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# Relativistic stochastic mechanics II:
Reduced Fokker-Planck equation in curved spacetime

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

The general covariant Fokker-Planck equations associated with the two
different versions of covariant Langevin equation in Part I of this series of
work are derived, both lead to the same reduced Fokker-Planck equation for the
non-normalized one particle distribution function (1PDF). The relationship
between various distribution functions is clarified in this process. Several
macroscopic quantities are introduced by use of the 1PDF, and the results
indicate an intimate connection with the description in relativistic kinetic
theory. The concept of relativistic equilibrium state of the heat reservoir is
also clarified, and, under the working assumption that the Brownian particle
should approach the same equilibrium distribution as the heat reservoir in the
long time limit, a general covariant version of Einstein relation arises.

##  1 Introduction

This is Part II of our series of works on relativistic stochastic mechanics.
Part I of this series has already been presented in [1]. The major subject of
concern in Part I is the construction of manifestly general covariant Langevin
equation from the observer’s perspective. Two different versions of
relativistic Langevin equation (denoted LEτ and LEt respectively) were
proposed, among which LEτ takes the proper time $\tau$ of the Brownian
particle as evolution parameter and LEt takes the proper time $t$ of some
prescribed observer as evolution parameter. It was shown that although LEτ
contains some conceptual issues from the point of view of the prescribed
observer, there is numerical evidence indicating that LEτ and LEt can produce
the same distribution in the same space of micro states (SoMS) for the case of
$(1+1)$-dimensional Minkowski spacetime, which in turn suggests that we may be
able to extract useful probability distributions from LEτ.

Besides Langevin equation, Fokker-Planck equation (FPE) is another important
equation in stochastic mechanics. The route leading from Langevin equation to
FPE can be regarded as a bridge from mechanics to statistical physics. The
study of FPE was initiated about a hundred years ago [2, 3], and the purpose
is to analyze the diffusion phenomena (in the configuration space) of
suspended particles in solution. Kolmogoroff [4] gave an explanation of the
equation of the same form from the perspective of stochastic processes,
therefore the corresponding equation is also called Kolmogoroff equation.
Later, Klein [5] and Kramers [6] generalized the equation to the phase space.
Chandrasekhar provided a detailed report on the relevant topics [7], and the
solution to the Klein-Kramers equation describing a relaxation process was
also given. All these works used the transition probability to study the
evolution of random variables. With the development of stochastic differential
equations, related topics have been extensively studied by use of Ito calculus
[8], and some more modern methods about this topic can be found in [9].

In the relativistic regime, there is no Markov process satisfying causality on
the spacetime manifold [10, 11]. The only choice is to study FPE on the SoMS —
a subspace of the future mass shell bundle. This means that the equation to be
considered needs to be of the Klein-Kramers type. However, with the usual
abuse of terminology, we still use the name FPE for convenience.

The study of relativistic stochastic process can be traced back to Dudley [10,
12] and Hakim [13, 11], who first discussed the space of states for stochastic
processes in a model independent way. The study of concrete relativistic
stochastic processes, e.g. the relativistic Ornstein-Uhlenbeck process, was
carried out by Debbasch et al in [14]. Barbachoux et al [15, 16] made some
discussions about the corresponding FPE (Kolmogoroff equation). Dunkel et
al[17, 18, 19, 20, 21] also studied similar topics in the special relativistic
context, and their model gave an intuitive understanding of the relativistic
Brownian motion. Herrmann [22] and Haba [23] extended the studies to general
relativistic context, with some emphasis placed on the manifest general
covariance.

It is necessary to point out that, in all previous works, the important role
played by the observer has not beed sufficiently addressed. In this work, we
shall show that properly addressing the role played by the observer is the
starting point in understanding different versions of general covariant FPE
that arise either directly or indirectly from the Langevin equations LEτ
and/or LEt proposed in [1]. In particular, the observer plays an important
role in the interpretation of various distribution functions that appear in
different versions of covariant FPE.

Another important aspect which has not been made sufficiently clear in
previous works is the state of the heat reservoir. The description for the
non-relativistic Brownian motion of a heavy particle inside a heat reservoir
relies on two basic assumptions. First, the heat reservoir should have reached
thermodynamic equilibrium, and the only impact of the reservoir on the
Brownian particle is provided through thermal fluctuations, of which the fast
and slow parts manifest respectively in the Langevin equation in the form of
stochastic and damping forces. Second, the stochastic motion of the Brownian
particle should be able to mimic a relaxation process, which means that, after
sufficiently long time, the probability distribution for the Brownian particle
should approach the same equilibrium distribution obeyed by the particles from
the heat reservoir. We shall see in Sec. 6 that the concept of equilibrium
state for the heat reservoir needs to be re-examined carefully in the
relativistic context.

This paper is organized as follows. In Sec. 2, we presents an introduction to
the SoMS for the Brownian particle and prepare the notations to be used in the
forthcoming sections. The description for the SoMS is placed on the same
ground as in relativistic kinetic theory [24, 25, 26], with the expectation
that the deep connection between these complementary approaches to non-
equilibrium statistical physics could be further elucidated. Such a treatment
is more appealing than the alternative approaches, e.g. those making use of
jet bundles. In Sec. 3, we deduce the covariant FPEs from the Langevin
equations with different evolution parameters introduced in [1]. Sec. 4 is
devoted to clarifying the relationship between different distribution
functions. In this section, we also introduce a new distribution, which is
identified to be the one particle distribution function (1PDF) in the sense of
relativistic kinetic theory, together with its evolution equation, i.e. the
reduced FPE. Sec. 5 introduces some thermodynamic quantities and thermodynamic
relations, and the formulation seems to indicate some deep connections between
the approaches of stochastic mechanics and relativistic kinetic theory. In
Sec. 6, we clarify the meaning of the equilibrium state of the heat reservoir,
and, by assuming that the 1PDF should approach the intrinsic equilibrium
distribution of the heat reservoir, we deduce a general relativistic version
of the Einstein relation. Finally, in Sec. 7, we present a brief summary of
the results.

##  2 The SoMS and its geometry

Since this work is a followup to Ref.[1], we use exactly the same notations
and conventions as in [1]. In particular, the spacetime manifold $\mathcal{M}$
is taken to be a curved pseudo-Riemannian manifold of dimension $(d+1)$ with a
mostly positive signature. The future mass shell bundle $\Gamma^{+}_{m}$ over
$\mathcal{M}$ is defined as

$\displaystyle\Gamma^{+}_{m}:=\\{(x,p)\in
T\mathcal{M}~{}|~{}g_{\mu\nu}(x)p^{\mu}p^{\nu}=-m^{2}\ \text{and}\
p^{\mu}Z_{\mu}(x)<0\\},$ (1)

in which $Z^{\mu}(x)$ denotes the proper velocity of some observer field.
Later on, we shall omit the word “future” and simply refer to $\Gamma^{+}_{m}$
as the mass shell bundle. The momentum space of the Brownian particle at the
event $x$ is identified as the intersection of the tangent space
$T_{x}\mathcal{M}$ with the mass shell bundle and is referred to simply as the
mass shell at $x$,

$\displaystyle\left(\Gamma^{+}_{m}\right)_{x}:=T_{x}\mathcal{M}\cap\Gamma^{+}_{m},$
(2)

and the configuration space is labeled by the proper time $t$ of a single
prescribed observer, Alice, as the level set

$\mathcal{S}_{t}:=\\{x\in{\mathcal{M}}|t(x)=t={\rm const}.\\},$

where $t(x)$ is an extension of the proper time $t$ over $\mathcal{M}$ as a
scalar field. Denoting the proper velocity of Alice also by $Z^{\mu}$ should
produce no confusions. The SoMS of the Brownian particle is then given by

$\displaystyle\Sigma_{t}:=\bigcup_{x\in\mathcal{S}_{t}}(\Gamma^{+}_{m})_{x}=\\{(x,p)\in\Gamma^{+}_{m}|x\in\mathcal{S}_{t}\\},$
(3)

which is clearly observer-dependent. The above specification for the SoMS of
the Brownian particle naturally falls inline with the tangent bundle formalism
of relativistic kinetic theory [24, 25, 26]. This will certainly benefit for
the communication between the two important branches of non-equilibrium
relativistic statistical physics. An immediate benefit is to adopt the Sasaki
metric [27] for describing the local geometry of the tangent bundle (and
subspaces thereof).

Before proceeding, let us introduce our conventions on indices. We use both
concrete and abstract index notations, however with omissions of the abstract
indices when there no confusions could arise. Lower-case Greek letters
$\alpha,\beta,\mu,\nu,\rho,...$ denote concrete indices and range from $0$ to
$d$. Latin capital letters $A,B,...$ and some lower-case Latin letters, such
as $i,j,...$, also denote concrete indices. The upper-case Latin indices range
from $0$ to $2d$ and is associated with tensors on the mass shell bundle,
while the lower-case Latin indices $i,j,...$ range from $1$ to $d$. The other
lower-case Latin letters $a,b,...$ denote abstract indices. Additionally, we
use the calligraphy fonts, like $\mathcal{F}$, $\mathcal{R}$ and
$\mathcal{K}$, to denote tensors on the momentum space $(\Gamma_{m}^{+})_{x}$,
and the cursive fonts, like $\mathscr{N}$, $\mathscr{Z}$ and $\mathscr{L}$, to
denote tensors on the mass shell bundle $\Gamma_{m}^{+}$.

Since $\Gamma^{+}_{m}$, $(\Gamma^{+}_{m})_{x}$ and $\Sigma_{t}$ are all
subspaces of the tangent bundle $T\mathcal{M}$, it is appropriate to begin by
describing the relevant geometric structures on $T\mathcal{M}$. What really
matters is the tangent space of the tangent bundle, which can be subdivided
into the direct sum of horizontal and vertical subspaces [28, 29],

$\displaystyle T_{(x,p)}(T\mathcal{M})=H_{(x,p)}\oplus V_{(x,p)},$ (4)

where $H_{(x,p)}$ is spanned by

$\displaystyle e_{\mu}=\frac{\partial}{\partial
x^{\mu}}-\Gamma^{\alpha}{}_{\mu\beta}p^{\beta}\frac{\partial}{\partial
p^{\alpha}},$ (5)

and $V_{(x,p)}$ is spanned by $\partial/\partial p^{\mu}$. Here
$\Gamma^{\alpha}{}_{\mu\beta}$ represents the usual Christoffel connection
associated with the spacetime metric $g_{\mu\nu}$.

