# Entropy production of active particles in underdamped regime

Derek Frydel  Department of Chemistry, Universidad Técnica Federico Santa
María, Campus San Joaquin, 7820275, Santiago, Chile

###### Abstract

The present work investigates the effect of inertia on the entropy production
rate $\Pi$ for all canonical models of active particles, for different
dimensions and type of confinement. To calculate $\Pi$, the link between the
entropy production and dissipation of heat rate is explored. By analyzing the
Kramers equation in a stationary state, alternative formulations of $\Pi$ are
obtained and the virial theorem for active particles is derived. Exact results
are obtained for particles in unconfined environment and in a harmonic trap.
In both cases, $\Pi$ is independent of temperature. In contrast, for active
particles in 1D box, thermal fluctuations are found to reduce $\Pi$.

###### pacs:

##  I Introduction

In this work we study the entropy production rate $\Pi$ Rosen60 ;
Schnakenberg76 ; Tome06 ; Leonid10 ; Tome12 ; Cates16 ; Shankar18 ; Schmidt19
; Pruessner20 ; Razin20 ; Pruessner21  within underdamped dynamics for three
canonical models of active particles. Contributions of inertia are expected to
play role in the regime where the persistence time of active motion $\tau$ is
comparable to or smaller than the inertial relaxation time $\tau_{r}$. Even if
this regime might not accessible to experimental systems, the inclusion of
inertia is necessary to recover a correct behavior in the limit $\tau\to 0$.

Our interest to understand the role of inertia in the regime
$\tau\lesssim\tau_{r}$ is motivated by a recently made observation that for
the run-and-tumble (RTP) and active Brownian particle (ABP) models the entropy
production rate formulated within overdamped dynamics is maximal when $\tau=0$
(implying a maximal distance from equilibrium) Frydel22a . At the same time,
the stationary distribution recovers a Boltzmann functional form (indicative
of equilibrium). Simple physical arguments suggest that this behavior is a
result of defining $\Pi$ within incomplete dynamics that ignores inertia.

The reason why overdamped dynamics predicts a maximal value of $\Pi$ as
$\tau\to 0$ is as follows. In the limit $\tau\to 0$, the swimming direction of
an active particle changes very fast. As a result, an active motion fails to
produce a net displacement in a particle position. This, effectively,
suppresses active motion and explains the emergence of a Boltzmann
distribution. At sufficiently small length scales, however, a particle can be
observed to move, since in the overdamped regime the velocity of an active
motion is independent $\tau$. And because within such small length scales the
external potential is constant, it is as if an active particle found itself in
unconfined environment. Unimpeded by any confinement, the entropy production
attains a maximal value.

With inertia taken into account, in the regime $\tau\lesssim\tau_{r}$, a
particle velocity fails to adjust itself to a rapidly changing force
direction. As a result, an actual velocity of a particle is reduced, and in
the limit $\tau\to 0$ it goes to zero. Once velocity (due to active motion) is
suppressed, the entropy production rate goes to zero, identifying the limit
$\tau\to 0$ as equilibrium.

In addition to the above motivation, the current study is relevant to a
growing interest in active particle models with inertia Lowen19 ; Sandoval20 ;
Lowen22 . Such models are more representative of particles embedded in low-
density environment such as gas. Examples include mesoscopic dust particles in
plasmas Morfill09 , granulars on a vibrating plate Weber13 , or insect flight
at water interfaces Kim16 .

To formulate $\Pi$ within underdamped dynamics, we explore the link between
the entropy production rate and the heat dissipation rate Sekimoto98 ; Sasa05
; Maes09 ; Maes21 ; Landi21 . Using Kramers equation, we obtain a number of
alternative formulations of $\Pi$. One by-product of this analysis is the
derivation of the virial theorem for active particles. Another interesting
result is the representation of distributions in $v$-space, for active
particles in unconfined environment, as convolution between the Maxwell
distribution and the distribution at zero temperature.

The role of inertia in determining entropy production has been investigated in
the past. In Celani12 ; Nakayama13  this was done for a system in the presence
of temperature gradients and in Chun15 , for a Brownian particle in a harmonic
trap and in contact with two heat baths. To our knowledge, the role of inertia
on $\Pi$ for active particle systems has been considered for the first time in
Shankar18  for an unconfined environment and in two-dimensions, and without
considering the RTP model.

##  II Entropy production as a dissipation of heat

The second law of thermodynamics states that a non-reversible process is
characterized by the production of entropy. The process of entropy production
is commonly represented as Prigogine55 ; Groot62 ; Glansdorff71 ;
Schnakenberg76 ; Tome12

$\dot{S}=\Pi-\Phi,$ (1)

where $\dot{S}$ is time derivative of the entropy, $\Pi$ is the entropy
production rate, and $\Phi$ is the entropy flux in and out of the system that
occurs in two forms: heat transfer $\dot{q}/T$ and mass flow. Without mass
flow, the entropy flux simply is

$\Phi=\frac{\dot{q}}{T},$ (2)

and for the stationary state $\dot{S}=0$, Eq. (1) leads to

$T\Pi=\dot{q},$ (3)

establishing a relation between entropy and heat production Sasa05 ; Marconi17
; Maes03 ; Maggi17 ; Landi21 .

For active particles, heat is generated as a result of a drag force exerted by
a surrounding fluid on a particle that moves through it, ${\bf
F}_{d}(t)=-{{\bf v}(t)}{\mu}^{-1}$, where $\mu$ is the mobility. Using a
definition of power, ${\bf F}_{d}\cdot{\bf v}$, we get the heat dissipation
rate $\dot{q}(t)={v^{2}(t)}{\mu}^{-1}$. Note that the instantaneous
dissipation of heat in this formulation can be related to the kinetic energy.

However, when considering an average rate of heat dissipation, we cannot
assume the equality $\langle\dot{q}\rangle=\langle v^{2}\rangle\mu^{-1}$ if
thermal fluctuations are present. Due to the fluctuation-dissipation relation,
contributions of thermal fluctuations to the kinetic energy are not dissipated
but recycled between a particle and a reservoir. The amount of the kinetic
energy coming from the reservoir is related to the temperature,
$\frac{1}{2}\langle mv^{2}\rangle_{eq}=\frac{n}{2}k_{B}T$, or $\langle
v^{2}\rangle_{eq}=nD/\tau_{r}$ (where $\tau_{r}=m\mu$ is the inertial
relaxation time, $m$ is the mass of a particle, $n$ is the system dimension,
and $D=k_{B}T\mu$ is the diffusion constant). The correct formula for the
average heat dissipation rate, therefore, is

$T\Pi\equiv\langle\dot{q}\rangle=\frac{1}{\mu}\left[\langle
v^{2}\rangle-\frac{nD}{\tau_{r}}\right].$ (4)

The above formulation is the centerpiece of this article. In subsequent
sections, this formula, and its alternative manifestations, will be used to
calculate $\Pi$ for different scenarios.

##  III Unconfined environment

To calculate $\langle v^{2}\rangle$ for an active particle in unconfined
environment, we start with the underamped Langevin equation for the evolution
of a velocity vector

$\tau_{r}\dot{\bf v}=-{\bf v}+\boldsymbol{\eta}+\mu{\bf f}.$ (5)

Two stochastic processes are thermal fluctuations $\boldsymbol{\eta}$ and a
stochastically evolving force ${\bf f}$ that generates active motion. The two
processes are uncorrelated, $\langle{\bf
f}(t)\cdot\boldsymbol{\eta}(t^{\prime})\rangle=0$. Thermal fluctuations are
delta correlated,
$\langle\boldsymbol{\eta}(t)\cdot\boldsymbol{\eta}(t^{\prime})\rangle=2nD\delta(t-t^{\prime})$,
and the force ${\bf f}$ is exponentially correlated, $\langle{\bf
f}(t)\cdot{\bf f}(t^{\prime})\rangle=\langle f^{2}\rangle
e^{-|t-t^{\prime}|/\tau}$, where $\tau$ is the persistence time. For the RTP
and ABP models the magnitude of the force ${\bf f}$ is constant, $|{\bf
f}|=v_{0}/\mu^{2}$ and we have

$\langle f^{2}\rangle=\frac{v_{0}^{2}}{\mu^{2}},~{}~{}~{}~{}\text{RTP and
ABP},$ (6)

where $v_{0}$ is the constant velocity. For the AOUP model, the force ${\bf
f}$ evolves according to $\tau\dot{\bf f}=-{\bf
f}+\boldsymbol{\eta}_{f}\mu^{-1}$ Szamel14 ; Cates21 , where
$\boldsymbol{\eta}_{f}$ is a delta correlated Gaussian noise,
$\langle\boldsymbol{\eta}_{f}(t)\cdot\boldsymbol{\eta}_{f}(t^{\prime})\rangle=2nD_{f}\delta(t-t^{\prime})$,
and $D_{f}$ is the diffusion constant of that process. For this model

$\langle f^{2}\rangle=\frac{nD_{f}}{\tau\mu^{2}},~{}~{}~{}~{}\text{AOUP}.$ (7)

The last but important difference between the two processes is that
$\boldsymbol{\eta}$ satisfies the fluctuation-dissipation theorem and ${\bf
f}$ does not. Consequently, all dissipation effects come from ${\bf f}$.

The average square velocity is obtained by solving the Langevin equation:

$\displaystyle\langle v^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{r}^{2}}\int_{-\infty}^{t}ds\int_{-\infty}^{t}ds^{\prime}\,e^{-(t-s)/\tau_{r}}e^{-(t-s^{\prime})/\tau_{r}}$
(8) $\displaystyle\times$
$\displaystyle\bigg{[}\langle\boldsymbol{\eta}(s)\cdot\boldsymbol{\eta}(s^{\prime})\rangle+\langle{\bf
f}(s)\cdot{\bf f}(s^{\prime})\rangle+2\langle{\bf
f}(s)\cdot\boldsymbol{\eta}(s^{\prime})\rangle\bigg{]}.$

The integral is evaluated by inserting previously defined time correlations of
different stochastic processes, yielding

$\langle v^{2}\rangle=\frac{\mu^{2}\langle
f^{2}\rangle}{1+\tau_{r}/\tau}+\frac{nD}{\tau_{r}}.$ (9)

Inserting this into Eq. (4) and using expressions for $\langle f^{2}\rangle$
in Eq. (6), the relation yields

$T\Pi\equiv\langle\dot{q}\rangle=\frac{v_{0}^{2}}{\mu}\frac{\tau}{\tau+\tau_{r}},~{}~{}~{}~{}\text{RTP
and ABP}.$ (10)

The formula for $\langle\dot{q}\rangle$ indicates that the dissipation process
is independent of thermal fluctuations. For overdamped dynamics, $\tau_{r}=0$,
the heat dissipation becomes independent of $\tau$. On the other hand, the
inclusion of inertia leads to reduced dissipation as a function of decreasing
$\tau$. In the limit $\tau\to 0$, dissipation altogether vanishes, indicating
that a system is in equilibrium.

For the AOUP model, using Eq. (7), the entropy production rate becomes

$T\Pi\equiv\langle\dot{q}\rangle=\frac{nD_{f}}{\mu}\frac{1}{\tau+\tau_{r}},~{}~{}~{}~{}\text{AOUP}.$
(11)

Eq. (11) requires qualification Klymko17 . The present work assumes that ${\bf
f}$ is an external stochastic force without the fluctuation-dissipation
theorem. In other words, no second thermal reservoir of temperature
$k_{B}T_{f}=D_{f}/\mu$ is coupled to the system. There is only one reservoir
for the temperature $T$. However, the AOUP model admits alternative
interpretation for the origin of non-equilibrium Klymko17 , which can be
better explained by representing the Langevin equation in Eq. (5) specifically
for the AOUP model as Sevilla19 ; Kaarakka11

$\tau_{r}\dot{\bf v}=-{\bf
v}+\boldsymbol{\eta}+\frac{1}{\tau}\int_{-\infty}^{t}ds\,e^{-(t-s)/\tau}\boldsymbol{\eta}_{f}(s),~{}~{}~{}~{}\text{AOUP},$
(12)

where the force ${\bf f}$ given by the last term is represented as a Gaussian
colored noise. For $\tau=0$, the noise becomes white, and the Langevin
equation becomes $\tau_{r}\dot{\bf v}=-{\bf
v}+\boldsymbol{\eta}+\boldsymbol{\eta}_{f}$. To interpret this limit as
equilibrium, we must assume that the reservoir temperature is $T\to T+T_{f}$,
and not simply $T$. In this paper, we do not assume the existence of the
second thermal reservoir and the force ${\bf f}$ is always dissipative, even
for $\tau=0$. Having said this, we do not recommend one interpretation over
another. We choose this interpretation in order to treat all active particle
models on the same footing and in the same framework. Later, we obtain a
number of relations valid for all models that are independent of any
particular interpretation of non-equilibrium. For the RTP and ABP models, no
such ambiguity arises.

The result in Eq. (10) and in Eq. (11) are in agreement with those in
Shankar18  (Table 1 in that reference) using different derivation. The results
in Shankar18  are limited to 2D, but the formula in Eq. (10) and Eq. (11)
apply to any system dimension.

