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# Fluctuation theorems in general relativistic
stochastic thermodynamics

Yifan Cai, Tao Wang, and Liu Zhao
School of Physics, Nankai University, Tianjin 300071, China email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>author, email<EMAIL_ADDRESS>
###### Abstract

Based on the recently proposed framework of general relativistic stochastic
mechanics [1, 2] and stochastic thermodynamics [3] at the ensemble level, this
work focuses on general relativistic stochastic thermodynamics at the
trajectory level. The first law of stochastic thermodynamics is reformulated
and the fluctuation theorems are proved on this level, with emphasis on
maintaining fully general covariance and on the choice of observers.

Keywords: Langevin equation, fluctuation theorem, time-reversal symmetry,
general relativity

##  1 Introduction

One of the central problem in modern statistical physics is the origin of
irreversibility in macroscopic and mesoscopic systems. This problem can be
traced back to Boltzmann’s efforts in proving the second law of thermodynamics
starting from deterministic mechanics, which has resulted in the well-known
H-theorem [4]. However, the debates about the validity of (the assumptions of)
the H-theorem have lasted for more than a century. The most acute critics to
the H-theorem is reflected by Loschmidt paradox [5, 6], which roughly states
that the macroscopic arrow of time cannot possibly arise from the underlying
microscopic mechanics obeying time reversal symmetry (TRS). This paradox
remains unresolved until the 1990s, when numerous works [7, 8, 9] emerged,
revealing that the forward and reversed processes are not probabilistically
equally likely, provided certain dissipative effect exists on the mechanical
level. These results, known as fluctuation theorems, largely resolved the
debates and paradoxes related to the H-theorem. In particular, the most
questioned molecular chaos hypothesis adopted in proving the H-theorem is
completely avoided in proving the fluctuation theorems.

Almost at the same time, Sekimoto [10] established the first law of stochastic
thermodynamics on the trajectory level by use of the Langevin equation. This
formulation establishes a connection between stochastic mechanics and
fluctuation theorems. In 2005, Seifert [11] presented a version of fluctuation
theorem based on the overdamped Langevin equation. Subsequently, several
fluctuation theorems based on non-relativistic stochastic mechanics were
presented [12, 13, 14, 15], making it clear that stochastic mechanics provides
an ideal starting point for constructing fluctuation theorems and interpreting
the origin of irreversibility.

Most of the works mentioned above are carried out in the non-relativistic
regime. Nowadays, it is widely acknowledged that thermodynamics and relativity
are both concentrated on the universal principles which every physical system
must obey. It is important to establish fluctuation theorems based on these
universal principle theories. However, since the spacetime symmetry in
relativity imposes stronger protection to the time reversal invariance, the
extension of fluctuation theorems to the relativistic regime proves to be more
difficult. The central difficulty lies in how to incorporate the breaking of
time reversal invariance while still maintaining relativistic covariance.
Refs. [16, 17, 18, 19, 20, 21] considered the extension of fluctuation
theorems to the special relativistic regime. However, the long awaited general
relativistic extension is still beyond our human’s sight.

Recently, we developed a framework for dealing with stochastic mechanics on
curved spacetime and investigated the general relativistic stochastic
thermodynamics based on this framework [1, 2, 3]. Meanwhile, we also
established a version of fluctuation theorem on an arbitrary curved Riemannian
manifold [15]. The aim of the present work is to employ the framework
established in [1, 2, 3] and make use of the technique introduced in [15] to
construct a version of fluctuation theorem based on a fully general
relativistic description of stochastic mechanics and stochastic
thermodynamics.

In our framework of relativistic stochastic mechanics [1, 2], it is important
to liberate the observer from the coordinate system. Rather than fixing the
zeroth component of the coordinate system, we utilize the observer’s proper
time $t$ to to label the configuration space $\mathcal{S}_{t}$ and the space
of microstates $\Sigma_{t}$, enabling our theory to possess general
covariance. For this purpose, we first clarify the geometry of the space of
$\mathcal{S}_{t}$ and $\Sigma_{t}$ in Sec. 2. To make the construction more
self-contained, we provide a brief review of the basics of our framework of
relativistic stochastic mechanics and relativistic stochastic thermodynamics
in Sec. 3, and the first law of relativistic stochastic thermodynamics at
trajectory level is established in this section also. The separation of the
observer from the coordinate system is also important in describing the time
reversal transformation (TRT) in curved spacetime. In contrast to the usual
practice in special relativistic theories (including special relativistic
field theories) in which the TRT is often described as a coordinate
transformation $(t,x^{i})\mapsto(-t,x^{i})$, our general covariant framework
calls for interpreting the TRT as a transformation from a future-directed
observer to a past-directed observer. This concept is elucidated in detail in
Sec. 4. Ref.[1] presents a method to establish the covariant relativistic
Langevin equation from the perspective of the observer. Based on this
approach, the forward and reversed processes are respectively defined as
stochastic processes from the perspectives of the future-directed and past-
directed observers. In Sec. 5, we demonstrate that the forward and reversed
processes possess distinct probabilities, thus breaking the TRS and leading to
a version of fluctuation theorem on curved spacetime manifold. Finally, in
Sec. 6, we provide brief concluding remarks.

We maintain the notations and conventions consistent with the previous works
[1, 2]. To distinguish random variables from their realizations, the former
are labeled with extra tildes. For instance, $\tilde{x}$ represents a random
variable, while $x$ denotes its realization. Several manifolds of different
dimensions will be relevant in our discussion. These include the
$(d+1)$-dimensional spacetime manifold $\mathcal{M}$ with metric
$g_{\mu\nu}(x)$ of signature $(-,+,\cdots,+)$, its tangent bundle
$T\mathcal{M}$ of dimension $(2d+2)$, and certain submanifolds within the
tangent bundle. To distinguish tensors on these different manifolds, we
introduce distinct indices. Lower-case Greek letters such as
$\alpha,\beta,\mu,\nu,\rho,\ldots$, are used as concrete indices and range
from $0$ to $d$. Lower-case Latin letters like $i,j,k,l,m,\ldots$ are also
used as concrete indices, which range from $1$ to $d$. Lastly, lower-case
Latin letters $a,b,c,\ldots$ are used as abstract indices. This paper is
intended to be as mathematically rigorous as possible. A more concise summary
of the main results can be found in Ref.[22].

##  2 Geometry of the space of microstates

Statistical physics is built on top of the space of microstates. For systems
consisting of classical massive particles, the space of microstates can be
subdivided into the configuration space and the momentum space. In
relativistic context, the configuration space is a subspace of the spacetime
manifold $\mathcal{M}$ consisting of simultaneous events at a given instance
of time, while the momentum space for each individual particle is a subspace
of the tangent (or cotangent) space of the spacetime at a given event which
obeys the mass shell condition

$\displaystyle\mathcal{H}(x,p):=g_{\mu\nu}(x)p^{\mu}p^{\nu}+m^{2}=0.$ (1)

Please be reminded that, due to the non-degeneracy of the spacetime metric
$g_{\mu\nu}(x)$, the tangent and cotangent spaces are dual to each other and
both are equally well for describing the momentum space. In this work, we
adopt the tangent space description. When cotangent space variables appear,
they are considered to be linear functions of the tangent space variables,
_e.g._ $p_{\mu}=g_{\mu\nu}(x)p^{\nu}$.

In principle, the space of microstates for a single particle should be
considered as a submanifold of the mass shell bundle

$\displaystyle\Gamma_{m}:=\\{(x,p)\in T\mathcal{M}|\mathcal{H}(x,p)=0\\},$ (2)

with the configuration space taken to be a subspace of the spacetime manifold
$\mathcal{M}$ consisting of simultaneous events. To clarify the concept of
simultaneous events, we need to introduce an arbitrary observer field which is
encoded by a normalized timelike vector field $Z^{\mu}$ obeying
$g_{\mu\nu}Z^{\mu}Z^{\nu}=-1$. For convenience, we shall refer to this
arbitrary observer field as Alice. When considering the motion of a Brownian
particle inside a heat reservoir, there is a particular observer field that
comoves with the reservoir. This particular observer field will be referred to
as Bob.

Consider the worldline $x_{\tau}$ of a massive relativistic particle of which
$\tau$ is its proper time. If the time orientations of Alice and the particle
align, _i.e._ $g_{\mu\nu}p^{\mu}_{\tau}Z^{\nu}<0$, where
$p^{\mu}_{\tau}:=m{\rm d}x^{\mu}_{\tau}/{\rm d}\tau$, the part of the mass
shell bundle in which the phase trajectory lies is defined as the future mass
shell bundle relative to Alice and is denoted as $\Gamma^{+}_{m}$,

$\displaystyle\Gamma^{+}_{m}:=\\{(x,p)\in T\mathcal{M}|\mathcal{H}(x,p)=0,\
g_{\mu\nu}p^{\mu}Z^{\nu}<0\\}.$ (3)

For notational convenience, the phase trajectory is denoted as
$X_{\tau}=(x_{\tau},p_{\tau})$, which is the uplift of the particle’s
worldline $x_{\tau}$ into the bundle $\Gamma^{+}_{m}$.

Figure 1: The worldline of a particle is lifted to different regions of the
mass shell bundle by different observers.

If we consider another observer field with proper velocity $C^{\mu}=-Z^{\mu}$,
referred to as Carol, whose time orientation is opposite to that of Alice, the
definition of the future mass shell bundle relative to Carol will be opposite
to that relative to Alice. To avoid confusion, we will designate Alice as the
future-directed observer and Carol as the past-directed observer, and
throughout this paper, the future and past mass shell bundles are always
defined relative to Alice. This designation is arbitrary because, in the
presence of TRS, the future- and the past-directed observer fields are
indistinguishable on the level of deterministic mechanics.

The future mass shell bundle is also a fiber bundle based on $\mathcal{M}$.
Its fiber, denoted by $(\Gamma^{+}_{m})_{x}$, is the momentum space of the
relativistic particle. We use the calligraphy fonts, like $\mathcal{F}$,
$\mathcal{R}$ and $\mathcal{K}$, to denote tensors on $(\Gamma_{m}^{+})_{x}$,
and the cursive fonts, like $\mathscr{N}$, $\mathscr{Z}$ and $\mathscr{L}$, to
denote tensors on $\Gamma_{m}^{+}$.

The future mass shell bundle still is not the space of microstates, because
the base manifold $\mathcal{M}$ is not the configuration space. There are
different ways to drop the temporal dimension of $\mathcal{M}$, e.g. either by
fixing the zeroth component $x^{0}$ of coordinates or by fixing the proper
time $t$ of Alice. The first approach lacks apparent general covariance. The
core idea of our framework for relativistic stochastic mechanics is to keep
the manifest general covariance and emphasize the role of observer choice. The
details can be found in Ref.[1]. Here we list some of the key points:

  1. 1.

The proper time $t$ of Alice can be extended into a scalar field $t(x)$ on
$\mathcal{M}$;

  2. 2.

The configuration space relative to Alice is defined as a constant proper time
slice of $\mathcal{M}$, _i.e._ a spacelike hypersurface
$\mathcal{S}_{t}:=\\{x\in\mathcal{M}|t(x)=t\\}$ in $\mathcal{M}$. Moreover,
the proper velocity field $Z^{\mu}$ of Alice is the unit normal vector field
of $\mathcal{S}_{t}$;

  3. 3.

The space of microstates is defined as a hypersurface in $\Gamma^{+}_{m}$ as a
constant time slice, _i.e._
$\Sigma_{t}^{+}:=\\{(x,p)\in\Gamma^{+}_{m}|t(x)=t\\}$;

  4. 4.

Since the gradient of $t$ must be a normal co-vector of $\mathcal{S}_{t}$,
$Z_{\mu}$ must be collinear with $\nabla_{\mu}t$. Denoting $|\nabla_{\mu}t|$
as $\lambda$, we have $\partial_{\mu}t=-\lambda Z_{\mu}$.

Since $Z$ is the unit future-directed normal vector field of
$\mathcal{S}_{t}$, the volume element on $\mathcal{S}_{t}$ can be written as
the interior derivative of the volume element on $\mathcal{M}$ along
$Z^{\mu}$:

$\displaystyle\eta_{\mathcal{S}_{t}}:=\iota_{Z}\eta_{\mathcal{M}},$ (4)

where $\iota$ represents interior derivative, and
$\eta_{\mathcal{M}}=g^{1/2}{\rm d}x^{0}\wedge{\rm
d}x^{1}\wedge\cdots\wedge{\rm d}x^{d}$ is the volume element on $\mathcal{M}$,
with $g=|{\rm det}(g_{\mu\nu})|$. Since $\iota_{Z}({\rm d}x^{\mu})=Z^{\mu}$
and ${\rm d}x^{0}=-\partial_{i}t{\rm d}x^{i}/\partial_{0}t$ on
$\mathcal{S}_{t}$, we have

$\displaystyle\eta_{\mathcal{S}_{t}}$ $\displaystyle=g^{1/2}\iota_{Z}({\rm
d}x^{0}\wedge{\rm d}x^{1}\wedge\cdots\wedge{\rm d}x^{d})$
$\displaystyle=g^{1/2}\sum_{\rho=0}^{d}(-1)^{\rho}\iota_{Z}({\rm
d}x^{\rho}){\rm d}x^{0}\wedge\cdots\wedge{\rm d}x^{\rho-1}\wedge{\rm
d}x^{\rho+1}\wedge\cdots\wedge{\rm d}x^{d}$ $\displaystyle=g^{1/2}Z^{0}{\rm
d}x^{1}\wedge\cdots\wedge{\rm d}x^{d}-g^{1/2}\sum_{i=1}^{d}Z^{i}{\rm
d}x^{1}\wedge\cdots\wedge{\rm d}x^{0}\wedge\cdots\wedge{\rm d}x^{d}$
$\displaystyle=g^{1/2}\left[Z^{0}+\frac{1}{\partial_{0}t}\partial_{i}tZ^{i}\right]{\rm
d}x^{1}\wedge\cdots\wedge{\rm d}x^{d}$ $\displaystyle=-\frac{\lambda
g^{1/2}}{\partial_{0}t}{\rm d}x^{1}\wedge\cdots\wedge{\rm d}x^{d}.$ (5)

We shall also need to make use of the volume elements on $\Gamma^{+}_{m}$ and
$\Sigma_{t}^{+}$. Since both of them are submanifolds of $T\mathcal{M}$, it is
appropriate to begin from the geometry of $T\mathcal{M}$. The non-degenerate
metric on $T\mathcal{M}$ is known as the Sasaki metric [23] $\hat{g}_{ab}$,
which is determined by the metric $g_{\mu\nu}$ of the base manifold
$\mathcal{M}$,

$\displaystyle\hat{g}_{ab}:=g_{\mu\nu}{\rm d}x^{\mu}{}_{a}{\rm
d}x^{\nu}{}_{b}+g_{\mu\nu}\theta^{\mu}{}_{a}\theta^{\nu}{}_{b},\quad\hat{g}^{ab}:=g^{\mu\nu}e_{\mu}{}^{a}e_{\nu}{}^{b}+g^{\mu\nu}\left(\frac{\partial}{\partial
p^{\mu}}\right)^{a}\left(\frac{\partial}{\partial p^{\nu}}\right)^{b},$ (6)

where

$\theta^{\mu}:={\rm d}p^{\mu}+\varGamma^{\mu}{}_{\alpha\beta}p^{\alpha}{\rm
d}x^{\beta},\quad e_{\mu}:=\frac{\partial}{\partial
x^{\mu}}-\varGamma^{\alpha}{}_{\mu\beta}p^{\beta}\frac{\partial}{\partial
p^{\alpha}},$

and $\varGamma^{\mu}{}_{\alpha\beta}$ is the Christoffel connection associated
with $g_{\mu\nu}$. The corresponding volume element reads