The metric on the tangent bundle $T\mathcal{M}$ is given by the Sasaki metric,
which can be written as a direct sum of metrics on the two subspaces,

$\displaystyle\hat{g}_{ab}:=\underbrace{g_{\mu\nu}{\rm d}x^{\mu}_{\ a}{\rm
d}x^{\nu}_{\ b}}_{\text{the metric of
}H_{(x,p)}}+\underbrace{g_{\mu\nu}\,\theta^{\mu}_{\ a}\theta^{\nu}_{\
b}}_{\text{the metric of }V_{(x,p)}},$ (6)

where

$\displaystyle\theta^{\mu}={\rm
d}p^{\mu}+\Gamma^{\mu}{}_{\alpha\beta}p^{\alpha}{\rm d}x^{\beta}.$ (7)

$\\{e_{\mu},\partial/\partial p^{\mu}\\}$ and $\\{{\rm
d}x^{\mu},\theta^{\mu}\\}$ are dual bases on the tangent and cotangent spaces
of the tangent bundle respectively. As a hypersurface on the tangent bundle,
the mass shell bundle is naturally equipped with an induced metric

$\displaystyle\hat{h}_{ab}$
$\displaystyle:=\hat{g}_{ab}+\hat{N}_{a}\hat{N}_{b},\qquad\hat{N}^{a}:=(m)^{-1}p^{\mu}\left(\frac{\partial}{\partial
p^{\mu}}\right)^{a},$ (8)

where $\hat{N}^{a}$ is the unit normal vector of the mass shell bundle. The
metric of the mass shell bundle can also be written as the direct sum of the
metrics of the horizontal subspace and the momentum space,

$\displaystyle\hat{h}_{ab}=\underbrace{g_{\mu\nu}{\rm d}x^{\mu}{}_{a}{\rm
d}x^{\nu}{}_{b}}_{\text{the metric of
}H_{(x,p)}}+\underbrace{\Delta_{\mu\nu}(p)\theta^{\mu}{}_{a}\theta^{\nu}{}_{b}}_{\text{the
metric of }T_{(x,p)}(\Gamma^{+}_{m})_{x}},$ (9)

where

$\Delta_{\mu\nu}(p):=g_{\mu\nu}+m^{-2}p_{\mu}p_{\nu}$

is the orthogonal projection tensor associated with $p^{\mu}$. The inverse of
the metric $\hat{h}$ reads

$\displaystyle\hat{h}^{ab}=g^{\mu\nu}e_{\mu}{}^{a}e_{\nu}{}^{b}+\Delta^{\mu\nu}(p)\left(\dfrac{\partial}{\partial
p^{\mu}}\right)^{a}\left(\dfrac{\partial}{\partial p^{\nu}}\right)^{b}.$ (10)

It is obvious that the metric on the momentum space $(\Gamma_{m}^{+})_{x}$ and
its inverse are respectively

$\displaystyle
h_{ab}=\Delta_{\mu\nu}(p)\theta^{\mu}{}_{a}\theta^{\nu}{}_{b},\qquad
h^{ab}=\Delta^{\mu\nu}(p)\left(\dfrac{\partial}{\partial
p^{\mu}}\right)^{a}\left(\dfrac{\partial}{\partial p^{\nu}}\right)^{b}.$ (11)

Please remember that we use $\hat{h}_{ab}$ for the metric on the mass shell
bundle and $h_{ab}$ for the metric on the fiber space alone.

Since $\Sigma_{t}$ is a hypersurface on the mass shell bundle, there is a
normal vector field. This normal vector field is given by

$\displaystyle\mathscr{Z}^{a}=Z^{\mu}e_{\mu}{}^{a}.$ (12)

$\mathscr{Z}^{a}$ is actually an up-lift of the observer field onto the mass
shell bundle.

Using the above metrics, it is easy to find the invariant volume elements on
$T\mathcal{M}$, $\Gamma^{+}_{m}$ and $(\Gamma^{+}_{m})_{x}$, respectively
[25],

$\displaystyle\eta_{T\mathcal{M}}$ $\displaystyle=g\,{\rm d}x^{0}\wedge{\rm
d}x^{1}\wedge...\wedge{\rm d}x^{d}\wedge{\rm d}p^{0}\wedge...\wedge{\rm
d}p^{d},$ (13) $\displaystyle\eta_{\Gamma^{+}_{m}}$
$\displaystyle=\frac{g}{p_{0}}{\rm d}x^{0}\wedge{\rm
d}x^{1}\wedge...\wedge{\rm d}x^{d}\wedge{\rm d}p^{1}\wedge...\wedge{\rm
d}p^{d},$ (14) $\displaystyle\eta_{(\Gamma^{+}_{m})_{x}}$
$\displaystyle=\frac{\sqrt{g}}{p_{0}}{\rm d}p^{1}\wedge...\wedge{\rm d}p^{d},$
(15)

where we have introduced $g=|\det(g_{\mu\nu})|$.

As mentioned above, $\\{e_{\mu},\partial/\partial p^{\mu}\\}$ is the basis of
$T_{(x,p)}(T\mathcal{M})$, so an arbitrary tangent vector on $T\mathcal{M}$
can be written as

$\displaystyle\mathscr{V}^{a}=V^{\mu}e_{\mu}^{\
a}+\mathcal{V}^{\mu}\left(\dfrac{\partial}{\partial p^{\mu}}\right)^{a}.$ (16)

The vectors with vanishing components $V^{\mu}$ can also be treated as tangent
vectors on the tangent space, and these will be denoted as

$\displaystyle\mathcal{V}^{a}=\mathcal{V}^{\mu}\left(\frac{\partial}{\partial
p^{\mu}}\right)^{a}.$ (17)

For tangent vectors on the mass shell $(\Gamma^{+}_{m})_{x}$, it is convenient
to introduce the following vector basis,

$\displaystyle\left(\dfrac{\partial}{\partial\breve{p}^{i}}\right)^{a}:=\left(\dfrac{\partial}{\partial{p}^{i}}\right)^{a}-\dfrac{p_{i}}{p_{0}}\left(\dfrac{\partial}{\partial{p}^{0}}\right)^{a}.$
(18)

Notice that, due to the mass shell condition, $(\Gamma^{+}_{m})_{x}$ has one
less dimension than $T_{x}\mathcal{M}$, and so are their respective tangent
spaces. Since $(\Gamma^{+}_{m})_{x}$ is a hypersurface in $T_{x}\mathcal{M}$
with normalized normal vector $\hat{N}^{a}$ given in eq.(8), any tangent
vector on $(\Gamma^{+}_{m})_{x}$ is automatically a tangent vector on
$T_{x}\mathcal{M}$. Therefore, we can also write the tangent vectors on
$(\Gamma^{+}_{m})_{x}$ in terms of the basis $\\{(\partial/\partial
p^{\mu})^{a}\\}$. In other words, any tangent vector $\mathcal{V}^{a}$ on
$(\Gamma^{+}_{m})_{x}$ acquires two different component representations

$\mathcal{V}^{a}=\mathcal{V}^{i}\left(\dfrac{\partial}{\partial\breve{p}^{i}}\right)^{a}\quad\text{and}\quad\mathcal{V}^{a}=\mathcal{V}^{\mu}\left(\frac{\partial}{\partial
p^{\mu}}\right)^{a}.$

It is straightforward to check that these two representations are equivalent,

$\displaystyle\mathcal{V}^{i}\left(\frac{\partial}{\partial\breve{p}^{i}}\right)^{a}=\mathcal{V}^{i}\left(\frac{\partial}{\partial
p^{i}}\right)^{a}-\mathcal{V}^{i}\frac{p_{i}}{p_{0}}\left(\frac{\partial}{\partial
p^{0}}\right)^{a}=\mathcal{V}^{i}\left(\frac{\partial}{\partial
p^{i}}\right)^{a}+\mathcal{V}^{0}\left(\frac{\partial}{\partial
p^{0}}\right)^{a}=\mathcal{V}^{\mu}\left(\dfrac{\partial}{\partial
p^{\mu}}\right)^{a},$ (19)

wherein we have used the orthogonal condition
$\mathcal{V}^{\mu}p_{\mu}=\mathcal{V}^{0}p_{0}+\mathcal{V}^{i}p_{i}=0$.
Similarly, the inverse metric on the momentum space can be expressed in two
different bases,

$\displaystyle h^{ab}=\Delta^{\mu\nu}(p)\left(\dfrac{\partial}{\partial
p^{\mu}}\right)^{a}\left(\dfrac{\partial}{\partial
p^{\nu}}\right)^{b}=\Delta^{ij}(p)\left(\dfrac{\partial}{\partial\breve{p}^{i}}\right)^{a}\left(\dfrac{\partial}{\partial\breve{p}^{j}}\right)^{b}.$
(20)

In order to describe the different versions of FPE, it is customary to
introduce the covariant derivatives on each of the relevant manifolds using
the standard conventions with the aid of the metrics introduced above.
However, this step can be skiped, because we only need the covariant
divergences. For a vector $\mathscr{F}^{A}=(F^{\mu},\mathcal{F}^{i})$ on the
mass shell bundle, the covariant divergence is simply given by

$\displaystyle\hat{\nabla}^{(\hat{h})}_{A}\mathscr{F}^{A}=\frac{p_{0}}{g}\frac{\partial}{\partial
x^{\mu}}\left(\frac{g}{p_{0}}F^{\mu}\right)+p_{0}\frac{\partial}{\partial\breve{p}^{i}}\left(\frac{1}{p_{0}}\mathcal{F}^{i}\right),$
(21)

where $\hat{\nabla}^{(\hat{h})}$ is the covariant derivative on the mass shell
bundle. If $F^{\mu}=0$, $\mathscr{F}$ reduces into a vector on the momentum
space, and the above equation becomes

$\displaystyle\hat{\nabla}^{(\hat{h})}_{A}\mathscr{F}^{A}={\nabla}^{(h)}_{i}\mathcal{F}^{i},$
(22)

which is automatically the covariant divergence on the the momentum space,
with ${\nabla}^{(h)}$ being the corresponding covariant derivative.