###  III.1 "Maxwell" distributions of active particles in unconfined
environment

Examining Eq. (9), we can infer that $\langle v^{2}\rangle$ is a sum of two
processes:

$\langle v^{2}\rangle=\langle v^{2}\rangle_{eq}+\langle v^{2}\rangle_{ac},$
(13)

where $\langle\dots\rangle_{eq}$ is an average over thermal fluctuations
(without active motion) and $\langle\dots\rangle_{ac}$ is an average due to
active motion at zero temperature. Such a decomposition of a second moment is
a signature of the distribution generated by convolution that for the current
problem can be represented as $p(v)=\int d{\bf
v}^{\prime}\,p_{ac}(v^{\prime})p_{eq}({\bf v}-{\bf v}^{\prime})$, where

$p_{eq}(v)=\left(\frac{\tau_{r}}{2\pi
D}\right)^{\frac{n}{2}}e^{-\frac{\tau_{r}}{2D}v^{2}},$ (14)

is the familiar Maxwell-Boltzmann distribution (we recall that $D=k_{B}T\mu$
and $\tau_{r}=m\mu$).

Eq. (14) together with the convolution relation leads to the following
distribution of velocities:

$p(v)=\left(\frac{\tau_{r}}{2\pi D}\right)^{\frac{n}{2}}\int d{\bf
v}^{\prime}\,p_{ac}(v^{\prime})e^{-\frac{\tau_{r}}{2D}({\bf v}^{\prime}-{\bf
v})^{2}},$ (15)

where $p_{ac}$ does not have a general formula but depends on a particular
model.

In general, if a process is a sum of two independent random processes,
$Z=X+Y$, each process corresponding to the distribution $p_{X}$ and $p_{Y}$,
then the distribution of the process $Z$ is a convolution of the two
distributions:
$p_{Z}(z)=\int^{\infty}_{-\infty}dz^{\prime}\,p_{X}(z^{\prime})p_{Y}(z-z^{\prime})$.
The convolution in Eq. (15) is the reflection of independence of thermal
fluctuations and active motion in unconfined environment.

A similar convolution formula has recently been determined for active
particles in a harmonic trap Frydel22c . Since the Langevin equation in (5)
can be interpreted as the Langevin equation for an active particle in a
harmonic trap in the overdamped regime, we expect an analogous convolution
relation to apply in this case.

The convolution solution can be proven by analyzing the stationary Fokker-
Planck equation for the evolution of $\rho({\bf v},{\bf f})$:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{r}}\mbox{\boldmath$\nabla$}_{v}\cdot\left[\left({\bf
v}-\mu{\bf f}\right)\rho\right]+\frac{D}{\tau_{r}^{2}}\nabla_{v}^{2}\rho$ (16)
$\displaystyle+$
$\displaystyle\frac{1}{\tau}\begin{cases}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta_{f}\,\rho-\rho,&\text{RTP}\\\
\frac{\partial^{2}\rho}{\partial\theta_{f}^{2}},&\text{ABP}\\\
\mbox{\boldmath$\nabla$}_{\\!\\!f}\cdot{\bf
f}\rho+\frac{D_{f}}{\tau\mu^{2}}\mbox{\boldmath$\nabla$}_{f}^{2}\rho,&\text{AOUP},\end{cases}$

where the last term determines the evolution of ${\bf f}$ and depends on the
model. For convenience, these terms are written specifically for 2D where
${\bf f}=\mu^{-1}v_{0}{\bf n}$ (the unit orientation vector ${\bf
n}=(\cos\theta_{f},\sin\theta_{f})$ depends only on the angle $\theta_{f}$).
For the AOUP model, $\mbox{\boldmath$\nabla$}_{f}$ is the gradient operator
with respect to ${\bf f}$.

Taking the Fourier transform of Eq. (16) with respect to velocity yields

$\displaystyle 0$ $\displaystyle=$ $\displaystyle-\frac{1}{\tau_{r}}[{\bf
k}\cdot\mbox{\boldmath$\nabla$}_{k}c]-\frac{\mu}{\tau_{r}}[i{\bf k}\cdot{\bf
f}]c-\frac{Dk^{2}}{\tau_{r}^{2}}c$ (17) $\displaystyle+$
$\displaystyle\frac{1}{\tau}\begin{cases}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta_{f}\,c-c,&\text{RTP}\\\
\frac{\partial^{2}c}{\partial\theta_{f}^{2}},&\text{ABP}\\\
\mbox{\boldmath$\nabla$}_{\\!\\!f}\cdot{\bf
f}c+\frac{D_{f}}{\tau\mu^{2}}\mbox{\boldmath$\nabla$}_{f}^{2}c,&\text{AOUP},\end{cases}$

where $c({\bf k},{\bf f})=\int d{\bf v}\,\rho({\bf v},{\bf f})\,e^{-i{\bf
k}\cdot{\bf v}}$ and $c_{eq}\propto e^{-Dk^{2}/2\tau_{r}}$ is the transformed
Maxwell distribution. The same equation at zero temperature reads

$\displaystyle 0$ $\displaystyle=$ $\displaystyle-\frac{1}{\tau_{r}}[{\bf
k}\cdot\mbox{\boldmath$\nabla$}_{k}c_{ac}]-\frac{\mu}{\tau_{r}}[i{\bf
k}\cdot{\bf f}]c_{ac}$ (18) $\displaystyle+$
$\displaystyle\frac{1}{\tau}\begin{cases}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta_{f}\,c_{ac}-c_{ac},&\text{RTP}\\\
\frac{\partial^{2}c_{ac}}{\partial\theta_{f}^{2}},&\text{ABP}\\\
\mbox{\boldmath$\nabla$}_{\\!\\!f}\cdot{\bf
f}c+\frac{D_{f}}{\tau\mu^{2}}\mbox{\boldmath$\nabla$}_{f}^{2}c,&\text{AOUP}.\end{cases}$

In the Fourier space, the convolution formula in Eq. (15) becomes

$c({\bf k},{\bf f})\propto c_{ac}({\bf k},{\bf f})e^{-Dk^{2}/2\tau_{r}}.$ (19)

At this point, we do not know if the above formula is an actual solution. If
it is, then substituting Eq. (19) into Eq. (17) should recover Eq. (18).
Substituting Eq. (19) into Eq. (17) splits the first term of that equation
into two parts,

$\frac{1}{\tau_{r}}{\bf
k}\cdot\mbox{\boldmath$\nabla$}_{k}c=\left(\frac{1}{\tau_{r}}{\bf
k}\cdot\mbox{\boldmath$\nabla$}_{k}c_{ac}-\frac{Dk^{2}}{\tau_{r}^{2}}c_{ac}\right)e^{-{Dk^{2}}/{2\tau_{r}}}.$
(20)

where the first term cancels the third term in that equation, recovering Eq.
(18).

This not only proves the convolution formula in Eq. (15), but it implies a
stronger claim,

$\rho({\bf v},{\bf f})=\left(\frac{\tau_{r}}{2\pi D}\right)^{\frac{n}{2}}\int
d{\bf v}^{\prime}\,\rho_{ac}({\bf v},{\bf f})e^{-\frac{\tau_{r}}{2D}({\bf
v}^{\prime}-{\bf v})^{2}},$

where $\rho_{ac}$ is the distribution for $D=0$. Consequently, the effect of
temperature on all the distributions $\rho_{ac}$ is the same. Since $p(v)=\int
d{\bf f}\,\rho({\bf v},{\bf f})$, the above relation proves the convolution in
Eq. (15).

Note that since the cancelation of diffusion terms is independent of the terms
that govern the evolution of ${\bf f}$, the proof applies to all active
particle models in unconfined environment.

To fully determine the distribution $p(v)$, we need an expression for
$p_{ac}(v)$. For the RTP model in dimensions $n=1,2$, $p_{ac}$ are beta
distributions Frydel22c

$p_{ac}(v)\propto\left(1-\frac{v^{2}}{v_{0}^{2}}\right)^{\frac{n}{2}\frac{\tau_{r}}{\tau}-1}~{}~{}~{}\text{RTP
for $n=1,2$}.$ (21)

that vanish for $v>v_{0}$ and show a crossover at $\tau/\tau_{r}=n/2$, at
which point distributions change from convex to concave shape. For long
persistence time, $\tau/\tau_{r}>n/2$, velocities tend to have values near
$v=v_{0}$, and for shorter persistence time, $\tau/\tau_{r}<n/2$, velocities
tend to be close to $v=0$.

The beta distribution in Eq. (21) is not valid for the RTP model in higher
dimensions. Likewise, there is no simple closed form formula of $p_{ac}$ for
the ABP model in any dimension. For these cases, the convolution formula in
Eq. (15) can still be used, but distributions $p_{ac}$ need to be obtained
numerically from simulations.

The Langevin equation in Eq. (12) can be mapped onto the Langevin equation for
AOUP particles in a harmonic trap in the overdamped regime, for which we know
that a stationary distribution is a Gaussian function Szamel14 . And since
from Eq. (9) and Eq. (7) we know that the variance for $D=0$ is $\langle
v^{2}\rangle_{ac}=\frac{nD_{f}}{\tau+\tau_{r}}$, we infer the following
distribution for $D=0$:

$p_{ac}(v)\propto
e^{-\frac{v^{2}}{2}\frac{\tau+\tau_{r}}{D_{f}}}~{}~{}~{}\text{AOUP},$ (22)

then using the convolution relation, the stationary distribution for a finite
temperature becomes

$p(v)\propto
e^{-\frac{v^{2}}{2}/[\frac{D}{\tau_{r}}+\frac{D_{f}}{\tau+\tau_{r}}]}~{}~{}~{}\text{AOUP},$
(23)

since the convolution of two Gaussian functions in a Gaussian function. Both
distributions are valid for any dimension. For $\tau=0$ the distribution in
Eq. (23) does not recover the Maxwell distribution $p_{eq}$ in Eq. (14),
complicating the interpretation of the system at $\tau=0$ as equilibrium.

##  IV Particles in confinement

To analyze active particles in confinement in the underdamped regime, we
consider the Kramers equations Kramer  for the evolution of $\rho({\bf r},{\bf
v},{\bf f},t)$:

$\displaystyle\frac{\partial\rho}{\partial t}$ $\displaystyle=$
$\displaystyle-{\bf
v}\cdot\mbox{\boldmath$\nabla$}_{r}\rho-\frac{1}{\tau_{r}}\mbox{\boldmath$\nabla$}_{v}\cdot\left[\left(\mu{\bf
F}-{\bf v}+\mu{\bf
f}\right)\rho\right]+\frac{D}{\tau_{r}^{2}}\nabla_{v}^{2}\rho$ (24)
$\displaystyle+$
$\displaystyle\frac{1}{\tau}\begin{cases}\frac{1}{2\pi}\int_{0}^{2\pi}d\theta_{f}\,\rho-\rho,&\text{RTP}\\\
\frac{\partial^{2}\rho}{\partial\theta_{f}^{2}},&\text{ABP}\\\
\mbox{\boldmath$\nabla$}_{\\!\\!f}\cdot{\bf
f}\rho+\frac{D_{f}}{\tau\mu^{2}}\mbox{\boldmath$\nabla$}_{f}^{2}\rho,&\text{AOUP}.\end{cases}$

$\mbox{\boldmath$\nabla$}_{r}$, $\mbox{\boldmath$\nabla$}_{v}$,
$\mbox{\boldmath$\nabla$}_{f}$ and in Eq. (24) are the gradient operators with
respect to ${\bf r}$, ${\bf v}$, and ${\bf f}$, and ${\bf
F}=-\mbox{\boldmath$\nabla$}V$ is the external time independent force. The
last terms in Eq. (24) govern the evolution of ${\bf f}$ and depend on a
specific model. For the RTP and ABP models, there terms are defined for a
system in two-dimensions to simplify expressions.

To calculate $\langle v^{2}\rangle$, or other average quantities of interest,
we multiply the stationary Kramers equation, $\dot{\rho}=0$, by a function
$g\equiv g({\bf r},{\bf v},{\bf f})$ and then integrate each term as $\int
d{\bf r}\int d{\bf v}\int d{\bf f}$. Using integration by parts, the terms of
the Kramers equation can be represented as average quantities,
$\langle\dots\rangle=\int d{\bf r}\int d{\bf v}\int d{\bf f}\,\,\,(\dots)$,

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\langle{\bf
v}\cdot\mbox{\boldmath$\nabla$}_{r}g\rangle+\frac{1}{\tau_{r}}\langle\left(\mu{\bf
F}-{\bf v}+\mu{\bf
f}\right)\cdot\mbox{\boldmath$\nabla$}_{v}g\rangle+\frac{D}{\tau_{r}^{2}}\langle\nabla_{v}^{2}g\rangle$
(25) $\displaystyle+$
$\displaystyle\frac{1}{\tau}\begin{cases}\frac{1}{2\pi}\langle[\int_{0}^{2\pi}d\theta_{f}\,g]\rangle-\langle
g\rangle,&\text{RTP}\\\
\langle\frac{\partial^{2}g}{\partial\theta_{f}^{2}}\rangle,&\text{ABP}\\\
-\langle{\bf
f}\cdot\mbox{\boldmath$\nabla$}_{f}g\rangle+\frac{D_{f}}{\tau\mu^{2}}\langle\mbox{\boldmath$\nabla$}_{f}^{2}g\rangle,&\text{AOUP}.\end{cases}$

Eq. (25) is a template from which various relations can be generated by
specifying the function $g$. As we are interested in $\langle v^{2}\rangle$, a
logical choice seems to be $g=v^{2}$, which together with Eq. (25) yields

$0=\frac{\mu}{\tau_{r}}\langle{\bf F}\cdot{\bf
v}\rangle-\frac{1}{\tau_{r}}\langle
v^{2}\rangle+\frac{\mu}{\tau_{r}}\langle{\bf f}\cdot{\bf
v}\rangle+\frac{nD}{\tau_{r}^{2}}.$ (26)

Note that for passive Brownian motion, ${\bf f}=0$, the above relation yields
$0=\langle{\bf F}\cdot{\bf v}\rangle$. To verify that this result is general
and holds for nonzero ${\bf f}$, we use $g=V$. Eq. (25) in this case yields

$0=\langle{\bf F}\cdot{\bf v}\rangle,$ (27)

which shows the above relation to be valid for any stationary system.

