Editing Fokker–Planck equation (section)

==The path integral formulation==

Every Fokker–Planck equation is equivalent to a [[Path integral formulation|path integral]]. The path integral formulation is an excellent starting point for the application of field theory methods.<ref>{{Cite book|author=Zinn-Justin, Jean |title=Quantum field theory and critical phenomena |publisher=Clarendon Press |location=Oxford |year=1996 |isbn=978-0-19-851882-2 }}</ref> This is used, for instance, in [[Critical phenomena#Critical dynamics|critical dynamics]].

The derivation of the path integral is similar to that used in quantum mechanics. A derivation for the Fokker–Planck equation with one variable <math>x</math> follows. Inserting a [[delta function]] and integrating by parts gives:

<math display="block">\begin{align}
\frac{\partial }{\partial t} p{\left( x', t\right)}
& = - \frac{\partial }{\partial x'} \left[ D_1(x',t) p(x',t) \right] +  \frac{\partial^2 }{\partial {x'}^2} \left[ D_2(x',t) p(x',t) \right] \\[1ex]
& = \int_{-\infty}^{\infty} dx\left[ \left( D_{1}{\left( x,t\right)} \frac{\partial }{\partial x} + D_2{ \left( x,t\right)} \frac{\partial^2}{\partial x^2}\right) \delta{\left( x' -x\right)} \right] p(x,t).
\end{align}</math>

The <math>x</math>-derivatives act only on the <math>\delta</math>-function, not on <math>p(x,t)</math>. Performing an integral over a time interval <math>\varepsilon</math> gives

<math display="block">p(x', t + \varepsilon) =\int_{-\infty}^\infty \, \mathrm{d}x\left(\left( 1+\varepsilon \left[ D_1(x,t) \frac \partial {\partial x} + D_2(x,t) \frac{\partial^2}{\partial x^2}\right]\right) \delta(x' - x) \right) p(x,t)+O(\varepsilon^2).</math>

The Dirac <math>\delta</math>-function can be represented by the [[Fourier integral]] as

<math display="block">\delta{\left( x' - x\right)} = \int_{-\infty}^{\infty} \frac{\mathrm{d} k}{2\pi} e^{-ik {\left( x - x'\right)}}</math>

which yields
<math display="block">\begin{align}
p(x', t+\varepsilon) & = \int_{-\infty}^\infty \mathrm{d}x \int_{-\infty}^{\infty} \frac{\mathrm{d}k}{2\pi} \left(1-\varepsilon \left[ ik D_1(x,t) +k^2 D_2(x,t) \right] \right) e^{-ik (x - x')}p(x,t) +O(\varepsilon^2) \\[5pt]
& =\int_{-\infty}^\infty \mathrm{d}x \int_{-\infty}^{\infty} \frac{\mathrm{d}k}{2\pi}\exp \left( -\varepsilon \left[ ik\frac{(x'- x)}{\varepsilon} +ik D_1(x,t) +k^2 D_2(x,t) \right] \right) p(x,t) +O(\varepsilon^2).
\end{align}</math>

This equation expresses <math>p(x', t+\varepsilon)</math> as functional of <math>p(x,t)</math>. Iterating <math>(t'-t)/\varepsilon</math> times and performing the limit <math>\varepsilon \rightarrow 0</math> gives a path integral with [[Action (physics)|action]]

<math display="block">S=-\int \mathrm{d}t\left[ ik D_1 (x,t) + k^2 D_2 (x,t) +ik\frac{\partial x}{\partial t} \right].</math>

The variable <math>k</math> conjugate to <math>x</math> is called the "response variable".<ref name="Janssen">{{Cite journal | last=Janssen |first=H. K. |title=On a Lagrangean for Classical Field Dynamics and Renormalization Group Calculation of Dynamical Critical Properties |journal=Z. Phys. |volume=B23 |issue= 4|pages=377–380 |year=1976 |doi=10.1007/BF01316547 |bibcode = 1976ZPhyB..23..377J |s2cid=121216943 }}</ref>

Although formally equivalent, different problems may be solved more easily in the Fokker–Planck equation or the path integral formulation. The equilibrium distribution for instance may be obtained more directly from the Fokker–Planck equation.