Finally, let us make some remarks on the notations and conventions. For any
vector field $\mathscr{V}^{a}$ and any scalar field $\Phi$, the map from
$\Phi$ to $\mathscr{V}^{a}$ is denoted as $\mathscr{V}^{a}[\Phi]$. On the
contrary, the action of the vector field $\mathscr{V}^{a}$ on $\Phi$ is
denoted as $\mathscr{V}(\Phi)$. It is crucial to distinguish these two
notations in the following text.

##  3 Covariant FPEs

In this section, we shall derive the FPE associated with each version of the
Langevin equation presented in [1] and try to make sense of the corresponding
probability distribution functions (PDFs). In practice, there are different
ways to obtain FPE from Langevin equation [30, 31]. To highlight the geometric
interpretation, we will adopt the diffusion operator method [32]. A brief
review of the method is presented in Appendix A, and the construction below
will be made as brief as possible in order to focus on the physical
interpretations.

###  3.1 FPE associated with LEτ

The Langevin equation LEτ is given as follows,

$\displaystyle{\rm d}\tilde{x}^{\mu}_{\tau}$
$\displaystyle=\frac{\tilde{p}^{\mu}_{\tau}}{m}{\rm d}\tau,$ (23)
$\displaystyle{\rm d}\tilde{p}^{\mu}_{\tau}$
$\displaystyle=\left[\mathcal{R}^{\mu}{}_{\mathfrak{a}}\circ_{S}{\rm
d}\tilde{w}_{\tau}^{\mathfrak{a}}+\mathcal{F}^{\mu}_{\text{add}}{\rm
d}\tau\right]+\mathcal{K}^{\mu\nu}U_{\nu}{\rm
d}\tau-\frac{1}{m}\Gamma^{\mu}{}_{\alpha\beta}\tilde{p}^{\alpha}_{\tau}\tilde{p}^{\beta}_{\tau}{\rm
d}\tau,$ (24)

and the meaning of each term is described in detail in [1]. Since the
stochastic forces arise from thermal fluctuations from the heat reservoir, it
is natural to expect that

$\mathcal{R}^{\mu}{}_{\mathfrak{a}}\to
0,\qquad\mathcal{F}^{\mu}_{\text{add}}\to 0$

in the low temperature limit.

Since LEτ preserves the mass shell condition, not all components of
$\tilde{p}^{\mu}$ could be viewed as independent, and it is appropriate to
take only $\tilde{p}^{i}$ as independent random variables. One can introduce a
corresponding probability distribution function (PDF)

$\displaystyle\Phi_{\tau}(x^{\mu},p^{i}):=\Pr[\tilde{x}^{\mu}_{\tau}=x^{\mu},\tilde{p}^{i}_{\tau}=p^{i}]$
(25)

which describes the probability for the Brownian particle to appear at the
position $x^{\mu}$ in the spacetime and meanwhile has the momentum $p^{i}$ at
the proper time $\tau$ of the Brownian particle itself. This PDF is
pathological for two reasons. First, $\Phi_{\tau}(x^{\mu},p^{i})$ depends on
two time variables $\tau$ and $x^{0}$, which makes it hard to assign a
physical interpretation; Second, $\Phi_{\tau}(x^{\mu},p^{i})$ is not a
distribution on the SoMS $\Sigma_{t}$, but rather on the full mass shell
bundle $\Gamma^{+}_{m}$. However, there is no technical obstacle which
prevents us from constructing the FPE obeyed by $\Phi_{\tau}(x^{\mu},p^{i})$.

In order to get the desired FPE, we need to construct the diffusion operator
of eq.(24). For Stratonovich type Langevin equation, the diffusion operator
can always be written in the form

$\displaystyle\mathbf{A}=\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}L_{\mathfrak{a}}L_{\mathfrak{b}}+L_{0}.$
(26)

In the case of eq.(24), we have

$\displaystyle L_{\mathfrak{a}}$
$\displaystyle=\mathcal{R}^{\mu}{}_{\mathfrak{a}}\frac{\partial}{\partial
p^{\mu}}=\mathcal{R}^{i}{}_{\mathfrak{a}}\frac{\partial}{\partial\breve{p}^{i}},$
(27) $\displaystyle L_{0}$
$\displaystyle=\frac{p^{\mu}}{m}\frac{\partial}{\partial
x^{\mu}}-\frac{1}{m}\varGamma^{\mu}{}_{\alpha\beta}p^{\alpha}p^{\beta}\frac{\partial}{\partial
p^{\mu}}+\left(\mathcal{F}^{\mu}_{\text{add}}+\mathcal{K}^{\mu\nu}U_{\nu}\right)\frac{\partial}{\partial
p^{\mu}}$
$\displaystyle=\frac{1}{m}\mathscr{L}+\left(\mathcal{F}^{i}_{\text{add}}+\mathcal{K}^{i\nu}U_{\nu}\right)\frac{\partial}{\partial\breve{p}^{i}},$
(28)

where $\mathscr{L}=p^{\mu}e_{\mu}$ is the Liouville vector field [25]. Using
the volume element of mass shell bundle eq.(14), the adjoint of coordinate
derivative operators can be obtained straightforwardly,

$\displaystyle\left(\frac{\partial}{\partial x^{\mu}}\right)^{*}$
$\displaystyle=-\frac{p_{0}}{g}\frac{\partial}{\partial
x^{\mu}}\frac{g}{p_{0}},$ (29)
$\displaystyle\left(\frac{\partial}{\partial\breve{p}^{i}}\right)^{*}$
$\displaystyle=-\frac{p_{0}}{g}\frac{\partial}{\partial\breve{p}^{i}}\frac{g}{p_{0}}=-\frac{p_{0}}{\sqrt{g}}\frac{\partial}{\partial\breve{p}^{i}}\frac{\sqrt{g}}{p_{0}}.$
(30)

With these operators, the adjoint of $L_{\mathfrak{a}}$ and $L_{0}$ can be
obtained, which read

$\displaystyle
L^{*}_{\mathfrak{a}}=-{\nabla}^{(h)}_{i}\mathcal{R}^{i}{}_{\mathfrak{a}},\qquad
L^{*}_{0}=-\frac{1}{m}\mathscr{L}-{\nabla}^{(h)}_{i}\left(\mathcal{F}^{i}_{\text{add}}+\mathcal{K}^{i\nu}U_{\nu}\right),$
(31)

where we have used $\mathscr{L}^{*}=-\mathscr{L}$. The FPE can then be written
as

$\displaystyle\partial_{\tau}\Phi_{\tau}$
$\displaystyle=\mathbf{A}^{*}\Phi_{\tau}$
$\displaystyle=\left(\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}L^{*}_{\mathfrak{a}}L^{*}_{\mathfrak{b}}+L^{*}_{0}\right)\Phi_{\tau}$
$\displaystyle=\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}{\nabla}^{(h)}_{i}\mathcal{R}^{i}{}_{\mathfrak{a}}{\nabla}^{(h)}_{j}\mathcal{R}^{j}{}_{\mathfrak{b}}\Phi_{\tau}-{\nabla}^{(h)}_{i}\left(\mathcal{F}^{i}_{\text{add}}\Phi_{\tau}+\mathcal{K}^{i\nu}U_{\nu}\Phi_{\tau}\right)-\frac{1}{m}\mathscr{L}(\Phi_{\tau})$
$\displaystyle={\nabla}^{(h)}_{i}\left[\frac{1}{2}\mathcal{D}^{ij}{\nabla}^{(h)}_{j}\Phi_{\tau}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\left(\mathcal{R}^{i}{}_{\mathfrak{a}}{\nabla}^{(h)}_{j}\mathcal{R}^{j}{}_{\mathfrak{b}}\right)\Phi_{\tau}-\mathcal{F}^{i}_{\text{add}}\Phi_{\tau}-\mathcal{K}^{i\nu}U_{\nu}\Phi_{\tau}\right]-\frac{1}{m}\mathscr{L}(\Phi_{\tau}),$
(32)

where we have introduced the diffusion tensor
$\mathcal{D}^{\mu\nu}:=\mathcal{R}^{\mu}{}_{\mathfrak{a}}\mathcal{R}^{\nu}{}_{\mathfrak{a}}$.
Defining the vector field

$\displaystyle\mathcal{I}^{a}[\Phi_{\tau}]:=\left[\frac{1}{2}\mathcal{D}^{ij}{\nabla}^{(h)}_{j}\Phi_{\tau}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\left(\mathcal{R}^{i}{}_{\mathfrak{a}}{\nabla}^{(h)}_{j}\mathcal{R}^{j}{}_{\mathfrak{b}}\right)\Phi_{\tau}-\mathcal{F}^{i}_{\text{add}}\Phi_{\tau}-\mathcal{K}^{i\nu}U_{\nu}\Phi_{\tau}\right]\left(\frac{\partial}{\partial\breve{p}^{i}}\right)^{a},$
(33)

the FPE for $\Phi_{\tau}$ can be written in more concise form

$\displaystyle\partial_{\tau}\Phi_{\tau}={\nabla}^{(h)}_{i}\mathcal{I}^{i}[\Phi_{\tau}]-\frac{1}{m}\mathscr{L}(\Phi_{\tau}).$
(34)

Eq.(34) can be viewed as a continuity equation for the PDF $\Phi_{\tau}$ and
its associated probability flow $\mathscr{J}[\Phi_{\tau}]$, which is defined
as

$\displaystyle\mathscr{J}[\Phi_{\tau}]:=\frac{\Phi_{\tau}}{m}\mathscr{L}-\mathcal{I}[\Phi_{\tau}].$
(35)

Here, the term proportional to the Liouville vector field corresponds to the
contribution from the free motion of the Brownian particle, while
$\mathcal{I}[\Phi_{\tau}]$ represents the contribution from the interaction
between the Brownian particle and the heat reservoir.

