Editing Fokker–Planck equation (section)

==Examples==
The Fokker–Planck equation encompasses a variety of more specific situations and contexts, which appear as special cases.

===Wiener process===

A standard scalar [[Wiener process]] is generated by the [[stochastic differential equation]]

<math display="block">dX_t = dW_t.</math>

Here the drift term is zero and the diffusion coefficient is 1/2. Thus the corresponding Fokker–Planck equation is

<math display="block">
\frac{\partial p(x,t)}{\partial t} = \frac{1}{2} \frac{\partial^2 p(x,t)}{\partial x^2},
</math>

which is the simplest form of a [[diffusion equation]]. If the initial condition is <math>p(x,0) = \delta(x)</math>, the solution is

<math display="block"> p(x,t) = \frac{1}{\sqrt{2 \pi t}}e^{-{x^2}/({2t})}.</math>

=== Boltzmann distribution at the thermodynamic equilibrium ===
The [[Brownian dynamics|overdamped Langevin equation]]
:<math>dX_t = -\frac{1}{k_\text{B}T} \left(\nabla_x U\bigg\vert_{x=X_t}\right) dt + dW_t</math>

leads to
:<math>\partial_t p = \frac 1 2 \nabla\cdot \left(\frac{p}{k_\text{B}T} \nabla U + \nabla p\right).</math>

The Boltzmann distribution
:<math>p(x) \propto e^{- U(x)/k_\text{B} T}</math>

is an equilibrium distribution, and assuming <math>U</math> grows sufficiently rapidly (that is, the potential well is deep enough to confine the particle), the Boltzmann distribution is the unique equilibrium.

===Ornstein–Uhlenbeck process===

The [[Ornstein–Uhlenbeck process]] is a process defined as

<math display="block">dX_t = -a X_t \, dt + \sigma \, dW_t.</math>

with <math>a>0</math>. Physically, this equation can be motivated as follows: a particle of mass <math> m </math> with velocity <math> V_t</math> moving in a medium, e.g., a fluid, will experience a friction force which resists motion whose magnitude can be approximated as being proportional to particle's velocity <math> -a V_t</math> with <math> a = \mathrm{constant} </math>.  Other particles in the medium will randomly kick the particle as they collide with it and this effect can be approximated by a white noise term; <math> \sigma (d W_t/dt) </math>. Newton's second law is written as

<math display="block"> m \frac{dV_t}{dt}=-a V_t +\sigma \frac{dW_t}{dt}. </math>

Taking <math> m = 1</math> for simplicity and changing the notation as <math> V_t\rightarrow X_t</math> leads to the Ornstein–Uhlenbeck form. The corresponding Fokker–Planck equation is
<math display="block">
\frac{\partial p(x,t)}{\partial t} = a \frac{\partial}{\partial x}\left(x \,p(x,t)\right) + \frac{\sigma^2}{2} \frac{\partial^2 p(x,t)}{\partial x^2},
</math>

The stationary solution <math>(\partial_t p = 0)</math> is
<math display="block">p_{\text{ss}}(x) = \sqrt{\frac{a}{\pi \sigma^2}} e^{-{ax^2}/{\sigma^2}}.</math>

===Plasma physics===

In plasma physics, the [[Distribution function (physics)|distribution function]] <math>p_s (\mathbf{x},\mathbf{v},t)</math> for a particle species <math>s</math> takes the place of the [[probability density function]]. The corresponding Boltzmann equation is given by

<math display="block">\frac{\partial p_s}{\partial t} + \mathbf{v} \cdot \boldsymbol{\nabla} p_s + \frac{Z_s e}{m_s} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right) \cdot \boldsymbol{\nabla}_v p_s = -\frac{\partial}{\partial v_i} \left(p_s \langle\Delta v_i\rangle\right) + \frac{1}{2} \frac{\partial^2}{\partial v_i \, \partial v_j} \left(p_s \langle\Delta v_i \, \Delta v_j\rangle\right),</math>

where the third term includes the particle acceleration due to the [[Lorentz force]] and the Fokker–Planck term at the right-hand side represents the effects of particle collisions. The quantities <math>\langle\Delta v_i\rangle</math> and <math>\langle\Delta v_i \, \Delta v_j\rangle</math> are the average change in velocity a particle of type <math>s</math> experiences due to collisions with all other particle species in unit time. Expressions for these quantities are given elsewhere.<ref name="Rosenbluth">{{Cite journal|last=Rosenbluth |first=M. N. |title=Fokker–Planck Equation for an Inverse-Square Force |journal=Physical Review |volume=107 |issue= 1|pages=1–6 |year=1957 |doi=10.1103/physrev.107.1|bibcode = 1957PhRv..107....1R |url=https://escholarship.org/uc/item/2gk1s1v8 }}</ref> If collisions are ignored, the Boltzmann equation reduces to the [[Vlasov equation]].

