar^{2}}{2mr^{2}}$
$None$

decreases with the increases of the radius of the quantization $r$. The energy
difference $\Delta E_{n+1,n}\approx 5\ 10^{-27}\ J$ of single electron with
the mass $m=9\ 10^{-31}\ kg$ in a ring with the real radius $r=10^{-6}\ m=1\
\mu m$ corresponds to very low temperature $T<\Delta E_{n+1,n}/k_{B}\approx
0.0004\ K$. Therefore the persistent current of electrons in normal metal
rings was reliably observed relatively recently PC2009Science ; Moler2009 ,
almost forty years after the prediction Kulik1970  of this paradoxical quantum
phenomenon.

The persistent current Cooper pairs in a superconductor with a non-zero
resistance $R>0$ was observed first LP1962  almost fifty years before the one
in normal metal since the spectrum of superconducting condensate is much more
discrete than the one of electrons FPP2008 . Cooper pairs, being bosons unlike
electrons, are at the same level $n$ and have the same velocity (7).
Therefore, the persistent current of Cooper pairs

$I_{p}=qsn_{s}v_{n}=q\frac{N_{s}}{2\pi
r}v_{n}=\frac{\Phi_{0}}{L_{k}}(n-\frac{\Phi}{\Phi_{0}})$ $None$

is much greater than the persistent current of electrons and the discreteness
of the energy spectrum

$E_{k}=N_{s}\frac{mv_{n}^{2}}{2}=\frac{L_{k}I_{p}^{2}}{2}=I_{p,A}\Phi_{0}(n-\frac{\Phi}{\Phi_{0}})^{2}$
$None$

is greater by a factor equal to the number of Cooper pairs $N_{s}=2\pi
rsn_{s}$ FPP2008  in the ring with the pair density $n_{s}$, the radius $r$
and the small cross-section $s\ll\lambda_{L}^{2}$. $L_{k}=m2\pi
r/sq^{2}n_{s}=(\lambda_{L}^{2}/s)\mu_{0}2\pi r$ is the kinetic inductance of
the ring; $\lambda_{L}=(m/\mu_{0}q^{2}n_{s})^{0.5}$ is the quantity generally
referred to as the London penetration depth. Its typical value equals
$\lambda_{L}=50\ nm=5\ 10^{-8}\ m$.

The persistent current oscillates in magnetic field $B$ with the period
$B_{0}=\Phi_{0}/\pi r^{2}$ corresponding to the magnetic flux
$\Phi_{0}=B_{0}\pi r^{2}$ inside the ring and the amplitude
$I_{p,A}=\Phi_{0}/2L_{k}$ JETP2007  when the quantum number $n$ takes the
value corresponding to the minimum of the kinetic energy (10). The quantum
oscillations are observed JETP2007 ; Moler2007 ; Letters2007  since the energy
(10) difference $|E_{k}(n+1)-E_{k}(n)|\approx I_{p,A}\Phi_{0}$ is very large.
The energy difference at $\Phi=n\Phi_{0}$ corresponds the temperature
$I_{p,A}\Phi_{0}/k_{B}\approx 2000\ K$ NanoLet2017  at a typical value of the
amplitude of the persistent current $I_{p,A}\approx 10\ \mu A$ observed in a
superconducting ring even with the small cross-section $s\ll\lambda_{L}^{2}$
JETP2007 . The discreteness increases with the increase of all three sizes of
the ring $|E_{k}(n+1)-E_{k}(n)|\approx n_{s}s2\pi
r(\hbar^{2}/2mr^{2})\propto(s/r)$ since the energy difference is proportional
to the number of Cooper pairs and, thus, the volume of the ring
$|E_{k}(n+1)-E_{k}(n)|\propto N_{s}=2\pi rsn_{s}\propto 2\pi rs$ FPP2008 .
That is why superconductivity is a macroscopic quantum phenomenon.

###  IV.4 The average value of the Langevin force can be nonzero due to the
quantization

The quantization (6) of angular momentum $pr=n\hbar$ was postulated in 1913 by
Bohr for the explanation of the discrete spectrum of electrons in atom. The
discreteness in atom is much larger than in a real ring because of the
smallness of the radius of the quantization. Normal metal ring is
fundamentally different from an atom, since the electrons in the metal are
scattered by defects and phonons, unlike atomic orbits, and therefore the ring
has a non-zero resistance $R>0$. The value of the resistance $R$ determines
the value of a relaxation time $\tau_{RL}=L/R$ during which any electric
current must decay in the ring with the total inductance $L=L_{f}+L_{k}$. The
electric current should not decay only in the case of the perfect
conductivity, when the resistance $R=0$. In accordance with the
electrodynamics law $L_{k}dI_{p}/dt=-d\Phi/dt$ the increase of magnetic flux
inside a ring with $R=0$ from $\Phi=0$ to $\Phi=BS-L_{f}I_{p}$ should induce
the electric current $I_{p}=-\Phi/L_{k}$ which cannot decay arbitrarily long
time, until $R=0$. The electric current does not decay indeed arbitrarily long
time in a superconducting ring.

The experimental results Moler2007 ; Letters2007 ; PC2009Science ; Moler2009 ;
LP1962  testify that the persistent current can be observed arbitrarily long
time not only at $R=0$ but also at $R>0$. In order to understand the cause of
this paradoxical phenomenon one should recall that the electric current
$I_{p}=-\Phi/L_{k}$ can appear in a superconducting ring in two ways: 1) first
the temperature is lowered below the temperature of superconducting transition
$T<T_{c}$ in zero magnetic field $H=0$ and thereafter the external magnetic
field is increased up to $H=B/\mu_{0}$; 2) first the magnetic field is
increase up to $H=B/\mu_{0}$ at $T>T_{c}$ and thereafter the temperature is
lower down to $T<T_{c}$.

The current $I_{p}=-\Phi/L_{k}$ appears on the first way under the influence
of the Faraday electromotive force $-d\Phi/dt\approx SdB/dt$ because of the
increase the magnetic field in time. But the current induced this force at
$T>T_{c}$ will decay during a short relaxation time $\tau_{RL}=L/R$ on the
second way because of a non-zero resistance $R>0$ in the normal state.
According to the electrodynamics laws lowering the temperature from $T>T_{c}$
to $T<T_{c}$ in a magnetic field $H=B/\mu_{0}$ constant in time $dB/dt=0$
should not induce a current in the ring. Nevertheless all measurements give
experimental evidence that the electric current appears not only on the first
way but also on the second way.