Please be reminded that we use the term “flow” instead of “current” to refer
to the spatial components of the objects which obey the continuity equation.
The term “current” is reserved for the full object, including the temporal
component. Using the definition (35), eq. (34) can be rewritten in the form

$\displaystyle\partial_{\tau}\Phi_{\tau}+\hat{\nabla}_{A}^{(\hat{h})}\mathscr{J}^{A}[\Phi_{\tau}]=0.$
(36)

Eq.(36) implies that the surface integral

$\displaystyle-\int_{\Sigma}\eta_{\Sigma_{t}}\mathscr{Z}_{A}\mathscr{J}^{A}[\Phi_{\tau}]$
(37)

should be the probability that the Brownian particle passes through the
subarea $\Sigma$ in the SoMS $\Sigma_{t}$ per unit proper time of the Brownian
particle. Although it looks puzzling to understand eq.(36) as a continuity
equation because of the presence of two time variables, this equation still
plays a key role while making connection to the alternative distribution
function to be introduced shortly.

###  3.2 FPE associated with LEt

The second Langevin equation proposed in [1], i.e. LEt, arises from a
reparametrization of LEτ. The concrete form of LEt reads

$\displaystyle{\rm d}\tilde{y}^{\mu}_{t}$
$\displaystyle=\frac{\tilde{k}^{\mu}_{t}}{m}\gamma^{-1}{\rm d}t,$ (38)
$\displaystyle{\rm d}\tilde{k}^{\mu}_{t}$
$\displaystyle=\left[\hat{\mathcal{R}}^{\mu}{}_{\mathfrak{a}}\circ_{S}{\rm
d}\tilde{W}_{t}^{\mathfrak{a}}+\hat{\mathcal{F}}^{\mu}_{\text{add}}{\rm
d}t\right]+\hat{\mathcal{K}}^{\mu\nu}U_{\nu}{\rm
d}t-\frac{1}{m}\Gamma^{\mu}{}_{\alpha\beta}\tilde{k}^{\alpha}_{t}\tilde{k}^{\beta}_{t}\gamma^{-1}{\rm
d}t,$ (39)

where $t$ represents the proper time of Alice, the prescribed observer, and

$\tilde{y}^{\mu}(t)=\tilde{x}^{\mu}(\tau(t)),\qquad\tilde{k}^{\mu}(t)=\tilde{p}^{\mu}(\tau(t)).$

$\hat{\mathcal{R}}^{\mu}{}_{\mathfrak{a}},\hat{\mathcal{K}}^{\mu\nu}$ and
$\hat{\mathcal{F}}_{\text{add}}^{\mu}$ are connected with their respective un-
hatted counterparts via

$\displaystyle\hat{\mathcal{R}}^{\mu}{}_{\mathfrak{a}}:=\gamma^{-1/2}\mathcal{R}^{\mu}{}_{\mathfrak{a}},\quad\hat{\mathcal{K}}^{\mu\nu}:=\gamma^{-1}{\mathcal{K}}^{\mu\nu},\quad\hat{\mathcal{F}}_{\text{add}}^{\mu}:=\gamma^{-1}\mathcal{F}^{\mu}_{\text{add}}-\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\mathcal{R}^{\mu}{}_{\mathfrak{a}}\mathcal{R}^{j}{}_{\mathfrak{b}}(\gamma^{-1/2}\hat{\nabla}_{j}\gamma^{-1/2}),$
(40)

and

$\displaystyle\lambda:=|\nabla
t|,\qquad\gamma(\tilde{x},\tilde{p}):=-\frac{\lambda
Z_{\mu}\tilde{p}^{\mu}}{m}.$ (41)

$\gamma(\tilde{x},\tilde{p})$ plays the role of a local Lorentz factor, i.e.
${\rm d}\tau=\gamma^{-1}{\rm d}t$, which is also random-valued because of the
random motion of the Brownian particle. The fact that the proper time $\tau$
of the Brownian particle becomes a random variable from the observer’s
perspective is the reason why the reparametrization leading from LEτ to LEt is
unavoidable.

The PDF for the Brownian particle described by eqs.(38)-(39) is

$\displaystyle\Psi_{t}(x^{\mu},p^{i}):=\Pr[\tilde{y}_{t}^{\mu}=x^{\mu},\tilde{k}_{t}^{i}=p^{i}].$
(42)

Apparently, this PDF is also a two-time distribution, just like
$\Phi_{\tau}(x^{\mu},p^{i})$ given in eq.(25), which is hard to understand
physically. However, the PDF $\Psi_{t}(x^{\mu},p^{i})$ actually encodes the
physical PDF $f(x^{\mu},p^{i})$ on $\Sigma_{t}$ in the following manner.
Recall that $\Sigma_{t}$ can be regarded as a hypersurface on the mass shell
bundle with normal vector field $\mathscr{Z}^{a}$. This relationship allows us
to introduce an invariant volume form on $\Sigma_{t}$, i.e.

$\displaystyle(\eta_{\Sigma_{t}})_{a_{1},...,a_{2d}}:=\mathscr{Z}^{a_{0}}(\eta_{\Gamma^{+}_{m}})_{a_{0},a_{1},...,a_{2d}}.$
(43)

Since $t$ is the proper time of Alice, there is no randomness in $t$,
therefore, using the co-area formula [33, 34] of geometric measure theory, we
can write

$\displaystyle\Psi_{t}(x^{\mu},p^{i})=\lambda\delta(t(x)-t)f(x^{\mu},p^{i}),$
(44)

in which $f(x^{\mu},p^{i})$ is the desired physical PDF on $\Sigma_{t}$. Let
us stress that the volume elements associated with $\Psi_{t}(x^{\mu},p^{i})$
and $f(x^{\mu},p^{i})$ are, respectively, $\eta_{\Gamma^{+}_{m}}$ and
$\eta_{\Sigma_{t}}$.

Following a similar procedure which leads to the FPE (32), we can get the FPE
for $\Psi_{t}(x^{\mu},p^{i})$, which is associated with LEt,

$\displaystyle\frac{\partial}{\partial
t}\Psi_{t}+\frac{1}{m}\mathscr{L}(\gamma^{-1}\Psi_{t})=\nabla^{(h)}_{i}\left[\frac{1}{2}\hat{\mathcal{D}}^{ij}\nabla^{(h)}_{j}\Psi_{t}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\left(\hat{\mathcal{R}}^{i}_{\mathfrak{a}}\nabla^{(h)}_{j}\hat{\mathcal{R}}^{j}_{\mathfrak{b}}\right)\Psi_{t}-\hat{\mathcal{F}}^{i}_{\text{add}}\Psi_{t}-\hat{\mathcal{K}}^{i\nu}U_{\nu}\Psi_{t}\right],$
(45)

where $\hat{\mathcal{D}}^{\mu\nu}:=\gamma^{-1}\mathcal{D}^{\mu\nu}$.

Now since

$\displaystyle\left[\frac{\partial}{\partial
t}+\gamma^{-1}\frac{p^{\mu}}{m}\frac{\partial}{\partial
x^{\mu}}\right]\delta(t(x)-t)$
$\displaystyle=\left[-1+\gamma^{-1}\frac{p^{\mu}}{m}\frac{\partial t}{\partial
x^{\mu}}\right]\delta^{\prime}(t(x)-t)$
$\displaystyle=\left[-1+\gamma^{-1}\frac{{\rm d}t}{{\rm
d}\tau}\right]\delta^{\prime}(t(x)-t)=0,$ (46)

substituting eq.(44) into the left hand side of eq.(45) yields

$\displaystyle\frac{\partial}{\partial
t}\Psi_{t}+\frac{1}{m}\mathscr{L}(\gamma^{-1}\Psi_{t})$
$\displaystyle=\delta(t(x)-t)\frac{1}{m}\mathscr{L}(\gamma^{-1}\lambda f).$
(47)

On the other hand, the first three terms in the square bracket on the right
hand side of eq.(45) can be rearranged in the form

$\displaystyle\frac{1}{2}\hat{\mathcal{D}}^{ij}\nabla^{(h)}_{j}\Psi_{t}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\left(\hat{\mathcal{R}}^{i}_{\mathfrak{a}}\nabla^{(h)}_{j}\hat{\mathcal{R}}^{j}_{\mathfrak{b}}\right)\Psi_{t}-\hat{\mathcal{F}}^{i}_{\text{add}}\Psi_{t}$
$\displaystyle\qquad=\frac{1}{2}{\mathcal{D}}^{ij}\nabla^{(h)}_{j}(\gamma^{-1}\Psi_{t})+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\left(\mathcal{R}^{i}_{\mathfrak{a}}\nabla^{(h)}_{j}{\mathcal{R}}^{j}_{\mathfrak{b}}\right)(\gamma^{-1}\Psi_{t})-{\mathcal{F}}^{i}_{\text{add}}(\gamma^{-1}\Psi_{t}).$
(48)

Therefore, the substitution of eq.(44) into eq.(45) yields

$\displaystyle\frac{1}{m}\mathscr{L}(\gamma^{-1}\lambda
f)=\nabla^{(h)}_{i}\mathcal{I}^{i}[\gamma^{-1}\lambda f],$ (49)

where $\mathcal{I}^{i}[\gamma^{-1}\lambda f]$ is defined in a similar fashion
as in eq.(34).

Notice that the FPEs (34) and (49) have a similar form. By dropping the time
derivative term $\partial_{\tau}\Phi_{\tau}$ in eq.(34) and replacing
$\Phi_{\tau}$ with $\gamma^{-1}\lambda f$, eq.(34) can be changed into
eq.(49). This is certainly not a coincidence, and we will demonstrate in the
next section how eq.(34) is intimately related to eq.(49).