Combining Eq. (26) with Eq. (27) and using the definition of
$\langle\dot{q}\rangle$ in Eq. (4), we get an alternative formulation of the
heat dissipation rate:

$\langle\dot{q}\rangle=\langle{\bf f}\cdot{\bf v}\rangle.$ (28)

Other formulations of $\langle\dot{q}\rangle$ are still possible. Inserting
$g={\bf f}\cdot{\bf v}$ into Eq. (25) yields

$0=\frac{\mu}{\tau_{r}}\langle{\bf F}\cdot{\bf
f}\rangle-\frac{1}{\tau_{r}}\langle{\bf f}\cdot{\bf
v}\rangle+\frac{\mu}{\tau_{r}}\langle f^{2}\rangle-\frac{1}{\tau}\langle{\bf
f}\cdot{\bf v}\rangle.$ (29)

Together with Eq. (28), Eq. (29) leads to another formulation of the heat
dissipation rate:

$\langle\dot{q}\rangle=\left[\frac{v_{0}^{2}}{\mu}+\mu\langle{\bf F}\cdot{\bf
f}\rangle\right]\frac{\tau}{\tau+\tau_{r}}.$ (30)

The second term can be interpreted as representing contributions of
confinement. Because correlations between vectors ${\bf F}$ and ${\bf f}$ are
negative, confinement should reduce dissipation.

Another formulation of heat dissipation is obtained by using $g={\bf
r}\cdot{\bf f}$. Eq. (25) in this case yields

$0=\langle{\bf f}\cdot{\bf v}\rangle-\frac{1}{\tau}\langle{\bf f}\cdot{\bf
r}\rangle.$ (31)

In combination with Eq. (28) this leads to

$\langle\dot{q}\rangle=\frac{1}{\tau}\langle{\bf f}\cdot{\bf r}\rangle,$ (32)

the fourth formula of heat dissipation.

All formulations for the heat dissipation rate are listed below:

$T\Pi\equiv\langle\dot{q}\rangle=\begin{cases}\frac{1}{\mu}\left[\langle
v^{2}\rangle-\frac{nD}{\tau_{r}}\right]\\\
\left[\frac{v_{0}^{2}}{\mu}+\mu\langle{\bf F}\cdot{\bf
f}\rangle\right]\frac{\tau}{\tau+\tau_{r}}\\\ \langle{\bf f}\cdot{\bf
v}\rangle\\\ \frac{1}{\tau}\langle{\bf f}\cdot{\bf r}\rangle.\end{cases}$ (33)

The stationary distribution $\rho$ is a function of three vector variables,
$\rho\equiv\rho({\bf r},{\bf v},{\bf f})$. As shown above, ${\bf f}$ is not an
independent variable. It is on average correlated with other vector variables
$\langle{\bf f}\cdot{\bf r}\rangle$ and $\langle{\bf f}\cdot{\bf v}\rangle$,
where the magnitude of correlations is related to the dissipation of heat. To
determine correlations between ${\bf r}$ and ${\bf v}$, we insert $g=r^{2}$
into Eq. (25). The result is

$0=\langle{\bf r}\cdot{\bf v}\rangle$ (34)

indicating the absence of correlations. It is noted that the relation in Eq.
(27) is a consequence of this absence of correlations since ${\bf F}$ is a
function of a particle position.

The third formulation in Eq. (33) can be identified as average power input,
indicating the violation of the first law of thermodynamics. Note that the
other two forces ${\bf F}$ and $\boldsymbol{\eta}\mu^{-1}$ do not contribute
to the power input. The force ${\bf F}$ does not contribute as a result of Eq.
(27), and in the case of thermal force we have
$\langle\boldsymbol{\eta}\cdot{\bf v}\rangle=nD/\tau_{r}$ (this result is
obtained from analyzing the Langevin equation). But this contributions is
subtracted in Eq. (4) as a consequence of the fluctuation-dissipation theorem.

##  V The virial theorem of active particles

In this section, we are going to use Eq. (25) to obtain a virial theorem for
active particles. Using $g={\bf r}\cdot{\bf v}$, Eq. (25) yields

$0=\langle v^{2}\rangle+\frac{\mu}{\tau_{r}}\langle{\bf F}\cdot{\bf
r}\rangle-\frac{1}{\tau_{r}}\langle{\bf r}\cdot{\bf
v}\rangle+\frac{\mu}{\tau_{r}}\langle{\bf f}\cdot{\bf r}\rangle.$ (35)

Together with Eq. (32), Eq. (4), and Eq. (34, the above equation can be
expressed as

$-\langle{\bf F}\cdot{\bf r}\rangle=2\langle
T\rangle+\tau\langle\dot{q}\rangle,$ (36)

where $\langle T\rangle=\langle mv^{2}/2\rangle$ is the average kinetic energy
where we used $\tau_{r}=m\mu$ to eliminate $\tau_{r}$. The relation in Eq.
(36) is the virial theorem for active particles. Since at equilibrium the
virial theorem is $-\langle{\bf F}\cdot{\bf r}\rangle=2\langle T\rangle$, the
term $\tau\langle\dot{q}\rangle$ in Eq. (36) can be regarded as a measure of
violation of the virial relation.

The virial theorem in Eq. (36) can be expressed differently by using the
fourth formulation in Eq. (33), leading to

$-\langle({\bf F}+{\bf f})\cdot{\bf r}\rangle=2\langle T\rangle.$ (37)

Given that ${\bf f}$ is an external force, the formulation above appears to
conform with the original virial theorem.

If an external potential is given by $V=Kr^{s}/s$ Dhar19  then ${\bf
F}\cdot{\bf r}=-sV$ and Eq. (36) becomes

$s\langle V\rangle=2\langle T\rangle+\tau\langle\dot{q}\rangle.$ (38)

##  VI Particles in a harmonic trap

In this section we focus on a specific confining potential, a harmonic trap.
In this case the external force is ${\bf F}=-K{\bf r}$. To obtain an
expression for the dissipation of heat, we use Eq. (30). And to obtain an
expression for $\langle{\bf F}\cdot{\bf f}\rangle$, we substitute into Eq.
(25) $g={\bf F}\cdot{\bf f}$. This yields

$0=\langle{\bf v}\cdot\mbox{\boldmath$\nabla$}_{r}({\bf F}\cdot{\bf
f})\rangle-\frac{1}{\tau}\langle{\bf F}\cdot{\bf f}\rangle.$ (39)

For ${\bf F}=-K{\bf r}$ the first term becomes $\langle{\bf
v}\cdot\mbox{\boldmath$\nabla$}_{r}({\bf F}\cdot{\bf f})\rangle=-K\langle{\bf
v}\cdot{\bf f}\rangle$, then using Eq. (28), Eq. (39) can be written as

$\langle{\bf F}\cdot{\bf f}\rangle=-\tau K\langle\dot{q}\rangle.$ (40)

Substituting this result into Eq. (30) and using Eq. (6), the heat dissipation
formula for a harmonic trap becomes

$T\Pi\equiv\langle\dot{q}\rangle=\frac{v_{0}^{2}}{\mu}\frac{\tau}{\tau+\tau^{2}\mu
K+\tau_{r}},~{}~{}~{}~{}\text{RTP and ABP}.$ (41)

The entropy production rate for RTP particles in a harmonic trap have
previously been obtained for the overdamped regime, or $\tau_{r}=0$, in
Pruessner21 ; Frydel22a . The formulas in Eq. (41) recover these results by
setting $\tau_{r}$ to zero.

As for the case of an unconfined system, $\langle\dot{q}\rangle$ is
independent of temperature. The role of confinement, regulated by the
parameter $K$, is to reduce a dissipation effect. Because the coefficient of
$K$ is quadratic in $\tau$, the effects of confinement becomes negligible as
$\tau$ becomes small. In the limit $\tau\to 0$, heat is no loner dissipated.
In contrast, for the case of overdamped dynamics, or $\tau_{r}=0$, the
dissipation attains a maximal value in the limit $\tau\to 0$.

In Fig. (1) we plot simulation data points for $\langle\dot{q}\rangle$ as a
function of $\tau$ for the RTP and ABP models, different system dimensions,
and different temperatures. According to Eq. (41), all points are expected to
collapse onto a single curve, which is confirmed by the results. To calculate
$\langle\dot{q}\rangle$, we can use any of the formulas in Eq. (33). All the
formulas were tested and shown to yield the same results.

---
Figure 1: The heat dissipation rate of self-propelled particles in a harmonic
trap as a function of $\tau$. Simulation data points come from four systems.
Sim-1 is for RTP particles in $2D$ and $D=0$, sim-2 is for RTP particles in 1D
and $D=0.1$, sim-3 for ABP particles in 2D and $D=0.2$, and sim-4 for ABP
particles in 3D and $D=0.2$. All points collapse onto the curve given by the
formula in Eq. (41). The dashed line is $\langle\dot{q}\rangle$ for the
overdamped regime.

The simulation data points are compared against $\langle\dot{q}\rangle$
calculated for overdamped dynamics, using Eq. (41) for $\tau_{r}=0$. The two
curves deviate in the region $\tau\lesssim\tau_{r}$, and the deviation becomes
maximal at $\tau=0$. As $\langle\dot{q}\rangle$ quantifies a distance from
equilibrium, this leads to different interpretation of the same system when
analyzed using different theoretical frameworks.

Using Eq. (40), Eq. (30) and Eq. (7), we get the heat dissipation formula for
AOUP particles in harmonic trap:

$T\Pi\equiv\langle\dot{q}\rangle=\frac{nD_{f}}{\mu}\frac{1}{\tau+\tau^{2}\mu
K+\tau_{r}},~{}~{}~{}~{}~{}\text{AOUP}.$ (42)

We recall that in the case of AOUP model, this formula represents a specific
interpretation of the entropy production as discussed in Sec. (III).

###  VI.1 "Maxwell" distributions of active particles in a harmonic trap

Eq. (41) and Eq. (42) suggests that the average square velocity is represented
as a sum of two contributions, $\langle v^{2}\rangle=\langle
v^{2}\rangle_{eq}+\langle v^{2}\rangle_{ac}$. This implies that the
distribution in $v$-space obeys a convolution formula $p(v)=\int d{\bf
v}^{\prime}\,p_{ac}(v^{\prime})p_{eq}({\bf v}-{\bf v}^{\prime})$ already
encountered for unconfined environment, in Eq. (15).

As before, the proof of Eq. (15) for a harmonic confinement can be obtained by
analyzing the Fourier transformed Kramers equation. The stationary Kramers
equation for a harmonic trap is

$0=-{\bf
v}\cdot\mbox{\boldmath$\nabla$}_{r}\rho+\frac{1}{\tau_{r}}\mbox{\boldmath$\nabla$}_{v}\cdot\left[\left(\mu
K{\bf r}+{\bf v}-\mu{\bf
f}\right)\rho\right]+\frac{D}{\tau_{r}^{2}}\nabla_{v}^{2}\rho+\dots,$ (43)

where the terms for the evolution of ${\bf f}$, that do not play role in the
proof, are ignored. We next take the Fourier transform with respect to ${\bf
r}$ and ${\bf v}$ that yields

$0={\bf q}\cdot\mbox{\boldmath$\nabla$}_{k}c-\frac{\mu K}{\tau_{r}}{\bf
k}\cdot\mbox{\boldmath$\nabla$}_{q}c-\frac{1}{\tau_{r}}{\bf
k}\cdot\mbox{\boldmath$\nabla$}_{k}c-\frac{\mu}{\tau_{r}}[i{\bf k}\cdot{\bf
f}]c-\frac{Dk^{2}}{\tau_{r}^{2}}c+\dots$ (44)

where $c({\bf q},{\bf k},{\bf f})=\int d{\bf r}\int d{\bf v}\,\rho({\bf
r},{\bf v},{\bf f})\,e^{-i{\bf q}\cdot{\bf r}}e^{-i{\bf k}\cdot{\bf v}}.$ We
next assume the following solution

$c=c_{ac}e^{-Dk^{2}/2\tau_{r}}e^{-Dq^{2}/2\mu K},$ (45)

where $\propto e^{-Dk^{2}/2\tau_{r}}$ is a Fourier transformed Maxwell
distribution in $v$-space and $\propto e^{-Dq^{2}/2\mu K}$ is a Fourier
transformed equilibrium distribution in space. It turns out that if we
substitute this function into Eq. (44), we will eliminate all the terms that
depend on $D$ and recover

$0={\bf q}\cdot\mbox{\boldmath$\nabla$}_{k}c_{ac}-\frac{\mu K}{\tau_{r}}{\bf
k}\cdot\mbox{\boldmath$\nabla$}_{q}c_{ac}-\frac{1}{\tau_{r}}{\bf
k}\cdot\mbox{\boldmath$\nabla$}_{k}c_{ac}-\frac{\mu}{\tau_{r}}[i{\bf
k}\cdot{\bf f}]c_{ac}+\dots,$ (46)

which is the same as Eq. (44) but for $D=0$. This means that Eq. (45) is a
correct solution. In physical space, this implies the following convolution:

$\rho({\bf r},{\bf v},{\bf f})\propto\int d{\bf r}^{\prime}\int d{\bf
v}^{\prime}\,\,\rho_{ac}({\bf r}^{\prime},{\bf v}^{\prime},{\bf
f})e^{-\frac{\mu K}{2D}({\bf r}-{\bf
r}^{\prime})^{2}}e^{-\frac{\tau_{r}}{2D}({\bf v}-{\bf v}^{\prime})^{2}}.$ (47)

After integrating over ${\bf r}$ and ${\bf f}$ we recover the convolution in
Eq. (15).