The electric current appears on the second way because of the quantization of
the the canonical momentum $p=mv+qA$ (6) since the velocity of Cooper pairs
(7) cannot be equal zero at $\Phi\neq n\Phi_{0}$. This current is persistent
since superconducting state with zero current is forbidden at $\Phi\neq
n\Phi_{0}$. The quantization replaces the Faraday electromotive force on the
second way and as consequence the same current $I_{p}=-\Phi/L_{k}$ appears on
the both ways at $|\Phi|\approx|BS|<\Phi_{0}/2$ when the quantum number
corresponding to the minimum energy equals zero $n=0$. We consider here rings
with the small cross-section $s\ll\lambda_{L}^{2}$ used in the experimental
works JETP2007 ; Moler2007 ; Letters2007 . The magnetic inductance $L_{f}$ of
such rings is smaller than the kinetic inductance $L_{f}\approx\mu_{0}2\pi
r\ll L_{k}=(\lambda_{L}^{2}/s)\mu_{0}2\pi r$ and the magnetic flux
$L_{f}I_{p}$ induced by the persistent current $I_{p}$ is small
$L_{f}I_{p}\ll\Phi_{0}$.

The velocity of the mobile charge carriers changes on the same value on the
both ways at $|\Phi|\approx|BS|<\Phi_{0}/2$ whereas the canonical momentum
$p=mv+qA$ changes only on the second way since the velocity on the first way
increases in accordance the Newton second law $mdv/dt=qE=-qdA/dt$ according to
which $dp/dt=mdv/dt+qdA/dt=0$. The canonical momentum should change on the
second way on the value $\Delta p=(\hbar/r)(n-\Phi/\Phi_{0})$ since the
angular momentum equals $rp=rqA=\hbar\Phi/\Phi_{0}$ in the normal state at
$H=B/\mu_{0}=\Phi/\mu_{0}S\neq 0$ because of the zero velocity $v=0$ and
should have the quantum value $pr=n\hbar$ in the superconducting state. This
change of the canonical momentum and the velocity because of the quantization
(6) should occur at each comeback of the ring in the superconducting state
when $\Phi\neq n\Phi_{0}$. This change because of the quantization during a
time unity at the switching of the ring between normal and superconducting
states with a frequency $f_{sw}=N_{sw}/\Theta$

$F_{q}=\hbar\frac{(\overline{n}-\Phi/\Phi_{0})}{r}f_{sw}$ $None$

was called ’quantum force’ in PRB2001 .
$\overline{n}=\int_{\Theta}dtn/\Theta=\sum_{i=1}^{i=N_{sw}}n_{i}/N_{sw}$ is
the average value of the quantum number after $N_{sw}\gg 1$ comeback of the
ring into the superconducting state during a long time $\Theta\gg 1/f_{sw}$.

The ring can be switched between superconducting and normal states because of
different reasons: temperature change, non-equilibrium noise, external ac
current and other external factors. But only thermal fluctuations can switch
the ring at thermodynamic equilibrium in a narrow temperature region $\Delta
T_{fl}$ near the superconducting transition $T\approx T_{c}$ Skospol1975 ;
Tinkham1996 . The persistent current of Cooper pairs is observed at a non-zero
resistance $R>0$ Moler2007 ; Letters2007 ; LP1962  in this and only in this
fluctuation region: at $T>T_{c}+\Delta T_{fl}$, above this region $R=R_{n}>0$
but $I_{p}=0$ whereas at $T<T_{c}-\Delta T_{fl}$, below this region $I_{p}\neq
0$ but $R=0$. This experimental fact reveals that this paradoxical phenomenon
is a fluctuation phenomena, i.e. a type of the circular Brownian motion, like
the Nyquist noise current in a metal ring (5).

The action of the Langevin force in this Brownian motion is manifested in the
increase in the velocity of $N_{s}$ mobile charge carriers up to the permitted
value (7) after switching the ring by fluctuations to the superconducting
state with the number of Cooper pairs $N_{s}$. The average velocity of these
mobile charge carriers decreases down to zero under the action of the
dissipation force after their switching by fluctuations to the normal state.
The angular momentum of each mobile charge carrier changes on
$\hbar(n-\Phi/\Phi_{0})$ at each switching in the superconducting state.
Therefore the average value of the Langevin force should be equal
$\overline{\chi}\approx\overline{N_{s}}F_{q}$, where $\overline{N_{s}}=2\pi
rs\overline{n_{s}}$ is the average number of the Cooper pairs in the ring
which decreases rapidly with the temperature increase at $T\approx T_{c}$
Skospol1975 ; Tinkham1996 , but remains a large number even at $T>T_{c}$,
where the persistent current was predicted KulikJETP  and observed
Moshchalkov1992 .

The spectrum of the permitted states remains discrete even in the fluctuation
region where $R>0$. The energy (10) difference $|E_{k}(n+1)-E_{k}(n)|\approx
I_{p,A}\Phi_{0}$ at the amplitude $I_{p,A}\approx 0.1\ \mu A$ of the
oscillations of the persistent current observed Moler2007 ; Letters2007  in
the fluctuation region at $T\approx T_{c}$ corresponds to the temperature
$I_{p,A}\Phi_{0}/k_{B}\approx 20\ K$. Thus, the persistent current is observed
at a non-zero resistance $0<R<R_{n}$ near superconducting transition Moler2007
; Letters2007 ; LP1962  since it is the directed Brownian motion PRB2001 . The
average value of the Langevin force
$\overline{\chi}\approx\overline{N_{s}}F_{q}$ is not zero because of the
discreteness of the permitted state spectrum even in the fluctuation region.

###  IV.5 The persistent voltage

The persistent current observed at $R>0$ can be used, in contrast to the
Nyquist noise (5), for an useful work since its power $RI_{p}$ is not zero at
the zero frequency $\omega=0$. It is well known that the potential difference

$V=0.5(R_{n}-R_{w})I$ $None$

is observed on the halves of the ring with different resistance $R_{B}>R_{A}$
when the conventional circular electric current $I$ is induced by the Faraday
electric field $RI=(R_{n}+R_{w})I=-d\Phi/dt$. It was shown firstly in LTP1998
that quantum mechanics predicts a possibility to observe the potential
difference with a directed component $V_{dc}$ proportional to the average
value of the persistent current
$\overline{I_{p}}(\Phi)=2I_{p,A}(\overline{n}-\Phi/\Phi_{0})$ when an
asymmetric ring or its segments are switched between superconducting and
normal states, see also Physica2021 . Such magnetic oscillations of the dc
voltage
$V_{dc}(\Phi)\propto\overline{I_{p}}(\Phi)=2I_{p,A}(\overline{n}-\Phi/\Phi_{0})$
were observed firstly thirty years before this prediction at the measurements
of an asymmetric dc SQUID, i.e. a superconducting loop with two Josephson
junctions Physica1967

A.Th.A.M. De Waele et al. Physica1967  have not understood the importance of
the paradoxical effect they discovered because of the belief in the law of
entropy increase. They were concerned only with the question about a source of
the power $V_{dc}\overline{I_{p}}=V_{dc}^{2}/R$ that they observed at
$\Phi\neq n\Phi_{0}$ and $\Phi\neq(n+0.5)\Phi_{0}$. They assumed that the
source of this power is the emitted electromagnetic radiation of broadcasting
stations and have confirmed that the visible dc voltage
$V_{dc}(\Phi)\propto\overline{I_{p}}(\Phi)$ disappears when all parts of the
equipment are shielded more carefully Physica1967 .