##  4 Reduced FPE

In Part I of this series of research [1], we used numerical method to
investigate whether the random paths generated by LEτ and LEt produce the same
physical PDF on the SoMS $\Sigma_{t}$. The results in the example case of
$(1+1)$-dimensional Minkowski spacetime indicate that nearly identical
distributions arise from the two Langevin equations LEτ and LEt. In this
section, we will provide an analytical proof in generic spacetimes. During
this proof, we will introduce a new distribution function, $\varphi$, together
with its evolution equation, which we call the reduced FPE.

Recall from eq.(37) that the integral
$\displaystyle-\int_{\Sigma}\eta_{\Sigma_{t}}\mathscr{Z}_{A}\mathscr{J}^{A}[\Phi_{\tau}]$
represents the probability that the Brownian particle passes through the
subregion $\Sigma$ in the SoMS $\Sigma_{t}$ per unit proper time of the
Brownian particle. From the observer’s perspective, the condition “per unit
proper time of the Brownian particle” is irrelevant, the actual probability
that the Brownian particle passes through the subarea $\Sigma$ should read

$\displaystyle\Pr[\text{The particle passes through
}\Sigma]=-\int_{\mathbb{R}}{\rm
d}\tau\int_{\Sigma}\eta_{\Sigma_{t}}\mathscr{Z}_{A}\mathscr{J}^{A}[\Phi_{\tau}]$
$\displaystyle\qquad=-\int_{\Sigma}\eta_{\Sigma_{t}}\mathscr{Z}_{A}\mathscr{J}^{A}[\varphi],$
(50)

where we have introduced

$\displaystyle\varphi(x,p):=\int_{\mathbb{R}}{\rm d}\tau\,\Phi_{\tau}(x,p).$
(51)

Since $\Sigma$ is an arbitrary subregion in the SoMS $\Sigma_{t}$, the
integrand $-\mathscr{Z}_{A}\mathscr{J}^{A}[\varphi]$ in eq.(4) should be the
PDF for the intersection points of the random paths with the SoMS
$\Sigma_{t}$, i.e.

$\displaystyle{f}=-\mathscr{Z}_{A}\mathscr{J}^{A}[\varphi]=-\dfrac{1}{m}Z_{\mu}p^{\mu}\varphi=\gamma\lambda^{-1}\varphi.$
(52)

Now let us consider a scenario in which the random paths of the Brownian
particles are infinitely stretched, i.e. extending from $\tau=-\infty$ to
$\tau=\infty$. It is natural to introduce the boundary conditions

$\displaystyle\Phi_{-\infty}(x,p)=\Phi_{+\infty}(x,p)=0$ (53)

for the PDF $\Phi_{\tau}$, because otherwise $\Phi_{\tau}$ will not be
normalizable. Then, by integrating eq.(34) with respect to $\tau$, we can get

$\displaystyle\frac{1}{m}\mathscr{L}(\varphi)=\nabla^{(h)}_{i}\mathcal{I}^{i}[\varphi],$
(54)

where again $\mathcal{I}^{i}[\varphi]$ is defined in a similar way as in
eq.(34).

Since $\varphi$ differs from the true PDF $f$ by the scalar factor
$\gamma\lambda^{-1}$, it cannot be a normalized PDF. Therefore, the equation
(54) obeyed by $\varphi$ will not be referred to as FPE, but rather as reduced
FPE. Bearing in mind the relationship (52), one can easily recognize that
eq.(54) is actually identical to eq.(49). In other words, both the FPEs (34)
and (45) give rise to the same reduced FPE. This fact gives an analytical
evidence for the correctness of the numerical test presented in [1].

Some remarks are in due here.

(1) Since the reduced FPE (54) is homogeneous in $\varphi$ and there is no
need to normalize $\varphi$, there is a freedom to multiply $\varphi$ with a
constant factor which preserves eq.(54). This freedom will be used in the next
section while defining the particle number density of the Brownian particle.

(2) There is a common misconception about the role of $\Phi_{\tau}$ regarding
a particular case, i.e. the stationary distribution
$\Phi_{\tau}(x,p):=\Phi(x,p)$, which is often considered to be identical to
the equilibrium distribution for the particles of the heat reservoir, i.e. the
Jüttner distribution. Technically it is true that, when $\Phi_{\tau}$ is
independent of $\tau$, eq. (34) will take the same form as eq.(54). However,
this coincidence does not imply that $\varphi(x,p)$ is identical to the
stationary distribution $\Phi(x,p)$. There are two primary reasons for this
difference: (i) The stationary distribution is a distribution which does not
change with the time of some stationary observer, rather than of the Brownian
particle; (ii) The identification of $\varphi(x,p)$ with the stationary
distribution $\Phi(x,p)$ implies that the reduced FPE can only describe the
stationary states, whereas it can actually describe the whole relaxation
process, as will be demonstrated in Sec. 5 and Sec. 6.

Following a similar fashion with eq.(35), we can introduce a current
associated with $\varphi$,

$\mathscr{J}[\varphi]:=\frac{\varphi}{m}\mathscr{L}-\mathcal{I}[\varphi].$

Then the reduced FPE (54) could be rewritten as the current conservation
equation

$\displaystyle\hat{\nabla}^{(\hat{h})}_{A}\mathscr{J}^{A}[\varphi]=0$ (55)

on the mass shell bundle. Let us stress that $\mathscr{J}[\varphi]$ is now
interpreted as a current, rather than flow, because only a single time
variable is present in the above equation which is hidden behind the index
$A$. The conservation of the current $\mathscr{J}^{A}[\varphi]$ does not
correspond to conservation of probability, but rather to conservation of
matter. More details on this point will be presented in the next section.

##  5 Macroscopic quantities and interpretation of $\varphi(x,p)$

Let $\mathcal{S}$ be an arbitrary subregion in the configuration space
$\mathcal{S}_{t}$, and $\Sigma:=\\{(x,p)\in\Gamma^{+}_{m}|x\in\mathcal{S}\\}$
is the corresponding subregion in the SoMS. When Alice is not bound together
with the coordinate system, the proper time $t$ will be different from the
coordinate time $x^{0}$, which means that $\mathcal{S}_{t}$ is not the
coordinate hypersurface with fixed $x^{0}$, but rather a tilted hypersurface
with mixtures between $x^{0}$ and $x^{i}$. Nevertheless, since the PDF
$f(x,p)$ is by definition the probability density on $\Sigma_{t}$, and that
$\eta_{\Sigma_{t}}=\eta_{\mathcal{S}_{t}}\wedge\eta_{(\Gamma^{+}_{m})_{x}}$,
we can calculate the probability for the Brownian particle to appear in
$\mathcal{S}$ at the time $t$ as

$\displaystyle\Pr[\mathcal{S}]$
$\displaystyle=\Pr[\Sigma]=\int_{\Sigma}\eta_{\Sigma_{t}}f$
$\displaystyle=-\int_{\Sigma}\eta_{\Sigma_{t}}\mathscr{Z}_{a}\mathscr{J}^{a}[\varphi]=-\int_{\mathcal{S}}\eta_{\mathcal{S}_{t}}Z_{\mu}\int_{(\Gamma_{m}^{+})_{x}}\eta_{(\Gamma_{m}^{+})_{x}}\dfrac{p^{\mu}}{m}\varphi.$
(56)

The change from $f$ to $\varphi$ in the integrand of the last equality
reflects the tiltedness of $\mathcal{S}_{t}$ in the spacetime.

Now consider the case with $N$ non-interacting Brownian particles coexisting
in the same heat reservoir. By putting an extra factor $N$ in front of the
integrals in eq.(56) and enlarging $\Sigma$ into $\Sigma_{t}$, we should get
$N$ as the final value of the integration. Therefore, by dropping the integral
over $\mathcal{S}$, we get the particle number density in the configuration
space

$\displaystyle\bar{n}=-NZ_{\mu}\int_{(\Gamma_{m}^{+})_{x}}\eta_{(\Gamma_{m}^{+})_{x}}\dfrac{p^{\mu}}{m}\varphi.$
(57)

Recall that the particle number density should be defined as

$\bar{n}=-Z_{\mu}N^{\mu}[\varphi],$

wherein $N^{\mu}$ denotes the particle number current. At present, the
particle number current reads

$\displaystyle
N^{\mu}[\varphi]=\int_{(\Gamma_{m}^{+})_{x}}\eta_{(\Gamma_{m}^{+})_{x}}\dfrac{p^{\mu}}{m}N\varphi.$
(58)

It is remarkable that the above form of the particle number current is
identical to that given in relativistic kinetic theory (except for the
constant factor $N$), provided that $\varphi$ is identified with the 1PDF
which obeys the relativistic Boltzmann equation. This resemblance reminds us
that there may be some deep connections between the approaches of relativistic
stochastic mechanics and of relativistic kinetic theory.