$\displaystyle\eta_{T\mathcal{M}}$ $\displaystyle=g~{}{\rm d}x^{0}\wedge{\rm
d}x^{1}\wedge\cdots\wedge{\rm
d}x^{d}\wedge\theta^{0}\wedge\cdots\wedge\theta^{d}$ $\displaystyle=g~{}{\rm
d}x^{0}\wedge{\rm d}x^{1}\wedge\cdots\wedge{\rm d}x^{d}\wedge{\rm
d}p^{0}\wedge\cdots\wedge{\rm d}p^{d}.$ (7)

As a hypersurface in $T\mathcal{M}$, $\Gamma^{+}_{m}$ has the unit normal
(co)vector

$\displaystyle\hat{N}_{a}=\frac{1}{|{\rm d}\mathcal{H}|}{\rm
d}\mathcal{H}_{a}=\frac{p_{\mu}}{m}\theta^{\mu}{}_{a},\qquad\hat{N}^{a}=\hat{g}^{ab}\hat{N}_{b}=\frac{p^{\mu}}{m}\left(\frac{\partial}{\partial
p^{\mu}}\right)^{a},$ (8)

giving rise to the induced metric

$\displaystyle\hat{h}_{ab}$
$\displaystyle:=\hat{g}_{ab}+\hat{N}_{a}\hat{N}_{b}=g_{\mu\nu}{\rm
d}x^{\mu}{}_{a}{\rm
d}x^{\nu}{}_{b}+\Delta_{\mu\nu}(p)\theta^{\mu}{}_{a}\theta^{\nu}{}_{b},$ (9)
$\displaystyle\Delta_{\mu\nu}(p)$
$\displaystyle=g_{\mu\nu}+\frac{1}{m^{2}}p_{\mu}p_{\nu},$ (10)

where the second term is the induced metric on $(\Gamma^{+}_{m})_{x}$:

$\displaystyle
h_{ab}:=\Delta_{\mu\nu}(p)\theta^{\mu}{}_{a}\theta^{\nu}{}_{b}.$ (11)

Consequently, we get the volume element

$\displaystyle\eta_{\Gamma^{+}_{m}}$
$\displaystyle:=\iota_{\hat{N}}\eta_{T\mathcal{M}}=-\frac{m}{p_{0}}g~{}{\rm
d}x^{0}\wedge\cdots\wedge{\rm d}x^{d}\wedge{\rm d}p^{1}\wedge\cdots\wedge{\rm
d}p^{d}.$ (12)

It is easy to see that the volume element $\eta_{\Gamma^{+}_{m}}$ can be
factorized into the wedge product of $\eta_{\mathcal{M}}$ and
$\eta_{(\Gamma^{+}_{m})_{x}}$,

$\displaystyle\eta_{\Gamma^{+}_{m}}=\eta_{\mathcal{M}}\wedge\eta_{(\Gamma_{m}^{+})_{x}},$
(13)

where

$\displaystyle\eta_{(\Gamma^{+}_{m})_{x}}:=-\frac{m}{p_{0}}g^{1/2}{\rm
d}p^{1}\wedge\cdots\wedge{\rm d}p^{d}$ (14)

is the volume element on the fiber space $(\Gamma^{+}_{m})_{x}$. Finally,
since $\Sigma^{+}_{t}=\bigcup_{x\in\mathcal{S}_{t}}(\Gamma_{m}^{+})_{x}$, the
volume element on $\Sigma^{+}_{t}$ also has a factorized form,

$\displaystyle\eta_{\Sigma^{+}_{t}}=\eta_{\mathcal{S}_{t}}\wedge\eta_{(\Gamma_{m}^{+})_{x}}=\frac{m\lambda
g}{p_{0}\partial_{0}t}{\rm d}x^{1}\wedge\cdots{\rm d}x^{d}\wedge{\rm
d}p^{1}\wedge\cdots\wedge{\rm d}p^{d}.$ (15)

##  3 General relativistic Langevin systems

This section is intended for a brief review of the framework for general
relativistic stochastic mechanics and thermodynamics [1, 2, 3] in order to fix
the notations and make the forthcoming presentation for the proof of
fluctuation theorem self-contained.

###  3.1 Covariant Langevin equations

Let us consider a relativistic Brownian particle carrying an electric charge
$q$ and moving in a heat reservoir residing in the curved spacetime
$\mathcal{M}$ and subjecting to an external electromagnetic field
$F=F_{\mu\nu}{\rm d}x^{\mu}\wedge{\rm d}x^{\nu}$. A version of the
corresponding general covariant Langevin equation (referred to as LEτ) that
takes the proper time $\tau$ of the particle as evolution parameter reads

$\displaystyle{\rm d}\tilde{x}_{\tau}^{\mu}$
$\displaystyle=\frac{\tilde{p}_{\tau}^{\mu}}{m}{\rm d}\tau,$ (16)
$\displaystyle{\rm d}\tilde{p}_{\tau}^{\mu}$
$\displaystyle=\xi^{\mu}_{\tau}{\rm d}\tau+\mathcal{F}^{\mu}_{\text{dp}}{\rm
d}\tau+\mathcal{F}^{\mu}_{\text{em}}{\rm
d}\tau-\frac{1}{m}\varGamma^{\mu}{}_{\alpha\beta}\tilde{p}_{\tau}^{\alpha}\tilde{p}_{\tau}^{\beta}{\rm
d}\tau,$ (17)

where $\mathcal{F}^{\mu}_{\text{dp}}:=\mathcal{K}^{\mu\nu}U_{\nu}$ is the
damping force with $\mathcal{K}^{\mu\nu}$ being the damping coefficient which
transforms as a tensor under general coordinate transformations, $U^{\mu}$ is
the proper velocity of the heat reservoir,
$\mathcal{F}^{\mu}_{\text{em}}:=\frac{q}{m}F^{\mu}{}_{\nu}\tilde{p}^{\nu}_{\tau}$
is the electromagnetic force, and

$\displaystyle\xi_{\tau}^{\mu}:=\mathcal{R}^{\mu}{}_{\mathfrak{a}}\circ_{S}{\rm
d}\tilde{w}^{\mathfrak{a}}_{\tau}/{\rm d}\tau+\mathcal{F}_{\text{add}}^{\mu}$
(18)

is the stochastic force, which is consisted of a random force term
$\mathcal{R}^{\mu}{}_{\mathfrak{a}}\circ_{S}{\rm
d}\tilde{w}^{\mathfrak{a}}_{\tau}/{\rm d}\tau$ encoding the Stratonovich type
coupling between the stochastic amplitudes
$\mathcal{R}^{\mu}{}_{\mathfrak{a}}$ with a set of independent Gaussian noises
${\rm d}\tilde{w}^{\mathfrak{a}}_{\tau}$ obeying the probability distribution

$\displaystyle\Pr[{\rm d}\tilde{w}^{\mathfrak{a}}_{\tau}={\rm
d}w^{\mathfrak{a}}]=\frac{1}{(2\pi{\rm
d}\tau)^{d/2}}\exp\left(-\frac{1}{2}\frac{\delta_{\mathfrak{a}\mathfrak{b}}{\rm
d}w^{\mathfrak{a}}{\rm d}w^{\mathfrak{b}}}{{\rm d}\tau}\right),$ (19)

and an additional stochastic force term

$\displaystyle\mathcal{F}_{\text{add}}^{\mu}=\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\mathcal{R}^{\mu}{}_{\mathfrak{a}}\nabla^{(h)}_{i}\mathcal{R}^{i}{}_{\mathfrak{b}}$
(20)

in which $\nabla^{(h)}_{i}$ denotes the spatial components of the covariant
derivative associated to the metric (11) on the momentum space. Both the
random force and the additional stochastic force terms arise from the
interaction of the Brownian particle with the heat reservoir. The additional
stochastic force is required in order for the Brownian particle to be able to
reach equilibrium in the long time limit [24]. Each component of
$\mathcal{R}^{\mu}{}_{\mathfrak{a}}$ is assumed to be smoothly dependent on
the coordinates on $\Sigma_{t}^{+}$, and for each fixed
$\mathfrak{a}=1,2,\cdots d$, $\mathcal{R}^{\mu}{}_{\mathfrak{a}}$ transforms
as a vector under general coordinate transformations. As did in Refs. [1, 2],
we use tilded and un-tilded symbols to denote the random variables and their
realizations.

Although LEτ is perfectly general covariant and encodes all necessary factors
that affect the motion of the Brownian particle, there is still some drawbacks
which call for an alternative version of the covariant Langevin equation. The
problem is connected to the choice of evolution parameter $\tau$. Since

$\displaystyle{\rm d}t=\partial_{\mu}t{\rm d}\tilde{x}^{\mu}=-\lambda
Z_{\mu}{\rm d}\tilde{x}^{\mu}=-\lambda Z_{\mu}\frac{{\rm
d}\tilde{x}^{\mu}}{{\rm d}\tau}{\rm
d}\tau=-\lambda\frac{Z_{\mu}\tilde{p}^{\mu}}{m}{\rm
d}\tau=\gamma(\tilde{x},\tilde{p}){\rm d}\tau,$ (21)

we have ${\rm d}\tau=\gamma^{-1}(\tilde{x},\tilde{p}){\rm d}t$. Therefore,
from the perspective of the observer Alice, the proper time $\tau$ of the
Brownian particle becomes a random variable. To avoid this inconvenience, a
reparametrization scheme is adopted in [1], with

$\tilde{X}_{\tau}=(\tilde{x}_{\tau},\tilde{p}_{\tau})~{}\mapsto~{}\tilde{Y}_{t}=(\tilde{y}_{t},\tilde{k}_{t}),$

where

$\tilde{y}_{t}:=\tilde{x}_{\tilde{\tau}_{t}},\quad\tilde{k}_{t}:=\tilde{p}_{\tilde{\tau}_{t}}.$

This leads to the following alternative version of covariant Langevin equation
which is referred to as LEt for short,

$\displaystyle{\rm d}\tilde{y}_{t}^{\mu}$
$\displaystyle=\frac{\tilde{k}^{\mu}_{t}}{m}\gamma^{-1}{\rm d}t,$ (22)
$\displaystyle{\rm d}\tilde{k}_{t}^{\mu}$
$\displaystyle=\hat{\xi}_{t}^{\mu}\gamma^{-1}{\rm
d}t+\mathcal{F}^{\mu}_{\text{dp}}\gamma^{-1}{\rm
d}t+\mathcal{F}^{\mu}_{\text{em}}\gamma^{-1}{\rm
d}t-\frac{1}{m}\varGamma^{\mu}{}_{\alpha\beta}\tilde{k}^{\alpha}_{t}\tilde{k}^{\beta}_{t}\gamma^{-1}{\rm
d}t.$ (23)

The new stochastic force $\hat{\xi}^{\mu}_{t}$ reads

$\displaystyle\hat{\xi}^{\mu}_{t}:=\gamma^{1/2}\mathcal{R}^{\mu}{}_{\mathfrak{a}}\circ_{S}{\rm
d}\tilde{W}_{t}^{\mathfrak{a}}/{\rm
d}t+\mathcal{F}^{\mu}_{\text{add}}-\frac{1}{2}\mathcal{D}^{\mu
i}\gamma^{1/2}\nabla^{(h)}_{i}\gamma^{-1/2},$ (24)

in which

$\displaystyle{\rm
d}\tilde{W}^{\mathfrak{a}}_{t}=\gamma^{1/2}(\tilde{Y}_{t}){\rm
d}\tilde{w}^{\mathfrak{a}}_{\tilde{\tau}_{t}}$

are still Gaussian noises but with the variance changed from ${\rm d}\tau$ to
${\rm d}t$, and
$\mathcal{D}^{\mu\nu}:=\mathcal{R}^{\mu}{}_{\mathfrak{a}}\delta^{\mathfrak{a}\mathfrak{b}}\mathcal{R}^{\nu}{}_{\mathfrak{b}}$
is the diffusion tensor.

In this work, we assume that the diffusion tensor has rank $d$, so that
$\mathcal{D}^{ij}$ is a full-rank symmetric matrix. This also requires that
$\mathcal{R}^{i}{}_{\mathfrak{a}}$ is a $d\times d$ full-rank matrix. The
above assumption is necessary and sufficient to ensure that the Brownian
particle couples to the heat reservoir in every spatial direction. Please keep
in mind that the choice for the stochastic amplitudes
$\mathcal{R}^{\mu}{}_{\mathfrak{a}}$ is non-unique. Different choices
correspond to different Langevin systems. The result of the present work does
not require the explicit values for the stochastic amplitudes and should be
valid for any choices obeying the above assumption.

###  3.2 Reduced Fokker-Planck equation

Using the diffusion operator method [27], the reduced Fokker-Planck equation
(RFPE) associated with LEτ or LEt is obtained in Ref. [2],

$\displaystyle\dfrac{1}{m}\mathscr{L}_{F}(\varphi)=\nabla_{i}^{(h)}\mathcal{I}^{i}[\varphi],$
(25)

where

$\displaystyle\mathscr{L}_{F}:=p^{\mu}e_{\mu}+qF^{\mu}{}_{\nu}p^{\nu}\frac{\partial}{\partial
p^{\mu}}$ (26)

is the Liouville vector field for a charged particle,

$\displaystyle\mathcal{I}[\varphi]=\left[\frac{1}{2}\mathcal{D}^{ij}\nabla^{(h)}_{j}\varphi-\mathcal{K}^{i\nu}U_{\nu}\varphi\right]\frac{\partial}{\partial\breve{p}^{i}}$
(27)

is a vector field which is connected to the heat transfer rate from the heat
reservoir to the Brownian particle via [1, 3]

$\displaystyle
Q[\varphi]:=\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}Z_{\nu}\mathcal{I}^{\nu}[\varphi],$
(28)

and the definition for the derivative operator
$\displaystyle\frac{\partial}{\partial\breve{p}^{i}}$ is provided in Appendix
A. The round and square brackets around $\varphi$ have different meanings:
$\mathscr{L}_{F}(\varphi)$ represents the action of the vector field
$\mathscr{L}_{F}$ on the scalar $\varphi$, while $\mathcal{I}^{i}[\varphi]$
implies that the vector field $\mathcal{I}^{i}$ is dependent on $\varphi$.
Such convention will be used throughout this paper.

It is important to point out that the one particle distribution function
(1PDF) $\varphi$ appearing in the RFPE is not a probability distribution in
$\Sigma^{+}_{m}$. To see this, we recall that the probability current
associated with the RFPE (25) is

$\displaystyle\mathscr{J}[\varphi]=\frac{\varphi}{m}\mathscr{L}_{F}-\mathcal{I}[\varphi].$
(29)

Therefore, the probability distribution function on $\Sigma^{+}_{t}$ should be

$\displaystyle
f:=-{\mathscr{Z}_{a}}\mathscr{J}^{a}[\varphi]=-\frac{1}{m}p^{\mu}Z_{\mu}\varphi=\gamma\lambda^{-1}\varphi,$
(30)

where $\mathscr{Z}=Z^{\mu}e_{\mu}$ is the unit normal vector of
$\Sigma_{t}^{+}$.