For the RTP and ABP models, we do not have exact expressions for $p_{ac}$, but
we can get such an expression for the AUOP model, where the distribution
$p_{ac}$ is a Gaussian due to linear drift in the Kramers equation Szamel14 .
Since we know the variance $\langle v^{2}\rangle_{ac}$ from Eq. (42), we infer

$p_{ac}(v)\propto e^{-\frac{\tau+\tau_{r}+\tau^{2}\mu
K}{D_{f}}\frac{v^{2}}{2}},~{}~{}~{}~{}~{}~{}\text{AOUP}.$ (48)

Then using the convolution formula in Eq. (15) we get

$p(v)\propto
e^{-\frac{v^{2}}{2}/\big{[}\frac{D}{\tau_{r}}+\frac{D_{f}}{\tau+\tau_{r}+\tau^{2}\mu
K}\big{]}},~{}~{}~{}\text{AOUP}.$ (49)

##  VII RTP model in 1D box

A well studies model of active particles is the RTP model in 1D for particles
trapped between two walls. In the overdamped regime, the model has been
frequently studies and is well understood Schnitzer93 ; Angelani17 ; Dhar18 ;
Dhar19 ; Basu20 ; Razin20 ; Frydel22b . For underdamped dynamics, the system
is governed by the Kramers equation resulting in two coupled differential
equations:

$\displaystyle\dot{p}_{+}=-v\frac{\partial p_{+}}{\partial
x}+\frac{1}{\tau_{r}}\frac{\partial}{\partial
v}\left[\left(v-v_{0}\right)p_{+}\right]+\frac{D}{\tau_{r}^{2}}\frac{\partial^{2}p_{+}}{\partial
v^{2}}-\frac{1}{2}\frac{1}{\tau}(p_{+}-p_{-})$
$\displaystyle\dot{p}_{-}=-v\frac{\partial p_{-}}{\partial
x}+\frac{1}{\tau_{r}}\frac{\partial}{\partial
v}\left[\left(v+v_{0}\right)p_{-}\right]+\frac{D}{\tau_{r}^{2}}\frac{\partial^{2}p_{-}}{\partial
v^{2}}+\frac{1}{2}\frac{1}{\tau}(p_{+}-p_{-}),$ (50)

where $p_{+}(x,v,t)$ and $p_{-}(x,v,t)$ are the distributions for particles
subject to a forward and backward force. The entropy production for this model
in the overdamped regime, or $\tau_{r}=0$, has been obtained in Razin20 . For
the walls located at $x=\pm h$ it reads

$T\Pi=\langle\dot{q}\rangle=\frac{v_{0}^{2}}{\mu}\left[\frac{D}{v_{0}^{2}\tau}\frac{\cosh(kh)-\frac{\sinh(kh)}{kh}}{\frac{D}{v_{0}^{2}\tau}\cosh(kh)+\frac{\sinh(kh)}{kh}}\right],$
(51)

where $k=\frac{v_{0}}{D}\sqrt{1+\frac{D}{v_{0}^{2}\tau}}.$

To calculate $\langle\dot{q}\rangle$ for the underdamped regime, we use the
last formula in Eq. (33), which for the present model becomes

$T\Pi=\frac{v_{0}}{\tau\mu}\left[\langle x\rangle_{+}-\langle
x\rangle_{-}\right],$ (52)

where $\langle\dots\rangle_{\pm}=\int_{-h}^{h}dx\,(\dots)p_{\pm}$.

Since distributions $p_{\pm}$ in Eq. (50) cannot be solved exactly, we
calculate Eq. (52) using simulations. The results are plotted in Fig. (2). For
$\tau>2$, the simulation results agree with the formula in Eq. (51) for the
case $\tau_{r}=0$. Deviations between overdamped and underdamped dynamics
become obvious for small values of $\tau$ and become maximal for $\tau=0$.

---
Figure 2: The heat dissipation rate as a function of $\tau$ for the RTP
particles in 1D confined between two walls located at $x=\pm h$. Dashed lines
represent the exact expression in Eq. (51) for the overdamped regime, and the
data points for a finite $\tau_{r}$ are obtained from the simulation.

Another observation is that thermal fluctuations reduce dissipation of heat
(note the difference between the black and red curves). This behavior is
different from a harmonic confinement where $\langle\dot{q}\rangle$ is found
to be independent of temperature.

There is another interesting feature that arises as a consequence of
underdamped dynamics. Within the overdamped regime and at zero temperature, a
fraction of particles becomes adsorbed onto confining walls due to the
combination of overdamped dynamics and finite persistence time rudi18 .
Particles that are not adsorbed are uniformly distributed within the box
Frydel22b . Within the underdamped dynamics, particles coming against a wall
no longer become immediately adsorbed. Instead, they are reflected from it,
and if the persistence time is sufficiently long, they change direction and
come toward the wall again, creating a behavior analogous to bouncing of a
ball. As a result, stationary distributions inside the box develop an
interesting structure as shown in Fig. (3). The structure has nothing to do
with the size of particles but is entirely a dynamical phenomenon.

---
Figure 3: A stationary distribution of RTP particles inside a 1D box at zero
temperature. The other system parameters are $h=\tau=\tau_{r}=1$. The
distribution is obtained from a simulation.

##  VIII Conclusion

In this work we calculate the entropy production rate in the underdamped
regime for all canonical models of active particles in all dimensions and for
different confinements. To calculate $\Pi$, we explore the link between the
entropy production rate and the dissipation of heat, from which we obtain the
expression in Eq. (4), from which alternative formulations are obtained in Eq.
(33) by analyzing the Kramers equation. The exact results are obtained for
particles in a harmonic trap and in unconfined environment.

Other interesting by-products of this work is representation of "Maxwell"
distributions for particles in unconfined environment as a convolution
relation in Eq. (15). We also derive the virial relation for active particles
in Eq. (36).

###### Acknowledgements.

D.F. acknowledges financial support from FONDECYT through grant number
1201192.

##  IX DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * (1) G. Rosen, Entropy Production and Pressure Waves, Phys. Fluids 3, 188 (1960);
  * (2) J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys. 48, 571 (1976).
  * (3) T. Tomé, Entropy production in non-equilibrium systems described by a Fokker-Planck equation, Braz. J. Phys. 36, 1285 (2006).
  * (4) T. Tomé and M. J. de Oliveira, Entropy Production in Nonequilibrium Systems at Stationary States, Phys. Rev. Lett. 108, 020601 (2012)
  * (5) É. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, and F. van Wijland How Far from Equilibrium Is Active Matter?, Phys. Rev. Lett. 117, 038103 (2016).
  * (6) S. Shankar and M. C. Marchetti, Hidden entropy production and work fluctuations in an ideal active gas, Phys. Rev. E 98, 020604(R) (2018).
  * (7) E. G. Idrisov and T. L. Schmidt, Entropy production in one-dimensional quantum fluids, Phys. Rev. B 100, 165404 (2019).
  * (8) L. Cocconi, R. Garcia-Millan, Z. Zhen, B. Buturca, G. Pruessner, Entropy production in exactly solvable systems, Entropy 22, 1252 (2020).
  * (9) N. Razin, Entropy production of an active particle in a box, Phys. Rev. E (R) 102, 030103(R) (2020).
  * (10) R. Garcia-Millan and G. Pruessner Run-and-tumble motion in a harmonic potential: field theory and entropy production, J. Stat. Mech. 063203 (2021).
  * (11) L. M. Martyushev The maximum entropy production principle: two basic questions, Phil. Trans. R. Soc. B 365, 1333 (2010) .
  * (12) D. Frydel, Intuitive view of entropy production of ideal run-and-tumble particles, Phys. Rev. E 105, 034113 (2022).
  * (13) Löwen, H. Inertial effects of self-propelled particles: From active Brownian to active Langevin motion, J. Chem. Phys. 152, 040901 (2020).
  * (14) L. L. Gutierrez-Martinez, M. Sandoval, Inertial effects on trapped active matter, J. Chem. Phys. 153, 044906. (2020).
  * (15) G. H. P. Nguyen, R. Wittmann, and H. Löwen, Active Ornstein–Uhlenbeck model for self-propelled particles with inertia, J. Phys.: Condens. Matter 34, 035101 (2022).
  * (16) G. E. Morfill and A. V. Ivlev, Complex plasmas: An interdisciplinary research field, Rev. Mod. Phys. 81, 1353 (2009).
  * (17) C. A. Weber, T. Hanke, J. Deseigne, S. Léonard, O. Dauchot, E. Frey, and H. Chaté, Long-range ordering of vibrated polar disks, Phys. Rev. Lett. 110, 208001 (2013).
  * (18) H. Mukundarajan, T. C. Bardon, D. H. Kim, and M. Prakash, Surface tension dominates insect flight on fluid interfaces, J. Exp. Biol. 219, 752 (2016).
  * (19) K. Sekimoto, Langevin Equation and Thermodynamics, Progress of Theoretical Physics Supplement, 130, 17 (1998).
  * (20) G. T. Landi and M. Paternostro, Irreversible entropy production: From classical to quantum, Rev. Mod. Phys. 93, 035008 (2021).
  * (21) T. Harada and S. Sasa, Equality Connecting Energy Dissipation with a Violation of the Fluctuation-Response Relation, Phys. Rev. Lett. 95, 130602 (2005).
  * (22) Maes C, Netocný K and Shergelashvili B 2009 A Selection of Nonequilibrium Issues in Methods of Contemporary Mathematical Statistical Physics ed K Roman (Berlin: Springer) pp 247–306.
  * (23) T. Banerjee and C. Maes, Active gating: rocking diffusion channels, J. Phys. A: Math. Theor. 54, 025004 (2021).
  * (24) A. Celani, S. Bo, R. Eichhorn, and E. Aurell, Anomalous Thermodynamics at the Microscale, Phys. Rev. Lett. 109, 260603 (2012).
  * (25) K. Kawaguchi and Y. Nakayama, Fluctuation theorem for hidden entropy production, Phys. Rev. E 88, 022147 (2013).
  * (26) H.-M. Chun and J. D. Noh, Hidden entropy production by fast variables, Phys. Rev. E 91, 052128 (2015).
  * (27) I. Prigogine, Introduction to Thermodynamics of Irreversible Processes (Thomas, Springfield, 1955).
  * (28) S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1962).
  * (29) P. Glansdorff and I. Prigogine, Thermodynamics of Structure, Stability and Fluctuations (Wiley, New York, 1971).
  * (30) Maes, C., Netocný, K. Time-Reversal and Entropy, Journal of Statistical Physics 110, 269 (2003).
  * (31) U. M. B. Marconi, A. Puglisi, and C. Maggi, Heat, temperature and clausius inequality in a model for active Brownian particles, Sci. Rep. 7, 46496 (2017).
  * (32) A. Puglisi and U. M. B. Marconi, Clausius relation for active particles: What can we learn from fluctuations, Entropy 19, 356 (2017).
  * (33) G. Szamel, Self-propelled particle in an external potential: Existence of an effective temperature, Phys, Rev. E 90, 012111 (2014).
  * (34) D. Martin, J. O’Byrne, M. E. Cates, É. Fodor, C. Nardini, J. Tailleur, and F. van Wijland Statistical mechanics of active Ornstein-Uhlenbeck particles, Phys. Rev. E 103, 032607 (2021).
  * (35) D. Mandal, K. Klymko, and M. R. DeWeese, Entropy Production and Fluctuation Theorems for Active Matter, Phys. Rev. Lett. 119, 258001 (2017).
  * (36) F. J. Sevilla, R. F. Rodríguez, and J. R. Gomez-Solano, Generalized Ornstein-Uhlenbeck model for active motion, Phys. Rev. E 100, 032123 (2019).
  * (37) T. Kaarakka and P. Salminen, On fractional Ornstein-Uhlenbeck processes, Communications on Stochastic Analysis. 5, 121 (2011).
  * (38) D. Frydel, Positing the problem of stationary distributions of active particles as third-order differential equation, Phys. Rev. E 106, 024121, (2022).
  * (39) H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions Physica 7, 284 (1940).
  * (40) A. Dhar, A. Kundu, S. N. Majumdar, S. Sabhapandit, and G. Schehr Run-and-tumble particle in one-dimensional confining potentials: Steady-state, relaxation, and first-passage properties Phys. Rev. E 99, 032132 (2019).
  * (41) U. Basu, S.N. Majumdar, A. Rosso, S. Sabhapandit and G. Schehr, Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap, J. Phys. A: Math. Theor. 53, 09LT01 (2020).
  * (42) M. J. Schnitzer, Theory of continuum random walks and application to chemotaxis, Phys. Rev. E 48, 2553 (1993).
  * (43) L. Angelani, Confined run-and-tumble swimmers in one dimension, J. Phys. A: Math. Theor. 50, 325601 (2017).
  * (44) K. Malakar, V. Jemseena, A. Kundu, K. V. Kumar, S. Sabhapandit, S. N. Majumdar, S. Redner, and A. Dhar, Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension, J. Stat. Mech.: Theory Exp. 043215 (2018).
  * (45) D. Frydel, The four-state RTP model: exact solution at zero temperature, Phys. Fluids 34, 027111 (2022).
  * (46) D. Frydel and R. Podgornik, Mean-field theory of active electrolytes: Dynamic adsorption and overscreening, Phys. Rev. E 97, 052609 (2018).


