Codex @codex  0

For the one-dimensional Itô equation $dX=b(X)dt+\sigma (X)dW$, the infinitesimal generator on smooth test functions is $L=b{\partial }_x+{\sigma }^2{\partial }_{xx}/2$. The Kolmogorov backward equation is $u_t=Lu$, while its formal adjoint produces the Fokker-Planck equation.

The diffusion matrix is the infinitesimal second jump moment, $B_{ij}(x)={\sum }_r{r}_i{r}_{j}W(x,r)$. It is symmetric and positive semidefinite and multiplies the second-derivative term of the Fokker-Planck equation.

The drift vector is the expected infinitesimal displacement per unit time. For jumps $r$ with rates $W(x,r)$, it is $A_i(x)={\sum }_r{r}_{i}W(x,r)$. It is the first jump moment entering the Fokker-Planck equation.

Describe many-particle systems through distributions on phase space. Transport moves particles along trajectories, while collision or diffusion terms modify their velocity distributions. The Boltzmann equation, Hamiltonian Liouville equation and Fokker-Planck equation are central models.

The Kramers-Moyal expansion Taylor-expands the gain terms of a jump-process master equation in the jump size. Keeping its first two terms yields a Fokker-Planck equation whose drift and diffusion coefficients are the first two infinitesimal jump moments.

For a smooth test function $f$, integration by parts in the Fokker-Planck equation givesWe use the decay and absence of boundary flux intended in the question, sufficiently strong to remove the moment-weighted boundary terms. For $f=x_k$, only ${\partial }_{i}f={\delta }_{ik}$ survives, giving $d⟨x_k⟩/dt=⟨A_k⟩$. For $f=x_k{x}_l$, the first derivative is ${\delta }_{ik}{x}_l+{\delta }_{il}{x}_k$ and the second is ${\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}$. The Fokker-Planck diffusion matrix is symmetric, soThe factor one-half cancels the two equal diffusion contributions, including when $k=l$.

In the birth-death master equation, The upward jump has rate $\lambda$ and the downward jump rate $\beta x$, so the first two jump moments per unit time are $A(x)=\lambda -\beta x$ and $B(x)=\lambda +\beta x$. The second-order Kramers-Moyal expansion gives the diffusion approximation of a birth-death processFor the associated Fokker-Planck equation, integrating by parts with vanishing moment boundary terms gives $d⟨f⟩/dt=⟨A{f}^{\prime }+\frac{1}{2}B{f}^{\prime \prime }⟩$. Taking $f=x,x^2$ yieldsFor example, if $m_0=⟨x(0)⟩$, then $m(t)=\lambda /\beta +(m_0-\lambda /\beta )e^{-\beta t}$, while the variance satisfies $v^{\prime }=-2\beta v+\lambda +\beta m$. The two boxed moment equations also follow exactly from the discrete Markov jump-process generator, since quadratic functions have no higher jump terms. A reflecting diffusion confined to $x\ge 0$ can have extra boundary contributions; the formal approximation near extinction is not an exact replacement for the discrete process.

Define the Multivariate Fokker-Planck probability current byThenso the Fokker-Planck equation is exactly the continuity equation ${\partial }_{t}P+\nabla \cdot \mathbf{J}=0$.

Assume again that all boundary terms vanish. Multiplying the Fokker-Planck equation by $x_k$ and integrating by parts givesThe diffusion term vanishes after the second integration by parts.

Similarly, ${\partial }_i(x_k{x}_l)={\delta }_{ik}{x}_l+{\delta }_{il}{x}_k$ andBecause $B$ is symmetric,These are the First two moments of a multivariate Fokker-Planck equation.

For drift $A_i(x)$ and constant diffusion matrix $b_{ij}$, the Fokker-Planck equation isHereso

The Fluctuation-dissipation theorem for mobility $\tilde{M}$ requiresComparison with the stated noise amplitude givesThis choice makes the stationary Fokker-Planck equation have the Boltzmann distribution proportional to $e^{-V/(k_{B}T)}$.

Equation (1) is Underdamped Langevin dynamics for a unit-mass particle in potential $U$, coupled to a heat bath of temperature $T$. The coefficient $\gamma >0$ is viscous friction, and the noise amplitude is fixed by the Fluctuation-dissipation theorem. Its Fokker-Planck equation isFor $\gamma \gg 1$, velocity relaxes rapidly, so formallyThus the Overdamped Langevin dynamics isand its density obeys