Since $\mathcal{I}[\varphi]$ is proportional to the heat transfer rate, the
condition for the Brownian particle to reach detailed thermal equilibrium with
the reservoir is $\mathcal{I}[\varphi_{\text{eq}}]=0$, which yields the
equilibrium distribution

$\displaystyle\varphi_{\text{eq}}=\operatorname{e}^{-\alpha+\beta_{\mu}p^{\mu}},\quad\beta_{\mu}:=\beta
U_{\mu},$ (31)

provided that the covariant Einstein relation

$\displaystyle\mathcal{D}^{\mu\nu}=2\beta^{-1}\mathcal{K}^{\mu\nu}$ (32)

holds and that $\alpha$ and $\beta_{\mu}$ obey the following equations

$\displaystyle\nabla_{\mu}\alpha+q\beta^{\nu}F_{\mu\nu}=0,\qquad\nabla_{(\mu}\beta_{\nu)}=0,$
(33)

which are simple consequences of the Liouville equation
$\mathscr{L}_{F}(\varphi_{\text{eq}})=0$. Eq. (33) implies that $\beta_{\mu}$
is a Killing vector field, while the relation $\beta_{\mu}=\beta U_{\mu}$
implies that it is timelike. Therefore, $\beta_{\mu}$ must be timelike
Killing. This leads to the conclusion that the existence of the equilibrium
distribution (31) requires the spacetime to be at least stationary.

As we have argued in Ref. [2], the equilibrium state is intrinsic to the
system which is independent of the choice of observer. However, the parameters
that characterize the equilibrium state is indeed observer-dependent. It has
been shown [29, 2] that the parameters $\alpha$ and $\beta$ are related to the
chemical potential and the temperature observed by Bob via

$\displaystyle\beta=\frac{1}{T_{\rm B}},\qquad\alpha=-\frac{\mu_{\rm
B}}{T_{\rm B}}.$ (34)

The equilibrium distribution (31) is recognized to be precisely the Jütnner
distribution which is also obeyed by particles of the heat reservoir.

###  3.3 Thermodynamic relations

The definition of the energy of a charged relativistic particle is non-unique.
For instance, both the kinematic momentum $p^{\mu}$ and canonical momentum
$P^{\mu}:=p^{\mu}+qA^{\mu}$ can be used for defining the energy [25]:

$\displaystyle E:=-Z_{\mu}p^{\mu},\qquad H:=-Z_{\mu}P^{\mu}.$ (35)

As an analogy of non-relativistic case, $E$ can be viewed as kinematic energy
and $H$ can be viewed as the sum of kinematic energy and electromagnetic
potential energy $-qZ_{\mu}A^{\mu}$. The non-uniqueness for the definition of
energy also appeared in the non-relativistic stochastic thermodynamics [15].
However, such non-uniqueness does not affect the description of heat in the
first law of stochastic thermodynamics. We will show that the same situation
also occurs the relativistic case.

Since the microstate of the Brownian particle is described by a set of random
variables, the energy of the Brownian particle also depends on the same set of
random variables,

$\displaystyle\tilde{E}_{\tau}:=E(\tilde{x}_{\tau},\tilde{p}_{\tau}),\qquad\tilde{H}_{\tau}:=H(\tilde{x}_{\tau},\tilde{p}_{\tau}).$
(36)

Since LEτ is a system of Stratonovich-type stochastic differential equations,
the chain rule is available

$\displaystyle{\rm d}\tilde{E}_{\tau}$ $\displaystyle=\frac{\partial
E}{\partial x^{\mu}}{\rm d}\tilde{x}^{\mu}_{\tau}+\frac{\partial E}{\partial
p^{\mu}}{\rm d}\tilde{p}^{\mu}_{\tau}$
$\displaystyle=-Z_{\mu}\left[\xi_{\tau}^{\mu}+\mathcal{F}_{\text{dp}}^{\mu}\right]{\rm
d}\tau-\frac{\tilde{p}_{\tau}^{\mu}\tilde{p}_{\tau}^{\nu}}{m}\nabla_{\nu}Z_{\mu}{\rm
d}\tau-Z_{\mu}\mathcal{F}_{\text{em}}^{\mu}{\rm d}\tau.$ (37)

Similarly,

$\displaystyle{\rm d}\tilde{H}_{\tau}$ $\displaystyle={\rm
d}\tilde{E}_{\tau}-\frac{q}{m}\frac{\partial}{\partial
x^{\nu}}(Z_{\mu}A^{\mu})\tilde{p}^{\nu}_{\tau}{\rm d}\tau$ $\displaystyle={\rm
d}\tilde{E}_{\tau}+Z_{\mu}\mathcal{F}^{\mu}_{\text{em}}{\rm
d}\tau-\frac{q}{m}[A_{\mu}\nabla_{\nu}Z^{\mu}+Z^{\mu}\nabla_{\mu}A_{\nu}]\tilde{p}^{\nu}_{\tau}{\rm
d}\tau$
$\displaystyle=-Z_{\mu}\left[\xi_{\tau}^{\mu}+\mathcal{F}_{\text{dp}}^{\mu}\right]{\rm
d}\tau-\frac{\tilde{p}_{\tau}^{\mu}\tilde{p}_{\tau}^{\nu}}{m}\nabla_{\nu}Z_{\mu}{\rm
d}\tau-\frac{q}{m}\pounds_{Z}A_{\mu}\tilde{p}^{\mu}_{\tau}{\rm d}\tau,$ (38)

where $\pounds_{Z}A_{\mu}$ is the Lie derivative of $A^{\mu}$ along the vector
field $Z^{\mu}$. If the electromagnetic field is controlled by an external
protocol denoted by $\sigma$, the last term of the above equation can be
rewritten as

$\displaystyle{\rm
d}_{\sigma}\tilde{\mathcal{U}}:=-\frac{q}{m}\pounds_{Z}A_{\mu}\tilde{p}^{\mu}_{\tau}{\rm
d}\tau.$ (39)

In the realm of stochastic thermodynamics, the energy exchange between the
Brownian particle and the heat reservoir is considered as heat, while the
other part of the change of energy of the Brownian particle is considered as
work. Therefore, the heat received by the Brownian particle from the heat
reservoir is identified to be

$\displaystyle{\rm
d}\tilde{\mathcal{Q}}_{\tau}:=-Z_{\mu}\left[\xi_{\tau}^{\mu}+\mathcal{F}_{\text{dp}}^{\mu}\right]{\rm
d}\tau,$ (40)

and gravitational [26] and electromagnetic works are identified respectively
as

$\displaystyle{\rm
d}\tilde{\mathcal{P}}_{\tau}:=-\frac{\tilde{p}_{\tau}^{\mu}\tilde{p}_{\tau}^{\nu}}{m}\nabla_{\nu}Z_{\mu}{\rm
d}\tau,$ (41) $\displaystyle{\rm
d}\tilde{\mathcal{W}}_{\tau}:=-Z_{\mu}\mathcal{F}^{\mu}_{\text{em}}{\rm
d}\tau.$ (42)

Therefore, the first law of relativistic stochastic thermodynamics can be
realized either as

$\displaystyle{\rm d}\tilde{E}_{\tau}={\rm d}\tilde{\mathcal{Q}}_{\tau}+{\rm
d}\tilde{\mathcal{P}}_{\tau}+{\rm d}\tilde{\mathcal{W}}_{\tau},$ (43)

or as

$\displaystyle{\rm d}\tilde{H}_{\tau}={\rm d}\tilde{\mathcal{Q}}_{\tau}+{\rm
d}\tilde{\mathcal{P}}_{\tau}+{\rm d}_{\sigma}\tilde{\mathcal{U}}.$ (44)

The energy currents associated to the above two definitions of energy are
presented as follows,

$\displaystyle E^{\mu}[\varphi]$
$\displaystyle:=\int_{{(\Gamma^{+}_{m})_{x}}}\eta_{(\Gamma^{+}_{m})_{x}}\frac{p^{\mu}}{m}\varphi
E=-Z_{\nu}T^{\mu\nu}[\varphi],$ (45) $\displaystyle H^{\mu}[\varphi]$
$\displaystyle:=\int_{{(\Gamma^{+}_{m})_{x}}}\eta_{(\Gamma^{+}_{m})_{x}}\frac{p^{\mu}}{m}\varphi
H=E^{\mu}[\varphi]-qZ_{\nu}A^{\nu}N^{\mu}[\varphi],$ (46)

wherein the energy-momentum tensor $T^{\mu\nu}[\varphi]$ reads

$\displaystyle
T^{\mu\nu}[\varphi]:=\int_{{(\Gamma^{+}_{m})_{x}}}\eta_{(\Gamma^{+}_{m})_{x}}\frac{p^{\mu}p^{\nu}}{m}\varphi.$
(47)

In relativistic kinetic theory, the entropy current associated with classical
non-degenerate particles is defined as [28]

$\displaystyle
S^{\mu}[\varphi]=-\int_{{(\Gamma^{+}_{m})_{x}}}\eta_{(\Gamma^{+}_{m})_{x}}\frac{p^{\mu}}{m}\varphi\left(\ln\varphi-1\right).$
(48)

In the equilibrium distribution (31), we have the following relation [29]:

$\displaystyle S^{\mu}[\varphi_{\text{eq}}]$
$\displaystyle=\beta^{\mu}P-T^{\mu\nu}[\varphi_{\text{eq}}]\beta^{\nu}+\alpha
n^{\mu}[\varphi_{\text{eq}}],$ (49)

where $\displaystyle
P:=\frac{1}{d}\Delta_{\mu\nu}(U)T^{\mu\nu}[\varphi_{\text{eq}}]$ is the
pressure and $n^{\mu}[\varphi]$ is particle current. Following a standard
procedure as did in [29], it can be shown that
$T^{\mu\nu}[\varphi_{\text{eq}}]$ and $n^{\mu}[\varphi_{\text{eq}}]$ can both
be expressed in terms of the proper velocity $U^{\mu}$ of Bob:
$n^{\mu}[\varphi_{\text{eq}}]$ is simply proportional to $U^{\mu}$, while
$T^{\mu\nu}[\varphi_{\text{eq}}]$ takes the form of the energy-momentum tensor
of a perfect fluid.

Now we would like to reinterpret eq.(49) in terms of the densities of various
physical quantities as measured by Alice. This can be achieved by contracting
each term in eq.(49) with $Z_{\mu}$. Recalling the fact $\beta^{\mu}=\beta
U^{\mu}$, the relationship between the proper velocities of Alice and Bob, can
be written as

$\displaystyle U^{\mu}=\gamma_{\rm A}(Z^{\mu}+z^{\mu}),\qquad\gamma_{\rm
A}=-U^{\mu}Z_{\mu},\qquad z^{\mu}Z_{\mu}=0,$ (50)

where $\gamma_{\rm A}$ represents the local Lorentz factor arising from the
relative motion between Alice and Bob. Consequently, we have

$\displaystyle e_{\rm A}$ $\displaystyle=(\gamma_{\rm A}\beta)^{-1}s_{\rm
A}-\alpha(\gamma_{\rm A}\beta)^{-1}n_{\rm A}-P+{\mathcal{T}}_{\rm A},$ (51)

or alternatively

$\displaystyle h_{\rm A}$ $\displaystyle=(\gamma_{\rm A}\beta)^{-1}s_{\rm
A}-[qZ_{\mu}A^{\mu}+\alpha(\gamma_{\rm A}\beta)^{-1}]n_{\rm
A}-P+{\mathcal{T}}_{\rm A},$ (52)

where

$\displaystyle e_{\rm A}:=T^{\mu\nu}[\varphi_{\text{eq}}]Z_{\mu}Z_{\nu},\quad
h_{\rm A}:=e_{\rm A}-qZ_{\mu}A^{\mu}n_{\rm A},\quad n_{\rm
A}:=-Z_{\mu}n^{\mu}[\varphi_{\text{eq}}],$ (53) $\displaystyle s_{\rm
A}:=-Z_{\mu}S^{\mu}[\varphi_{\text{eq}}],\quad{\mathcal{T}}_{\rm
A}:=-T^{\rho\nu}[\varphi_{\text{eq}}]z_{\mu}Z_{\nu}.$ (54)

$e_{\rm A}$ is the density of the energy $E$, $h_{\rm A}$ is the density of
the energy $H$, $n_{\rm A}$ is the particle number density, $s_{\rm
A}:=-Z_{\mu}S^{\mu}[\varphi_{\text{eq}}]$ is the entropy density, and
${\mathcal{T}}_{\rm A}$ is the relative kinematic energy density arising from
the relative motion between Alice and Bob. The subscript $\rm A$ in the
notation for all these density quantities indicates that they are all defined
with respect to the observer Alice. Eq. (51) can be viewed as the localized
version of the Euler relation in which the coefficients of $s_{\rm A}$ and
$n_{\rm A}$ should be interpreted as the temperature and chemical potential as
measured by Alice. Therefore we have

$\displaystyle T_{\rm A}:=(\gamma_{\rm A}\beta)^{-1},\qquad\mu_{\rm
A}:=-\alpha(\gamma_{\rm A}\beta)^{-1}.$ (55)

Similarly, from eq. (52), we can read off

$\displaystyle\hat{\mu}_{\rm A}:=-[qZ_{\mu}A^{\mu}+\alpha(\gamma_{\rm
AB}\beta)^{-1}]=\mu_{\rm A}-qZ_{\mu}A^{\mu},$ (56)

which is recognized to be the electrochemical potential. Inserting eq.(34)
into eq. (55), the transformation rule for the temperature and chemical
potential presented in Ref.[29] can be easily recovered,

$T_{\rm A}:=(\gamma_{\rm A})^{-1}T_{\rm B},\qquad\mu_{\rm A}:=(\gamma_{\rm
A})^{-1}\mu_{\rm B}.$

We can also substitute eqs. (55) and (56) into eqs. (51)-(52) to make the
appearance of Euler relation simpler,

$\displaystyle e_{\rm A}$ $\displaystyle=T_{\rm A}s_{\rm A}+\mu_{\rm A}n_{\rm
A}-P+{\mathcal{T}}_{\rm A},$ (57) $\displaystyle h_{\rm A}$
$\displaystyle=T_{\rm A}s_{\rm A}+\hat{\mu}_{\rm A}n_{\rm
A}-P+{\mathcal{T}}_{\rm A}.$ (58)

The different choices of definition for the energy induce different chemical
potentials, while such choices have no influence on the temperature. Putting
these thermodynamic quantities into eq. (31), the equilibrium distribution can
be rewritten as

$\displaystyle\varphi_{\text{eq}}={\rm e}^{(E-\mu_{\rm
A}-z_{\nu}p^{\nu})/T_{\rm A}}={\rm e}^{(H-\hat{\mu}_{\rm
A}-z_{\nu}p^{\nu})/T_{\rm A}}.$ (59)

It remains to introduce the trajectory entropy for the Brownian particle which
plays an important role in the formulation of fluctuation theorem. Denoting
the phase trajectory of the Brownian particle as
$\tilde{Y}_{t}=(\tilde{y}_{t},\tilde{k}_{t})$, the trajectory entropy is
defined as

$\displaystyle\tilde{S}_{t}=-\ln\varphi(\tilde{Y}_{t}),$ (60)

The entropy production of the Brownian particle in the time interval
$[t_{I},t_{F}]$ is related to the trajectory entropy via

$\displaystyle\Delta
S=\int_{\mathcal{S}_{t_{F}}}\eta_{\mathcal{S}_{t_{F}}}Z_{\mu}S^{\mu}[\varphi]-\int_{\mathcal{S}_{t_{I}}}\eta_{\mathcal{S}_{t_{I}}}Z_{\mu}S^{\mu}[\varphi]=\langle\tilde{S}_{t_{F}}\rangle-\langle\tilde{S}_{t_{I}}\rangle.$
(61)

This part of the entropy production is also referred to as the trajectory
entropy production.