=== Smoluchowski diffusion equation ===
The Smoluchowski diffusion equation is effectively equivalent to the [[convection–diffusion equation]]. Consider an overdamped Brownian particle under external force <math>F(r)</math>:<ref name=":0">{{Cite web|title=Smoluchowski Diffusion Equation|url=https://www.ks.uiuc.edu/~kosztin/PHYCS498NSM/|last=Ioan|first=Kosztin|date=Spring 2000|website=Non-Equilibrium Statistical Mechanics: Course Notes}}</ref>

:<math>m\ddot{r} = - \gamma \dot{r} + F(r) + \sigma \xi(t)</math>

where the <math>m\ddot r</math> term is negligible (the meaning of "overdamped"). Thus, it is just
:<math>\gamma \, dr = F(r)\, dt + \sigma \, dW_t.</math>

The Fokker–Planck equation for this particle is the Smoluchowski diffusion equation:
:<math>\partial_t P(r,t| r_0, t_0) = \nabla \cdot \left[D (\nabla - \beta F(r)) P(r,t| r_0, t_0)\right] </math>

Here, <math>D</math> is the diffusion constant and <math>\beta = 1 / k_\text{B} T</math>. The importance of this equation is it allows for both the inclusion of the effect of temperature on the system of particles and a spatially dependent diffusion constant.

{{Hidden begin| title = Derivation of the Smoluchowski Equation from the Fokker–Planck Equation}}
Starting with the [[Langevin equation|Langevin Equation]] of a Brownian particle in external field <math>F(r)</math>, where <math>\gamma</math> is the friction term, <math>\xi</math> is a fluctuating force on the particle, and <math>\sigma</math> is the amplitude of the fluctuation.

<math display="block">m\ddot{r} = - \gamma \dot{r} + F(r) + \sigma \xi(t)</math>

At equilibrium the frictional force is much greater than the inertial force, <math>\left\vert \gamma \dot{r} \right\vert \gg \left\vert m \ddot{r} \right\vert</math>. Therefore, the Langevin equation becomes,

<math display="block">\gamma \dot{r} = F(r) + \sigma \xi(t)</math>

Which generates the following Fokker–Planck equation,

<math display="block">\partial_t P(r,t|r_0,t_0) = \left(\nabla^2\frac{\sigma^2}{2 \gamma^2} - \nabla \cdot \frac{F(r)}{\gamma}\right) P(r,t|r_0,t_0)  </math>

Rearranging the Fokker–Planck equation,

<math display="block">\partial_t P(r,t|r_0,t_0)= \nabla \cdot \left( \nabla  D- \frac{F(r)}{\gamma}\right) P(r,t|r_0,t_0)</math>

Where <math>D =  \frac{\sigma^2}{2 \gamma^2}</math>. '''Note''', the diffusion coefficient may not necessarily be spatially independent if <math>\sigma</math> or <math>\gamma</math> are spatially dependent.

Next, the total number of particles in any particular volume is given by,

<math display="block">N_V (t| r_0, t_0) = \int_V dr P(r,t|r_0,t_0)</math>

Therefore, the flux of particles can be determined by taking the time derivative of the number of particles in a given volume, plugging in the Fokker–Planck equation, and then applying [[Divergence theorem|Gauss's Theorem]].

<math display="block">\partial_t N_V (t|r_0, t_0) = \int_V dV \nabla \cdot\left( \nabla  D- \frac{F(r)}{\gamma}\right) P(r,t|r_0, t_0)
= \int_{\partial V} d\mathbf{a} \cdot j(r,t|r_0, t_0)</math>

<math display="block">j(r,t|r_0, t_0) = \left( \nabla  D- \frac{F(r)}{\gamma}\right)P(r,t|r_0, t_0)</math>

In equilibrium, it is assumed that the flux goes to zero. Therefore, Boltzmann statistics can be applied for the probability of a particles location at equilibrium, where <math>F(r) = -\nabla U(r)</math> is a conservative force and the probability of a particle being in a state <math>r</math> is given as <math>P(r,t|r_0, t_0) = \frac{e^{-\beta U(r)}}{Z}</math>.

<math display="block">j(r,t|r_0, t_0) = \left( \nabla  D- \frac{F(r)}{\gamma}\right)\frac{e^{-\beta U(r)}}{Z} = 0</math>

<math display="block">\Rightarrow  \nabla D   = F(r) \left(\frac{1}{\gamma} - D \beta\right)</math>

This relation is a realization of the [[fluctuation–dissipation theorem]]. Now applying <math> \nabla \cdot \nabla </math> to <math>D P(r,t|r_0, t_0)</math> and using the Fluctuation-dissipation theorem,

<math display="block">\begin{align}
\nabla \cdot \nabla D P(r,t|r_0,t_0) &= \nabla \cdot D \nabla P(r,t|r_0,t_0)+ \nabla \cdot P(r,t|r_0,t_0) \nabla D \\
&=\nabla \cdot D \nabla P(r,t|r_0,t_0)+\nabla \cdot P(r,t|r_0,t_0) \frac{F(r)}{\gamma} - \nabla \cdot P(r,t|r_0,t_0) D \beta F(r)
\end{align}</math>

Rearranging,

<math display="block"> \Rightarrow \nabla \cdot \left( \nabla  D- \frac{F(r)}{\gamma}\right)P(r,t|r_0,t_0)=
\nabla \cdot D(\nabla-\beta F(r)) P(r,t|r_0,t_0)</math>

Therefore, the Fokker–Planck equation becomes the Smoluchowski equation,
<math display="block">\partial_t P(r,t| r_0, t_0) = \nabla \cdot D (\nabla - \beta F(r)) P(r,t| r_0, t_0) </math>
for an arbitrary force <math>F(r)</math>.{{Hidden end}}