Relativistic Stochastic Mechanics II: Reduced Fokker-Planck Equation in Curved Spacetime

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.

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

$$\begin{aligned} \textrm{d}{\tilde{X}}^\mu _t=F^\mu \textrm{d}t+ C^\mu {}_{\mathfrak {a}} \circ _I \textrm{d}\tilde{w}_t^{\mathfrak {a}}. \end{aligned}$$

(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:

$$\begin{aligned} \textrm{d}{\tilde{h}}_t=\left[ \frac{\partial h}{\partial x^{\mu }} F^{\mu } +\frac{\delta ^{\mathfrak {a}\mathfrak {b}}}{2} \frac{\partial ^2 h}{\partial x^{\mu } \partial x^{\nu }} C^{\mu }{}_{\mathfrak {a}} C^{\nu }{}_{\mathfrak {b}} \right] \textrm{d}t + \frac{\partial h}{\partial x^\mu }C^{\mu }{}_{\mathfrak {a}}\circ _I \textrm{d}\tilde{w}_t^\mathfrak {a}. \end{aligned}$$

(89)

Therefore, the expectation of $\textrm{d}{\tilde{h}}_t$ is

$$\begin{aligned} \langle \textrm{d}{\tilde{h}}_t\rangle =\left\langle \frac{\partial h}{\partial x^{\mu }} F^{\mu } +\frac{\delta ^{\mathfrak {a}\mathfrak {b}}}{2} \frac{\partial ^2 h}{\partial x^{\mu } \partial x^{\nu }} C^{\mu }{}_{\mathfrak {a}} C^{\nu }{}_{\mathfrak {b}} \right\rangle \textrm{d}t . \end{aligned}$$

(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

$$\begin{aligned} {\textbf{A}}=F^\mu \frac{\partial }{\partial x^\mu } +\frac{\delta ^{\mathfrak {a}\mathfrak {b}}}{2}C^\mu {}_{\mathfrak {a}} C^\nu {}_{\mathfrak {b}}\frac{\partial ^2}{\partial x^\mu \partial x^\nu }, \end{aligned}$$

(91)

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

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\langle {\tilde{h}}_t\rangle =\langle {\textbf{A}} \tilde{h}_t \rangle . \end{aligned}$$

(92)

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

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\int _M h \Phi _t \sqrt{g} \textrm{d}^n x =\int _M h \partial _t \Phi _t \sqrt{g} \textrm{d}^n x =\int _M \Phi _t {\textbf{A}} h \sqrt{g} \textrm{d}^n x =\int _M ({\textbf{A}}^* \Phi _t) h \sqrt{g} \textrm{d}^n x, \end{aligned}$$

(93)

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

$$\begin{aligned} \partial _t \Phi _t={\textbf{A}}^* \Phi _t, \end{aligned}$$

(94)

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

There are four rules for computing the adjoint operator:

• 1. $({\textbf{A}}+{\textbf{B}})^*={\textbf{A}}^*+{\textbf{B}}^*$.

• 2. $(\textbf{AB})^*={\textbf{B}}^*{\textbf{A}}^*$.

• 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. $(F^\mu )^*=F^\mu$.

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

$$\begin{aligned} {\textbf{A}}^*&=-\frac{1}{\sqrt{g}}\frac{\partial }{\partial x^\mu }\sqrt{g} F^\mu +\frac{\delta ^{\mathfrak {a}\mathfrak {b}}}{2}\frac{1}{\sqrt{g}}\frac{\partial ^2}{\partial x^\mu \partial x^\nu } \sqrt{g} C^\mu {}_{\mathfrak {a}} C^\nu {}_{\mathfrak {b}}. \end{aligned}$$

(95)

Since the Stratonovich type SDE

$$\begin{aligned} \textrm{d}{\tilde{X}}^\mu =F^\mu \textrm{d}t+ C^\mu {}_{\mathfrak {a}} \circ _S \textrm{d}{\tilde{w}}^{\mathfrak {a}} \end{aligned}$$

(96)

is equivalent to the Ito type SDE

$$\begin{aligned} \textrm{d}{\tilde{X}}^{\mu }=\left( F^{\mu }+\frac{\delta ^{\mathfrak {a}\mathfrak {b}}}{2}C^{\nu }{}_{\mathfrak {a}} \frac{\partial }{\partial x^{\nu }} C^{\mu }{}_{\mathfrak {b}}\right) \textrm{d}t + C^{\mu }{}_{\mathfrak {a}} \circ _I \textrm{d}{\tilde{w}}^{\mathfrak {a}}, \end{aligned}$$

(97)

the corresponding diffusion operation reads

$$\begin{aligned} {\textbf{A}}&=\left( F^\mu +\frac{\delta ^{\mathfrak {a}\mathfrak {b}}}{2}C^\nu {}_{\mathfrak {a}} \frac{\partial }{\partial x^\nu } C^\mu {}_{\mathfrak {b}}\right) \frac{\partial }{\partial x^\mu } +\frac{\delta ^{\mathfrak {a}\mathfrak {b}}}{2}C^\nu {}_{\mathfrak {a}} C^\mu {}_{\mathfrak {b}} \frac{\partial ^2}{\partial x^\mu \partial x^\nu }\nonumber \\&=F^\mu \frac{\partial }{\partial x^\mu } +\frac{\delta ^{\mathfrak {a}\mathfrak {b}}}{2}C^\nu {}_{\mathfrak {a}}\frac{\partial }{\partial x^\nu } C^\mu {}_{\mathfrak {b}} \frac{\partial }{\partial x^\mu }. \end{aligned}$$

(98)

Introducing the vector fields

$$\begin{aligned} L_0=F^\mu \frac{\partial }{\partial x^\mu }\qquad L_{\mathfrak {a}} =C^\mu {}_{\mathfrak {a}}\frac{\partial }{\partial x^\mu }, \end{aligned}$$

(99)

the diffusion operation can be written as simpler form

$$\begin{aligned} {\textbf{A}}=\frac{\delta ^{\mathfrak {a}\mathfrak {b}}}{2}L_{\mathfrak {a}} L_{\mathfrak {b}} +L_0. \end{aligned}$$

(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 ${\textbf{A}}$ on the PDF becomes

$$\begin{aligned} {\textbf{A}}^*\Phi _t&=\frac{\delta ^{\mathfrak {a}\mathfrak {b}}}{2}L^*_{\mathfrak {a}} L^*_{\mathfrak {b}}\Phi _t +L^*_0\Phi _t\nonumber \\&=\frac{\delta ^{\mathfrak {a}\mathfrak {b}}}{2}\nabla _\mu ( C^\mu {}_{\mathfrak {a}}(\nabla _\nu C^\nu {}_{\mathfrak {b}}\Phi _t)) -\nabla _\mu (F^\mu \Phi _t). \end{aligned}$$

(101)

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