Since there is no chemical reactions between the Brownian particles, the
particle current must be conserved. This fact can be proved using Stokes’
theorem. Let $V$ be an region in the spacetime manifold $\mathcal{M}$, and
$\Gamma=\\{(x,p)\in\Gamma^{+}_{m}|\ x\in V\\}$ is the corresponding region on
the mass shell bundle. Let $n^{\mu}$ be the unit normal vector field of
$\partial V$ which induces the unit normal vector field $\mathscr{N}$ of
$\partial\Gamma$. The Stokes’ theorem on the mass shell bundle reads

$\displaystyle\int_{\Gamma}\eta_{\Gamma^{+}_{m}}\hat{\nabla}^{(\hat{h})}_{A}\mathscr{J}^{A}[\varphi]=\int_{\partial\Gamma}\eta_{\partial\Gamma}\mathscr{N}_{A}\mathscr{J}^{A}[\varphi].$
(59)

Using the fact that $\partial\Gamma=\\{(x,p)\in\Gamma^{+}_{m}|\ x\in\partial
V\\}$ and that $\mathscr{N}^{a}=n^{\mu}e_{\mu}{}^{a}$, we can rewrite the
above equation as

$\displaystyle\int_{V}\eta_{\mathcal{M}}\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\hat{\nabla}^{(\hat{h})}_{A}\mathscr{J}^{A}[\varphi]=\int_{\partial
V}\eta_{\partial
V}\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\mathscr{N}_{A}\mathscr{J}^{A}[\varphi]$
$\displaystyle\qquad=\frac{1}{N}\int_{\partial V}\eta_{\partial
V}n_{\mu}N^{\mu}[\varphi]=\frac{1}{N}\int_{V}\eta_{\mathcal{M}}\nabla_{\mu}N^{\mu}[\varphi],$
(60)

where $\nabla_{\mu}$ denotes the usual covariant derivative on the spacetime
manifold. Due to the arbitrariness of $V$, we can drop the integration with
respect to the measure $\eta_{\mathcal{M}}$ and get

$\displaystyle\nabla_{\mu}N^{\mu}[\varphi]=N\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\hat{\nabla}^{(\hat{h})}_{A}\mathscr{J}^{A}[\varphi]=0,$
(61)

which means that $N^{\mu}[\varphi]$ is a conservation current on the
spacetime.

The energy of a single Brownian particle measured by Alice is defined as

$\displaystyle E:=-p^{\mu}Z_{\mu}.$ (62)

Thus the single particle contribution to the average energy flux through the
subregion $\Sigma$ in the SoMS $\Sigma_{t}$ should read

$\displaystyle\bar{E}[\Sigma]$
$\displaystyle:=\int_{\Sigma}\eta_{\Sigma}\mathscr{Z}_{A}\mathscr{J}^{A}[\varphi]p^{\mu}Z_{\mu}=\int_{\mathcal{S}}\eta_{\mathcal{S}}Z_{\mu}Z_{\nu}\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\frac{p^{\nu}p^{\mu}}{m}\varphi.$
(63)

The second integration factor, i.e.

$\displaystyle
T^{\mu\nu}[\varphi]:=\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\frac{p^{\nu}p^{\mu}}{m}\varphi,$
(64)

is recognized to be the single particle contribution to the energy-momentum
tensor, and

$\displaystyle\rho:=Z_{\mu}Z_{\nu}T^{\mu\nu}[\varphi]=Z_{\mu}Z_{\nu}\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\frac{p^{\nu}p^{\mu}}{m}\varphi$
(65)

is naturally the single particle contribution to the energy density.

The single particle contribution to the average energy-momentum vector of the
Brownian particle is defined as

$E^{\mu}[\varphi]:=-Z_{\nu}T^{\mu\nu}[\varphi].$

In general, $E^{\mu}[\varphi]$ is non-conserved because of the joint effects
of gravitational work and heat transfer from the heat reservoir. Since

$\displaystyle-\int_{V}\eta_{\mathcal{M}}\nabla_{\mu}(Z_{\nu}T^{\mu\nu}[\varphi])=-\int_{\partial
V}\eta_{\partial V}n_{\mu}Z_{\nu}T^{\mu\nu}[\varphi]=-\int_{\partial
V}\eta_{\partial
V}\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}(Z_{\nu}p^{\nu})n_{\mu}\frac{p^{\mu}}{m}\varphi$
$\displaystyle\qquad=\int_{\partial V}\eta_{\partial
V}\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\mathscr{N}_{A}\mathscr{J}^{A}[\varphi]E=\int_{V}\eta_{\mathcal{M}}\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\hat{\nabla}^{(\hat{h})}_{A}(E\mathscr{J}^{A}[\varphi]),$
(66)

we have

$\displaystyle\nabla_{\mu}E^{\mu}[\varphi]$
$\displaystyle=-\nabla_{\mu}\left(Z_{\nu}T^{\mu\nu}[\varphi]\right)=-\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\hat{\nabla}^{(\hat{h})}_{A}\left(p^{\nu}Z_{\nu}\mathscr{J}^{A}[\varphi]\right)$
$\displaystyle=-\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\mathscr{J}^{A}[\varphi]\hat{\nabla}^{(\hat{h})}_{A}\left(p^{\nu}Z_{\nu}\right)$
$\displaystyle=-\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\left(\frac{\varphi}{m}\mathscr{L}(Z_{\nu}p^{\nu})-Z_{\nu}\mathcal{I}^{\nu}[\varphi]\right)$
$\displaystyle=-T^{\mu\nu}[\varphi]\nabla_{\mu}Z_{\nu}+Z_{\nu}\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\mathcal{I}^{\nu}[\varphi].$
(67)

The first term on the right hand side of eq.(67) is the average gravitational
power acting on the Brownian particle, i.e.

$\displaystyle P_{\text{grav}}[\varphi]$
$\displaystyle:=-T^{\mu\nu}[\varphi]\nabla_{\mu}Z_{\nu}=\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}\mathcal{P}_{\text{grav}}(Z)\varphi,$
(68)

where

$\displaystyle\mathcal{P}_{\text{grav}}(Z)=-\frac{p^{\mu}p^{\nu}}{m}\nabla_{\mu}Z_{\nu}$
(69)

is the the gravitational power along a single trajectory of the particle [35]
as measured by Alice. Thus the second term on the right hand side of eq.(67)
should be interpreted as the heat transfer rate from the heat reservoir,

$\displaystyle
Q[\varphi]:=\int_{(\Gamma^{+}_{m})_{x}}\eta_{(\Gamma^{+}_{m})_{x}}Z_{\nu}\mathcal{I}^{\nu}[\varphi]=-Z_{\nu}\nabla_{\mu}T^{\mu\nu}[\varphi].$
(70)

In the end, we have

$\displaystyle\nabla_{\mu}E^{\mu}[\varphi]=P_{\text{grav}}[\varphi]+Q[\varphi],$
(71)

which is reminiscent to the first law of thermodynamics, but is presented in
terms of the divergence of the average energy-momentum vector, the
gravitational power and the heat transfer rate. Please note that the last
equation is valid for any observer. However, for different observers, the
values of $P_{\text{grav}}[\varphi]$ and $Q[\varphi]$ can be different.

##  6 Relativistic equilibrium state and Einstein relation

So far, we have not paid a word on the state of the heat reservoir, except for
the implicit assumption of an equilibrium state. This does not make any harm
to the formal construction of FPE. However, when the solution to the FPE is
concerned, an explicit description for the equilibrium state of the reservoir
becomes inevitable.

As mentioned in the introduction, there are two basic assumptions in the
description for the Brownian motion of a heavy particle in a heat reservoir.
In the relativistic context, these assumptions need to be re-examined.

The first problem one encounters is the proper definition for the equilibrium
state of the reservoir. It is well known that, in the presence of gravity, a
macroscopic system cannot reach the thermodynamic equilibrium in the usual
sense, i.e. the one with uniform temperature and chemical potential. The
reason lies in that there is a bilateral interaction between thermal and
gravitational effects. On the one hand, thermal energy as a form of energy
should generate gravity; on the other hand, gravity, as a long range
interaction, has nontrivial impact on the relaxation process, leading to the
final state with non-uniform temperature and chemical potential.

Meanwhile, the choice of observer also brings about some subtleties in
describing the state of the heat reservoir. The importance of the role of
observer can be revealed in two different aspects: i) According to the
equivalence principle, gravity is locally indistinguishable from acceleration.
Therefore, the strength of the gravitational force experienced by the observer
and by the macroscopic system being observed could be different, provided the
amounts of accelerations are different. ii) There has long been a dispute on
the of relativistic transformation rules of thermodynamic parameters, mostly
about the transform of temperature, but also include the transform of chemical
potential. According to the results of [36], these transformation rules are
related to the choice of observer.

Due to the above reasons, we need to answer the following questions in order
to clarify the state of the heat reservoir:

Q1.

What is a relativistic equilibrium state? Is the equilibrium state observer-
dependent?

Q2.

What is the equilibrium distribution for particles of the heat reservoir?

Q3.

Is this distribution observer-dependent?

Fortunately the answers to these questions can be inferred from the studies on
relativistic kinetic theory. To answer Q1, let us infer that equilibrium
states could be viewed as the final states of relaxation processes, and a
system carrying out a relaxation process should not care about who is
observing it. Therefore, the final state of the relaxation process should not
be affected by the choice of observer. Given an isolated system, there can
only be one intrinsic equilibrium state, i.e. the state in detailed balance,
which is characterized by several macroscopic features, including the absence
of entropy production rate and vanishing collision integral in the Boltzmann
equation.

From the point of view of the comoving observer, Bob, the equilibrium state
has one extra feature, i.e. the absence of transport flows. By definition, the
proper velocity of Bob is identical to the proper velocity $U^{\mu}$ of an
element of the heat reservoir viewed as a relativistic fluid. The same
$U^{\mu}$ also appeared in the damping force term in the Langevin equation.
According to [37, 38], the driving forces for the relativistic transports are
the generalized gradients for the temperature and chemical potential, which
are dependent on the proper velocity of the observer. For the comoving
observer Bob, the generalized gradients for the temperature and chemical
potential read

$\displaystyle\mathcal{D}_{\nu}T_{\rm B}=\nabla_{\nu}T_{\rm B}+T_{\rm
B}U^{\rho}\nabla_{\rho}U_{\nu}=0,\qquad\mathcal{D}_{\nu}\mu_{\rm
B}=\nabla_{\nu}\mu_{\rm B}+\mu_{\rm B}U^{\rho}\nabla_{\rho}U_{\nu}=0,$ (72)

where $T_{\rm B}$ and $\mu_{\rm B}$ respectively are the temperature and
chemical potential of the heat reservoir measured by Bob. One immediately sees
that the ordinary gradients $\nabla_{\nu}T_{\rm B}$ and $\nabla_{\nu}\mu_{\rm
B}$ are nonzero, unless Bob undergoes geodesic motion, i.e.
$U^{\rho}\nabla_{\rho}U_{\nu}=0$. In the latter case, $T_{\rm B}$ and
$\mu_{\rm B}$ becomes uniform, which is fully consistent with the definition
of equilibrium state in the non-relativistic thermodynamics.