Measurements carried out on asymmetric aluminum rings confirmed that below the
temperature of the superconducting transition $T<T_{c}$, quantum oscillations
of the dc voltage $V_{dc}(\Phi)\propto\overline{I_{p}}(\Phi)$ are induced by a
non-equilibrium noise NANO2002  and can be induced by an alternating current
Letter2003 . These measurements revealed that the dependence of the amplitude
$V_{A}$ of the DC voltage oscillations in magnetic field
$V_{dc}(\Phi)\propto\overline{I_{p}}(\Phi)$ on the amplitude $I_{0}$ of the AC
current $I_{ac}=I_{0}\sin\omega t$ or noise is non-monotonic: the DC voltage
$V_{dc}(\Phi)$ appears when the amplitude $I_{0}$ reaches the value of the
critical current of the ring $I_{0}\approx I_{c}(T)\approx
I_{c}(0)(1-T/T_{c})^{3/2}$ at the measurement temperature $T$, the amplitude
$V_{A}$ increases rapidly to a maximum with the increase of $I_{0}$, and
decreases with further increase of $I_{0}$.

Reducing the critical current $I_{c}(T)\approx I_{c}(0)(1-T/T_{c})^{3/2}$ to
zero as the temperature approaches the critical temperature $T\rightarrow
T_{c}$ means that arbitrarily weak noise can induce the DC voltage
$V_{dc}(\Phi)$. But the amplitude of the oscillations of the persistent
current $I_{p,A}(T)\approx I_{p,A}(0)(1-T/T_{c})$ reduces also at
$T\rightarrow T_{c}$. Therefore the visible oscillations of the DC voltage
$V_{dc}(\Phi)\propto\overline{I_{p}}(\Phi)$ are observed Letters2007 ;
PLA2012Ex ; APL2016  only a narrow temperature region at a give value of the
amplitude $I_{0}$ of noise or the AC current: below this region
$I_{0}<I_{c}(T)\approx I_{c}(0)(1-T/T_{c})^{3/2}$ and therefore the noise or
the AC current cannot switch the ring in the normal state whereas above this
region the DC voltage $V_{dc}(\Phi)\propto\overline{I_{p}}(\Phi)$ is not
visible because small value of the persistent current.

Measurements of single asymmetric ring NANO2002  and system of such identical
rings connected in series Letters2007 ; PLA2012Ex ; Kulik75  have revealed
that the reduction of the non-equilibrium noise with the help of careful
shielding reduces rather than annihilates the DC voltage
$V_{dc}(\Phi)\propto\overline{I_{p}}(\Phi)$. The amplitude $V_{A}$ of
$V_{dc}(\Phi)$ increases proportional to the number of the identical
asymmetric rings connected in series Letter2003  since the DC power adds up,
in contrast to the chaotic Nyquist voltage. For example, the oscillations
$V_{dc}(\Phi)$ with the maximum amplitude $V_{A}\approx 1.5\ \mu V$ was
observed on single asymmetric ring and with $V_{A}\approx 1500\ \mu V$ on
system of 1080 such rings connected in series at the same amplitude
$I_{0}\approx 3\ \mu A$ of the AC current and approximately the same
temperature Physica2019 . Therefore arbitrarily weak noise, up to equilibrium
noise, can be detected using a system with a sufficiently large number of
asymmetric superconducting rings APL2016 .

The oscillations $V_{dc}(\Phi)$ with the maximum amplitude $V_{A,1080}\approx
0.1\ \mu V=10^{-7}\ V$ were observed on the system of 1080 asymmetric rings
(the amplitude $V_{A,1}\approx 10^{-10}\ V$ on each ring) when the screening
of non-equilibrium noise was particularly thorough PLA2012Ex . The observation
of $V_{dc}(\Phi)$ in the temperature region corresponding to the resistive
transition where $R>0$ PLA2012Ex  allows to assume that this DC voltage was
induced by the switching of the rings between superconducting and normal state
at thermodynamic equilibrium. But it should be emphasized that the
observations Physica1967 ; NANO2002 ; Letters2007 ; PLA2012Ex ; Kulik75 ;
APL2016  of the DC power $V_{dc}\overline{I_{p}}=V_{dc}^{2}/R$ challenges to
the law of entropy increase in any case.

A.Th.A.M. De Waele et al. Physica1967  did not take into account that the DC
voltage $V_{dc}(\Phi)\propto\overline{I_{p}}(\Phi)$ is observed in a magnetic
field constant in time, i.e. without the Faraday electromotive force
$-d\Phi/dt$. Therefore the persistent current flows against the total electric
field $E_{t}=E_{p}=-\nabla V_{dc}$ in one of the ring halves QuanStud2016 , in
contrast to the conventional circular current $I$ (12) which flows against the
potential electric field $E_{p}=-\nabla V$ but not against the total electric
field $E_{t}=E_{p}+E_{F}=-\nabla V-dA/dt$. This paradox can be explain only if
we take into account that the quantum force (11) replaces the Faraday
electromotive force and counteract to the dissipation force PLA2012  when
rings are switched between superconducting and normal states.

The persistent current, having a macroscopic kinetic energy (10) $E_{k}\gg
k_{B}T$ at $T<T_{c}$, is damped with the generation of Joule heat during a
short relaxation time $\tau_{RL}=L/R$ after the switching of the ring into the
normal state. The dissipation of the kinetic energy (10) $E_{k}\gg k_{B}T$
into Joule heat $Q=E_{k}$ is an irreversible thermodynamic process at which
the entropy increases on a macroscopic value $\Delta S=Q/T=E_{k}/T\gg k_{B}$.
Therefore the appearance of the persistent current because of the quantization
(6) at each return of the ring in superconducting state is a process reverse
to the irreversible thermodynamic process during which Joule heat $Q$ is
converted back into the kinetic energy (10) of the persistent current (9) and
the entropy decreases on the macroscopic value $\Delta S=-Q/T=-E_{k}/T$. The
impossibility of such reverse process is the essence of the law of entropy
increase.