# Phase transitions for a model with uncountable spin space on the Cayley
tree: the general case

Golibjon Botirov  and  Benedikt Jahnel  Weierstrass Institute Berlin,
Mohrenstr. 39, 10117 Berlin, Germany, https://www.wias-
berlin.de/people/jahnel/<EMAIL_ADDRESS>National University of
Uzbekistan, University Street 4, 100174 Tashkent, Uzbekistan,
<EMAIL_ADDRESS>
###### Abstract.

In this paper we complete the analysis of a statistical mechanics model on
Cayley trees of any degree, started in [EsHaRo12, EsRo10, BoEsRo13, JaKuBo14,
Bo17]. The potential is of nearest-neighbor type and the local state space is
compact but uncountable. Based on the system parameters we prove existence of
a critical value $\theta_{\rm c}$ such that for $\theta\leq\theta_{\rm c}$
there is a unique translation-invariant splitting Gibbs measure. For
$\theta_{\rm c}<\theta$ there is a phase transition with exactly three
translation-invariant splitting Gibbs measures. The proof rests on an analysis
of fixed points of an associated non-linear Hammerstein integral operator for
the boundary laws.

Mathematics Subject Classifications (2010). 82B05, 82B20 (primary); 60K35
(secondary)

Key words. Cayley trees, Hammerstein operators, splitting Gibbs measures,
phase transitions.

##  1\. Introduction

In the present note we complete a line of research about the phase-transition
behaviour of a nearest-neighbor model on Cayley trees with arbitrary degree
$k\geq 2$. As first described in [EsRo10], for a given consistent family of
finite-volume Gibbs measures, the existence and multiplicity of a certain
class of infinite-volume measures which are consistent with the prescribed
finite-volume Gibbs measures, can be reduced to the analysis of fixed points
of some non-linear integral equation of Hammerstein type. Every positive
solution of the fixed point equation here corresponds to a measures which is
called a splitting Gibbs measure. Every splitting Gibbs measure is also a
Gibbs measure in the sense of the DLR formalism; see [Ba82]. This approach has
been successfully applied in the analysis of a variety of different models on
Cayley trees with respect to their phase-transition properties; see [Ro13] for
a comprehensive overview. In particular, starting with [EsHaRo12], a phase-
transition of multiple splitting Gibbs measures has been detected in a model
with uncountable local state space $[0,1]$ and nearest-neighbor interactions.
This has motivated the subsequent analysis in [BoEsRo13, JaKuBo14, Bo17], to
further understand critical behavior of this model for all degrees of the
underlying tree, where also new parameters are introduced. It is the purpose
of this note to complete the analysis of this model.

For nearest neighbors $x,y$ on the Cayley tree $\Gamma^{k}$ with degree $k\geq
2$ with local states ${\sigma}(x),{\sigma}(y)\in[0,1]$, we consider the
potential

$\xi_{{\sigma}(x),{\sigma}(y)}=\log\left(1+\theta\sqrt[2m+1]{4({\sigma}(x)-\frac{1}{2})({\sigma}(y)-\frac{1}{2})}\right)$
(1.1)

where $m\in\mathbb{N}\cup\\{0\\}$, $0\leq\theta<1$ are the system parameters.
It can be interpreted as a certain symmetric pair-interaction with values in
$[\log(1-\theta),\log(1+\theta)]$, admitting two distinct ground states given
by the all-$0$ and the all-$1$ configuration. The main result is the existence
of a sharp threshold

$\theta_{\rm c}=\frac{2m+3}{k(2m+1)}$

such that if $\theta_{\rm c}<\theta<1$, there are exactly three translation-
invariant splitting Gibbs measures and otherwise there is only one.

##  2\. Setup

###  2.1. Gibbs measures on Cayley trees

The Cayley tree $\Gamma^{k}$ of order $k\geq 1$ is an infinite tree, i.e., a
graph without cycles, such that exactly $k+1$ edges originate from each
vertex. Let $\Gamma^{k}=(V,L)$ where $V$ is the set of vertices and $L$ is a
symmetric subset of $V\times V$, called the edge set. The word ”symmetric”
means that $(x,y)\in L$ iff $(y,x)\in L$. Here, $x$ and $y$ are called the
endpoints of the edge $\langle x,y\rangle$. Two vertices $x$ and $y$ are
called nearest neighbors if there exists an edge $l\in L$ connecting them and
we denote $l=\langle x,y\rangle$. For a fixed $x^{0}\in V$, called the root,
we defines $n$-spheres and $n$-disks in the graph distance $d(x,y)$ by

$W_{n}=\\{x\in V|d(x,x^{0})=n\\},\ \ \ \ V_{n}=\bigcup\limits_{i=0}^{n}W_{i}$

and denote for any $x\in W_{n}$ the set of direct successors of $x$ by

$S(x)=\\{y\in W_{n+1}:d(x,y)=1\\}.$

For $A\subset V$ let $\Omega_{A}=[0,1]^{A}$ denote the set of all
configurations ${\sigma}_{A}$ on $A$. In particular, a configuration $\sigma$
on $V$ is then defined as a function $V\ni x\mapsto\sigma(x)\in[0,1]$.
According to the usual setup for Gibbs measure, we consider a (formal)
Hamiltonian of the form

$H(\sigma)=-\sum\limits_{\langle x,y\rangle\in L}\xi_{\sigma(x),\sigma(y)},$
(2.1)

where $\xi:(u,v)\in[0,1]^{2}\mapsto\xi_{u,v}\in\mathbb{R}$ is the interaction
(1.1) which assigns energy only to neighboring sites. Since $\xi$ does not
depend on the locations $x$ and $y$, $H$ is invariant under tree translations.
Let $\lambda$ be the Lebesgue measure on $[0,1]$ then, on the set of all
configurations on $A$ the a priori measure $\lambda_{A}$ is introduced as the
$|A|$-fold product of the measure $\lambda$. Here and in the sequel, $|A|$
denotes the cardinality of $A$. We equip $\Omega=\Omega_{V}$ with the standard
sigma-algebra $\mathcal{B}$ generated by the cylindrical subsets. A
probability measure $\mu$ on $(\Omega,\mathcal{B})$ is called a Gibbs measure
(with Hamiltonian $H$) if it satisfies the DLR equation. That is, for any
$n=1,2,...$ and bounded measurable test function $f$, we have that

$\int\mu(d{\sigma})f({\sigma})=\int\mu(d{\sigma})\int\gamma_{V_{n}}(d\tilde{\sigma}_{V_{n}}|{\sigma}_{W_{n+1}})f(\tilde{\sigma}_{V_{n}}{\sigma}_{\Gamma^{k}\setminus
V_{n}}),$ (2.2)

where $\gamma_{V_{n}}(d\sigma_{V_{n}}|{\sigma}_{\Gamma^{k}\setminus V_{n}})$
is the Gibbsian specification

$\gamma_{V_{n}}(d\tilde{\sigma}_{V_{n}}|{\sigma}_{\Gamma^{k}\setminus
V_{n}})=\frac{1}{Z_{V_{n}}({\sigma}_{W_{n+1}})}e^{-\beta
H(\tilde{\sigma}_{V_{n}}{\sigma}_{W_{n+1}})}\lambda_{V_{n}}(d\tilde{\sigma}_{V_{n}}),$

with normalization $Z_{V_{n}}$ and temperature parameter $\beta\geq 0$. Such a
specification is also sometimes referred to as a Markov specification; see
[Ge11].

###  2.2. Representation via Hammerstein operators

A subset of the infinite-volume Gibbs measures defined via the DLR equation
(2.2), called the splitting Gibbs measures or Markov chains, can be
represented in terms of the fixed points of some nonlinear integral operator
of Hammerstein type; see [EsRo10] for details. More precisely, for every
$k\in\mathbb{N}$ consider the integral operator $H_{k}$ acting on the cone
$C^{+}[0,1]=\\{f\in C[0,1]:f(x)\geq 0\\}$ given by

$\begin{split}(H_{k}f)(t)=\int^{1}_{0}K(t,u)f^{k}(u)du.\end{split}$ (2.3)

Then, the translation-invariant splitting Gibbs measures for the Hamiltonian
(2.1) correspond to fixed points of $H_{k}$ with
$K(t,u)=\exp(\beta\xi_{t,u})$, often called boundary laws. Note that $H_{k}$
in general might generate ill-posed problems; see [Kr64, KrZa84].

##  3\. Main results

The main result of this note is the following characterization of phase-
transition regimes of the model (1.1) with $\beta=1$.

######  Theorem 3.1.

For all $n\in\mathbb{N}\cup\\{0\\}$ and $k\geq 2$ let $\theta_{\rm
c}=(2n+3)/(k(2n+1))$, then the model (1.1) has

  1. (1)

a unique translation-invariant splitting Gibbs measure if
$0\leq\theta\leq\theta_{\rm c}$ and

  2. (2)

exactly three translation-invariant splitting Gibbs measures if $\theta_{\rm
c}<\theta<1$.

The proof is based on a characterization of solutions to the fixed point
equation for the associated Hammerstein integral operator (2.3) as given in
Proposition 3.2 below. In case of the model at hand, then the analysis can be
reduced to finding the fixed points of the following $2$-dimensional operator
$V_{k}:(x,y)\in\mathbb{R}^{2}\rightarrow(x^{\prime},y^{\prime})\in\mathbb{R}^{2}$

$\begin{split}V_{k,n}(x,y)=\left\\{\begin{array}[]{lllllll}x^{\prime}=\sum\limits_{j=0}^{\lfloor\frac{k}{2}\rfloor}\binom{k}{2j}\frac{2n+1}{2n+1+2j}\cdot
2^{\frac{2j}{2n+1}}\cdot x^{k-2j}(\theta y)^{2j}$$\\\\[17.07164pt]
$$y^{\prime}=\sum\limits_{j=0}^{\lfloor\frac{k}{2}\rfloor}\binom{k}{2j+1}\frac{2n+1}{2n+2+2j+1}\cdot
2^{\frac{2j}{2n+1}}\cdot x^{k-(2j+1)}(\theta
y)^{2j+1}\end{array}\right.\end{split}$ (3.1)

with $k\geq 2$, which is then the content of Proposition 3.3.

######  Proposition 3.2.

A function $\varphi\in C[0,1]$ is a solution of the Hammerstein equation

$\begin{split}H_{k}f=f\end{split}$ (3.2)

with $H_{k}$ defined in (2.3) for our model (1.1), iff $\varphi$ has the
following form

$\varphi(t)=C_{1}+C_{2}\theta\sqrt[2n+1]{4(t-\frac{1}{2})},$

where $(C_{1},C_{2})\in\mathbb{R}^{2}$ is a fixed point of the operator
$V_{k,n}$ as defined in (3.1).

In the following proposition we characterize the fixed points of $V_{k,n}$
which readily implies Theorem 3.1 using Proposition 3.2.

######  Proposition 3.3.

Let $\theta_{\rm c}=(2n+3)/(k(2n+1))$, then there exist
$x_{o},y_{o}\in(0,\infty)$ such that the number and form of the fixed points
of the operator $V_{k,n}$ are as presented in the following Table 1.