##  4 Time-reversal symmetry and its breaking

Before dwelling into the construction of fluctuation theorems in the framework
of general relativistic stochastic thermodynamics, it crucial to accurately
describe what the TRS is meant on curved spacetime.

In ordinary textbooks on special relativistic field theories, the time-
reversal transformation (TRT) is often represented by a coordinate
transformation $(x^{0},x^{i})\mapsto(-x^{0},x^{i})$ with different
transformation rules for temporal and spatial components of the coordinate
bases

$\displaystyle\frac{\partial}{\partial x^{0}}\mapsto-\frac{\partial}{\partial
x^{0}},\qquad\frac{\partial}{\partial x^{i}}\mapsto\frac{\partial}{\partial
x^{i}}.$ (62)

Since $x^{0}$ serves as the evolution parameter, the spatial components
$p^{i}$ of the momentum should reverse their signs, while the temporal
component of the momentum remains unchanged, which corresponds to the
particle’s energy. Consequently, the transformation under TRT for the
contraction of momentum components and the basis obeys

$\displaystyle p=p^{0}\frac{\partial}{\partial
x^{0}}+p^{i}\frac{\partial}{\partial
x^{i}}\mapsto-p^{0}\frac{\partial}{\partial
x^{0}}-p^{i}\frac{\partial}{\partial x^{i}}=-p.$ (63)

To maintain general covariance in our construction, it is better to treat the
TRT as a change of the observer’s time orientation, _i.e._ from the
perspective of Alice to that of Carol, rather than treating it as a coordinate
transformation. Meanwhile, we adopt the scalar expression like
$-Z_{\mu}p^{\mu}$ for defining the energy of a particle instead of using the
coordinate dependent definition $p^{0}$. Therefore, we employ an alternative
representation of the TRT, which reverses the sign of full momentum (_i.e._
$p^{\mu}\mapsto-p^{\mu}$) while keeping the spacetime coordinates intact. Such
a representation implies the reversal of the particle’s proper time
derivative: ${\rm d}/{\rm d}\tau\mapsto-{\rm d}/{\rm d}\tau$. If Alice
observes a particle evolving from $\tau_{I}$ to $\tau_{F}$ in its own proper
time during the time interval $t_{I}\rightarrow t_{F}$, then Carol will
observe the proper time of the charged particle evolves from $\tau_{F}$ to
$\tau_{I}$.

Since the momentum is defined only in the local tangent space on curved
spacetime, the TRT can be realized as an automorphism of $T_{x}\mathcal{M}$:

$\displaystyle I_{x}:p\mapsto-p,\qquad\text{for }\forall p\in
T_{x}\mathcal{M}.$ (64)

Furthermore, $I_{x}$ induces a homeomorphism between the future and past mass
shell bundles,

$\displaystyle I:\Gamma^{+}_{m}\to\Gamma^{-}_{m},$ (65) $\displaystyle
I:(x,p)\mapsto(x,I_{x}(p))=(x,-p).$ (66)

Therefore, the phase trajectories lifted by Alice and Carol are on two
disconnected regions of the mass shell bundle. For convenience, the trajectory
lifted by Alice will be referred to as the _forward trajectory_ , and the one
by Carol will be referred to as the _reversed trajectory_. For later
reference, we also list the induced action of the map $I$ on the coordinate
basis for the vector field on $T\mathcal{M}$,

$\displaystyle I^{*}\frac{\partial}{\partial
p^{\mu}}=-\frac{\partial}{\partial p^{\mu}},\qquad
I^{*}\frac{\partial}{\partial x^{\mu}}=\frac{\partial}{\partial x^{\mu}}.$
(67)

Figure 2: The evolution directions of the particle under the perspectives of
Alice and Carol.

If we discretize the time interval $[t_{I},t_{F}]$ into a sequence of $N+1$
equal-distance nodes $t_{I}=t_{0}<t_{1}<\cdots<t_{N}=t_{F}$, then the forward
trajectory can be written as a sequence $Y_{[t]}:=(Y_{0},Y_{1},\cdots,Y_{N})$
with $Y_{n}:=Y_{t_{n}}$. Consequently, the reversed trajectory can also be
written as a sequence $Y_{[t]}^{-}:=(Y_{0}^{-},Y_{1}^{-},\cdots,Y_{N}^{-})$ in
which

$\displaystyle Y^{-}_{n}=I(Y_{N-n}).$ (68)

The probability distribution $f=-p^{\mu}Z_{\mu}\varphi/m$ describes the
distribution of the intersections of the stochastic trajectories with the
space of microstates $\Sigma^{+}_{t}$, and it can be easily extended to the
past mass shell bundle by use of the relationship between the forward and the
reversed trajectories. Additionally, as a metric-preserving transformation,
the TRT $I$ preserves the volume elements on every submanifolds of the tangent
bundle $T\mathcal{M}$ mentioned in Sec. 2. Consequently, we have

$\displaystyle I^{*}\varphi=\varphi,\qquad
I^{*}\eta_{\Gamma^{+}_{m}}=\eta_{\Gamma^{-}_{m}},\qquad
I^{*}\eta_{(\Gamma^{+}_{m})_{x}}=\eta_{(\Gamma^{-}_{m})_{x}},\qquad
I^{*}\eta_{\Sigma^{+}_{t}}=\eta_{\Sigma_{t}^{-}},$ (69)

where $\Sigma_{t}^{-}$ is the space of microstates associated to the past
directed observer, Carol.

It remains to describe the TRT for the electromagnetic field. Consider the
motion of a massive charged particle subjecting to the electromagnetic field
$F^{\mu\nu}$. The equation of motion reads

$\displaystyle\frac{{\rm d}p^{\mu}}{{\rm
d}\tau}=\frac{q}{m}F^{\mu}{}_{\nu}p^{\nu}.$ (70)

The TRS implies that the above equation of motion is kept invariant. Since
$p^{\mu}$ is odd and hence $\displaystyle\frac{{\rm d}p^{\mu}}{{\rm d}\tau}$
is even under the TRT, the invariance of the equation of motion requires
either

$\displaystyle q\mapsto q,\qquad F^{\mu\nu}\mapsto-F^{\mu\nu}$ (71)

or

$\displaystyle q\mapsto-q,\qquad F^{\mu\nu}\mapsto F^{\mu\nu}.$ (72)

We will adopt the second convention which is known as Feynman’s convention in
the literature[30]. Therefore, we have

$\displaystyle I^{*}q=-q,\qquad
I^{*}\mathcal{F}_{\text{em}}^{\mu}=\mathcal{F}_{\text{em}}^{\mu}.$ (73)

Using the definition (26), it is not difficult to check that the induced
action of the TRT on the Liouville vector field is given by

$\displaystyle I^{*}\mathscr{L}_{F}=-\mathscr{L}_{F}.$ (74)

Bringing all the above conventions together, we will see that the equations
for the phase trajectory of a massive charged particle moving in a generic
curved spacetime and subjecting to an external electromagnetic field is kept
invariant under the TRT,

$\displaystyle\frac{{\rm d}x^{\mu}}{{\rm d}\tau}$
$\displaystyle=\frac{p^{\mu}}{m},$ (75) $\displaystyle\frac{{\rm
d}p^{\mu}}{{\rm d}\tau}$
$\displaystyle=\frac{q}{m}F^{\mu}{}_{\nu}p^{\nu}-\frac{1}{m}\varGamma^{\mu}{}_{\alpha\beta}p^{\alpha}p^{\beta}.$
(76)

Let us now bring the above system into a broader picture. Consider a scenario
in which there are a great number of massive charged particles of different
masses and charges moving together. The electromagnetic field is produced by
the particles themselves and the spacetime geometry is determined by the
masses and charges of the particles. We assume that the system is consisted of
different species of particles, each species carries different mass $m_{s}$
and charge $q_{s}$ and obey a different distribution $\Phi_{s}$, which are
differentiated from each other by the suffix $s$. We also assume that the
distributions $\Phi_{s}$ are invariant under the TRT, _i.e._
$I^{*}\Phi_{s}=\Phi_{s}$. Then the total electric current

$\displaystyle
J^{\mu}=\sum_{s}\int\eta_{(\Gamma_{m_{s}}^{+})_{x}}\frac{q_{s}p^{\mu}_{s}}{m_{s}}\Phi_{s}$
(77)

as well as the energy-momentum tensor contributed by the particles

$\displaystyle
T^{\mu\nu}_{\text{pa}}=\sum_{s}\int\eta_{(\Gamma_{m_{s}}^{+})_{x}}\frac{p^{\mu}_{s}p^{\nu}_{s}}{m_{s}}\Phi_{s}$
(78)

should be both TRT invariants. Consequently, the Maxwell equation

$\displaystyle\nabla_{\nu}F^{\mu\nu}=J^{\mu}$ (79)

that determines the electromagnetic field as well as the Einstein equation

$\displaystyle R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R=8\pi
G\left(T^{\mu\nu}_{\text{pa}}+T^{\mu\nu}_{\text{em}}\right)$ (80)

that determines the spacetime geometry should all be invariant under the TRT,
wherein

$\displaystyle
T^{\mu\nu}_{\text{em}}=F^{\mu\rho}F^{\nu}{}_{\rho}-\frac{1}{4}g^{\mu\nu}F^{\rho\sigma}F_{\rho\sigma}$
(81)

is the energy-momentum tensor of the electromagnetic field, whose TRT
invariance is self-evident.

Stochastic mechanics can be regarded as an effective theory for describing a
complete mechanical system within specific spatial and temporal scales [31,
32, 33]. Let us consider a heavy particle in the above system, assuming that
the remaining particles constitute a heat reservoir which has already reached
equilibrium. The electromagnetic interaction acting on the heavy particle can
be divided into two parts: the coarse-grained averaging effects at larger
spatial and temporal scales, and the stochastic remnants at smaller scales.
Consequently, eqs. (75)-(76) can be approximated by LEτ [1], and this heavy
particle is the Brownian particle.

However, such a coarse-grained description violates the TRS. Let us first
consider the damping force. If the forward and reversed trajectories
simultaneously satisfy following equation

$\displaystyle\frac{1}{m}p^{\nu}\nabla_{\nu}p^{\mu}=\mathcal{K}^{\mu\nu}U_{\nu},$
(82)

the damping tensor must reverse its sign under the TRT, since the velocity of
the heat reservoir should reverse its sign. The sign change in
$\mathcal{K}^{\mu\nu}$ implies that the damping force reverses its role from a
decelerating force to an accelerating force, which makes a difference between
future- and past-directed observers. On the other hand, the Einstein relation
[2]

$\displaystyle\mathcal{D}^{\mu\nu}=2T_{\rm B}\mathcal{K}^{\mu\nu}$ (83)

requires that the damping tensor should be kept invariant under the TRT,
because the diffusion tensor
$\mathcal{D}^{\mu\nu}=\mathcal{R}^{\mu}{}_{\mathfrak{a}}\mathcal{R}^{\nu}{}_{\mathfrak{a}}$
is a quadratic form in $\mathcal{R}^{\mu}{}_{\mathfrak{a}}$ whose eigenvalues
must be non-negative. Therefore, the assumption that eq.(82) is invariant
under the TRT has to be wrong. The correct behaviors of the damping and
diffusion tensors and the stochastic amplitudes are provided as follows,

$\displaystyle I^{*}\mathcal{K}^{\mu\nu}=\mathcal{K}^{\mu\nu},\qquad
I^{*}\mathcal{D}^{\mu\nu}=\mathcal{D}^{\mu\nu},\qquad
I^{*}\mathcal{R}^{\mu}{}_{\mathfrak{a}}=\pm\mathcal{R}^{\mu}{}_{\mathfrak{a}}.$
(84)

The sign of the stochastic amplitude $\mathcal{R}^{\mu}{}_{\mathfrak{a}}$ is
of little significance, since the probability distribution of the Gaussian
noise is an even function. For convenience, we adopt
$I^{*}\mathcal{R}^{\mu}{}_{\mathfrak{a}}=\mathcal{R}^{\mu}{}_{\mathfrak{a}}$.
For the sake of mathematical consistency, we define a vector field $V^{a}$ on
$T\mathcal{M}$, which satisfies

$\displaystyle V^{a}|_{(x,p)}=\begin{cases}\displaystyle
U^{\mu}|_{x}\frac{\partial}{\partial
p^{\mu}}\bigg{|}_{(x,p)}&(x,p)\in\Gamma^{+}_{m}\\\
\displaystyle-U^{\mu}|_{x}\frac{\partial}{\partial
p^{\mu}}\bigg{|}_{(x,p)}&(x,p)\in\Gamma^{-}_{m}\end{cases}.$ (85)

Then the damping force and the right-hand side of eq. (27) can be rewritten as

$\displaystyle\mathcal{F}_{\text{dp}}^{a}=\mathcal{K}^{ab}V_{b},\qquad\mathcal{I}^{a}[\varphi]=\frac{1}{2}\mathcal{D}^{ab}\nabla^{(h)}_{b}\varphi-\mathcal{K}^{ab}V_{b}\varphi,$
(86)

and it is easy to check that their transformation rules under the TRT are

$\displaystyle I^{*}\mathcal{F}_{\text{dp}}=\mathcal{F}_{\text{dp}},\qquad
I^{*}\mathcal{I}[\varphi]=\mathcal{I}[\varphi].$ (87)

Therefore, the probability current (29) of the Brownian particle can be
divided into the even and odd parts under the TRT, _i.e._

$\displaystyle\mathscr{J}_{\text{r}}[\varphi]=\dfrac{\varphi}{m}\mathscr{L}_{F},\qquad\mathscr{J}_{\text{d}}[\varphi]=-\mathcal{I}[\varphi],$
(88)

with

$\displaystyle
I^{*}\mathscr{J}_{\text{r}}[\varphi]=-\mathscr{J}_{\text{r}}[\varphi],\qquad
I^{*}\mathscr{J}_{\text{d}}[\varphi]=\mathscr{J}_{\text{d}}[\varphi].$ (89)

It is evident that $\mathscr{J}_{\text{d}}[\varphi]$ violates the TRS. The
entropy production is always closely related to the breaking of the TRS, and
we have proved [3] that, on the ensemble level,

$\displaystyle\nabla_{\mu}S^{\mu}[\varphi]=-\int\eta_{(\Gamma_{m}^{+})_{x}}\varphi^{-1}\frac{\partial\varphi}{\partial
p^{\mu}}\mathscr{J}^{\mu}_{\mathrm{d}}[\varphi],$ (90)
$\displaystyle\nabla_{\mu}S_{R}^{\mu}=\int\eta_{(\Gamma_{m}^{+})_{x}}\beta_{\mu}\mathscr{J}^{\mu}_{\text{d}}[\varphi],$
(91)

where $S^{\mu}[\varphi]$ and $S_{R}^{\mu}$ respectively denote the entropy
currents of the Brownian particle and of the heat reservoir. Eq. (91) is
actually the relativistic version of the celebrated Clausius’ identity

$\displaystyle\nabla_{\mu}S^{\mu}_{R}=-\frac{Q_{\rm B}[\varphi]}{T_{\rm
B}}=\frac{Q_{R}}{T_{\rm B}},$ (92)

where $Q_{\rm B}[\varphi]$ is the heat transfer rate from Bob’s perspective,
and $Q_{R}$ is that of the heat reservoir. It is worth noticing that the
Clausius’ identity holds only from the perspective of Bob.