According to the Carnot equation (1) a temperature difference is needed in
order to create the directional movement with the help of a heat engine. But
the persistent current $\overline{I_{p}}(\Phi)$ is already the directional
movement observed Moler2007 ; Letters2007 ; PC2009Science ; Moler2009  at
thermodynamic equilibrium in the absence of a temperature difference. The
persistent voltage $V_{dc}(\Phi)\propto\overline{I_{p}}(\Phi)$ observed on the
halves of the asymmetric rings can be used to perform a useful work in the
absence of a temperature difference due to a non-zero value of its power
$P_{dc}=V_{dc}\overline{I_{p}}=V_{dc}^{2}/R$ at the zero frequency $\omega=0$.
The Nyquist noise cannot be used for a useful work without a temperature
difference since each element of the electrical circuit has the same power
$P_{Nyq}=k_{B}T\Delta\omega$ in any frequency range $\Delta\omega$.

Therefore, the power $P_{Nyq}=k_{B}T\Delta\omega$ will always flow from a hot
body to a cold body, in accordance with the law of entropy increase, no matter
what frequency filters are used. The DC power can be transferred from a cold
body to a hot body when the filters pass only the DC current. The DC voltage
$V_{dc}(\Phi)\propto\overline{I_{p}}(\Phi)$ induced on 1080 asymmetric rings
at the temperature $T\approx 1\ K$ is observed at the room temperature
$T\approx 300\ K$ in the experimental work Physica2019 . The DC power
$P_{dc}=V_{dc}\overline{I_{p}}=V_{dc}^{2}/R$ is observed in Physica2019  since
the directed electric current $\overline{I_{p}}$ flows against the directed
electric field $E=-\nabla V_{dc}$ in one of the halves of asymmetric
superconducting ring due to the violation of the assumption of molecular
disorder in the quantum phenomenon of the persistent current.

###  IV.6 The persistent current observed at a nonzero resistance refutes
experimentally the assumption of molecular disorder

The observation of the persistent current in rings with nonzero resistance at
thermodynamic equilibrium Moler2007 ; Letters2007 ; PC2009Science ; Moler2009
; LP1962  is an obvious experimental refutation of the assumption of molecular
disorder, the validity of which was questioned by Planck Planck . This quantum
phenomenon was first interpreted as a directed Brownian motion in the paper
PRB2001 . The directed Brownian motion is observed, contrary to the proof of
its impossibility by Smolukhovsky Smoluchowski  and Feynman Feynman1963 ,
since the discreteness of the spectrum is not parts of a mechanical device
that are subject to chaotic thermal motion FQMT2004 .

The electric current $I$ decays during a short relaxation time $\tau_{RL}=L/R$
at a non-zero resistance $R>0$ since its kinetic energy dissipates into Joule
heat with the power $RI^{2}$. The electric current $I$ can be observed at
$R>0$ during a long time $t\gg\tau_{RL}$ if a power source $RI^{2}$
compensates the energy dissipation with the power $RI^{2}$. Therefore the
observations Moler2007 ; Letters2007 ; PC2009Science ; Moler2009 ; LP1962  of
the persistent current $I_{p}\neq 0$ at $R>0$ during a long time
$t\gg\tau_{RL}$ proves the existence of a DC power source $RI^{2}$ at
thermodynamic equilibrium. The experimental evidence Moler2007 ; Letters2007 ;
PC2009Science ; Moler2009 ; LP1962  of the existence of the DC power source
$RI^{2}$ at thermodynamic equilibrium refutes the second law of
thermodynamics.

There is only one way to save the faith in this law, to claim that the
persistent current is a special electric current that can flow in rings with
non-zero resistance $R>0$ without energy dissipation. This is exactly what
A.C. Bleszynski-Jayich et al. PC2009Science  have made. They claim that the
persistent current, they observed in normal metal rings, is dissipationless
and write: ”A dissipationless equilibrium current flowing through a resistive
circuit is counterintuitive, but it has a familiar analog in atomic physics:
Some atomic species electronic ground states possess nonzero orbital angular
momentum, which is equivalent to a current circulating around the atom”
PC2009Science .

The analogy with the atom is obviously false. Firstly, it is hardly possible
to think that an electron creates a current in an atomic orbit, since an atom
is centrally symmetric, unlike a ring. Secondly, there are no defects and
phonons in the atom on which electrons are scattered in the metal ring. The
electron’s elastic scattering length $\approx 40\ nm$, found by A.C.
Bleszynski-Jayich et al. PC2009Science  in their normal metal rings, is less
than the circumference of the rings $2\pi r\approx 2000\div 5000\ nm$. A.C.
Bleszynski-Jayich et al. PC2009Science  admit that the persistent current is
produced by electrons diffusing around the ring. But how electrons can diffuse
in atom? The analogy and the claim of A.C. Bleszynski-Jayich et al.
PC2009Science  also contradict the experimental fact that the amplitude of the
persistent current oscillations, measured by them, decreases exponentially
with increasing temperature. Electronic orbits in atoms do not depend on
temperature and dissipationless current cannot depend on temperature since the
absence of energy dissipation means the absence of energy exchange with the
environment.

N.O. Birge agrees with A.C. Bleszynski-Jayich et al. PC2009Science , but
recognizes: ”The idea that a normal, non-superconducting metal ring can
sustain a persistent current - one that flows forever without dissipating
energy - seems preposterous. Metal wires have an electrical resistance, and
currents passing through resistors dissipate energy” Birge2009 . This idea not
only is preposterous but also contradicts to elementary mathematics. A.C.
Bleszynski-Jayich et al. PC2009Science  observe $I_{p}\neq 0$ and measure
$R>0$, but they claim that $RI_{p}^{2}=0$. If A.C. Bleszynski-Jayich et al.
and N.O. Birge PC2009Science ; Birge2009  have paid attention to the reason
for Planck’s doubts about the Boltzmann H - theorem, then perhaps they would
not make the preposterous claim contradicting to the elementary mathematics
and experimental results. Only blind faith in the law of entropy increase
could force A.C. Bleszynski-Jayich et al. and N.O. Birge PC2009Science ;
Birge2009  to make this preposterous claim. This faith has led physicists much
earlier to make obvious mistakes and even to create false theories.

##  V The Meissner effect

According to almost universal belief from Maxwell’s time to the present day
Lebowitz1999  ”the second law is drawn from our experience of bodies
consisting of an immense number of molecules” Maxwell1878 . Quantization of
electron orbits in atom cannot challenge the second law of thermodynamics
since thermodynamics does not describe individual atoms. The rings measured in
the works JETP2007 ; Moler2007 ; Letters2007 ; PC2009Science ; Moler2009  are
bodies consisting of an immense number of atoms, electrons and/or Cooper
pairs. Therefore the quantization (6) in the rings challenges the law of
entropy increase. The discreteness of the spectrum of a superconducting ring
(10) is proportional to the number $N_{s}=2\pi rsn_{s}$ of Cooper pairs in it.
Therefore, quantization is observed in macroscopic superconducting structures
and superconductivity is a macroscopic quantum phenomenon.