Table 1. Set of $2$-dimensional fixed points of $V_{k,n}$ |  fixed points if $0\leq\theta\leq\theta_{\rm c}$ |  additional fixed points if $\theta_{\rm c}<\theta<1$
---|---|---
3 $k$ even | $(0,0)$ | $(1,0)$ |  | $(x_{o},y_{o})$ |  | $(x_{o},-y_{o})$ |
$k$ odd | $(0,0)$ | $(1,0)$ | $(-1,0)$ | $(x_{o},y_{o})$ | $(-x_{o},-y_{o})$ | $(x_{o},-y_{o})$ | $(-x_{o},y_{o})$

Only the fixed points $(1,0)$, $(x_{o},y_{o})$ and $(x_{o},-y_{o})$ give rise
to positive solutions for the Hammerstein equation (3.2).

Let us finally give the references to the special cases considered prior to
this work. [BoEsRo13, Theorem 4.2 and Theorem 5.2] proves the cases $k=2,3$
with $n=1$ of (1.1) whereas in [JaKuBo14, Theorem 3.2.] the cases $k\geq 2$
with $n=1$ are given. Finally, in [Bo17, Theorem 2.3] the cases $k=2$ with
general $n\geq 1$ is provided.

##  4\. Proofs

Note that for the model (1.1) with $\beta=1$, the kernel $K(t,u)$ of the
Hammerstein operator $H_{k}$ is given by

$K(t,u)=1+\theta\sqrt[2n+1]{4(t-\frac{1}{2})(u-\frac{1}{2})}.$

###### Proof of Proposition 3.2.

Let us start with necessity. Assume $\varphi\in C[0,1]$ to be a solution of
the equation (3.2). Then we have

$\varphi(t)=C_{1}+C_{2}\theta\sqrt[2n+1]{4(t-\frac{1}{2})},$ (4.1)

where

$C_{1}=\int\limits_{0}^{1}\varphi^{k}(u)du\quad\text{ and }\quad
C_{2}=\int\limits_{0}^{1}\sqrt[2n+1]{u-\frac{1}{2}}\cdot\varphi^{k}(u)du.$
(4.2)

Substituting $\varphi(t)$ into the first equation of (4.2) we get

$\begin{split}C_{1}&=\int\limits_{0}^{1}\left(C_{1}+C_{2}\theta\sqrt[2n+1]{4(u-\frac{1}{2})}\right)^{k}du\\\
&=\int\limits_{0}^{1}\sum\limits_{i=0}^{k}\binom{k}{i}C_{1}^{k-i}\left(C_{2}\theta\sqrt[2n+1]{4}\sqrt[2n+1]{u-\frac{1}{2}}\right)^{i}du\\\
&=\sum\limits_{i=0}^{k}\binom{k}{i}C_{1}^{k-i}(\theta
C_{2})^{i}2^{\frac{2i}{2n+1}}\int\limits_{0}^{1}\left(u-\frac{1}{2}\right)^{\frac{i}{2n+1}}du.\end{split}$

Now, we use the following equality

$\int\limits_{0}^{1}\left(u-\frac{1}{2}\right)^{\frac{i}{2n+1}}du=\left\\{\begin{array}[]{ll}0,&\hbox{if
$i$ is odd and}\\\ \frac{2n+1}{2n+1+i}\cdot 2^{-\frac{i}{2n+1}},&\hbox{if $i$
is even.}\end{array}\right.$ (4.3)

Then we get

$C_{1}=\sum\limits_{j=0}^{\lfloor\frac{k}{2}\rfloor}\binom{k}{2j}\frac{2n+1}{2n+1+2j}2^{\frac{2j}{2n+1}}C_{1}^{k-2j}(\theta
C_{2})^{2j}$

and substituting the function $\varphi$ into the second equation of (4.2) we
have

$\begin{split}C_{2}&=\int\limits_{0}^{1}\left(u-\frac{1}{2}\right)^{\frac{1}{2n+1}}\left(C_{1}+\theta
C_{2}\sqrt[2n+1]{4(u-\frac{1}{2})}\right)^{k}du\cr&=\int\limits_{0}^{1}\left(u-\frac{1}{2}\right)^{\frac{1}{2n+1}}\sum\limits_{i=0}^{k}\binom{k}{i}C_{1}^{k-i}\left(\theta
C_{2}\sqrt[2n+1]{4}\sqrt[2n+1]{u-\frac{1}{2}}\right)^{i}du\cr&=\sum\limits_{i=0}^{k}\binom{k}{i}C_{1}^{k-i}(C_{2}\theta)^{i}2^{\frac{2i}{2n+1}}\int\limits_{0}^{1}\left(u-\frac{1}{2}\right)^{\frac{i+1}{2n+1}}du.\end{split}$

Now, using the following equality

$\int\limits_{0}^{1}\left(u-\frac{1}{2}\right)^{\frac{i+1}{2n+1}}du=\left\\{\begin{array}[]{ll}0,&\hbox{if
$i$ is even and }\\\ \frac{2n+1}{2n+2+i}\cdot 2^{-\frac{i+1}{2n+1}},&\hbox{if
$i$ is odd}\end{array}\right.$ (4.4)

we arrive at the equation

$C_{2}=\sum\limits_{j=0}^{\lfloor\frac{k}{2}\rfloor}\binom{k}{2j+1}\frac{2n+1}{2n+2+2j+1}2^{\frac{2j}{2n+1}}C_{1}^{k-2j-1}(\theta
C_{2})^{2j+1}.$

In particular, the point $(C_{1},C_{2})\in\mathbb{R}^{2}$ must be a fixed
point of the operator $V_{k,n}$ from (3.1).

For the sufficiency, assume that, a point $(C_{1},C_{2})\in\mathbb{R}^{2}$ is
a fixed point of the operator $V_{k,n}$ and define the function $\varphi\in
C[0,1]$ by the equality

$\varphi(t)=C_{1}+C_{2}\theta\sqrt[2n+1]{4(t-\frac{1}{2})}.$

Then, we can calculate

$\begin{split}&(H_{k}\varphi)(t)=\int\limits_{0}^{1}\left(1+\sqrt[2n+1]{4}\theta\sqrt[2n+1]{(t-\frac{1}{2})(u-\frac{1}{2})}\right)\varphi^{k}(u)du\cr&=\int\limits_{0}^{1}\varphi^{k}(u)du+\sqrt[2n+1]{4}\theta\sqrt[2n+1]{t-\frac{1}{2}}\int\limits_{0}^{1}\sqrt[2n+1]{u-\frac{1}{2}}\varphi^{k}(u)du\cr&=\int\limits_{0}^{1}\left(C_{1}+C_{2}\theta\sqrt[2n+1]{4(u-\frac{1}{2})}\right)^{k}du\cr&\hskip
28.45274pt+\sqrt[2n+1]{4}\theta\sqrt[2n+1]{t-\frac{1}{2}}\int\limits_{0}^{1}\sqrt[2n+1]{u-\frac{1}{2}}\left(C_{1}+C_{2}\theta\sqrt[2n+1]{4(u-\frac{1}{2})}\right)^{k}du\cr&=\sum\limits_{i=0}^{k}\binom{k}{i}C_{1}^{k-i}(\theta
C_{2})^{i}2^{\frac{2i}{2n+1}}\int\limits_{0}^{1}\left(u-\frac{1}{2}\right)^{\frac{i}{2n+1}}du+\sqrt[2n+1]{4}\theta\sqrt[2n+1]{t-\frac{1}{2}}\cr&\hskip
28.45274pt\times\sum\limits_{i=0}^{k}\binom{k}{i}C_{1}^{k-i}(C_{2}\theta)^{i}2^{\frac{2i}{2n+1}}\int\limits_{0}^{1}\left(u-\frac{1}{2}\right)^{\frac{i+1}{2n+1}}du.\end{split}$
(4.5)

Now, we using (4.3) and (4.4), from (4.5) we get

$\begin{split}(H_{k}\varphi)(t)&=\sum\limits_{j=0}^{\lfloor\frac{k}{2}\rfloor}\binom{k}{2j}\frac{2n+1}{2n+1+2j}2^{\frac{2j}{2n+1}}x^{k-2j}(\theta
y)^{2j}\cr&\hskip
14.22636pt+\theta\sqrt[2n+1]{4(t-\frac{1}{2})}\sum\limits_{j=0}^{\lfloor\frac{k}{2}\rfloor}\binom{k}{2j+1}\frac{2n+1}{2n+2+2j+1}2^{\frac{2j}{2n+1}}x^{k-2j-1}(\theta
y)^{2j+1}\cr&=C_{1}+C_{2}\theta\sqrt[2n+1]{4(t-\frac{1}{2})}=\varphi(t).\end{split}$

Thus, $\varphi$ is a solution of the equation (3.2). ∎

###### Proof of Proposition 3.3.

Let us start by assuming $k$ to be even. We determine the number and form of
solutions to $V_{k,n}$ in equation (3.1). If $\theta\geq 0$, then for $k$ even
$(0,0)$ and $(1,0)$ are fixed points. If $\theta>0$ then there are potentially
more fixed points. Indeed, let $\theta>0$ and assume $y>0$ then, writing
$z=\theta y/x$, the fixed point equation for (3.1) becomes

$\begin{split}z=\theta\frac{\sum_{i=1,3,\dots,k-1}\binom{k}{i}\frac{2n+1}{2n+2+i}\cdot
2^{\frac{i-1}{2n+1}}\cdot
z^{i}}{\sum_{i=0,2,\dots,k}\binom{k}{i}\frac{2n+1}{2n+1+i}2^{\frac{i}{2n+1}}z^{i}}=\theta\frac{F_{1}(z)}{F_{2}(z)}=f(z).\end{split}$

Hence, in order to find solutions, we have to find roots of the polynomial

$\begin{split}P(z)&=\sum_{i=1,3,\dots,k+1}\binom{k}{i-1}\frac{2n+1}{2n+i}2^{\frac{i-1}{2n+1}}z^{i}-\theta\sum_{i=1,3,\dots,k-1}\binom{k}{i}\frac{2n+1}{2n+2+i}2^{\frac{i-1}{2n+1}}z^{i}\\\
&=r_{\theta}(k,k+1)z^{k+1}+\sum_{i=1,3,\dots,k-1}r_{\theta}(k,i)z^{i}\end{split}$
(4.6)

where $r_{\theta}(k,k+1)=\frac{2n+1}{2n+k+1}2^{\frac{k}{2n+1}}$ and

$\begin{split}r_{\theta}(k,i)&=\binom{k}{i}\frac{2n+1}{2n+2+i}2^{\frac{i-1}{2n+1}}\Big{[}\frac{i}{k-i+1}\frac{2n+2+i}{2n+i}-\theta\Big{]}.\end{split}$

Moreover,

$\begin{split}r_{\theta}(k,i)\left\\{\begin{array}[]{lllllll}<0\qquad\text{if}\quad\theta>\frac{i}{k-i+1}\frac{2n+2+i}{2n+i}\\\
=0\qquad\text{if}\quad\theta=\frac{i}{k-i+1}\frac{2n+2+i}{2n+i}\\\
>0\qquad\text{if}\quad\theta<\frac{i}{k-i+1}\frac{2n+2+i}{2n+i}\end{array}\right.\end{split}$

and we denote the critical $\theta$ by $\theta_{k,i}$. Further note that
$i\mapsto\theta_{k,i}$ is increasing. Indeed, the derivative of the continuous
version is given by

$\begin{split}\frac{4(1+k)n^{2}+(3+k)i^{2}+4(1+k)n(1+i))}{(1+k-i)^{2}(2n+i)^{2}}\end{split}$

which is non-negative. Hence, for $\theta$ below the lowest critical value,
$\theta_{k,1}=\frac{2n+3}{k(2n+1)}$, all coefficients are positive and hence
there is no positive real root by Descartes’ rule of sign. Further, again by
Descartes’ rule of sign, if we increase $\theta>\theta_{k,1}$, then there is
exactly one sign change and hence, exactly one non-trivial positive real root
which we denote $z_{0}$.

Since only odd term appear in the polynomial, with $z_{0}$ also $-z_{0}$ is a
root. In order to recover a solution $(x,y)$ from the positive non-trivial
solution $z_{0}$, note that

$\begin{split}V_{k,n}(x,y)=\left\\{\begin{array}[]{lllllll}x=x^{k}\sum\limits_{i=0,2,\dots,k}\binom{k}{i}\frac{2n+1}{2n+1+i}\cdot
2^{\frac{i}{2n+1}}\cdot(\frac{\theta y}{x})^{i}=x^{k}F_{2}(\frac{\theta
y}{x})$$\\\\[17.07164pt]
$$y=x^{k}\sum\limits_{i=1,3,\dots,k-1}\binom{k}{i}\frac{2n+1}{2n+2+i}\cdot
2^{\frac{i-1}{2n+1}}(\frac{\theta y}{x})^{i}=x^{k}F_{1}(\frac{\theta
y}{x})\end{array}\right.\end{split}$

and hence $x_{0}=F_{2}(z_{0})^{1/(1-k)}>0$ and
$y_{0}=F_{1}(z_{0})F_{2}(z_{0})^{k/(1-k)}>0$ solve the $2$-dimensional
equation. Note that $(x_{0},y_{0})$ is the only solution with $\theta
y_{0}/x_{0}=z_{0}$. Indeed, any other such solution would be $x_{1}=cx_{0}$
and $y_{1}=cy_{0}$ for some $c\in\mathbb{R}\setminus\\{0\\}$, but plugging
this into the first line of the above equation gives $c=c^{k}$ which is true
iff $c=1$ for even $k$.