##  5 Fluctuation theorem

Unlike the case of ordinary differential equations, there is no deterministic
solution for stochastic differential equation. The best one can do is to
determine the probability for a certain trajectory to be realized.
Consequently, the breaking of the TRS of the Langevin equation could be
described in terms of the non-equal probabilities for the forward and reversed
trajectories,

$\displaystyle\Pr[\tilde{Y}_{[t]}=Y_{[t]}]\neq\Pr[\tilde{Y}_{[t]}^{-}=Y_{[t]}^{-}],$
(93)

where $\tilde{Y}_{[t]}=(\tilde{Y}_{0},\tilde{Y}_{1},\cdots,\tilde{Y}_{N})$ and
$\tilde{Y}_{[t]}^{-}=(\tilde{Y}^{-}_{0},\tilde{Y}_{1}^{-},\cdots,\tilde{Y}_{N}^{-})$
denote the forward and reversed processes, and $Y_{[t]}$ and $Y^{-}_{[t]}$
denote the forward and reversed trajectories which are already described in
Sec. 4. Notice that “forward process” and “forward trajectory” are different
concepts: the latter is a concrete realization of the former. Unlike the
relation between forward and reversed trajectories, the only requirement on
the reversed process is that its initial state is identical to the TRT of the
final state of the forward process, _i.e._

$\displaystyle\tilde{Y}^{-}_{0}=I(\tilde{Y}_{N}).$ (94)

The Lorentz factors relative to Alice and Carol can be related via the TRT

$\displaystyle\gamma_{\rm A}=-\lambda Z_{\mu}p^{\mu}/m=\lambda
C_{\mu}p^{\mu}/m=I^{*}\gamma_{C}.$ (95)

These Lorentz factors can be merged into a single scalar field on the complete
mass shell bundle,

$\displaystyle\gamma:=\begin{cases}\gamma_{\rm A}|_{Y}&Y\in\Gamma^{+}_{m}\\\
\gamma_{C}|_{Y}&Y\in\Gamma^{-}_{m}\end{cases},$ (96)

which is even under the TRT, $I^{*}\gamma=\gamma$. Accounting for the
transformation rules under the TRT, we can rearrange the spatial components of
the LEt presented in eqs. (22)-(23) into the form

$\displaystyle{\rm d}\tilde{y}_{t}^{i}$
$\displaystyle=\frac{\tilde{k}^{i}_{t}}{m}\gamma^{-1}{\rm d}t,$ (97)
$\displaystyle{\rm d}\tilde{k}_{t}^{i}$
$\displaystyle=\hat{\mathcal{R}}^{i}{}_{\mathfrak{a}}\circ_{S}{\rm
d}\tilde{W}^{\mathfrak{a}}_{n}+F^{i}{\rm d}t+\bar{F}^{i}{\rm d}t,$ (98)

where
$\hat{\mathcal{R}}^{i}{}_{\mathfrak{a}}:=\gamma^{-1/2}\mathcal{R}^{i}{}_{\mathfrak{a}}$
and

$\displaystyle
F^{i}:=\gamma^{-1}\left(\mathcal{F}^{i}_{\text{em}}-\frac{1}{m}\varGamma^{i}{}_{\alpha\beta}\tilde{k}^{\alpha}\tilde{k}^{\beta}\right),\quad\bar{F}^{i}$
$\displaystyle:=\gamma^{-1}\left(\mathcal{F}_{\text{add}}^{i}-\frac{1}{2}\mathcal{D}^{ij}\gamma^{1/2}\nabla^{(h)}_{j}\gamma^{-1/2}+\mathcal{F}^{i}_{\text{dp}}\right),$
(99)

which have opposite behaviors under the TRT,

$\displaystyle
F^{i}|_{Y}=F^{i}|_{I(Y)},\qquad\bar{F}^{i}|_{Y}=-\bar{F}^{i}|_{I(Y)}.$ (100)

In order to describe the fluctuation theorems at the trajectory level, we need
a discretized version of LEt, which reads

$\displaystyle{\rm d}\tilde{y}_{n}^{i}$
$\displaystyle=\frac{\tilde{k}^{i}_{n}}{m}\gamma^{-1}|_{\tilde{Y}_{\bar{n}}}{\rm
d}t,$ (101) $\displaystyle{\rm d}\tilde{k}_{n}^{i}$
$\displaystyle=\hat{\mathcal{R}}^{i}{}_{\mathfrak{a}}|_{\tilde{Y}_{\bar{n}}}{\rm
d}\tilde{W}^{\mathfrak{a}}_{n}+F^{i}|_{\tilde{Y}_{\bar{n}}}{\rm
d}t+\bar{F}^{i}|_{\tilde{Y}_{\bar{n}}}{\rm d}t,$ (102)

where $\tilde{Y}_{\bar{n}}:=(\tilde{Y}_{n+1}+\tilde{Y}_{n})/2$ which comes
from the Stratonovich coupling. We also need to introduce the measures on the
space of forward and reversed trajectories which is defined to be the
successive wedge products of the volume element of the space of microstates at
each moment of the discrete time,

$\displaystyle\mathcal{D}[Y_{[t]}]:=\eta_{\Sigma_{0}}\wedge\eta_{\Sigma_{1}}\wedge\cdots\wedge\eta_{\Sigma_{N}},\qquad\mathcal{D}[Y^{-}_{[t]}]:=\eta_{\Sigma^{-}_{0}}\wedge\eta_{\Sigma^{-}_{1}}\wedge\cdots\wedge\eta_{\Sigma^{-}_{N}}.$
(103)

Eq. (69) implies that these measures are images of each other under the TRT,

$\displaystyle I^{*}\mathcal{D}[Y_{[t]}]=\mathcal{D}[Y^{-}_{[t]}].$ (104)

In other words, $Y_{[t]}\mapsto Y^{-}_{[t]}$ is a volume preserving map.

Since both the forward and reversed processes are Markovian, the trajectory
probabilities can be written as products of transition probabilities with the
initial probability,

$\displaystyle\Pr[\tilde{Y}_{[t]}=Y_{[t]}]$
$\displaystyle=\bigg{(}\prod_{n=0}^{N-1}\Pr[\tilde{Y}_{n+1}=Y_{n+1}|\tilde{Y}_{n}=Y_{n}]\bigg{)}\Pr[\tilde{Y}_{0}=Y_{0}],$
(105) $\displaystyle\Pr[\tilde{Y}^{-}_{[\bar{t}\,]}=Y^{-}_{[t]}]$
$\displaystyle=\bigg{(}\prod_{n=0}^{N-1}\Pr[\tilde{Y}^{-}_{n+1}=Y^{-}_{n+1}|\tilde{Y}^{-}_{n}=Y^{-}_{n}]\bigg{)}\Pr[\tilde{Y}^{-}_{0}=Y^{-}_{0}]$
$\displaystyle=\bigg{(}\prod_{n=0}^{N-1}\Pr[\tilde{Y}^{-}_{n+1}=I(Y_{N-n-1})|\tilde{Y}^{-}_{n}=I(Y_{N-n})]\bigg{)}\Pr[I(\tilde{Y}_{N})=I(Y_{N})]$
$\displaystyle=\bigg{(}\prod_{n=0}^{N-1}\Pr[\tilde{Y}^{-}_{N-n}=I(Y_{n})|\tilde{Y}^{-}_{N-n-1}=I(Y_{n+1})]\bigg{)}\Pr[\tilde{Y}_{N}=Y_{N}].$
(106)

Eq. (68) and eq. (94) are used in the second line of eq. (106), and the volume
preserving property of $I$ is used in the third line. Appendix B proves the
following continuum limit

$\displaystyle\lim_{N\rightarrow+\infty}\prod_{n=0}^{N-1}\frac{\Pr[\tilde{Y}_{n+1}=Y_{n+1}|\tilde{Y}_{n}=Y_{n}]}{\Pr[\tilde{Y}^{-}_{N-n}=I(Y_{n})|\tilde{Y}^{-}_{N-n-1}=I(Y_{n+1})]}$
$\displaystyle\qquad=\frac{(\lambda\gamma^{-1})|_{Y_{0}}}{(\lambda\gamma^{-1})|_{Y_{N}}}\exp\left[\int^{t_{F}}_{t_{I}}\gamma^{-1}{\rm
d}t\frac{1}{T_{\rm
B}}({m}a^{\mu}-\mathcal{F}^{\mu}_{\text{em}})U_{\mu}\right].$ (107)

Under the perspective of Bob, the complete differential of the energy is

$\displaystyle{\rm d}E_{\rm
B}=-\frac{p^{\nu}}{m}\nabla_{\nu}(p^{\mu}U_{\mu}){\rm
d}\tau=-ma^{\mu}U_{\mu}{\rm d}\tau+{\rm d}\mathcal{P},$ (108)

or

$\displaystyle{\rm d}H_{\rm
B}=-\frac{p^{\nu}}{m}\nabla_{\nu}(P^{\mu}U_{\mu}){\rm
d}\tau=-ma^{\mu}U_{\mu}{\rm d}\tau+{\rm d}\mathcal{P}+{{\rm
d}_{\sigma}\mathcal{U}}-{\rm d}\mathcal{W}.$ (109)

Comparing with eq. (43), it is clear that the integral on the exponent in eq.
(5) is actually the increase of the entropy of the heat reservoir

$\displaystyle\int\gamma^{-1}{\rm d}t\frac{1}{T_{\rm
B}}(ma^{\mu}-\mathcal{F}^{\mu}_{\text{em}})U_{\mu}=-\int\frac{{\rm
d}\mathcal{Q}}{T_{\rm B}}=\Delta S_{R},$ (110)

where the second equality encodes the Clausius equality at the level of
trajectories. Let us stress that, although the Clausius equality holds only
from the perspective of Bob, the total increase of the entropy of the heat
reservoir is actually observer-independent. This fact enables us to make use
of eq. (110) in addressing the fluctuation theorem from the perspective of an
arbitrary observer.

In the continuum limit, the logarithm of the ratio of the probabilities of the
forward and reversed trajectories is the sum of the trajectory entropy
production of the Brownian particle with the entropy increase of the heat
reservoir,

$\displaystyle\ln\frac{\Pr[\tilde{Y}_{[t]}=Y_{[t]}]}{\Pr[\tilde{Y}^{-}_{[t]}=Y^{-}_{[t]}]}$
$\displaystyle=\lim_{N\to\infty}\sum_{n=0}^{N-1}\ln\frac{\Pr[\tilde{Y}_{n+1}=Y_{n+1}|\tilde{Y}_{n}=Y_{n}]}{\Pr[\tilde{Y}^{-}_{N-n}=I(Y_{n})|\tilde{Y}^{-}_{N-n-1}=I(Y_{n+1})]}$
$\displaystyle\qquad+\ln\frac{(\lambda\gamma^{-1})|_{X_{0}}\Pr[\tilde{Y}_{0}=Y_{0}]}{(\lambda\gamma^{-1})|_{Y_{N}}\Pr[\tilde{Y}_{N}=Y_{N}]}$
$\displaystyle=-\int\frac{{\rm d}\mathcal{Q}}{T_{\rm
B}}+\ln\frac{\varphi(Y_{0})}{\varphi(Y_{N})}$ $\displaystyle=\Delta
S_{R}+\Delta S.$ (111)

Please be reminded that the relation eq. (30) between different distribution
functions and the definition (60) of the trajectory entropy production are
used here. Denoting the total entropy production at the trajectory level as
$\Sigma_{Y_{[t]}}:=\Delta S_{R}+\Delta S$, the usual form of the detailed
fluctuation theorem arises

$\displaystyle\frac{\Pr[\tilde{Y}_{[t]}=Y_{[t]}]}{\Pr[\tilde{Y}^{-}_{[t]}=Y^{-}_{[t]}]}={\rm
e}^{\Sigma_{Y_{[t]}}}.$ (112)

This theorem reveals how the breaking of the TRS is associated with the
entropy production at the trajectory level. Moreover, taking the expectation
value of ${\rm e}^{\Sigma_{X_{[t]}}}$ and using Jensen inequality, one gets
the integral fluctuation theorem

$\displaystyle{\rm
e}^{-\left\langle\Sigma_{\tilde{Y}_{[t]}}\right\rangle}\leq\left\langle{\rm
e}^{-\Sigma_{\tilde{Y}_{[t]}}}\right\rangle=\int\mathcal{D}[Y_{[t]}]\Pr[\tilde{Y}^{-}_{[t]}=Y^{-}_{[t]}]=\int\mathcal{D}[Y^{-}_{[t]}]\Pr[\tilde{Y}^{-}_{[t]}=Y^{-}_{[t]}]=1,$
(113)

which implies that the statistical expectation value of the total entropy
production must be no-negative, _i.e._
$\left\langle\Sigma_{\tilde{Y}_{[t]}}\right\rangle\geq 0$.

##  6 Concluding remarks

Fluctuation theorems not only are important in understanding the theoretical
origin of irreversibility, but also have found various applications in diverse
areas ranging from macroscopic to mesoscopic systems, and even to active
matter. The research carried our in the present work opens new area for the
potential applications of fluctuation theorems which embodies relativistic
gravity, as in typical cases of cosmological and astrophysical processes.

In our construction, it is important to decouple the choice of observers from
the choice of coordinate systems, which has several important consequences.
First, it allows us to parametrize the stochastic trajectory using the proper
time of the observer field, which helps to get rid of the random clock carried
by the Brownian particle. Second, it helps to properly describe the TRT in a
coordinate independent manner. Last, it is precisely such decoupling that
makes the construction fully general covariant.

Although the values of many thermodynamic quantities are observer dependent,
including _e.g._ the energy and its density, the temperature and chemical
potential, and even the Clausius’ identity etc., the total entropy production
is not among the observer dependent quantities. Meanwhile, the trajectory
probability is a pure mathematical entity which is also observer independent.
Therefore, it is not surprising that the form of the fluctuation theorems
obtained in this work is identical to that obtained from non-relativistic
stochastic thermodynamics.

## Acknowledhement

This work is supported by the National Natural Science Foundation of China
under the grant No. 12275138.

## Conflict of interests declaration

The authors declare no known conflict of interests.