###  V.1 Perfect conductivity and superconductivity

The first experimental evidence of this paradoxical fact is the Meissner
effect discovered in 1933 Meissner1933 . Superconductivity was discovered in
1911 as perfect conductivity when Heike Kamerlingh Onnes observed for the
first time that the resistance of some metals can abruptly decrease to zero at
low temperatures. Physicists were sure before 1933 that the electric current
can appear only in accordance with the Newton second law $mdv/dt=qE$ and the
electrodynamics laws. The current density $j=n_{s}qv$ in a perfect conductor
with a density $n_{s}$ of the mobile carriers of a charge $q$ should change in
time $dj/dt=(n_{s}q^{2}/m)E=E/\mu_{0}\lambda_{L}^{2}$ under the action of an
electric field $E$, according to these laws. According to the Maxwell
equations $rotE=-dB/dt$, $rotH=j$ and $B=\mu_{0}H$ the expression

$\lambda_{L}^{2}\bigtriangledown^{2}\frac{dH}{dt}=\frac{dH}{dt}$ $None$

should be valid. According to (13) a change in the magnetic field $H$ over
time can penetrate in the perfect conductor only up to the penetration depth
$\lambda_{L}$. The magnetic field inside a long cylinder with a macroscopic
radius $R\gg\lambda_{L}$ equals

$h=H\exp-\frac{R-r}{\lambda_{L}}$ $None$

after increasing the external magnetic field from $0$ to $H$ at $T<T_{c}$. The
density of the surface screening current equals

$j=j_{0}\exp-\frac{R-r}{\lambda_{L}}$ $None$

where $j_{0}=H/\lambda_{L}$ is the current density at $r=R$. The screening
current (15) has the kinetic energy, the density of which
$\varepsilon=n_{s}mv^{2}/2=\mu_{0}\lambda_{L}^{2}j^{2}/2$ corresponds to the
energy

$E_{k}=\mu_{0}H^{2}\lambda_{L}\pi RL=\mu_{0}H^{2}V\frac{\lambda_{L}}{R}$
$None$

of a macroscopic cylinder with the radius $R\gg\lambda_{L}$ and the length
$L$.

Physicists understood that the surface screening current should dissipate in
Joule heat within a short time after the transition in the normal state with a
resistivity $\rho>0$ and were sure that this current cannot reappear when the
cylinder will return to the superconducting state with $\rho=0$ in a magnetic
field $H$ constant in time $dH/dt=0$. Therefore the transition of a bulk
superconductor into the normal state in a magnetic field $H$ was considered as
an irreversible process at which ”the surface currents (15) associated with
the field are damped with the generation of Joule heat” Shoenberg1952 .

But W. Meissner and R. Ochsenfeld discovered in 1933 Meissner1933  that the
surface screening currents (15) appear at the transition into the
superconducting state in a magnetic field $H$ constant in time $dH/dt=0$,
contrary to the Newton second law and the electrodynamics laws. This
experimental fact, because of which a superconductor differs from a perfect
conductor, contradicts to the law of entropy increase if ”the surface currents
associated with the field are damped with the generation of Joule heat”
Shoenberg1952  when the superconductivity is destroyed. The analogy of the
screening current (15) with the movement of the car makes the contradiction
more obvious.

The increase of the screening current (15) with an increase in the field
$dj_{0}/dt=-\lambda_{L}^{-1}dH/dt$ under the influence of the Faraday electric
field $E=-(2\pi r)^{-1}d\Phi/dt=-\mu_{0}\lambda_{L}dH/dt$ can be compared with
the acceleration of a car by a driving force in the absence of a dissipation
force. The car, like the current (15), can move indefinitely in the absence of
the dissipation force and will quickly stop after the appearance of this
force, like the current (15) after the transition into the normal state. The
emergence of the persistent screening current (15) observed at the Meissner
effect is analog to if a car began to move in the absence of a driving force,
and uphill, since mobile charge carriers should accelerate against the action
of the Faraday electric field $E=-(2\pi r)^{-1}d\Phi/dt$ in order to push the
magnetic field $H$ out of the superconductor, see (14).

If the car started moving uphill in the absence of a driving force, then
everyone would consider it a miracle. But scientists do not believe in
miracles and therefore they do not consider the Meissner effect a miracle.
Physicists in 1933 did not believe also that the law of entropy increase can
be violated. Therefore, they concluded that the Meissner effect was able to
turn the irreversible transition in which Joule heat is generated into a
reversible transition in which no Joule heating can be. No one could explain
how the persistent currents can be annihilated without the generation of Joule
heat. Nevertheless, all experts were convinced after 1933 that the
superconducting transition is a phase transition, i.e. a reversible
thermodynamic process at which there no Joule heating can be. All theories of
superconductivity were created on the basis of this confidence.

###  V.2 The Meissner effect is a special case of the flux quantization

From the very beginning, the Meissner effect was considered a quantum
phenomenon, since this effect is impossible according to the laws of classical
physics. Lev Landau was first who proposed an explanation of the Meissner
effect as a quantum phenomenon. He proposed in 1941 Landau41  to describe
superconducting state with the help of the wave function
$\Psi_{GL}=|\Psi_{GL}|\exp{i\varphi}$ the square of the module of which
$|\Psi_{GL}|^{2}=n_{s}$ describes the density of all superconducting particles
whereas the phase gradient $\hbar\nabla\varphi=p=mv+qA$ describes the
canonical momentum of each particle. This wave function became the basis of
the famous Ginzburg-Landau (GL) theory GL1950 , which made it possible to
describe many effects observed in superconductors as macroscopic quantum
phenomena. The density of superconducting current equals

$j=\frac{q}{m}n_{s}(\hbar\nabla\varphi-qA)$ $None$

according to Landau41  and the GL theory GL1950 . The GL equation (17) allows
deduced the quantization (6) from the requirement that the complex wave
function $\Psi_{GL}=|\Psi_{GL}|\exp{i\varphi}$ must be single-valued at any
point in the superconductor. According to this requirement the phase $\varphi$
must change by integral multiples $2\pi$ following a complete turn along a
path $l$ of integration $\oint_{l}dl\nabla\varphi=n2\pi$ Huebener . Therefore
the relation

$\mu_{0}\oint_{l}dl\lambda_{L}^{2}j+\Phi=n\Phi_{0}$ $None$

between the current density $j$ along any closed path $l$ and a magnetic flux
$\oint_{l}dlA=\Phi$ inside this path must be valid. The expression deduced
from the wave function $\Psi_{GL}=|\Psi_{GL}|\exp{i\varphi}$ in 1941 Landau41
allows not only to explain the Meissner effect but also to predict the
quantization of a magnetic flux observed first in 1961 at measurement of
superconducting cylinder with thick wall $w\gg\lambda_{L}$ FluxQuant1961  and
the quantization of the persistent current (9) observed first in 1962 at
measurement of superconducting cylinder with a thin wall $w\ll\lambda_{L}$
LP1962 .