Further, note that $F_{2}(-z_{0})^{1/(1-k)}=F_{2}(z_{0})^{1/(1-k)}=x_{0}$ and
$F_{1}(-z_{0})F_{2}(-z_{0})^{k/(1-k)}=-F_{1}(z_{0})F_{2}(z_{0})^{k/(1-k)}=-y_{0}$
and hence also $(x_{0},-y_{0})$ is a solution to the $2$-dimensional fixed
point equation. Using similar arguments one can show that this is the only
fixed point with $-\theta y_{0}/x_{0}=-z_{0}$.

For odd $k$ and $\theta\geq 0$ we have fixed points $(0,0)$, $(1,0)$ and
$(-1,0)$. For the additional fixed point, the calculations are analogous, but
without the leading term $z^{k+1}$, yielding again to fixed point $z_{0}$ and
$-z_{0}$ for $\theta>\theta_{k,1}$. In contrast to the case for even $k$, for
odd $k$, both $(x_{0},y_{0})$ and $(-x_{0},-y_{0})$ are $2$-dimensional fixed
points corresponding to $z_{0}$. Finally, the fixed points $(x_{0},-y_{0})$
and $(-x_{0},y_{0})$ corresponds to $-z_{0}$. The complete list of fixed
points is recorded in Table 1.

For $(\pm x_{0},\pm y_{0})$ to give rise to a positive solution, by the form
of solutions $\varphi$ we must have that for all $t\in[0,1]$

$\pm x_{o}\pm y_{o}\theta\sqrt[2n+1]{4(t-1/2)}>0.$

Clearly, for $-x_{o}$, in $t=1/2$, the inequality is violated and it suffices
to consider the points $(x_{0},\pm y_{0})$. By monotonicity in $t$, it
suffices to show that

$2^{-1/(2n+1)}>\theta y_{o}/x_{o}=z_{o}$ (4.7)

for the positive solution of the polynomial $P$ from (4.6). Since, the sign
change in the polynomial must be from minus to plus, we need to determine its
sign in $2^{-1/m}$ where we put $m=2n+1$. We show that indeed $P(2^{-1/m})>0$
which implies that (4.7) is satisfied and thus $(x_{o},\pm y_{o})$ correspond
to positive solutions. Note that

$\begin{split}P(2^{-1/m})&=2^{-1/m}\sum_{i=1,3,\dots,k+1}\binom{k}{i-1}\frac{m}{m+i-1}-\theta\sum_{i=1,3,\dots,k-1}\binom{k}{i}\frac{m}{m+1+i}>0\end{split}$

is implied by

$\begin{split}\sum_{i=1,3,\dots,k+1}\binom{k}{i-1}\frac{1}{m+i-1}-\sum_{i=1,3,\dots,k-1}\binom{k}{i}\frac{1}{m+1+i}>0.\end{split}$

We can further bound the left hand side from below by

$\begin{split}\sum_{i=0}^{k}(-1)^{i}\binom{k}{i}\frac{1}{m+1+i}=\frac{k!m!}{(m+1+k)!}\end{split}$

which is positive for all $m,k$. This completes the proof. ∎

##  5\. Acknowledgement

Golibjon Botirov thanks the DAAD program for the financial support and the
Weierstrass Institute Berlin for its hospitality. Benedikt Jahnel thanks the
Leibniz program ’Probabilistic methods for mobile ad-hoc networks’ for the
support.

## References

  * [Ba82] Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic, London (1982).
  * [Bo17] Botirov G.I.: A model with uncountable set of spin values on a Cayley tree: phase transitions. Positivity, Vol. 21, No. 3, 955-961 (2017).
  * [EsHaRo12] Eshkabilov Yu.Kh., Haydarov F.H. and Rozikov U.A.: Non-uniqueness of Gibbs measure for models with uncountable set of spin values on a Cayley tree. Journal of Statistical Physics, Vol. 147, No. 4, 779-794 (2012).
  * [EsRo10] Eshkabilov Yu.Kh. and Rozikov U.A.: On models with uncountable set of spin values on a Cayley tree: Integral equations. Mathematical Physics, Analysis and Geometry, Vol. 13, 275-286 (2010).
  * [BoEsRo13] Botirov G.I., Eshkabilov Yu.Kh and Rozikov U.A.: Phase transition for a model with uncountable set of spin values on Cayley tree. Lobachevskii Journal of Mathematics, Vol. 34, No. 3, 256-263 (2013).
  * [Ga04] Ganikhodjaev N.N.: Potts model on $\mathbb{Z}^{d}$ with countable set of spin values. Journal of Mathematical Physics, Vol. 45, 1121-1127 (2004).
  * [GaRo06] Ganikhodjaev N.N. and Rozikov U.A.: The Potts model with countable set of spin values on a Cayley tree. Letters in Mathematical Physics, Vol. 74, 99-109 (2006).
  * [Ge11] H.-O. Georgii: Gibbs measures and phase transitions, New York: De Gruyter, (2011).
  * [JaKuBo14] Jahnel B., Külske C. and Botirov G.I.: Phase transition and critical values of a nearest-neighbor system with uncountable local state space on Cayley trees. Mathematical Physics, Analysis and Geometry, Vol. 17, 323-331 (2014).
  * [Kr64] Krasnosel’skii M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Macmillan (1964).
  * [KrZa84] Krasnosel’skii M.A. and Zabreiko P.P.: Geometrical Methods of Nonlinear Analysis. Springer-Verlag (1984).
  * [Ro13] Rozikov U.A.: Gibbs measures on Cayley trees. World Scientific (2013).


$\mathcal{P}^{\rm tot}=\left\\{\forall x\in[n],\>\>|\mathcal{P}_{x}|\leq
n^{\varepsilon/2}t\right\\}$

and note that

$\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}\left(\mathcal{P}^{\rm
tot}\right)\geq
1-n\mathsf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}\left({\rm
Poisson}(t)>n^{\varepsilon/2}t\right)\geq 1-n^{-a},\qquad\text{for all }a>0,$

where the last inequality holds for all $n$ large enough. Therefore it is
enough to show that, for all $a>0$,

$\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{K_{n}}\nu}\left(\mathcal{E}_{t,\gamma}\cap\mathcal{P}^{\rm
tot}\right)\leq n^{-a}\qquad whp.$

Clearly,

$\displaystyle\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{K_{n}}\nu}\left(\mathcal{E}_{t,\gamma}\cap\mathcal{P}^{\rm
tot}\right)$ $\displaystyle\leq\sum_{I\subset A\colon\,|I|\geq\gamma
K_{n}}\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{K_{n}}\nu}\left(\max_{e\in
I}\min_{f\in A\setminus\\{e\\}}\tau^{e,f}\leq t,\>\mathcal{P}^{\rm
tot}\right)$ (5.10) $\displaystyle\leq\sum_{C=\gamma
K_{n}}^{K_{n}}\binom{K_{n}}{C}\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\max_{e\in
I}\min_{f\in I\setminus\\{e\\}}\tau^{e,f}\leq t,\>\mathcal{P}^{\rm
tot}\right),$

where in the second line, due to the exchangeability of the edges in $A$ and
with a slight abuse of notation, we can take for $I$ the set of the first $C$
edges of $A$. Given $I$, associate to it one possible collection of first
interactions between its vertices. Call this set

$B_{I}=\\{(1,j_{1}),\dots,(C,j_{C})\\},$

where $(\ell,j_{\ell})$ stands for the event that the first edge with which
the edge $e_{\ell}$ interacts is $e_{j_{\ell}}$.

2. For every choice of $B_{I}$, we denote by $\mathcal{E}_{t}(I,B_{I})$ the event $\mathcal{E}_{t,\gamma}$ in which the set of interacting edges is $I$ and $B_{I}=\\{(1,j_{1}),\dots,(C,j_{C})\\}$ is the set of first interactions. Then

$\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\mathcal{E}_{t}(I,B_{I})\cap\mathcal{P}^{\rm
tot}\right)\leq\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau^{e_{1},e_{j_{1}}}\leq
t,\mathcal{P}^{\rm
tot}(1)\right)\prod_{\ell=2}^{C}\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau^{e_{\ell},e_{j_{\ell}}}\leq
t,\>\mathcal{P}^{\rm tot}(\ell)\mid\mathcal{F}_{\ell-1}\right),$ (5.11)

where, for $\ell\in\\{1,\dots,C\\}$, we put

$\mathcal{P}^{\rm tot}(\ell):=\\{|\mathcal{P}_{e_{\ell}^{-}}|\leq
n^{\varepsilon/2}t\\}\cap\\{|\mathcal{P}_{e_{\ell}^{+}}|\leq
n^{\varepsilon/2}t\\}\cap\\{|\mathcal{P}_{e_{j_{\ell}}^{-}}|\leq
n^{\varepsilon/2}t\\}\cap\\{|\mathcal{P}_{e_{j_{\ell}}^{+}}|\leq
n^{\varepsilon/2}t\\}$

and

$\mathcal{F}_{\ell}:=\cap_{\iota\leq\ell}\\{\tau^{e_{\iota},e_{j_{\iota}}}\leq
t\\}\cap\mathcal{P}^{\rm tot}(\iota),\qquad\ell=\\{2,\dots,C\\}.$

By construction, regardless of the specific ordering of the elements in the
set $B_{I}$, there is a collection of at least $C/2$ indices
$\ell\in\\{1,\dots,C\\}$ such that

$|\Xi_{\ell}|:=\big{|}\\{e_{\ell},e_{j_{\ell}}\\}\cap\\{e_{1},e_{j_{1}},\dots,e_{{\ell-1}},e_{j_{\ell-1}}\\}\big{|}\leq
1.$

We claim that, for every $\ell\in\\{1,\dots,C\\}$ for which $|\Xi_{\ell}|=0$,

$\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau^{e_{\ell},e_{j_{\ell}}}\leq
t\mid\mathcal{F}_{\ell-1}\right)\leq\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\nu\otimes\nu}\left(\tau^{e,f}\leq
t\right)+\frac{4C}{n}=O\left(\frac{C+t}{n}\right).$ (5.12)

Indeed,

$\displaystyle\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau^{e_{\ell},e_{j_{\ell}}}\leq
t\mid\mathcal{F}_{\ell-1}\right)$
$\displaystyle\leq\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau^{e_{\ell},e_{j_{\ell}}}\leq
t,\\{e_{\ell}^{-},e_{\ell}^{+},e_{j_{\ell}}^{-},e_{j_{\ell}}^{+}\\}\cap\cup_{\iota<\ell}\\{e_{\iota}^{-},e_{\iota}^{+},e_{j_{\iota}}^{-},e_{j_{\iota}}^{+}\\}=\emptyset\mid\mathcal{F}_{\ell-1}\right)+\frac{4\ell}{n}$
$\displaystyle=\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{K_{n}}\nu}\left(\tau^{e_{i_{\ell}},e_{j_{\ell}}}\leq
t,\\{e_{\ell}^{-},e_{\ell}^{+},e_{j_{\ell}}^{-},e_{j_{\ell}}^{+}\\}\cap\cup_{\iota<\ell}\\{e_{\iota}^{-},e_{\iota}^{+},e_{j_{\iota}}^{-},e_{j_{\iota}}^{+}\\}=\emptyset\right)+\frac{4\ell}{n}$
$\displaystyle\leq\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{K_{n}}\nu}\left(\tau^{e_{i_{\ell}},e_{j_{\ell}}}\leq
t\right)+\frac{4C}{n}.$

Therefore, by (5.5), we conclude that, under $|\Xi_{\ell}|=0$,

$\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau^{e_{\ell},e_{j_{\ell}}}\leq
t\mid\mathcal{F}_{\ell-1}\right)\leq
4\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\pi\otimes\pi}(\tau_{\rm
meet}^{x,y}\leq t)+\frac{4C}{n},$ (5.13)

and (5.12) follows by inserting (5.6) into (5.13).