## References

  * [1] Y.F. Cai, T. Wang, and L. Zhao.  “Relativistic stochastic mechanics I: Langevin equation from observer’s perspective”.  J. Stat. Phys., 190:193, 2023.[arXiv:2306.01982].
  * [2] Y.F. Cai, T. Wang, and L. Zhao.  “Relativistic stochastic mechanics II: Reduced Fokker-Planck equation in Curved Spacetime”.  J. Stat. Phys., 190:181, 2023.[arXiv:2307.07805].
  * [3] T. Wang, Y.F. Cai, L. Cui, and L. Zhao.  “General relativistic stochastic thermodynamics”, 2023.  [arXiv:2311.04025].
  * [4] L. Boltzmann.  Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, pages 115–225.  Vieweg+Teubner Verlag, Wiesbaden, 1970.  ISBN: 9783322849861.
  * [5] P. Ehrenfest and T. Ehrenfest.  The conceptual foundations of the statistical approach in mechanics.  Dover Publications, 2015.  ISBN: 9780486662503.
  * [6] T. Y. Wu.  “Boltzmann’s H theorem and the Loschmidt and the Zermelo paradoxes”.  Int. J. Theor. Phys., 14:289–294, 1975.
  * [7] D. J. Evans and D. J. Searles.  “Equilibrium microstates which generate second law violating steady states”.  Phys. Rev. E, 50(2):1645, 1994.
  * [8] G. Gallavotti and E. G. D. Cohen.  “Dynamical ensembles in nonequilibrium statistical mechanics”.  Phys. Rev. Lett., 74(14):2694, 1995.[arXiv:chao-dyn/9410007].
  * [9] G. E. Crooks.  “Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences”.  Phys. Rev. E, 60(3):2721, 1999.[arXiv:cond-mat/9901352].
  * [10] K. Sekimoto.  “Langevin equation and thermodynamics”.  Prog. Theor. Phys. Supp., 130:17–27, 1998.
  * [11] U. Seifert.  “Entropy production along a stochastic trajectory and an integral fluctuation theorem”.  Phys. Rev. Lett., 95(4):040602, 2005.[arXiv:cond-mat/0503686].
  * [12] A Imparato and L Peliti.  “Fluctuation relations for a driven Brownian particle”.  Phys. Rev. E, 74(2):026106, 2006.[arXiv:cond-mat/0603506].
  * [13] V. Y. Chernyak, M. Chertkov, and C. Jarzynski.  “Path-Integral analysis of fluctuation theorems for general Langevin processes”.  J. Stat. Mech. Theory Exp., 2006(08):P08001, 2006.[arXiv:cond-mat/0605471].
  * [14] T. Ohkuma and T. Ohta.  “Fluctuation theorems for non-linear generalized Langevin systems”.  J. Stat. Mech. Theory Exp., 2007(10):P10010, 2007.
  * [15] Y.F. Cai, T. Wang, and L. Zhao.  “Fluctuation theorem on a Riemann manifold”.  2024\.  [arXiv:2401.00046].
  * [16] F. Axel.  “Relativistic fluctuation theorems”.  C. R. Phys., 8(5):696–713, 2007.[arXiv:0705.3115].
  * [17] B. Cleuren, K. Willaert, A. Engel, and C. Van den Broeck.  “Fluctuation theorem for entropy production during diffusion of a relativistic ideal gas”.  Phys. Rev. E, 77(2):022103, 2008.[arXiv:0709.3801].
  * [18] Z.Y. Fei, J.N. Zhang, R. Pan, T. Qiu, and H. T. Quan.  “Quantum work distributions associated with the dynamical Casimir effect”.  Phys. Rev. A, 99(5):052508, 2019.[arXiv:1902.00155].
  * [19] A. Teixidó-Bonfill, A. Ortega, and E. Martín-Martínez.  “First law of quantum field thermodynamics”.  Phys. Rev. A, 102(5):052219, 2020.[arXiv:2008.09146].
  * [20] P. S. Pal and S. Deffner.  “Stochastic thermodynamics of relativistic Brownian motion”.  New J. Phys., 22(7):073054, 2020.[arXiv:2003.02136].
  * [21] J. H. Pei, J. F. Chen, and H. T. Quan.  “Special relativistic covariant fluctuation theorems”.  2024\.  [arXiv:2312.17621].
  * [22] Y.F. Cai, T. Wang, and L. Zhao.  “General relativistic fluctuation theorems”.  2024\.  [arXiv:2407.09912].
  * [23] S. Sasaki.  “On the differential geometry of tangent bundles of Riemannian” manifolds.  Tohoku Math. J., 10(3):338–354, 1958.
  * [24] Y. L. Klimontovich.  “Nonlinear Brownian motion”.  Phys-Usp, 37(8):737, 1994.
  * [25] L. D. Landau.  The classical theory of fields, volume 2.  Elsevier, 1975.  ISBN:  978-0-08-025072-4.
  * [26] S. F. Liu and L. Zhao.  “Work and work-energy theorem in curved spacetime”, 2021.  [arXiv:2010.13071].
  * [27] D. Bakry, I. Gentil, and M. Ledoux.  Analysis and geometry of Markov diffusion operators, volume 103\.  Springer, 2014.  ISBN: 9783319002279.
  * [28] C. Cercignani and G. M. Kremer.  The relativistic Boltzmann equation: Theory and applications, volume 22.  Springer Science & Business Media, 2002.  ISBN: 9783034881654.
  * [29] X. Hao, S. Liu, and L. Zhao.  “Relativistic transformation of thermodynamic parameters and refined Saha equation”.  Commun. Theor. Phys., 2022.[arXiv:2105.07313].
  * [30] R. Feynman.  Nobel lecture.  Nobel Lectures in Physics 1963-1970, pages 155–178, 1972.  ISBN: 9789810234041.
  * [31] G. W. Ford, M. Kac, and P. Mazur.  “Statistical mechanics of assemblies of coupled oscillators”.  J. Math. Phys., 6(4):504–515, 1965.
  * [32] H. Mori.  “Transport, collective motion, and Brownian motion”.  Prog. Theor. Phys., 33(3):423–455, 1965.
  * [33] R. Zwanzig.  “Nonlinear generalized Langevin equations”.  J. Stat. Phys., 9(3):215–220, 1973.
  * [34] O. Sarbach and T. Zannias.  “The geometry of the tangent bundle and the relativistic kinetic theory of gases”.  Class. Quant. Grav., 31(8):085013, 2014.[arXiv:1309.2036].

##  Appendix A Coordinate bases for tangent vectors

In our description of the tangent bundle $T\mathcal{M}$, we employ $2d+2$
independent coordinates $(x^{\mu},p^{\mu})$ with the corresponding coordinate
basis $\displaystyle\left(\left.\frac{\partial}{\partial
x^{\mu}}\right|_{T\mathcal{M}},\left.\frac{\partial}{\partial
p^{\mu}}\right|_{T\mathcal{M}}\right)$ for tangent vectors on $T\mathcal{M}$.
However, due to the mass shell constraint, the above coordinates are no longer
independent on the mass shell bundle $\Gamma^{+}_{m}$, rendering the
coordinate basis also redundant. The same also happens when we restrict
ourselves from the mass shell bundle to the space of microstates
$\Sigma_{t}^{+}$. The aim of this appendix is to resolve the redundancies in
the coordinate bases on various useful submanifolds of $T\mathcal{M}$.

To resolve the coordinate redundancy on $\Gamma^{+}_{m}$, we take the
coordinates on $\Gamma^{+}_{m}$ to be $(x^{\mu},p^{i})$ and view $p^{0}$ as a
function

$\displaystyle
p^{0}(x^{\mu},p^{i})=\frac{g_{0i}(x)p^{i}\pm\sqrt{\left[g_{0i}(x)p^{i}\right]^{2}-g_{00}(x)\left[m^{2}+g_{ij}(x)p^{i}p^{j}\right]}}{-g_{00}(x)}.$
(114)

This leads to the partial derivatives

$\displaystyle\frac{\partial p^{0}}{\partial
x^{\mu}}=-\frac{1}{2p_{0}}\frac{\partial g_{\alpha\beta}}{\partial
x^{\mu}}p^{\alpha}p^{\beta},\qquad\frac{\partial p^{0}}{\partial
p^{i}}=-\frac{p_{i}}{p_{0}}.$ (115)

It is essential to recognize that the coordinate basis for tangent vectors on
$\Gamma^{+}_{m}$ is different from the corresponding subset of coordinate
basis on $T\mathcal{M}$,

$\displaystyle\left.\frac{\partial}{\partial
x^{\mu}}\right|_{T\mathcal{M}}\neq\left.\frac{\partial}{\partial
x^{\mu}}\right|_{\Gamma^{+}_{m}},\quad\left.\frac{\partial}{\partial
p^{i}}\right|_{T\mathcal{M}}\neq\left.\frac{\partial}{\partial
p^{i}}\right|_{\Gamma^{+}_{m}}.$ (116)

To demonstrate this difference, let us consider a scalar field
$f(x^{\mu},p^{\mu})$ on $T\mathcal{M}$ and its restriction on
$\Gamma^{+}_{m}$, _i.e._

$\displaystyle\left.f\right|_{\Gamma^{+}_{m}}(x^{\mu},p^{i}):=f(x^{\mu},p^{0}(x^{\mu},p^{i}),p^{i}).$
(117)

The action of the coordinate basis vectors on
$\left.f\right|_{\Gamma^{+}_{m}}$ reads

$\displaystyle\left.\frac{\partial}{\partial
p^{i}}\right|_{\Gamma^{+}_{m}}\left.f\right|_{\Gamma^{+}_{m}}$
$\displaystyle=\left.\frac{\partial}{\partial
p^{i}}\right|_{T\mathcal{M}}f+\frac{\partial p^{0}}{\partial
p^{i}}\left.\frac{\partial}{\partial
p^{0}}\right|_{T\mathcal{M}}f=\left.\frac{\partial}{\partial\breve{p}^{i}}\right|_{T\mathcal{M}}f,$
(118) $\displaystyle\left.\frac{\partial}{\partial
x^{\mu}}\right|_{\Gamma^{+}_{m}}\left.f\right|_{\Gamma^{+}_{m}}$
$\displaystyle=\left.\frac{\partial}{\partial
x^{\mu}}\right|_{T\mathcal{M}}f+\frac{\partial p^{0}}{\partial
x^{\mu}}\left.\frac{\partial}{\partial
p^{0}}\right|_{T\mathcal{M}}f=\left.\frac{\partial}{\partial\breve{x}^{\mu}}\right|_{T\mathcal{M}}f,$
(119)

where the “breved” partial derivatives are defined as

$\displaystyle\left.\frac{\partial}{\partial\breve{p}^{i}}\right|_{T\mathcal{M}}:=\left[\frac{\partial}{\partial
p^{i}}-\frac{p_{i}}{p_{0}}\frac{\partial}{\partial
p^{0}}\right]_{T\mathcal{M}},\qquad\left.\frac{\partial}{\partial\breve{x}^{\mu}}\right|_{T\mathcal{M}}:=\left[e_{\mu}+\varGamma^{i}{}_{\beta\mu}p^{\beta}\frac{\partial}{\partial\breve{p}^{i}}\right]_{T\mathcal{M}}.$
(120)

Therefore, the partial derivatives on $\Gamma^{+}_{m}$ and $T\mathcal{M}$ are
related via

$\displaystyle i^{*}\left.\frac{\partial}{\partial
p^{i}}\right|_{\Gamma^{+}_{m}}=\left.\frac{\partial}{\partial\breve{p}^{i}}\right|_{T\mathcal{M}},\qquad
i^{*}\left.\frac{\partial}{\partial
x^{\mu}}\right|_{\Gamma^{+}_{m}}=\left.\frac{\partial}{\partial\breve{x}^{\mu}}\right|_{T\mathcal{M}},$
(121)

where $i:\Gamma^{+}_{m}\rightarrow T\mathcal{M}$ is the embedding map. From
the exterior geometric point of view, we can safely identify
$\\{\partial/\partial\breve{x}^{\mu},\partial/\partial\breve{p}^{i}\\}$ as the
coordinate basis on the mass shell bundle, omitting the subscripts.

The tangent vector $\mathcal{V}=\mathcal{V}^{\mu}(\partial/{\partial
p^{\mu}})$ of the momentum space must be orthogonal to the normal vector
$\hat{N}=p^{\mu}(\partial/\partial p^{\mu})/m$. The orthogonality condition
$\mathcal{V}^{\mu}p_{\mu}=0$ implies that

$\displaystyle\mathcal{V}^{\mu}\frac{\partial}{\partial
p^{\mu}}=\mathcal{V}^{i}\frac{\partial}{\partial
p^{i}}-\frac{\mathcal{V}^{i}p_{i}}{p_{0}}\frac{\partial}{\partial
p^{0}}=\mathcal{V}^{i}\frac{\partial}{\partial\breve{p}^{i}}.$ (122)

Therefore, the tangent vectors of the momentum space have two different
representations under the above two bases. We will use them interchangeably.
There is a similar property for the vectors on mass shell bundle. Let
$\mathscr{V}$ be a tangent vector of the phase trajectory
$X_{\tau}:=(x_{\tau},p_{\tau})$. As a vector on tangent bundle, $\mathscr{V}$
can be expanded as

$\displaystyle\mathscr{V}=\frac{p^{\mu}}{m}\frac{\partial}{\partial
x^{\mu}}+m\mathcal{A}^{\mu}\frac{\partial}{\partial
p^{\mu}}=\frac{p^{\mu}}{m}e_{\mu}+ma^{\mu}\frac{\partial}{\partial p^{\mu}},$
(123)

where $\mathcal{A}^{\mu}:=m^{-1}{\rm d}p^{\mu}/{\rm d}\tau$ is the coordinate
acceleration and
$a^{\mu}:=\mathcal{A}^{\mu}+\varGamma^{\mu}{}_{\alpha\beta}p^{\alpha}p^{\beta}/m^{2}$
is the covariant acceleration. The mass shell condition implies that the
covariant acceleration must always be orthogonal to the momentum, so that
$\mathscr{V}$ can be rewritten as

$\displaystyle\mathscr{V}=\frac{p^{\mu}}{m}e_{\mu}+ma^{i}\frac{\partial}{\partial\breve{p}^{i}}=\frac{p^{\mu}}{m}\frac{\partial}{\partial\breve{x}^{\mu}}+m\mathcal{A}^{i}\frac{\partial}{\partial\breve{p}^{i}}.$
(124)

The two representations for vectors on the mass shell bundle under the bases
$\\{\partial/\partial\breve{x}^{\mu},\partial/\partial\breve{p}^{i}\\}$ and
$\\{\partial/\partial x^{\mu},\partial/\partial p^{\mu}\\}$ will also be used
interchangeably.

