Fokker–Planck equation
Fokker–Planck equation
In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well.[1] It is named after Adriaan Fokker and Max Planck,[2][3] and is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered the concept in 1931.[4] When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski), and in this context it is equivalent to the convection–diffusion equation. The case with zero diffusion is known in statistical mechanics as the Liouville equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion.
The first consistent microscopic derivation of the Fokker–Planck equation in the single scheme of classical and quantum mechanics was performed by Nikolay Bogoliubov and Nikolay Krylov.[5][6]
The Smoluchowski equation is the Fokker–Planck equation for the probability density function of the particle positions of Brownian particles.[7]
One dimension
![{\displaystyle {\frac {\partial }{\partial t}}p(x,t)=-{\frac {\partial }{\partial x}}\left[\mu (x,t)p(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}\left[D(x,t)p(x,t)\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f66f454692d4179f5de829965a2237a65485940)
![{\displaystyle {\mathcal {L}}p(X_{t})=\lim _{\Delta t\to 0}{\frac {1}{\Delta t}}\left(\mathbb {E} {\big [}p(X_{t+\Delta t})\mid X_{t}=x{\big ]}-p(x)\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbac0ac010adeabfb6a94014fcae99d7f637597)
Note now that
which is a time derivative. Finally we arrive to
![{\displaystyle \int [{\mathcal {L}}p(x)]\mathbb {P} _{t,t'}(x\mid x')\,dx=\int p(x)\,\partial _{t}\mathbb {P} _{t,t'}(x\mid x')\,dx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3015cb16839e7e02e958ea511e2cd5041c52a25e)
![{\displaystyle \int [{\mathcal {L}}p(x)]\mathbb {P} _{t,t'}(x\mid x')\,dx=\int p(x)[{\mathcal {L}}^{\dagger }\mathbb {P} _{t,t'}(x\mid x')]\,dx,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d792d0f77b1cb0bdb5b405ed785eb3bda1873b3)
Then, for a particle subject to an Itō equation, using
it can be easily calculated, using integration by parts, that
which bring us to the Fokker–Planck equation:
While the Fokker–Planck equation is used with problems where the initial distribution is known, if the problem is to know the distribution at previous times, the Feynman–Kac formula can be used, which is a consequence of the Kolmogorov backward equation.
The stochastic process defined above in the Itō sense can be rewritten within the Stratonovich convention as a Stratonovich SDE:
![{\displaystyle dX_{t}=\left[\mu (X_{t},t)-{\frac {1}{2}}{\frac {\partial }{\partial X_{t}}}D(X_{t},t)\right]\,dt+{\sqrt {2D(X_{t},t)}}\circ dW_{t}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74f54b781333612a077d5fc2cb0f91f8f7fe60f2)
It includes an added noise-induced drift term due to diffusion gradient effects if the noise is state-dependent. This convention is more often used in physical applications. Indeed, it is well known that any solution to the Stratonovich SDE is a solution to the Itō SDE.
The zero-drift equation with constant diffusion can be considered as a model of classical Brownian motion:
![{\displaystyle {\frac {\partial }{\partial t}}p(x,t)=D_{0}{\frac {\partial ^{2}}{\partial x^{2}}}\left[p(x,t)\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9029fd21c2e3e7258c6d6c08f64914193bbbf6)
It has been shown[9] that in this case an analytical spectrum of solutions allows deriving a local uncertainty relation for the coordinate-velocity phase volume:
Higher dimensions
More generally, if
![{\displaystyle {\frac {\partial p(\mathbf {x} ,t)}{\partial t}}=-\sum _{i=1}^{N}{\frac {\partial }{\partial x_{i}}}\left[\mu _{i}(\mathbf {x} ,t)p(\mathbf {x} ,t)\right]+\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}\left[D_{ij}(\mathbf {x} ,t)p(\mathbf {x} ,t)\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75b4e22a5e4eff2c83ea5895484ee1fb39cf5ea5)
If instead of an Itō SDE, a Stratonovich SDE is considered,
the Fokker–Planck equation will read ([8] pag. 129):
![{\frac {\partial p(\mathbf {x} ,t)}{\partial t}}=-\sum _{i=1}^{N}{\frac {\partial }{\partial x_{i}}}\left[\mu _{i}(\mathbf {x} ,t)\,p(\mathbf {x} ,t)\right]+{\frac {1}{2}}\sum _{k=1}^{M}\sum _{i=1}^{N}{\frac {\partial }{\partial x_{i}}}\left\{\sigma _{ik}(\mathbf {x} ,t)\sum _{j=1}^{N}{\frac {\partial }{\partial x_{j}}}\left[\sigma _{jk}(\mathbf {x} ,t)\,p(\mathbf {x} ,t)\right]\right\}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66bb0ebd3cce662da5ddb4f025d1f08ab0a0b15c)
Examples
Wiener process
A standard scalar Wiener process is generated by the stochastic differential equation
Here the drift term is zero and the diffusion coefficient is 1/2. Thus the corresponding Fokker–Planck equation is
Ornstein–Uhlenbeck process
The Ornstein–Uhlenbeck process is a process defined as
- .
Plasma physics
Computational considerations
Solution
Particular cases with known solution and inversion
Fokker–Planck equation and path integral
Every Fokker–Planck equation is equivalent to a path integral. The path integral formulation is an excellent starting point for the application of field theory methods.[12] This is used, for instance, in critical dynamics.
A derivation of the path integral is possible in a similar way as in quantum mechanics. The derivation for a Fokker–Planck equation with one variable x is as follows. Start by inserting a delta function and then integrating by parts:
![{\displaystyle {\begin{aligned}{\frac {\partial }{\partial t}}p\!\left(x^{\prime },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]\\[5pt]&=\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^{\prime }-x\right)\right)p\!\left(x,t\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a7bcde74b70937f4b2c4186a36ad3651ca26f15)
![{\displaystyle p(x^{\prime },t+\varepsilon )=\int _{-\infty }^{\infty }\,dx\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^{\prime }-x)\right)p(x,t)+O(\varepsilon ^{2}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9958a13bc6ed99aa34014074996fd62c67dadaf8)
Insert the Fourier integral
![{\displaystyle {\begin{aligned}p(x^{\prime },t+\varepsilon )&=\int _{-\infty }^{\infty }dx\int _{-i\infty }^{i\infty }{\frac {d{\tilde {x}}}{2\pi i}}\left(1+\varepsilon \left[{\tilde {x}}D_{1}(x,t)+{\tilde {x}}^{2}D_{2}(x,t)\right]\right)e^{{\tilde {x}}(x-x^{\prime })}p(x,t)+O(\varepsilon ^{2})\\[5pt]&=\int _{-\infty }^{\infty }dx\int _{-i\infty }^{i\infty }{\frac {d{\tilde {x}}}{2\pi i}}\exp \left(\varepsilon \left[-{\tilde {x}}{\frac {(x^{\prime }-x)}{\varepsilon }}+{\tilde {x}}D_{1}f(x,t)+{\tilde {x}}^{2}D_{2}(x,t)\right]\right)p(x,t)+O(\varepsilon ^{2}).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17ab2d99e05d7e5925ace49f71b82934a7170ccd)
![{\displaystyle S=\int dt\left[{\tilde {x}}D_{1}(x,t)+{\tilde {x}}^{2}D_{2}(x,t)-{\tilde {x}}{\frac {\partial x}{\partial t}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b385884d6d5ef443c5f67e98c62ae45e29f43e00)
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.
See also
Kolmogorov backward equation
Boltzmann equation
Vlasov equation
Master equation
Mean field game theory
Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy of equations
Ornstein–Uhlenbeck process