The current density has the same value along the wall in the case of weak
screening at $w\ll\lambda_{L}$ according to (18) and therefore this expression
gives the expression (9) for the persistent current. In the opposite case of
strong screening at $w\gg\lambda_{L}$ there is a contour inside the thick wall
along which the current density is zero $j=0$ and therefore $\Phi=n\Phi_{0}$
according to (18). The quantum number $n$ can be non-zero if only a
singularity of the wave function $\Psi_{GL}=|\Psi_{GL}|\exp{i\varphi}$ is
inside the closed path $l$. A hole in the superconducting cylinder
FluxQuant1961  and the Abrikosov vortex Abrikosov  are such singularities. The
integral $\oint_{l}dl\nabla\varphi=n2\pi=0$ and the quantum number $n$ must
equal zero when the closed path $l$ can be reduced to a point in the region
inside $l$ without singularity. Thus, the Meissner effect may be considered as
a special case of the flux quantization $\Phi=n\Phi_{0}$ when the quantum
number $n=0$ FPP2008 .

According to both the GL theory GL1950  and the BCS theory BCS1957  the
Meissner effect is observed due to the condensation of mobile charge carriers
which are the Cooper pairs with the charge $q=2e$ according to the BCS theory
BCS1957 . The GL theory GL1950  was created within the framework of
equilibrium reversible thermodynamics and superconducting transition is the
phase transition according to this theory. But L.D. Landau and V.L. Ginzburg
ignored the concern of W. H. Keesom that ”it is essential that the persistent
currents have been annihilated before the material gets resistance, so that no
Joule-heat is developed” Keesom1934  in order the superconducting transition
could be the phase transition. Therefore the GL theory GL1950  is internally
inconsistent. Other theorists, including J. Bardeen, L.N. Cooper and J.R.
Schrieffer BCS1957 , were concerned with the problem of how electrons can
become bosons. They have forgotten, perhaps because of the complexity of this
problem, that Joule heating is irreversible thermodynamic process. Therefore
the BCS theory BCS1957  contradicts also to the law of entropy increase.

###  V.3 The Meissner effect puzzle

Jorge Hirsch Hirsch2020EPL ; Hirsch2020Physica ; Hirsch2020ModPhys  has drawn
reader’s attention to the internal inconsistency of the BCS theory BCS1957  in
order to prove the superiority of his theory of hole superconductivity
Hirsch2020book  over the conventional theory. He has been trying for more than
thirty years to convince the superconducting community that the conventional
theory of superconductivity is inadequate and only his theory can describe
superconducting phenomena without contradictions, see his numerous
publications on the website https://jorge.physics.ucsd.edu/jh.html .

For a long time he did not succeed. Hirsch compared the attitude of most
physicists to the conventional theory of superconductivity with the attitude
of the characters in Andersen’s fairy tale ’The Emperor’s New Clothes’ to the
Emperor’s new clothes in the publication Hirsch2020APS . He points out in
particular that the conventional theory of superconductivity GL1950 ; BCS1957
cannot explain the Meissner effect: ”’Heaven help me’, thought smart students
that couldn’t understand how BCS theory explains the Meissner effect. ’I can’t
possibly see how momentum conservation is accounted for and Faraday’s law is
not violated’. But they did not say so” Hirsch2020APS .

The conventional theory of superconductivity GL1950 ; BCS1957 , following
Landau Landau41 , considers the Meissner effect as a special case of the
quantization (6): the persistent currents, both (9) and (15) appear since
superconducting state with zero electric current is forbidden. But the theory
GL1950 ; BCS1957  says nothing about the force that could accelerate the
mobile charge carriers against the action of the Faraday electric field
$E=-(2\pi r)^{-1}d\Phi/dt$.

Hirsch expressed astonishment that this puzzle is ignored: ”Strangely, the
question of what is the ’force’ propelling the mobile charge carriers and the
ions in the superconductor to move in direction opposite to the
electromagnetic force in the Meissner effect was essentially never raised nor
answered to my knowledge, except for the following instances: London1935  (H.
London states: ”The generation of current in the part which becomes
supraconductive takes place without any assistance of an electric field and is
only due to forces which come from the decrease of the free energy caused by
the phase transformation,” but does not discuss the nature of these forces),
PRB2001  (A.V. Nikulov introduces a ’quantum force’ to explain the Little-
Parks effect in superconductors and proposes that it also explains the
Meissner effect)” Hirsch10Meis .

I should note that the ”quantum force” PRB2001  was deduced from the
conventional theory GL1950  and experimental results in order to explain why
the persistent current can be observed at a non-zero resistance. The deduction
of the quantum force PRB2001  does not go beyond the GL theory GL1950  and
therefore cannot claim to explain what is the ’force’ propelling the mobile
charge carriers when the persistent current appears in superconducting ring
(9) and because of the Meissner effect (15). Hirsch claims that his theory of
superconductivity can explain not only what force creates the persistent
current (15), but also how this current can be annihilated before the material
gets resistance, as W. H. Keesom wanted Keesom1934 .

Hirsch’s theory requires radial flow of charge during the transition between
normal and superconducting states. The persistent screening current (15)
appears and is annihilated under the action of the Lorenz force due to this
radial flow of charge, according to Hirsch’s theory. This explanation has two
fundamental flaws: 1) according to Hirsch’s theory, an electric field directed
from the center to the edge of a bulk superconductor must exist in the
superconducting state Hirsch10Meis ; 2) Hirsch’s theory cannot explain the
appearance of the persistent current (9) in a superconducting ring in which no
radial flow of charge is possible in principle.

Hirsch agrees with the opinion of most physicists that the Meissner effect was
able to turn the irreversible thermodynamic process into the reversible one
Hirsch2017 . His theory, as well as the conventional theory of
superconductivity GL1950 ; BCS1957 , was created within the framework of
equilibrium reversible thermodynamics. Therefore Hirsch ignores the
experimental evidences that the persistent currents are not annihilated not
only before but even after the material gets resistance Moler2007 ;
Letters2007  and that the persistent currents are observed even in normal
metal rings PC2009Science ; Moler2009 . He also ignores the obvious mistakes
and contradictions that physicists had to make after 1933 in order to preserve
their faith in the law of entropy increase.