The answer to Q2 is also provided by relativistic kinetic theory, and the
explicit 1PDF for the heat reservoir is given by the Jüttner-like
distribution[39]

$\displaystyle\varphi_{\rm HR}(x,p)=\frac{\mathfrak{g}}{\mathrm{e}^{\alpha-
B_{\mu}p^{\mu}}-\varsigma}=\frac{\mathfrak{g}}{\mathrm{e}^{\alpha-
U_{\mu}p^{\mu}/T_{\rm
B}}-\varsigma}=\frac{\mathfrak{g}}{\mathrm{e}^{(\varepsilon_{\rm B}-\mu_{\rm
B})/T_{\rm B}}-\varsigma}$ (73)

provided that the background spacetime is stationary, where $\varsigma=0,\pm
1$, $\mathfrak{g}$ denotes the quantum degeneracy, $\varepsilon_{\rm
B}=-U_{\mu}p^{\mu}$ is the energy of the particle measured by Bob, $\mu_{\rm
B}$ is the chemical potential of the heat reservoir, and $\alpha=-{\mu_{\rm
B}}/{T_{\rm B}}$ is a constant in spacetime. In order that the distribution
(73) indeed describes a state in detailed balance, the vector field
$B^{\mu}=U^{\mu}/T_{\rm B}$ is required to be timelike Killing, i.e.

$\displaystyle\nabla_{(\mu}B_{\nu)}=0.$ (74)

The existence of a timelike Killing field implies that the underlying
spacetime needs to be stationary.

We assume that the heat reservoir is consisted of purely classical particles.
In this case, the above 1PDF becomes the standard Jüttner distribution

$\displaystyle\varphi_{\rm HR}(x,p)=\mathrm{e}^{-\alpha+U_{\mu}p^{\mu}/T_{\rm
B}}=\mathrm{e}^{(\mu_{\rm B}-\varepsilon_{\rm B})/T_{\rm B}}.$ (75)

The 1PDF $\varphi_{\rm HR}(x,p)$ as presented in the form (75) contains the
proper velocity $U^{\mu}$ of Bob and the temperature $T_{\rm B}$ measured by
Bob, thus it is explicitly dependent on the choice of observer. This answers
Q3. It is remarkable that the distribution (75) has the same form as the non-
relativistic Boltzmann distribution. However, due to eq.(72), the above
distribution is in fact different from the Boltzmann distribution, because
$T_{\rm B}$ and $\mu_{\rm B}$ are now non-uniform.

It is interesting to ask what the distribution (75) would look like from the
point of view of Alice. To answer this question, let us remind that all
measurements in curved spacetime must be made on the spot. Therefore, to
consider the distributions of the same particles, Alice and Bob must appear at
the same spacetime event, and their proper velocities can differ by at most a
local Lorentz boost. Let $\gamma_{\rm AB}$ denotes the relative Lorentz factor
between Alice and Bob. Then the proper velocity $U^{\mu}$ of Bob can be
expressed as

$\displaystyle U^{\mu}=\gamma_{\rm
AB}\left(Z^{\mu}+z^{\mu}\right),\quad\gamma_{\rm AB}=-U_{\mu}Z^{\mu},\quad
z^{\mu}Z_{\mu}=0,$ (76)

The energy of the particle observed by Alice reads $\varepsilon_{\rm
A}=-Z_{\mu}p^{\mu}$. Denoting the temperature and chemical potential of the
heat reservoir measured by Alice as $T_{\rm A}$ and $\mu_{\rm A}$
respectively, we get, by inserting the eq.(76) into eq.(75), the following
distribution,

$\displaystyle\varphi_{\rm HR}(x,p)$
$\displaystyle=\mathrm{e}^{-\alpha+\gamma_{\rm
AB}(Z_{\mu}+z_{\mu})p^{\mu}/T_{\rm B}}=\mathrm{e}^{[\mu_{\rm B}+\gamma_{\rm
AB}(Z_{\mu}+z_{\mu})p^{\mu}]/T_{\rm B}}$
$\displaystyle=\mathrm{e}^{[(\gamma_{\rm AB})^{-1}\mu_{\rm
B}+Z_{\mu}p^{\mu}+z_{\mu}p^{\mu}]/T_{\rm A}}=\mathrm{e}^{(\mu_{\rm
A}-\varepsilon_{\rm A}+z_{\mu}p^{\mu})/T_{\rm A}},$ (77)

where the temperatures and the chemical potentials measured by both observers
are related as [36]

$\displaystyle T_{\rm A}=(\gamma_{\rm AB})^{-1}T_{\rm B},\qquad\mu_{\rm
A}=(\gamma_{\rm AB})^{-1}\mu_{\rm B}.$ (78)

Now let us proceed with the second basic assumption for the Brownian motion
un-altered, hence the the probability distribution for the Brownian particle
should approach the same form as the 1PDF for the heat reservoir after
sufficient long time, i.e.

$\displaystyle\varphi(x,p)\to\varphi_{\rm
HR}(x,p)=\mathrm{e}^{-\alpha+U_{\mu}p^{\mu}/T_{\rm B}}\quad{\rm as}\quad
t\to\infty.$ (79)

The above assumption also implies that the long time limit of the heat
transfer rate $Q[\varphi]$ should approach zero, because for the Jüttner
distribution $\varphi$, we always have $\nabla_{\mu}T^{\mu\nu}[\varphi]=0$ and
thus $Q[\varphi]=-Z_{\nu}\nabla_{\mu}T^{\mu\nu}[\varphi]=0$. This result is
independent of $Z_{\mu}$. It is worth noting that the condition $Q[\varphi]=0$
does not necessarily imply $Z_{\mu}\mathcal{I}^{\mu}[\varphi]=0$. When the
latter fails to vanish, it means that the Brownian particle is more likely to
absorb heat from in some states and is more likely to transfer heat to the
heat reservoir in some other states. Although the heat transfer between
different micro states cancels out, this may lead to a deviation from detailed
balance in the transition probabilities between different micro states,
causing a change in the momentum distribution of the Brownian particle.
Therefore, the detailed thermal equilibrium between the Brownian particle and
the heat reservoir should be given by the stronger condition
$Z_{\mu}\mathcal{I}^{\mu}[\varphi]=0$. Due to the arbitrariness in the choice
of $Z^{\mu}$, this condition can be further reduced to
$\mathcal{I}[\varphi]=0$, i.e.

$\displaystyle\frac{1}{2}\mathcal{D}^{ij}\nabla^{(h)}_{j}\varphi+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\left(\mathcal{R}^{i}{}_{\mathfrak{a}}\nabla^{(h)}_{j}\mathcal{R}^{j}{}_{\mathfrak{b}}\right)\varphi-\mathcal{F}^{i}_{\text{add}}\varphi-\mathcal{K}^{i\nu}U_{\nu}\varphi=0.$
(80)

Inserting eq.(79) into eq.(80) we have

$\displaystyle\mathcal{F}^{\mu}_{\text{add}}=\left[\frac{1}{2T_{\rm
B}}\mathcal{D}^{\mu\nu}-\mathcal{K}^{\mu\nu}\right]U_{\nu}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\mathcal{R}^{\mu}{}_{\mathfrak{a}}\nabla^{(h)}_{j}\mathcal{R}^{j}{}_{\mathfrak{b}}.$
(81)

In the low temperature limit, both $\mathcal{F}^{\mu}_{\text{add}}$ and
$\displaystyle\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\mathcal{R}^{\mu}{}_{\mathfrak{a}}\nabla^{(h)}_{j}\mathcal{R}^{j}{}_{\mathfrak{b}}$
tends to vanish at least as $\mathcal{O}(T_{\rm B})$. Therefore we get

$\displaystyle\mathcal{D}^{\mu\nu}=2T_{\rm
B}\mathcal{K}^{\mu\nu}+\mathcal{O}(T_{\rm B}^{2}),$ (82)

where the extra $\mathcal{O}(T_{\rm B}^{2})$ term is dependent on the choice
of damping model. When appropriate damping model is taken, e.g. like in [40],
this term can be removed completely, yielding

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

This relation is the general relativistic analogue of the celebrated Einstein
relation.

As a simple intuitive example case, let us consider the isotropic damping
model in which the diffusion tensor and tensorial damping coefficients are
given as

$\displaystyle\mathcal{K}^{\mu\nu}=\kappa\Delta^{\mu\nu}(p),\qquad\mathcal{D}^{\mu\nu}=D\Delta^{\mu\nu}(p),$
(84)

where $k$ and $D$ are both scalar functions, i.e. the scalar friction
coefficient and diffusion coefficient respectively. Then the Einstein relation
will degenerate into

$\displaystyle D=2\kappa T_{\rm B},$ (85)

which is formally identical to that arises from non-relativistic linear
response theory, except that $T_{\rm B}$ now could have a nonvanishing
ordinary gradient because of eq.(72). This result suggests that linear
response theory should still hold in the relativistic context, at least
locally.

When the Einstein relation (83) holds precisely, we have

$\displaystyle\mathcal{F}^{\mu}_{\text{add}}=\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\mathcal{R}^{\mu}_{\mathfrak{a}}\nabla^{(h)}_{j}\mathcal{R}^{j}_{\mathfrak{b}}.$
(86)

This result has already been adopted in [1] while constructing the general
covariant Langevin equations. Inserting eq.(86) into the definition of
$\mathcal{I}[\varphi]$ yields

$\displaystyle\mathcal{I}[\varphi]=\left[\frac{1}{2}\mathcal{D}^{ij}\nabla^{(h)}_{j}\varphi-\mathcal{K}^{i\nu}U_{\nu}\varphi\right]\frac{\partial}{\partial\breve{p}^{i}}.$
(87)

This formula provides a physical image of how the Brownian particle reaches
equilibrium after long time of relaxation. The damping force causes a heat
transfer from the Brownian particle to the heat reservoir, while the
stochastic force causes a heat transfer form the heat reservoir to the
Brownian particle. After long time of relaxation, the damping and stochastic
forces balances each other in the statistical sense.

