Alright, let's break this down step by step—first covering the general derivation, then applying it directly to the Langevin equation you provided.

General Derivation: Langevin → Fokker-Planck #

First, let's set up the general form of a Langevin equation for a vector of variables $\mathbf{x}(t)$:
$$
\frac{d\mathbf{x}}{dt} = \mathbf{A}(\mathbf{x}, t) + \mathbf{B}(\mathbf{x}, t) \cdot \boldsymbol{\eta}(t)
$$
Here:

• $\mathbf{A}(\mathbf{x}, t)$ is the drift term (deterministic force/velocity driving the system)
• $\mathbf{B}(\mathbf{x}, t)$ is the diffusion coefficient matrix (scales the magnitude of the noise)
• $\boldsymbol{\eta}(t)$ is Gaussian white noise, satisfying:
  • $\langle \eta_i(t) \rangle = 0$ (zero mean, no net noise contribution over time)
  • $\langle \eta_i(t) \eta_j(t') \rangle = \delta_{ij} \delta(t - t')$ (delta-correlated, uncorrelated across components)

Our goal is to find the evolution equation for the probability density function $P(\mathbf{x}, t)$ (the probability of finding the system in state $\mathbf{x}$ at time $t$). Here's how we get there:

1. Start with the Chapman-Kolmogorov Equation
   This core equation describes how probability density evolves over a small time step $\Delta t$:
   $$
   P(\mathbf{x}, t+\Delta t) = \int P(\mathbf{x}', t) W(\mathbf{x}' \to \mathbf{x}, \Delta t) d\mathbf{x}'
   $$
   where $W(\mathbf{x}' \to \mathbf{x}, \Delta t)$ is the transition probability density (chance of moving from state $\mathbf{x}'$ to $\mathbf{x}$ in $\Delta t$).

2. Expand State Change for Small $\Delta t$
   For tiny time steps, the change in state $\Delta\mathbf{x} = \mathbf{x} - \mathbf{x}'$ is approximately:
   $$
   \Delta\mathbf{x} = \mathbf{A}(\mathbf{x}', t)\Delta t + \mathbf{B}(\mathbf{x}', t) \cdot \int_t^{t+\Delta t} \boldsymbol{\eta}(s) ds
   $$

3. Calculate Moments of $\Delta\mathbf{x}$
   We only need up to the second moment (higher-order terms vanish as $\Delta t \to 0$):

   • First moment: $\langle \Delta x_i \rangle = A_i(\mathbf{x}', t)\Delta t$ (noise has zero mean, so only the drift contributes)
   • Second moment: $\langle \Delta x_i \Delta x_j \rangle = (BB^T)_{ij} \Delta t$ (integrating the white noise correlation gives this result)

4. Taylor Expand $P(\mathbf{x}, t+\Delta t)$
   Expand $P$ around $\mathbf{x}'$ (since $\mathbf{x} = \mathbf{x}' + \Delta\mathbf{x}$) to second order:
   $$
   P(\mathbf{x}, t+\Delta t) = P(\mathbf{x}', t) + \Delta x_i \frac{\partial P}{\partial x_i} + \frac{1}{2}\Delta x_i \Delta x_j \frac{\partial^2 P}{\partial x_i \partial x_j} + o(\Delta t)
   $$

5. Average and Take the Limit $\Delta t \to 0$
   Take the expectation of both sides (to account for random $\Delta\mathbf{x}$), subtract $P(\mathbf{x}', t)$, divide by $\Delta t$, and take the limit as $\Delta t$ approaches 0. This gives the general Fokker-Planck equation:
   $$
   \frac{\partial P}{\partial t} = -\sum_i \frac{\partial}{\partial x_i} \left( A_i P \right) + \frac{1}{2} \sum_{i,j} \frac{\partial^2}{\partial x_i \partial x_j} \left( (BB^T)_{ij} P \right)
   $$

Application to Your Specific Langevin Equation #

Your equation describes a damped velocity with thermal noise:
$$
\frac{dv}{dt} = -\frac{v}{\tau} + \sqrt{2c},\eta(t)
$$
This is a 1-dimensional case (only variable $v$), so we can simplify the general equation step by step:

1. Identify Drift and Diffusion Terms

   • Drift term: $A(v) = -\frac{v}{\tau}$ (damping force that slows the velocity over time)
   • Diffusion coefficient: $B(v) = \sqrt{2c}$ (constant scaling factor for the noise, no dependence on $v$)

2. Compute the Diffusion Matrix Term
   In 1D, $(BB^T)$ simplifies to $B^2$:
   $$
   B^2 = (\sqrt{2c})^2 = 2c
   $$

3. Plug into the 1D Fokker-Planck Equation
   The 1D version of the general equation is:
   $$
   \frac{\partial P(v,t)}{\partial t} = -\frac{\partial}{\partial v} \left( A(v) P(v,t) \right) + \frac{1}{2} \frac{\partial^2}{\partial v^2} \left( B^2 P(v,t) \right)
   $$

4. Expand the Terms

   • First term (drift contribution):
     $$
     -\frac{\partial}{\partial v} \left( -\frac{v}{\tau} P \right) = \frac{1}{\tau} \frac{\partial}{\partial v} \left( v P \right) = \frac{1}{\tau} \left( P + v \frac{\partial P}{\partial v} \right)
     $$
   • Second term (diffusion contribution):
     $$
     \frac{1}{2} \frac{\partial^2}{\partial v^2} \left( 2c P \right) = c \frac{\partial^2 P}{\partial v^2}
     $$

5. Combine into the Final Equation
   Putting it all together, the Fokker-Planck equation for your system is:
   $$
   \frac{\partial P(v,t)}{\partial t} = \frac{1}{\tau} \frac{\partial}{\partial v} \left( v P(v,t) \right) + c \frac{\partial^2 P(v,t)}{\partial v^2}
   $$
   If you prefer the expanded form, it's:
   $$
   \frac{\partial P(v,t)}{\partial t} = \frac{P(v,t)}{\tau} + \frac{v}{\tau} \frac{\partial P(v,t)}{\partial v} + c \frac{\partial^2 P(v,t)}{\partial v^2}
   $$

内容的提问来源于stack exchange，提问作者Drebin J.