###  V.4 Contradiction with elementary logic

The radical change of opinion about superconducting transition after 1933 fits
with the law of entropy increase, but contradicts elementary logic: the
process of the energy (16) dissipation of the electric current (15) in the
normal state cannot depend on how this current appeared in the superconducting
state. W.H. Keesom wrote in 1934: ”In passing (in the beginning of process 3)
the threshold value curve, the electromotive force on the solenoid that
maintains the magnetic field must do an amount of work equal to twice the
energy of the field that comes into existence in the metal. Till now we
imagined that the surplus work served to deliver the Joule-heat developed by
the persistent currents the metal getting resistance while passing to the non-
supraconductive condition. As, however, the conception of Joule-heat can
rather difficulty be reconciled with reversibility we think now that there
must be going on another process that absorbs energy” Keesom1934 .

It is extremely strange that even W.H. Keesom could not understand that the
surplus work can increase only the Joule heat, despite that ”the conception of
Joule-heat can rather difficulty be reconciled with reversibility”. W.H.
Keesom was understanding that the power source of the solenoid should perform
the work

$A=\int_{t}dtIV=\int_{t}dtId\Phi/dt=I\Phi=\pi R^{2}\mu_{0}H^{2}$ $None$

per unit length of the superconducting cylinder at its transition in the
normal state in order to maintain the external magnetic field $H=I$. Half of
this work is spent on creating the energy of the magnetic field
$VBH/2=V\mu_{0}H^{2}/2$ inside the volume $V=L\pi R^{2}$ of the cylinder. It
should be emphasized that the energy of the magnetic field $VBH/2$ is zero in
the superconducting state since the magnetic flux density $B=0$. W.H. Keesom
and other physicists did not doubt before 1933 that the surplus work, the
second half of the work (19), generates Joule heat as well as no physicist can
doubt that the second part of the work (19) generates Joule heat when a rapid
change in the magnetic field from $0$ to $H=I$ induces the screening current
in a normal metal cylinder.

W.H. Keesom and other physicists did not take into account, because of their
faith in the law of entropy increase, that superconducting state with the
persistent current (15) can be achieved in two ways. On the first way,
considered above (see section (4.4)), the power source of the solenoid does
not perform a work when the temperature decreases from $T>T_{c}$ to $T<T_{c}$
in the zero magnetic field and performs a small work equal to the kinetic
energy (16) of the persistent current (15) when the magnetic field increases
from $0$ to $H=I$ at $T<T_{c}$. The power source performs the main work (19)
at the transition of the cylinder with a macroscopic radius to the normal
state.

The kinetic energy (16) is $2\lambda_{L}/R\approx 0.0001$ times less than the
energy of the magnetic field in the normal state $\mu_{0}H^{2}/2$ at typical
values of the radius of the cylinder $R\approx 1\ mm$ and the London
penetration depth $\lambda_{L}\approx 50\ nm=0.00005\ mm$. Half of the work
(19) returns to the power source when the magnetic field decreases from $H=I$
to $0$ at $T>T_{c}$. Thus, the power source in one cycle performs the work
equal to the sum $\mu_{0}H^{2}\lambda_{L}\pi RL+\mu_{0}H^{2}/2$ of the kinetic
energy (16) in the superconducting state and the energy of the magnetic field
in the normal state. The cycle can be repeated many times. In which reservoir
can the energy generated by this work accumulate, if it does not turn into
Joule heat?

###  V.5 Contradiction with the law of energy conservation

It is surprising that none of the physicists who believed that the
superconducting transition became a phase transition after 1933 did not ask
this obvious question. Even more surprising is the belief in the equality of
the free energies at the critical magnetic field $H_{c}$ at which the
superconducting cylinder goes into the normal state. According to the
universal belief on which the theory of superconductivity GL1950 ; BCS1957  is
based superconducting transition is observed at $T=T_{c}$ since the free
energy of superconducting state is smaller than the one of the normal state
$F_{s0}<F_{n0}$ at $T<T_{c}$ and $F_{s0}>F_{n0}$ at $T>T_{c}$ in the absence
of a magnetic field $H=0$. It was found already before 1933 that bulk
superconductor goes in the normal state at $T<T_{c}$ when external magnetic
field reaches a critical value $H=H_{c}(T)$. Hirsch, following C. J. Gorter
Gorter1933 , writes in Hirsch2017  that the free energy of superconducting and
normal states should be equal $F_{s}=F_{n}$ at $H=H_{c}(T)$ in order the
transition in the magnetic field could be reversible and wrote the equality

$F_{n0}(T)-F_{s0}(T)=V\frac{\mu_{0}H_{c}^{2}(T)}{2}$ $None$

postulated by C.J. Gorter and others in 1933.

The equality (20) is written in almost all books on superconductivity
Tinkham1996 ; Shoenberg1952  and underlies both the conventional theory of
superconductivity GL1950 ; BCS1957  and Hirsch’s theory. But this equality
contradicts obviously to the law of energy conservation. The equality (20) is
deduced from the equality $F_{s}=F_{s0}+V\mu_{0}H_{c}^{2}(T)/2=F_{n}=F_{n0}$
of the free energy of superconducting $F_{s}$ and normal $F_{n}$ states when
the external magnetic field equals the critical magnetic field $H=H_{c}(T)$.
The equality of the free energy $F_{s}=F_{n}$ was postulated in 1933 in order
superconducting transition observed at $H=H_{c}(T)$ could be considered as the
phase transition. According to this equality the magnetic field $H=H_{c}(T)$
should increase the free energy of superconducting state
$F_{s}=F_{s0}+V\mu_{0}H_{c}^{2}(T)/2$ rather than the normal state
$F_{n}=F_{n0}$ since $F_{s0}<F_{n0}$ at $T<T_{c}$ and $H=0$. But this
postulate contradicts to the law of energy conservation since the power source
of the solenoid performs in superconducting state the small work equal the
kinetic energy (16) which is much smaller the work performed by the power
source in order to create the energy $V\mu_{0}H_{c}^{2}(T)/2$ of magnetic
field $H=H_{c}(T)$ in the normal state.

Hirsch seems to think that the energy $V\mu_{0}H_{c}^{2}(T)/2$ on the right
side of the equation (20) is the kinetic energy of the surface current (15) in
the superconducting state. In order to convince himself and others of this,
Hirsch deduced the equality (B13) from the equality (B7) in Hirsch2017 .
According to the equality (B7) and (B13) the energy of the magnetic field in
the volume $V$ of a bulk cylinder with a macroscopic radius $R\gg\lambda_{L}$
in the normal state $V\mu_{0}H^{2}/2$ equals the density of the kinetic energy
$\varepsilon=n_{s}mv^{2}/2=\mu_{0}\lambda_{L}^{2}j^{2}/2=\mu_{0}H^{2}/2$ of
the persistent current in the superconducting state at $r=R$. But the energy
cannot be equal the energy density in principle. Hirsch made obvious
mathematical mistake in Hirsch2017  in order to convince himself and others
that the equality (20) is correct and superconducting transition is the phase
transition.