The space of microstates $\Sigma^{+}_{t}$ is also an embedding submanifold of
the mass shell bundle defined by fixing $t(x)$ to be a constant. Therefore,
the coordinates on $\Gamma^{+}_{m}$ becomes redundant once again on
$\Sigma^{+}_{t}$. Such redundancy can also be eliminated by taking
$(x^{i},p^{i})$ as coordinate of $\Sigma^{+}_{t}$ and viewing $x^{0}(t,x^{i})$
as a function. The restriction of $f$ on $\Sigma^{+}_{t}$ is defined as

$\displaystyle\left.f\right|_{\Sigma_{t}^{+}}(x^{i},p^{i}):=\left.f\right|_{\Gamma^{+}_{m}}(x^{0}(t,x^{i}),x^{i},p^{i})=f(x^{0}(t,x^{i}),x^{i},p^{0}(x^{0}(t,x^{i}),x^{i},p^{i}),p^{i}).$
(125)

Correspondingly, the partial derivatives acting on $f|_{\Sigma^{+}_{t}}$ can
be evaluated to be

$\displaystyle\left.\frac{\partial}{\partial
x^{i}}\right|_{\Sigma_{t}^{+}}\left.f\right|_{\Sigma_{t}^{+}}$
$\displaystyle=\left.\frac{\partial}{\partial
x^{i}}\right|_{\Gamma^{+}_{m}}\left.f\right|_{\Gamma^{+}_{m}}+\frac{\partial
x^{0}}{\partial x^{i}}\left.\frac{\partial}{\partial
x^{0}}\right|_{\Gamma^{+}_{m}}\left.f\right|_{\Gamma^{+}_{m}}=\left.\frac{\partial}{\partial\breve{x}^{i}}\right|_{T\mathcal{M}}f-\frac{\partial_{i}t}{\partial_{0}t}\left.\frac{\partial}{\partial\breve{x}^{0}}\right|_{T\mathcal{M}}f,$
(126) $\displaystyle\left.\frac{\partial}{\partial
p^{i}}\right|_{\Sigma_{t}^{+}}\left.f\right|_{\Sigma^{+}_{t}}$
$\displaystyle=\left.\frac{\partial}{\partial
p^{i}}\right|_{\Gamma^{+}_{m}}\left.f\right|_{\Gamma^{+}_{m}}=\left.\frac{\partial}{\partial\breve{p}^{i}}\right|_{T\mathcal{M}}f,$
(127) $\displaystyle\frac{\partial}{\partial
t}\left.f\right|_{\Sigma^{+}_{t}}$ $\displaystyle=\frac{\partial
x^{0}}{\partial t}\left.\frac{\partial}{\partial
x^{0}}\right|_{\Gamma^{+}_{m}}\left.f\right|_{\Gamma^{+}_{m}}=\frac{1}{\partial_{0}t}\left.\frac{\partial}{\partial\breve{x}^{0}}\right|_{T\mathcal{M}}f,$
(128)

thanks to the relations (118)-(119). Let
$\pi_{t}:\Sigma_{t}^{+}\rightarrow\Gamma^{+}_{m}$ be the embedding map and
define

$\displaystyle\left.\frac{\partial}{\partial\hat{x}^{i}}\right|_{T\mathcal{M}}:=\left.\frac{\partial}{\partial\breve{x}^{i}}\right|_{T\mathcal{M}}-\frac{\partial_{i}t}{\partial_{0}t}\left.\frac{\partial}{\partial\breve{x}^{0}}\right|_{T\mathcal{M}},$
(129)

we have

$\displaystyle(i\circ\pi_{t})^{*}\left.\frac{\partial}{\partial
p^{i}}\right|_{\Sigma_{t}^{+}}=\left.\frac{\partial}{\partial\breve{p}^{i}}\right|_{T\mathcal{M}},\qquad(i\circ\pi_{t})^{*}\left.\frac{\partial}{\partial
x^{i}}\right|_{\Sigma_{t}^{+}}=\left.\frac{\partial}{\partial\hat{x}^{i}}\right|_{T\mathcal{M}}.$
(130)

Therefore, we can take
$\\{\partial/\partial\hat{x}^{i},\partial/\partial\breve{p}^{i}\\}$ as the
coordinate basis on $\Sigma^{+}_{t}$ without specifying the manifold with a
subscript.

Since the phase trajectory does not lie on a certain $\Sigma^{+}_{t}$, the
basis $\\{\partial/\partial\hat{x}^{i},\partial/\partial\breve{p}^{i}\\}$ is
insufficient to describe the tangent vector of the phase trajectory. The
missing dimension in the tangent vector of the phase trajectory is described
by $\partial/\partial t$ as described in eq. (128), which, from the exterior
geometric point of view, can be simply denoted

$\displaystyle\frac{\partial}{\partial
t}=\frac{1}{\partial_{0}t}\frac{\partial}{\partial\breve{x}^{0}}.$ (131)

Therefore, under the above vector basis, the tangent vector (124) to the phase
trajectory can be re-expressed as

$\displaystyle\mathscr{V}=\gamma\frac{\partial}{\partial
t}+\frac{p^{i}}{m}\frac{\partial}{\partial\hat{x}^{i}}+m\mathcal{A}^{i}\frac{\partial}{\partial\breve{p}^{i}},$
(132)

where

$\displaystyle\gamma=\frac{{\rm d}t}{{\rm
d}\tau}=\frac{1}{m}p^{\mu}\partial_{\mu}t$ (133)

is the local Lorentz factor between the particle and the observer Alice.

##  Appendix B Continuum limit

This appendix provides the details for proving eq. (5).

First, we introduce two mathematical lemmas.

1\. Let $A$ be a full rank square matrix and $B$ be an arbitrary matrix of the
same size. Then the determinant of $A+B{\rm d}t+o(t^{2})$ can be expanded into
power series in ${\rm d}t$,

$\displaystyle\det[A+B{\rm d}t+o({\rm
d}t^{2})]=\det[A]+\det[A]\mathrm{Tr}[A^{-1}B]{\rm d}t+o({\rm d}t^{2}).$ (134)

2\. Let $f(t)$ be a continuous function on $[t_{I},t_{F}]$, then

$\displaystyle\lim_{N\rightarrow+\infty}\prod_{n=0}^{N-1}[1+f(t_{n}){\rm
d}t+o({\rm d}t^{2})]=\exp\left[\int_{t_{I}}^{t_{F}}f(t){\rm d}t\right],$ (135)

where ${\rm d}t=(t_{F}-t_{I})/N$ and $t_{n}\in[t_{I}+n{\rm d}t,t_{I}+(n+1){\rm
d}t]$.

Combining the above two lemmas, the following corollary can be deduced,

$\displaystyle\lim_{N\rightarrow+\infty}\prod_{n=0}^{N-1}\frac{\det[A(t_{n})+B(t_{n}){\rm
d}t+o({\rm d}t^{2})]}{\det[A(t_{n})+C(t_{n}){\rm d}t+o({\rm
d}t^{2})]}=\exp\left[\int_{t_{I}}^{t_{F}}\mathrm{Tr}[A^{-1}(B-C)]{\rm
d}t\right],$ (136)

where $A(t)$, $B(t)$ and $C(t)$ are all time-dependent matrix functions.

Let $\mathscr{O}$ be a function on the mass shell bundle and
$Y_{t}=X_{\tau(t)}$ be a phase trajectory parametrized by the proper time of
Alice. Then

$\displaystyle\frac{\mathscr{O}(Y_{t_{F}})}{\mathscr{O}(Y_{t_{I}})}$
$\displaystyle=\frac{\mathscr{O}(X_{\tau_{F}})}{\mathscr{O}(X_{\tau_{I}})}=\lim_{N\rightarrow+\infty}\prod_{n=0}^{N-1}\frac{\mathscr{O}(X_{\tau_{n+1}})}{\mathscr{O}(X_{\tau_{n}})}$
$\displaystyle=\lim_{N\rightarrow+\infty}\prod_{n=0}^{N-1}\frac{\mathscr{O}(X_{\tau_{n}})+\mathscr{V}(\mathscr{O}){\rm
d}\tau+o({\rm d}\tau^{2})}{\mathscr{O}(X_{\tau_{n}})}$
$\displaystyle=\exp\left[\int_{\tau_{I}}^{\tau_{F}}\mathscr{O}^{-1}\mathscr{V}(\mathscr{O}){\rm
d}\tau\right]$
$\displaystyle=\exp\left[\int_{t_{I}}^{t_{F}}\gamma^{-1}\mathscr{O}^{-1}\mathscr{V}(\mathscr{O}){\rm
d}t\right],$ (137)

where $\mathscr{V}$ is the tangent vector along $Y_{t}$, as presented by eq.
(124) and/or eq. (132). It is worth noting that in this appendix, we will use
$(y,k)$ to represent the coordinates of the tangent bundle, rather than
$(x,p)$. This choice does not imply a coordinate transformation. It simply
emphasizes that the discretized time corresponds to Alice’s proper time $t$.

Eq. (B) can be used to evaluate the continuum limit of certain ratios which
are useful in proving the fluctuation theorem. For instance,

$\displaystyle\frac{k_{0}|_{Y_{N}}}{k_{0}|_{Y_{0}}}$
$\displaystyle=\exp\left[\int_{t_{I}}^{t_{F}}\frac{\gamma^{-1}}{k_{0}}\left\\{\frac{k^{\mu}}{m}\frac{\partial}{\partial
y^{\mu}}k_{0}+m\mathcal{A}^{\mu}\frac{\partial}{\partial
k^{\mu}}k_{0}\right\\}{\rm d}t\right]$
$\displaystyle=\exp\left[\int_{t_{I}}^{t_{F}}\frac{\gamma^{-1}}{k_{0}}\left\\{\frac{1}{m}\partial_{\mu}g_{0\nu}k^{\mu}k^{\nu}+m\mathcal{A}_{0}\right\\}{\rm
d}t\right]$
$\displaystyle=\exp\left[\int_{t_{I}}^{t_{F}}\frac{\gamma^{-1}}{k_{0}}\left\\{\frac{1}{2m}\partial_{0}g_{\mu\nu}k^{\mu}k^{\nu}+ma_{0}\right\\}{\rm
d}t\right],$ (138) $\displaystyle\frac{g|_{Y_{0}}}{g|_{Y_{N}}}$
$\displaystyle=\exp\left[-\int_{t_{I}}^{t_{F}}\gamma^{-1}g^{-1}\frac{k^{\mu}}{m}\frac{\partial}{\partial
y^{\mu}}g{\rm d}t\right],$ (139)

and

$\displaystyle\frac{\partial_{0}t|_{Y_{N}}}{\partial_{0}t|_{Y_{0}}}$
$\displaystyle=\exp\left[\int_{t_{I}}^{t_{F}}\frac{\gamma^{-1}}{\partial_{0}t}\frac{k^{\mu}}{m}\frac{\partial}{\partial\breve{y}^{\mu}}\frac{\partial}{\partial\breve{y}^{0}}t{\rm
d}t\right]$
$\displaystyle=\exp\left[\int_{t_{I}}^{t_{F}}\frac{1}{\partial_{0}t}\left\\{\frac{\partial}{\partial\breve{y}^{0}}\left(\gamma^{-1}\frac{k^{\mu}}{m}\partial_{\mu}t\right)-\partial_{\mu}t\frac{\partial}{\partial\breve{y}^{0}}\left(\gamma^{-1}\frac{k^{\mu}}{m}\right)\right\\}{\rm
d}t\right]$
$\displaystyle=\exp\left[\int_{t_{I}}^{t_{F}}\frac{1}{\partial_{0}t}\left\\{\gamma^{-2}\frac{k^{\mu}}{m}\partial_{\mu}t\frac{\partial}{\partial\breve{y}^{0}}\gamma-\frac{\gamma^{-1}}{m}\partial_{0}t\frac{\partial}{\partial\breve{y}^{0}}k^{0}\right\\}{\rm
d}t\right]$
$\displaystyle=\exp\left[\int_{t_{I}}^{t_{F}}\gamma^{-1}\frac{\partial}{\partial
t}\gamma{\rm
d}t\right]\exp\left[\int_{t_{I}}^{t_{F}}\frac{\gamma^{-1}}{k_{0}}\frac{1}{2m}\partial_{0}g_{\mu\nu}k^{\mu}k^{\nu}{\rm
d}t\right].$ (140)

Eq. (101) can be rewritten as

$\displaystyle\tilde{y}^{i}_{n+1}=h^{i}(\tilde{Y}_{n+1},\tilde{Y}_{n})=\frac{\tilde{k}^{i}_{\bar{n}}}{m}\gamma^{-1}|_{\tilde{Y}_{\bar{n}}}{\rm
d}t+\tilde{y}^{i}_{n},$ (141)

which implies that the realization of $\tilde{y}^{i}_{n+1}$ is determined by
the realization of $\tilde{Y}_{n}$ and $\tilde{k}^{i}_{n+1}$. The probability
distribution of a random variable $\tilde{x}$ obeying the constraint
$\tilde{x}=f(\tilde{x})$ is given by

$\displaystyle\Pr[\tilde{x}=x]=(1-f^{\prime}(x))\delta(x-f(x)).$ (142)

The generalization of this equation to general dimension is straightforward,
telling that the probability distribution of $\tilde{y}^{i}_{n+1}$ under the
conditions $\tilde{k}^{i}_{n+1}={k}^{i}_{n+1},\tilde{Y}_{n}=Y_{n}$ is given by

$\displaystyle\Pr[\tilde{y}_{n+1}=y_{n+1}|\tilde{k}_{n+1}={k}_{n+1},\tilde{Y}_{n}=Y_{n}]$
$\displaystyle\qquad=\det\left[\delta^{i}{}_{j}-\frac{\partial
h^{i}}{\partial\hat{y}_{n+1}^{j}}\right]\left|\frac{\partial_{0}t}{\lambda\sqrt{g}}\right|_{y_{n+1}}\delta^{d}(y_{n+1}-h({Y_{n+1},Y_{n}}))$
$\displaystyle\qquad=\left[1+\gamma^{-2}\frac{k^{i}}{2m}\frac{\partial}{\partial\hat{{y}}^{i}}\gamma{\rm
d}t+o({\rm
d}t^{2})\right]_{Y_{\bar{n}}}\left|\frac{\partial_{0}t}{\lambda\sqrt{g}}\right|_{y_{n+1}}\Delta(Y_{n+1},Y_{n}),$
(143)

where $y_{n}$ denotes the sequence $(y_{n}^{1},\cdots,y_{n}^{d})$ and we will
also use the notation $k_{n}=(k^{1}_{n},\cdots,k_{n}^{d})$.
$\Delta(Y_{n+1},Y_{n})$ is defined as
$\Delta(Y_{n+1},Y_{n}):=\delta^{d}(y_{n+1}-h(Y_{n+1},Y_{n}))$, which has the
property

$\displaystyle\Delta(Y_{n+1},Y_{n})=\Delta(I(Y_{n}),I(Y_{n+1})).$ (144)

The appearance of $\partial/\partial\hat{y}^{i}$ in the second line of eq. (B)
is due to the fact that $\tilde{y}^{i}_{n+1}$ only takes values in the
configuration space, and its probability density is a scalar field on
$\mathcal{S}_{n}$. The factor $|\partial_{0}t/(\lambda\sqrt{g})|$ arises from
the change of coordinate volume element into invariant volume element, as
indicated by eq. (2).