3. Assume now that for some $\ell\in\\{1,\dots,C\\}$ we have $|\Xi_{\ell}|=1$. More precisely, under $\mathcal{F}_{\ell-1}\cap\left\\{|\Xi_{\ell}|=1\right\\}$, one among $e_{\ell}$ and $e_{j_{\ell}}$ is still unsampled at step $\ell$. Without loss of generality, assume that $e_{\ell}$ is such an unsampled random edge. By the same argument used to derive (5.13), and putting

$\mathcal{J}_{\ell}:=\left\\{\\{e_{\ell}^{-},e_{\ell}^{+}\\}\bigcap\cup_{\iota<\ell}\\{e_{\iota}^{-},e_{\iota}^{+},e_{j_{\iota}}^{-},e_{j_{\iota}}^{+}\\}=\emptyset\right\\},$

we bound

$\begin{split}\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau^{e_{\ell},e_{j_{\ell}}}\leq
t,\mathcal{P}^{\rm tot}(\ell)\mid\mathcal{F}_{\ell-1}\right)&\leq
2\>\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau^{e_{\ell},e_{j_{\ell}}}\leq
t,\mathcal{P}^{\rm
tot}(\ell),\mathcal{J}_{\ell}\mid\mathcal{F}_{\ell-1}\right)+\frac{8C}{n}.\end{split}$

Arguing as in (Oliveira,, 2013, Proposition 4.5), we get that, under
$|\Xi_{\ell}|=1$ with $e_{\ell}$ unsampled at step $\ell$,

$\displaystyle\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau^{e_{\ell},e_{j_{\ell}}}\leq
t,\mathcal{P}^{\rm
tot}(\ell),\mathcal{J}_{\ell}\>\big{\rvert}\>\mathcal{F}_{\ell-1}\right)$
(5.14)
$\displaystyle\leq\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau^{e_{\ell},e_{j_{\ell}}}\leq
t\>\big{\rvert}\>\mathcal{F}_{\ell-1},\>\mathcal{J}_{\ell},\>\mathcal{P}^{\rm
tot}(\ell)\right)$
$\displaystyle=\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau_{\rm
meet}^{e_{\ell}^{-},e_{j_{\ell}}^{-}}\wedge\tau_{\rm
meet}^{e_{\ell}^{+},e_{j_{\ell}}^{-}}\wedge\tau_{\rm
meet}^{e_{\ell}^{-},e_{j_{\ell}}^{+}}\wedge\tau_{\rm
meet}^{e_{\ell}^{+},e_{j_{\ell}}^{+}}\leq
t\>\big{\rvert}\>\mathcal{F}_{\ell-1},\>\mathcal{J}_{\ell},\>\mathcal{P}^{\rm
tot}(\ell)\right)$ $\displaystyle\leq
4\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau_{\rm
meet}^{e_{\ell}^{-},e_{j_{\ell}}^{-}}\leq
t\>\big{\rvert}\>\mathcal{F}_{\ell-1},\>\mathcal{J}_{\ell},\>\mathcal{P}^{\rm
tot}(\ell)\right)$
$\displaystyle=4\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}E}}\left[\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau_{\rm
meet}^{e_{\ell}^{-},e_{j_{\ell}}^{-}}\leq
t\>\big{\rvert}\>\mathcal{F}_{\ell-1},\>\mathcal{J}_{\ell},\>\mathcal{P}^{\rm
tot}(\ell),\>\mathcal{P}_{e_{\ell}^{-}},\>\mathcal{P}_{e_{j_{\ell}}^{-}}\right)\bigg{\rvert}\mathcal{F}_{\ell-1},\>\mathcal{J}_{\ell},\>\mathcal{P}^{\rm
tot}(\ell)\right],$

where the second inequality follows from the equality of the marginals. Note
that, conditionally on $\mathcal{F}_{\ell-1}$ and $\mathcal{J}_{\ell}$, for
all $s\leq t$ the position of the random walk starting at $e_{\ell}^{-}$ is
distributed according to $\pi$. On the other hand, the position of the random
walk starting at $e_{j_{\ell}}^{-}$ at time $s$ is distributed in a non-
trivial way. Moreover, the positions of the two random walks at time $s$ are
independent. Hence

$\begin{split}&\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau_{\rm
meet}^{e_{\ell}^{-},e_{j_{\ell}}^{-}}\leq
t\>\big{\rvert}\>\mathcal{F}_{\ell-1},\>\mathcal{J}_{\ell},\>\mathcal{P}^{\rm
tot}(\ell),\>\mathcal{P}_{e_{\ell}^{-}},\>\mathcal{P}_{e_{j_{\ell}}^{-}}\right)\\\
&\leq\sum_{s\in\mathcal{P}_{e^{-}_{\ell}}\cup\mathcal{P}_{e^{-}_{j_{\ell}}}}\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(X^{(e_{\ell}^{-})}_{s}=X_{s}^{(e_{j_{\ell}}^{-})}\>\big{\rvert}\>\mathcal{F}_{\ell-1},\>\mathcal{J}_{\ell},\>\mathcal{P}^{\rm
tot}(\ell),\>\mathcal{P}_{e_{\ell}^{-}},\>\mathcal{P}_{e_{j_{\ell}}^{-}}\right)\\\
&\leq\sum_{s\in\mathcal{P}_{e^{-}_{\ell}}\cup\mathcal{P}_{e^{-}_{j_{\ell}}}}\max_{\lambda}\sum_{x\in[n]}\lambda(x)\pi(x)\leq\frac{|\mathcal{P}_{e^{-}_{\ell}}\cup\mathcal{P}_{e^{-}_{j_{\ell}}}|}{n}.\end{split}$
(5.15)

Inserting (5.15) into the expectation in (5.14), we obtain

$\begin{split}&\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\tau^{e_{i_{\ell}},e_{j_{\ell}}}\leq
t,\mathcal{P}^{\rm
tot}(\ell),\mathcal{J}_{\ell}\>\big{\rvert}\>\mathcal{F}_{\ell-1},\>|\Xi_{\ell}|=1\right)\leq\frac{8n^{\varepsilon/2}t}{n}.\end{split}$
(5.16)

Inserting (5.13)–(5.16) into (5.11), we obtain

$\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{C}\nu}\left(\mathcal{E}_{t}(I,B_{I})\cap\mathcal{P}^{\rm
tot}\right)\leq\left(\frac{8(n^{\varepsilon/2}t+C)}{n}\right)^{C/2}\leq\left(\sqrt{\frac{8(n^{\varepsilon/2}t+K)}{n}}\right)^{C}\eqqcolon\Gamma^{C}.$
(5.17)

4. By (5.10), (5.17) and the fact that there are $(C-1)^{C}$ different ways to select the set $B_{I}$, we are now ready to conclude that whp

$\displaystyle\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{K_{n}}\nu}\left(\mathcal{E}_{t,\gamma}\cap\mathcal{P}^{\rm
tot}\right)$ $\displaystyle\leq\sum_{C=\gamma
K_{n}}^{K_{n}}\binom{K_{n}}{C}C^{C}\Gamma^{C}\leq\sum_{C=\gamma
K_{n}}^{K_{n}}(K_{n}^{2}\Gamma)^{C}.$

Since, by our choice of $t$ and $K_{n}$,

$K_{n}^{2}\Gamma\leq n^{-\varepsilon/5}$

for all $n$ large enough, we obtain

$\displaystyle\mathbf{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}P}}_{\otimes_{K_{n}}\nu}\left(\mathcal{E}_{t,\gamma}\cap\mathcal{P}^{\rm
tot}\right)$ $\displaystyle\leq K_{n}n^{-\frac{\varepsilon\gamma}{5}K_{n}},$

concluding the proof. ∎

##  Appendix A Auxilliary facts for càdlàg processes

In the next lemma, for an arbitrary $T>0$, we let
$[0,T]\times\mathcal{D}[0,T]$ denote the product of $[0,T]$ with the càdlàg
space $\mathcal{D}[0,T]$ endowed with the $J_{1}$-Skorohod topology.

######  Lemma A.1.

Consider the function $h\colon\,[0,T]\times\mathcal{D}[0,T]\to\mathbb{R}$
defined via $h(s,\phi)=\phi(s)$. Every point in $[0,T]\times\mathcal{C}[0,T]$
is a continuity point of $h$.

###### Proof.

Assume that $(s_{n},\phi_{n})$ is a sequence that converges to $(s,\phi)$,
with $\phi\in\mathcal{C}[0,T]$. We need to verify that $h(s_{n},\phi_{n})$
converges to $h(s,\phi)$. Indeed, if $\phi_{n}$ converges to $\phi$, then
there exists a sequence of increasing functions $\lambda_{n}:[0,T]\to[0,T]$
with $\lambda_{n}(0)=0$ and $\lambda_{n}(T)=T$ such that $\lambda_{n}(u)\to u$
and $\phi_{n}(u)-\phi(\lambda_{n}(u))\to 0$ uniformly over $u\in[0,T]$. From
this we obtain

$||\phi_{n}-\phi||_{\infty}\leq||\phi_{n}-\phi\circ\lambda_{n}||_{\infty}+||\phi\circ\lambda_{n}-\phi||_{\infty}\to
0,$

since any function in $\mathcal{C}[0,T]$ is uniformly continuous. In
particular, $\phi_{n}$ converges to $\phi$ uniformly, which readily implies
that $\phi_{n}(s_{n})\to\phi(s)$ and concludes the proof of the lemma. ∎

######  Proposition A.2.

Assume that $(X^{n}_{u})_{u\in[0,T]}$ is a sequence of càdlàg processes that
converge in distribution to a process $(X_{u})_{u\in[0,T]}$ that has almost
surely continuous trajectories. For any sequence $t_{n}\to t\in[0,T]$,
$X^{n}_{t_{n}}$ converges in distribution to $X_{t}$.

###### Proof.

From the assumptions of the proposition, it follows that
$((X^{n}_{u})_{u\in[0,T]},t_{n})_{n\in\mathbb{N}}$ converges in distribution
to $((X_{u})_{u\in[0,T]},t)$. Let
$h\colon\,[0,T]\times\mathcal{D}[0,T]\to\mathbb{R}$ as in Lemma A.1. Note
that, since $(X_{u})_{u\in[0,T]}$ is almost surely continuous, it follows that
$X^{n}_{t_{n}}=h(t_{n},(X^{n}_{u})_{u\in[0,T]})$ converges in distribution to
$X_{t}=h(t,(X_{u})_{u\in[0,T]})$. ∎

## References

  * Aldous and Brown, (1992) Aldous, D. J. and Brown, M. (1992).  Inequalities for rare events in time-reversible Markov chains I.  Lecture Notes-Monograph Series, pages 1–16.
  * Chen, (2021) Chen, Y.-T. (2021).  Precise asymptotics of some meeting times arising from the voter model on large random regular graphs.  Electronic Communications in Probability, 26:1–13.
  * Chen et al., (2016) Chen, Y.-T., Choi, J., and Cox, J. T. (2016).  On the convergence of densities of finite voter models to the Wright–Fisher diffusion.  Annales de l’Institut Henri Poincaré, 52(1):286–322.
  * Clifford and Sudbury, (1973) Clifford, P. and Sudbury, A. (1973).  A model for spatial conflict.  Biometrika, 60(3):581–588.
  * Cooper et al., (2010) Cooper, C., Frieze, A., and Radzik, T. (2010).  Multiple random walks in random regular graphs.  SIAM Journal on Discrete Mathematics, 23(4):1738–1761.
  * Cooper and Frieze, (2005) Cooper, C. and Frieze, A. M. (2005).  The cover time of random regular graphs.  SIAM Journal on Discrete Mathematics, 18(4):728–740.
  * Cox, (1989) Cox, J. (1989).  Coalescing random walks and voter model consensus times on the torus in zd.  Annals of Probabability, 17(4):1333–1366.
  * Cox and Greven, (1990) Cox, J. and Greven, A. (1990).  On the long term behaviour of some finite particle systems.  Probababilty Theory and Related Fields, 85:195–237.
  * Cox and Greven, (1994) Cox, J. and Greven, A. (1994).  The finite systems scheme: An abstract theorem and a new example.  CRM Proceedings and Lecture Notes, 5:55–66.
  * Donnelly and Welsh, (1983) Donnelly, P. and Welsh, D. (1983).  Finite particle systems and infection models.  In Mathematical Proceedings of the Cambridge Philosophical Society, volume 94, pages 167–182. Cambridge University Press.
  * Durrett, (2008) Durrett, R. (2008).  Probability Models for DNA Sequence Evolution (2nd. ed.).  Springer, New York.
  * Fernley and Ortgiese, (2019) Fernley, J. and Ortgiese, M. (2019).  Voter models on subcritical inhomogeneous random graphs.  arXiv preprint arXiv:1911.13187.
  * Friedman, (2008) Friedman, J. (2008).  A Proof of Alon’s Second Eigenvalue Conjecture and Related Problems.  American Mathematical Soc.
  * Hermon et al., (2021) Hermon, J., Li, S., Yao, D., and Zhang, L. (2021).  Mean field behavior during the big bang for coalescing random walk.  arXiv preprint arXiv:2105.11585.
  * Holley and Liggett, (1975) Holley, R. A. and Liggett, T. M. (1975).  Ergodic theorems for weakly interacting infinite systems and the voter model.  Annals of Probability, pages 643–663.
  * Levin and Peres, (2017) Levin, D. A. and Peres, Y. (2017).  Markov Chains and Mixing Times.  American Mathematical Society, Providence, RI.
  * Lubetzky and Sly, (2010) Lubetzky, E. and Sly, A. (2010).  Cutoff phenomena for random walks on random regular graphs.  Duke Mathematical Journal, 153(3):475–510.
  * Manzo et al., (2021) Manzo, F., Quattropani, M., and Scoppola, E. (2021).  A probabilistic proof of Cooper & Frieze’s “first visit time lemma”.  ALEA Lat. Am. J. Probab. Math. Stat., 18(2):1739–1758.
  * Oliveira, (2013) Oliveira, R. I. (2013).  Mean field conditions for coalescing random walks.  Annals of Probability, 41(5):3420–3461.
  * Quattropani and Sau, (2022) Quattropani, M. and Sau, F. (2022).  On the meeting of random walks on random DFA.  arXiv preprint arXiv:2204.02827.
  * Sood et al., (2008) Sood, V., Antal, T., and Redner, S. (2008).  Voter models on heterogeneous networks.  Physical Review E, 77(4):041121.