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The even pair potential $u$ represents repulsion: because it decreases for positive separation, the force $-u^{\prime }(x_i-x_j)$ pushes particles apart. Marginalizing the $N$-particle Fokker-Planck equation givesUnder the mean-field approximation, propagation of chaos gives $P_2(x_1,x_2)\simeq p(x_1)p(x_2)$. Taking $N\to \infty$ yields the closed nonlocal equationThe convolution term is the collective repulsive drift generated by the population density.

For a sufficiently regular Itô diffusion with $D=\sigma {\sigma }^T/2$, a density $\pi$ is stationary if and only if it solves the stationary Fokker-Planck equationwith integrable Fokker-Planck probability current and boundary conditions that make its outward flux vanish. Here $(\nabla \cdot (D\pi ){)}_i={\sum }_j{\partial }_j(D_{ij}\pi )$.

For the displayed parametrization, assumewhere $Z<\infty$, that $D(x)$ is symmetric positive semidefinite, and that $Q(x)$ is antisymmetric. Substituting $\nabla \pi =-\pi \nabla H$ and the stated $\Gamma$ into the probability current cancels all $D$ terms. The remaining divergence isbecause the second derivatives are symmetric in $i,j$ whereas $Q_{ij}=-Q_{ji}$. Thus these conditions, together with the boundary and regularity assumptions, imply stationarity. More generally, the divergence equation above is the exact necessary and sufficient condition; within this construction, $\pi \propto e^{-H}$ and antisymmetric $Q$ are the standard way to satisfy it.

The Fokker-Planck equation corresponds to the Itô diffusionabsorbed at $0$ and reflected at $L$.

For $v(x)=x$, an Euler--Maruyama proposal isIf $Y\le 0$, kill the path. If $X_n>0$ and $Y>0$, an endpoint-only test can miss a crossing. Conditional on the endpoints, the local Brownian bridge crossing probability isKill the path with this probability; otherwise impose reflection at $L$ by replacing an overshoot $Y>L$ with $2L-Y$ and set $X_{n+1}=Y$. Repeated reflection handles very rare multiple overshoots, and the approximation converges as $\Delta t\to 0$.

This is a birth-death process with birth and death ratesIts transition diagram has the two arrowsThe gain into state $n$ comes from a death in state $n+1$ or a birth in state $n-1$, while the loss is the sum of both rates out of state $n$. The birth-death master equation is consequently

Multiplying by $n$, summing over the nonnegative integers, and shifting the summation indices shows that each birth contributes $+1$ and each death contributes $-1$. Thus the first-moment equation of a birth-death process givesAt a stationary state, use the variance identity $⟨n^2⟩=⟨n{⟩}^2+(\Delta n{)}^2$. Solving the resulting quadratic equation givesThe minus root is inadmissible whenever it is negative, namely when $(\Delta n{)}^2<\gamma /\alpha$. Equality would give zero mean, which is also incompatible with a positive immigration rate $\gamma$, so for $\gamma >0$ the minus branch requires $(\Delta n{)}^2>\gamma /\alpha$ even to be a possible mean.

For a continuum approximation, write $b(n)=\gamma +\beta n$ and $d(n)=\alpha n(n-1)$. Applying the Kramers-Moyal expansion to the two gain terms and retaining derivatives through second order gives the Fokker-Planck equationwhere the negative drift and infinitesimal jump variance areThis truncation requires the typical population and the scale on which $P$ varies to be much larger than the unit jump size, with the rates varying smoothly across that scale.

The positive zero of $g$ isso in particular $n_{\ast }\sim \beta /\alpha$ as $\alpha /\beta \to 0$. Put $x=n-n_{\ast }$. The linear noise approximation useswhereand, because $b(n_{\ast })=d(n_{\ast })$,In a stationary state with zero Fokker-Planck probability current,Its normalized solution is the normal distributionIt follows that

These estimates agree with the exact stationary first-moment relation on its plus branch to leading order: inserting $(\Delta n{)}^2\sim \beta /\alpha$ gives $⟨n⟩\sim \beta /\alpha$. Moreover,Thus the stationary mass lies far from the boundary $n=0$, is narrow relative to its mean, yet changes across many lattice sites. These are precisely the large-population and slow-variation conditions needed for the diffusion approximation.

The two-dimensional Fokker-Planck equation iswith Fokker-Planck probability currentThe point initial condition iswhere $\delta$ is the Dirac delta function. The reflecting boundary condition for a diffusion iswith $\mathbf{n}$ the outward unit normal. This zero-flux condition conserves the integral of the probability density function over the square.

When jumps are small relative to a large population and moments exist, the Kramers-Moyal expansion through second order givesThis Fokker-Planck equation is a conservation law with vanishing boundary flux; $u$ is the drift and $D$ the diffusivity. Equal constant rates for jumps $r=\pm 1$ give pure diffusion.

A probability density left unchanged by the Fokker-Planck equation ${\partial }_{t}f=L^{\ast }f$. Equivalently its Fokker-Planck probability current has zero divergence; the current itself need not vanish. For the Ornstein-Uhlenbeck Fokker-Planck equation, the normalized standard Gaussian density has zero current and is stationary.