The single step transition probability can be rewritten as

$\displaystyle\Pr[\tilde{Y}_{n+1}|\tilde{Y}_{n}]=\Pr[\tilde{y}_{n+1}|\tilde{k}_{n+1},\tilde{Y}_{n}]\Pr[\tilde{k}_{n+1}|\tilde{Y}_{n}],$
(145)

where the second factor remains to be evaluated. Defining the function

$\displaystyle{\rm d}W^{\mathfrak{a}}(Y_{n+1},Y_{n})$
$\displaystyle:=(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{i}|_{Y_{\bar{n}}}\left[k^{i}_{n+1}-k^{i}_{n}-F^{i}|_{Y_{\bar{n}}}{\rm
d}t-\bar{F}^{i}|_{Y_{\bar{n}}}{\rm d}t\right]$
$\displaystyle=(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{i}|_{Y_{\bar{n}}}\left[\gamma^{-1}m\mathcal{A}^{i}_{n}-F^{i}-\bar{F}^{i}\right]_{Y_{\bar{n}}}{\rm
d}t,$ (146)

where
$\mathcal{A}^{i}_{n}:=\gamma|_{Y_{\bar{n}}}(k^{i}_{n+1}-k^{i}_{n})/(m{\rm
d}t)$ is the coordinate acceleration, then eq. (102) can be rewritten as

$\displaystyle{\rm d}\tilde{W}^{\mathfrak{a}}_{n}={\rm
d}W^{\mathfrak{a}}(\tilde{Y}_{n+1},\tilde{Y}_{n}).$ (147)

In the case that the realization of $\tilde{Y}_{n}$ is given,
$\tilde{y}_{n+1}$ is determined by $\tilde{k}_{n+1}$. Therefore, eq. (147)
gives the relation between ${\rm d}\tilde{W}^{\mathfrak{a}}_{n}$ and
$\tilde{k}^{i}_{n+1}$, which can be used to calculate the conditional
probability

$\displaystyle\Pr[\tilde{k}_{n+1}=k_{n+1}|\tilde{Y}_{n}=Y_{n}]=\left|\frac{k_{0}}{m\sqrt{g}}\right|_{Y_{n+1}}\det\left[T^{\mathfrak{a}}{}_{i}(Y_{n+1},Y_{n})\right]\Pr[{\rm
d}\tilde{W}^{\mathfrak{a}}_{n}={\rm d}W^{\mathfrak{a}}(Y_{n+1},Y_{n})],$ (148)

where

$\displaystyle
T^{\mathfrak{a}}{}_{i}(Y_{n+1},Y_{n}):=\frac{\partial}{\partial\breve{k}^{i}_{n+1}}{\rm
d}W^{\mathfrak{a}}(Y_{n+1},Y_{n}),$ (149)

and $y_{n+1}$ appearing on the right hand side is regarded as a function
$y_{n+1}(k_{n+1},Y_{n})$ determined by eq. (141). Using the implicit
relationship (141) between $y_{n+1}$ and $k_{n+1}$, we can get the following
relation by use of a differentiation with respect to $k_{n+1}^{i}$,

$\displaystyle\left[\delta^{i}{}_{\ell}-\frac{\partial\gamma^{-1}}{\partial\hat{{y}}_{n+1}^{\ell}}{\rm
d}t\right]\frac{\partial
y_{n+1}^{\ell}}{\partial\breve{k}_{n+1}^{j}}=\left[\frac{1}{2m}\delta^{i}{}_{j}\gamma^{-1}|_{Y_{\bar{n}}}+\frac{k^{i}_{\bar{n}}}{2m}\frac{\partial\gamma^{-1}}{\partial\breve{{k}}^{j}_{n+1}}\right]{\rm
d}t,$ (150)

which indicates that $\partial y^{\ell}_{n+1}/\partial\breve{k}^{j}_{n+1}\sim
o({\rm d}t)$. Therefore,

$\displaystyle T^{\mathfrak{a}}{}_{i}({Y_{n+1},Y_{n}})=\frac{\partial{\rm
d}W^{\mathfrak{a}}}{\partial\breve{k}^{i}_{n+1}}+\frac{\partial
y^{k}_{n+1}}{\partial\breve{k}^{i}_{n+1}}\frac{\partial{\rm
d}W^{\mathfrak{a}}}{\partial\hat{y}^{k}_{n+1}}=\frac{\partial{\rm
d}W^{\mathfrak{a}}}{\partial\breve{k}^{i}_{n+1}}+o({\rm d}t^{2})$
$\displaystyle\qquad=(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{i}|_{Y_{\bar{n}}}+\frac{1}{2}\left\\{m\gamma^{-1}\frac{\partial}{\partial\breve{k}^{i}}(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{j}\mathcal{A}^{j}_{n}-\frac{\partial}{\partial\breve{k}^{i}}[(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{j}F^{j}]-\frac{\partial}{\partial\breve{k}^{i}}[(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{j}\bar{F}^{j}]\right\\}_{Y_{\bar{n}}}{\rm
d}t,$ (151)

where terms of order $o({\rm d}t^{2})$ and higher have been omitted. Finally,
the transition probability of the forward process can be expressed as

$\displaystyle\Pr[\tilde{Y}_{n+1}=Y_{n+1}|\tilde{Y}_{n}=Y_{n}]$
$\displaystyle\qquad=\left[1+\frac{k^{i}}{2m}\gamma^{-2}\frac{\partial}{\partial\hat{y}^{i}}\gamma{\rm
d}t+o({\rm
d}t^{2})\right]_{Y_{\bar{n}}}\left|\frac{k_{0}\partial_{0}t}{m\lambda
g}\right|_{Y_{n+1}}\Delta(Y_{n+1},Y_{n})\det\left[T^{a}{}_{i}(Y_{n+1},Y_{n})\right]$
$\displaystyle\qquad\quad\times\Pr[{\rm d}\tilde{W}^{\mathfrak{a}}_{n}={\rm
d}W^{\mathfrak{a}}{(Y_{n+1},Y_{n})}].$ (152)

The single step transition probability in the reversed process can be
evaluated following a similar procedure, yielding

$\displaystyle\Pr[\tilde{Y}^{-}_{N-n}=I(Y_{n})|\tilde{Y}^{-}_{N-n-1}=I(Y_{n+1})]$
$\displaystyle\qquad=\left[1+\frac{k^{i}}{2m}\gamma^{-2}\frac{\partial}{\partial\hat{{y}}^{i}}\gamma{\rm
d}t+o({\rm
d}t^{2})\right]_{I(Y_{\bar{n}})}\left|\frac{k_{0}\partial_{0}t}{m\lambda
g}\right|_{I(Y_{n})}$
$\displaystyle\qquad\quad\times\Delta(I(Y_{n}),I(Y_{n+1}))\det\left[T^{a}{}_{i}(I(Y_{n}),I(Y_{n+1}))\right]\Pr[{\rm
d}\tilde{W}^{\mathfrak{a}}_{n}={\rm d}W^{\mathfrak{a}}(I(Y_{n}),I(Y_{n+1}))]$
$\displaystyle\qquad=\left[1-\frac{k^{i}}{2m}\gamma^{-2}\frac{\partial}{\partial\hat{y}^{i}}\gamma{\rm
d}t+o({\rm
d}t^{2})\right]_{Y_{\bar{n}}}\left|\frac{k_{0}\partial_{0}t}{m\lambda
g}\right|_{Y_{n}}\Delta(Y_{n+1},Y_{n})\det\left[T^{a}{}_{i}(I(Y_{n}),I(Y_{n+1}))\right]$
$\displaystyle\qquad\quad\times\Pr[{\rm d}\tilde{W}^{\mathfrak{a}}_{n}={\rm
d}W^{\mathfrak{a}}(I(Y_{n}),I(Y_{n+1}))].$ (153)

The Jacobian matrix $T^{a}{}_{i}(I(Y_{n}),I(Y_{n+1}))$ in the reversed process
reads

$\displaystyle T^{\mathfrak{a}}{}_{i}(I(Y_{n}),I(Y_{n+1}))$
$\displaystyle\qquad=(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{i}|_{I(Y_{\bar{n}})}+\frac{1}{2}\left\\{m\gamma^{-1}\frac{\partial}{\partial\breve{k}^{i}}(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{j}\mathcal{A}^{j}_{n}-\frac{\partial}{\partial\breve{k}^{i}}[(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{j}F^{j}]-\frac{\partial}{\partial\breve{k}^{i}}[(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{j}\bar{F}^{j}]\right\\}_{I(Y_{\bar{n}})}{\rm
d}t$
$\displaystyle\qquad=(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{i}|_{Y_{\bar{n}}}+\frac{1}{2}\left\\{-m\gamma^{-1}\frac{\partial}{\partial\breve{k}^{i}}(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{j}\mathcal{A}^{j}_{n}+\frac{\partial}{\partial\breve{k}^{i}}[(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{j}F^{j}]-\frac{\partial}{\partial\breve{k}^{i}}[(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{j}\bar{F}^{j}]\right\\}_{Y_{\bar{n}}}{\rm
d}t.$ (154)

In the continuum limit, we have

$\displaystyle\lim_{N\rightarrow+\infty}\prod_{n=0}^{N-1}\frac{\det[T^{a}{}_{i}(Y_{n+1},Y_{n+1})]}{\det[T^{a}{}_{i}(I(Y_{n}),I(Y_{n+1}))]}$
$\displaystyle\qquad=\exp\left[\int_{t_{I}}^{t_{F}}\hat{\mathcal{R}}^{i}{}_{\mathfrak{a}}\left\\{m\gamma^{-1}\frac{\partial}{\partial\breve{k}^{i}}(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}{j}\mathcal{A}^{j}-\frac{\partial}{\partial\breve{k}^{i}}[(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{j}\bar{F}^{j}]\right\\}{\rm
d}t\right]$
$\displaystyle\qquad=\exp\left[\int_{t_{I}}^{t_{F}}\gamma^{-1}\hat{\mathcal{R}}^{i}{}_{\mathfrak{a}}\frac{\partial}{\partial\breve{k}^{i}}(\hat{\mathcal{R}}^{-1})^{\mathfrak{a}}{}_{j}[ma^{j}-\mathcal{F}^{j}_{\text{em}}]{\rm
d}t\right]\exp\left[-\int_{t_{I}}^{t_{F}}\frac{\partial}{\partial\breve{k}^{i}}F^{i}{\rm
d}t\right],$ (155)

where the exponent in the last term can be expanded as

$\displaystyle-\frac{\partial}{\partial\breve{k}^{i}}F^{i}=$
$\displaystyle-\left(\mathcal{F}^{i}_{\text{em}}-\frac{1}{m}\varGamma^{i}{}_{\alpha\beta}k^{\alpha}k^{\beta}\right)\frac{\partial}{\partial\breve{k}^{i}}\gamma^{-1}+\gamma^{-1}\frac{k^{\mu}}{m}g^{-1}\frac{\partial}{\partial
y^{\mu}}g$
$\displaystyle-\frac{\gamma^{-1}}{k_{0}}\left[\frac{1}{m}\partial_{0}g_{\alpha\beta}k^{\alpha}k^{\beta}+(\mathcal{F}_{\text{em}})_{0}\right].$
(156)

Meanwhile, we also have

$\displaystyle\lim_{N\rightarrow+\infty}\prod_{n=0}^{N-1}\frac{\left[1+\frac{k^{i}}{2m}\gamma^{-2}\frac{\partial}{\partial\hat{y}^{i}}\gamma{\rm
d}t+o({\rm
d}t^{2})\right]_{Y_{\bar{n}}}}{\left[1-\frac{k^{i}}{2m}\gamma^{-2}\frac{\partial}{\partial\hat{y}^{i}}\gamma{\rm
d}t+o({\rm
d}t^{2})\right]_{Y_{\bar{n}}}}=\exp\left[\int_{t_{I}}^{t_{F}}\gamma^{-2}\frac{k^{i}}{m}\frac{\partial}{\partial\hat{y}^{i}}\gamma{\rm
d}t\right].$ (157)

Moreover, the continuum limit of the ratio of the probabilities of the
Gaussian increments reads

$\displaystyle\lim_{N\rightarrow+\infty}\prod_{n=0}^{N-1}\frac{\Pr[{\rm
d}\tilde{W}^{\mathfrak{a}}_{n}={\rm
d}W^{\mathfrak{a}}(Y_{n+1},Y_{n})]}{\Pr[{\rm
d}\tilde{W}^{\mathfrak{a}}_{n}={\rm
d}W^{\mathfrak{a}}(I(Y_{n}),I(Y_{n+1}))]}=\exp\left[\int_{t_{I}}^{t_{F}}2\gamma^{-2}(\hat{\mathcal{D}}^{-1})_{ij}(m\mathcal{A}^{i}-\gamma
F^{i})\gamma\bar{F}^{j}{\rm d}t\right],$ (158)

where the integrads can also be expanded as

$\displaystyle
2\gamma^{-2}(\hat{\mathcal{D}}^{-1})_{ij}(m\mathcal{A}^{i}-\gamma
F^{i})\gamma\bar{F}^{j}$
$\displaystyle\qquad=2\gamma^{-1}(\mathcal{D}^{-1})_{ij}(ma^{i}-\mathcal{F}^{i}_{\text{em}})\left(\mathcal{F}^{j}_{\text{dp}}+\mathcal{F}^{j}_{\text{add}}-\frac{1}{2}\mathcal{D}^{jk}\gamma^{1/2}\nabla_{k}^{(h)}\gamma^{-1/2}\right)$
$\displaystyle\qquad=\gamma^{-1}\frac{1}{T_{\rm
B}}(ma^{\mu}-\mathcal{F}^{\mu}_{\text{em}})U_{\mu}+\gamma^{-1}(\mathcal{R}^{-1})^{a}_{j}\nabla_{i}^{(h)}\mathcal{R}^{i}{}_{a}(ma^{j}-\mathcal{F}^{j}_{\text{em}})$
$\displaystyle\qquad\quad+\frac{1}{2}\gamma^{-2}(ma^{i}-\mathcal{F}^{i}_{\text{em}})\nabla_{i}^{(h)}\gamma$
$\displaystyle\qquad=\gamma^{-1}\frac{1}{T_{\rm
B}}(ma^{\mu}-\mathcal{F}^{\mu}_{\text{em}})U_{\mu}+\gamma^{-1}(\hat{\mathcal{R}}^{-1})^{a}{}_{j}\frac{\partial}{\partial\breve{k}^{i}}\hat{\mathcal{R}}^{i}{}_{a}(ma^{j}-\mathcal{F}^{j}_{\text{em}})$
$\displaystyle\qquad\quad+\gamma^{-2}(ma^{i}-\mathcal{F}^{i}_{\text{em}})\frac{\partial}{\partial\breve{k}^{i}}\gamma-\frac{\gamma^{-1}}{k_{0}}(ma_{0}-(\mathcal{F}_{\text{em}})_{0}).$
(159)

In the end, by combining all the continuum limit together, we obtain the ratio
of the conditional probabilities for the forward and reversed processes,

$\displaystyle\lim_{N\rightarrow+\infty}\prod_{n=0}^{N-1}\frac{\Pr[\tilde{Y}_{n+1}=Y_{n+1}|\tilde{Y}_{n}=Y_{n}]}{\Pr[\tilde{Y}^{-}_{N-n}=I(Y_{n})|\tilde{Y}^{-}_{N-n-1}=I(Y_{n+1})]}$
$\displaystyle\qquad=\frac{\lambda|_{Y_{0}}}{\lambda|_{Y_{N}}}\times\text{eq.~{}}\eqref{rate-p0}\times\text{eq.~{}}\eqref{rate-g}\times\text{eq.~{}}\eqref{rate-t0}\times\text{eq.~{}}\eqref{rate-T}\times\text{eq.~{}}\eqref{rate-
gamma}\times\text{eq.~{}}\eqref{rate-dw}$
$\displaystyle\qquad=\frac{\lambda|_{Y_{0}}}{\lambda|_{Y_{N}}}\exp\left[\int_{t_{I}}^{t_{F}}\gamma^{-1}\frac{1}{T_{\rm
B}}(ma^{\mu}-\mathcal{F}^{\mu}_{\text{em}})U_{\mu}{\rm d}t\right]$
$\displaystyle\qquad\quad\times\exp\left[\int_{t_{I}}^{t_{F}}\gamma^{-2}\left(\frac{k^{i}}{m}\frac{\partial}{\partial\hat{y}^{i}}\gamma+\gamma\frac{\partial}{\partial
t}\gamma+m\mathcal{A}^{i}\frac{\partial}{\partial\breve{k}^{i}}\gamma\right){\rm
d}t\right]$
$\displaystyle\qquad=\frac{\lambda|_{Y_{0}}}{\lambda|_{Y_{N}}}\exp\left[\int_{t_{I}}^{t_{F}}\gamma^{-1}\frac{1}{T_{\rm
B}}(ma^{\mu}-\mathcal{F}^{\mu}_{\text{em}})U_{\mu}{\rm
d}t\right]\exp\left[\int_{t_{I}}^{t_{F}}\gamma^{-2}\mathscr{V}(\gamma){\rm
d}t\right]$
$\displaystyle\qquad=\frac{(\lambda\gamma^{-1})|_{Y_{0}}}{(\lambda\gamma^{-1})|_{Y_{N}}}\exp\left[\int_{t_{I}}^{t_{F}}\gamma^{-1}\frac{1}{T_{\rm
B}}(ma^{\mu}-\mathcal{F}^{\mu}_{\text{em}})U_{\mu}{\rm d}t\right],$ (160)

which is exactly eq. (5) appeared in the main text.