The mathematical mistake made by Hirsch in Hirsch2017  has provoked a delusion
of H. Koizumi Koizumi2020  who thinks that the kinetic energy of the
persistent current in the superconducting state equals the energy of the
magnetic field in the normal state because of his belief that Hirsch deduced
correctly the equality (B13) from the equality (B7) in the paper Hirsch2017 .
Therefore he writes that ”The reversible process implies $\Delta F_{m}+\Delta
F_{k}=0$” Koizumi2020 , i.e. that the kinetic energy $F_{k}$, see(2) in
Koizumi2020 , is transformed in the energy of the magnetic field $F_{m}$
see(1) in Koizumi2020 , at the transition of a bulk superconductor to the
normal state. Such a transition is certainly reversible. Not only an
irreversible increase in entropy, which occurs during Joule heating, but even
a reversible increase in entropy, which occurs during the phase transition of
the first kind is absent when the kinetic energy is transformed in the energy
of the magnetic field. But such transformation cannot be possible since the
kinetic energy in superconducting state (16) is $2\lambda_{L}/R$ times less
than the energy of the magnetic field in the normal state MyComment2021 .

Unfortunately, not only H. Koizumi Koizumi2020  does not know how a reversible
thermodynamic transition differs from an irreversible transition. Therefore,
this difference is explained in the Comment MyComment2021  on the article
Koizumi2020 . The blind faith in the law of entropy increase provoked not only
the contradiction with the law of energy conservation and mathematical mistake
but also profanation of physics. Only this blind faith could force physicists
to forget that the energy density $VBH/2$ of the magnetic field $H$ in the
volume $V$ of a bulk superconductor is zero when the magnetic flux density is
$B=0$ zero rather than when $B=\mu_{0}H$.

###  V.6 No work can be performed during any phase transition

The blind faith has forced physicists also to forget that the change in the
magnetic flux $\Phi$ in a bulk superconductor during the superconducting
transition induces the Faraday electromotive force $-d\Phi/dt$ that can
perform work on external bodies. Or could physicists have forgotten that no
work can be performed during the phase transition when they concluded that the
Meissner effect was able to turn the irreversible transition into the phase
transition? No work can be performed because of the equality of the free
energy $F_{s}=F_{n}$ postulated at any phase transition. The free energy by
definition $F=H-ST$ is a part of the internal energy $H$ that is able to
perform work in the absence of a temperature difference. Therefore the free
energy should decrease when a work $A$ is perform during the superconducting
transition and the equality $F_{s}=F_{n}$ cannot be possible according to the
law of energy conservation.

A visual example of the work performed during the superconducting transition
is the levitation of a small magnet over the surface of a superconductor. The
potential energy of the magnet with a mass $m$ in the gravitational field
increases on the value $gmh$ when the magnet lifts to a height $h$ due to the
transition of a superconductor into the superconducting state. This well-known
example is enough to refute the belief of several generations of physicists
that the superconducting transition is the phase transition. The free energy
of the superconductor must be reduced on the value of the work $A=gmh$,
performed in order to increase the potential energy of the magnet. Here we may
recall that Smoluchowski argued in 1914 Smoluchowski  the impossibility of a
useful perpetuum mobile, proving that the potential energy of a particle at
the Brownian motion cannot increase by an amount much greater than the energy
of thermal fluctuations $k_{B}T$. Now we see that the potential energy of the
magnet increases by an amount much greater than this energy $gmh\gg k_{B}T$
due to the Meissner effect.

##  VI Conclusion

The problem of violation of the law of entropy increase has both fundamental
and practical importance. Elliott Lieb and Jakob Yngvason wrote that this ”law
has caught the attention of poets and philosophers and has been called the
greatest scientific achievement of the nineteenth century. Engels disliked it,
for it supported opposition to dialectical materialism, while Pope Pius XII
regarded it as proving the existence of a higher being” PhysRep1999 . The
history of this law shows that our ideas about Nature are based not only on
experience and logic, but also on faith. The supporters of the thermodynamic-
energy worldview denied at the end of the nineteenth century the existence of
atoms and their perpetual thermal motion because of their faith in this law.

The experience of the Brownian motion and other fluctuation phenomena,
investigated at the beginning of the twentieth century, convinced most of the
supporters of this worldview in the existence of the perpetual thermal motion
of atoms and other particles. But the faith of twentieth-century physicists
turned out to be more blind than the faith of nineteenth-century scientists.
Only the blind faith could force W.H. Keesom and others to postulate the false
equality of free energies (20), which became the basis for understanding
superconductivity, despite the fact that they knew that the power source of
the solenoid performs work during the transition of the bulk superconductor to
the normal state, and not during the reverse transition.

The main psychological argument of believers in the law of entropy increase is
the assertion that no one has managed to make a perpetuum mobile for several
centuries. But, for example, a nuclear reactor also could not be made for many
centuries. A nuclear reactor and a perpetuum mobile differ first of all in the
time of the appearance of the task of their creation. The task of creating a
nuclear reactor could have arisen only in the twentieth century, when nuclear
fission was discovered, while from time immemorial, everyone could be
convinced by experience that the cart would stop if it was not pushed. The
laziness inherent in people made them dream about a perpetuum mobile. But not
every dream comes true right away. Now first of all, the belief in the
impossibility of a perpetuum mobile prevents the dream of a perpetuum mobile
from coming true. It is possible the perpetuum mobile would have been created
before the nuclear reactor without this belief.

Most, but not all, scientists believe in the impossibility of a perpetuum
mobile. Published articles Physica2021 ; Vlada1999 ; Daniel2000 ; Theo2000 ;
Mystique2004 ; Lee2021  and even book Book2005  in which this belief is
questioned. Several conferences devoted to the problem of the universality of
the law of entropy increase have been held, the results of which have been
published DanielCon2002 ; DanielCon2007 ; DanielCon2011 . The Special Issue of
Entropy Entropy2004  was dedicated to this problem. P.D. Keefe considered in
the articles Peter2002 ; Peter2004 ; Peter2005 ; Peter2010  a contradiction of
the Meissner effect with the law of entropy increase. Nevertheless, the law of
entropy increase remains a matter of faith rather than understanding for most
scientists.

Of course, not all doubts in this law are well-founded and even scientific.
But these doubts should be refuted by scientific arguments, and not banned
from publication only on the basis of the belief of the majority in the
impossibility of a perpetuum mobile. Most scientists have to understand that
the supreme position of the second law of thermodynamics has no scientific
justification. This position is determined rather by the psychology of
scientists who find it difficult to recognize that the centuries-old belief of
most scientists can be false. Possibility of violation of the second law of
thermodynamics should be openly discussed in scientific journals.

This work was made in the framework of State Task No 075-00355-21-00.

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