As a final remark, let us mention that, due to the transformation rule (78),
the Einstein relation rewritten in terms of the temperature measured by Alice
should read

$\mathcal{D}^{\mu\nu}=2\gamma_{\rm AB}T_{\rm A}\mathcal{K}^{\mu\nu}.$

##  7 Conclusion

The major results of the present work can be summarized as follows.

1) The general covariant FPEs associated with both versions of the general
relativistic Langevin equation proposed in Ref.[1] are presented, both give
rise to the same reduced FPE obeyed by the 1PDF $\varphi(x,p)$ for the
Brownian particle. The relationship between different distribution functions
is clarified.

2) Several important macroscopic quantities and the quantitative relationships
between them are obtained with the aid of the 1PDF which obeys the reduced
FPE. These quantities and relationships reveal a close connection between the
approaches of stochastic mechanics and relativistic kinetic theory.

3) The meaning of the relativistic equilibrium state of the heat reservoir is
properly addressed, and, by assuming that the long time relaxation result for
the 1PDF of the Brownian particle should be identical to the 1PDF of the heat
reservoir, we derive a general covariant version of the Einstein relation.

These results resolve several common confusions which exist in the literature.
Moreover, we hope to use these results as the starting point for further
exploring some important subjects in relativistic macroscopic systems, e.g.
the origin of irreversibility in relativistic systems, the area law of near
horizon entropies, etc. More on these topics will come about later.

## Appendex A: Diffusion operator approach to the FPE

In order to derive the FPE from a stochastic differential equation (SDE), we
need to use Ito’s lemma to calculate the differential of an arbitrary scalar
function, and perform integration by parts twice. When the SDE is defined on a
manifold, this procedure can be very complicated.

There is a simpler approach, i.e. the diffusion operator approach [32], for
obtaining the FPE on a manifold. Here we give a brief review of this
alternative method.

The Ito type SDE on Riemannian manifold or Pseudo-Riemannian manifold $(M,g)$
can be written as

$\displaystyle{\rm d}\tilde{X}^{\mu}_{t}=F^{\mu}{\rm
d}t+C^{\mu}{}_{\mathfrak{a}}\circ_{I}{\rm d}\tilde{w}_{t}^{\mathfrak{a}}.$
(88)

Let $h$ be an arbitrary scalar field on $M$, then the time differential of
$\tilde{h}_{t}:=h(\tilde{X}_{t})$ can be derived by Ito’s lemma:

$\displaystyle{\rm d}\tilde{h}_{t}=\left[\frac{\partial h}{\partial
x^{\mu}}F^{\mu}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\frac{\partial^{2}h}{\partial
x^{\mu}\partial
x^{\nu}}C^{\mu}{}_{\mathfrak{a}}C^{\nu}{}_{\mathfrak{b}}\right]{\rm
d}t+\frac{\partial h}{\partial x^{\mu}}C^{\mu}{}_{\mathfrak{a}}\circ_{I}{\rm
d}\tilde{w}_{t}^{\mathfrak{a}}.$ (89)

Therefore, the expectation of ${\rm d}\tilde{h}_{t}$ is

$\displaystyle\langle{\rm d}\tilde{h}_{t}\rangle=\left\langle\frac{\partial
h}{\partial
x^{\mu}}F^{\mu}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\frac{\partial^{2}h}{\partial
x^{\mu}\partial
x^{\nu}}C^{\mu}{}_{\mathfrak{a}}C^{\nu}{}_{\mathfrak{b}}\right\rangle{\rm
d}t.$ (90)

This means $\langle\tilde{h}_{t}\rangle$ is differentiable with respect to
time in spite of the fact that $\tilde{h}_{t}$ isn’t differentiable. Defining
the diffusion operator as

$\displaystyle\mathbf{A}=F^{\mu}\frac{\partial}{\partial
x^{\mu}}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}C^{\mu}{}_{\mathfrak{a}}C^{\nu}{}_{\mathfrak{b}}\frac{\partial^{2}}{\partial
x^{\mu}\partial x^{\nu}},$ (91)

the derivative of $\langle\tilde{h}_{t}\rangle$ can be written as

$\displaystyle\frac{{\rm d}}{{\rm
d}t}\langle\tilde{h}_{t}\rangle=\langle\mathbf{A}\tilde{h}_{t}\rangle.$ (92)

Let $\Phi_{t}(x):=\Pr[\tilde{X}_{t}=x]$ be a PDF associated with the invariant
volume element $\sqrt{g}{\rm d}^{n}x$ of $M$, above equation actually means

$\displaystyle\frac{{\rm d}}{{\rm d}t}\int_{M}h\Phi_{t}\sqrt{g}{\rm
d}^{n}x=\int_{M}h\partial_{t}\Phi_{t}\sqrt{g}{\rm
d}^{n}x=\int_{M}\Phi_{t}\mathbf{A}h\sqrt{g}{\rm
d}^{n}x=\int_{M}(\mathbf{A}^{*}\Phi_{t})h\sqrt{g}{\rm d}^{n}x,$ (93)

where $\mathbf{A}^{*}$ is the adjoint of $\mathbf{A}$. Since $h$ is arbitrary,
the above equation implies

$\displaystyle\partial_{t}\Phi_{t}=\mathbf{A}^{*}\Phi_{t},$ (94)

which is the FPE associated with the SDE (88).

There are four rules for computing the adjoint operator:

  1. 1.

$(\mathbf{A}+\mathbf{B})^{*}=\mathbf{A}^{*}+\mathbf{B}^{*}$.

  2. 2.

$(\mathbf{AB})^{*}=\mathbf{B}^{*}\mathbf{A}^{*}$.

  3. 3.

$\displaystyle\left(\frac{\partial}{\partial
x^{\mu}}\right)^{*}=-\frac{1}{\sqrt{g}}\frac{\partial}{\partial
x^{\mu}}\sqrt{g}$, where the right hand side needs to be understood as a right
associative operator.

  4. 4.

$(F^{\mu})^{*}=F^{\mu}$.

Using these rules, the adjoint of the diffusion operator (91) is evaluated to
be

$\displaystyle\mathbf{A}^{*}$
$\displaystyle=-\frac{1}{\sqrt{g}}\frac{\partial}{\partial
x^{\mu}}\sqrt{g}F^{\mu}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\frac{1}{\sqrt{g}}\frac{\partial^{2}}{\partial
x^{\mu}\partial
x^{\nu}}\sqrt{g}C^{\mu}{}_{\mathfrak{a}}C^{\nu}{}_{\mathfrak{b}}.$ (95)

Since the Stratonovich type SDE

$\displaystyle{\rm d}\tilde{X}^{\mu}=F^{\mu}{\rm
d}t+C^{\mu}{}_{\mathfrak{a}}\circ_{S}{\rm d}\tilde{w}^{\mathfrak{a}}$ (96)

is equivalent to the Ito type SDE

$\displaystyle{\rm
d}\tilde{X}^{\mu}=\left(F^{\mu}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}C^{\nu}{}_{\mathfrak{a}}\frac{\partial}{\partial
x^{\nu}}C^{\mu}{}_{\mathfrak{b}}\right){\rm
d}t+C^{\mu}{}_{\mathfrak{a}}\circ_{I}{\rm d}\tilde{w}^{\mathfrak{a}},$ (97)

the corresponding diffusion operation reads

$\displaystyle\mathbf{A}$
$\displaystyle=\left(F^{\mu}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}C^{\nu}{}_{\mathfrak{a}}\frac{\partial}{\partial
x^{\nu}}C^{\mu}{}_{\mathfrak{b}}\right)\frac{\partial}{\partial
x^{\mu}}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}C^{\nu}{}_{\mathfrak{a}}C^{\mu}{}_{\mathfrak{b}}\frac{\partial^{2}}{\partial
x^{\mu}\partial x^{\nu}}$ $\displaystyle=F^{\mu}\frac{\partial}{\partial
x^{\mu}}+\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}C^{\nu}{}_{\mathfrak{a}}\frac{\partial}{\partial
x^{\nu}}C^{\mu}{}_{\mathfrak{b}}\frac{\partial}{\partial x^{\mu}}.$ (98)

Introducing the vector fields

$\displaystyle L_{0}=F^{\mu}\frac{\partial}{\partial x^{\mu}}\qquad
L_{\mathfrak{a}}=C^{\mu}{}_{\mathfrak{a}}\frac{\partial}{\partial x^{\mu}},$
(99)

the diffusion operation can be written as simpler form

$\displaystyle\mathbf{A}=\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}L_{\mathfrak{a}}L_{\mathfrak{b}}+L_{0}.$
(100)

It is easy to see that $L_{0}$ provides the drift term of FPE and
$L_{\mathfrak{a}}$ provides the diffusion term. Notice that the adjoint of the
coordinate derivative operator looks like the covariant divergence operator
when acting on a vector field. Therefore, the action of the adjoint of
$\mathbf{A}$ on the PDF becomes

$\displaystyle\mathbf{A}^{*}\Phi_{t}$
$\displaystyle=\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}L^{*}_{\mathfrak{a}}L^{*}_{\mathfrak{b}}\Phi_{t}+L^{*}_{0}\Phi_{t}$
$\displaystyle=\frac{\delta^{\mathfrak{a}\mathfrak{b}}}{2}\nabla_{\mu}(C^{\mu}{}_{\mathfrak{a}}(\nabla_{\nu}C^{\nu}{}_{\mathfrak{b}}\Phi_{t}))-\nabla_{\mu}(F^{\mu}\Phi_{t}).$
(101)

Inserting this result into eq.(94) gives rise to the Fokker-Planck equation
associated with the Stratonovich type SDE (96).

## Acknowledgement

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

## Data Availability Statement

This research has no associated data.

## Declaration of competing interest

The authors declare no competing interest.

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