Fokker-Planck-Kramers equation is a stochastic differential function describing the motion in phase space. It can degenerate to diffusion function (in coordinate space) or Fokker-Planck function (in velocity space).
The post describes the derivation of FPK equation from moment generating function.
flowchart TB conservation["mass conservation (flow)"] continuity["continuity equation (density)"] LE["Langevin stochastic differential equation"] solution["solution for SDE with operators"] FPK["Fokker-Planck-Kramers equation"] conservation --> |Gauss theorem| continuity continuity --> solution LE --> solution solution --> |ensemble average in the form of \n moment generating function| FPK
Solution to the stochastic differential equation
Continuity equation
For a conserved system, the change of mass with respect to time is equal to the flow arround the surface,
\[ \iiint \partial_{t} \rho \ dV = - \iint \dot{\pmb{r}} \rho \ dS \]
According to Guass theorem, we get the continuity equation,
\[ \begin{equation} \partial_{t} \rho = - \nabla \cdot ( \dot{\pmb{r}} \rho ) \label{continuity} \end{equation} \]
Langevin SDE
Inserting Langevin stochastic differential equation,
\[ \begin{equation} \dot{\pmb{r}} = \pmb{f}(\pmb{r}) + \pmb{g}(\pmb{r}) \eta(t) \label{Langevin} \end{equation} \]
we get,
\[ \partial_{t} \rho = - \nabla \cdot [ \pmb{f}(\pmb{r}) \rho ] - \eta(t) \nabla \cdot [ \pmb{g}(\pmb{r}) \rho ] = ( \mathcal{L}_{d} + \mathcal{L}_{s} ) \rho \]
where the operator \(\mathcal{L}_{d} = - \nabla \cdot [ \pmb{f}(\pmb{r}) ]\) is deterministic and the operator \(\mathcal{L}_{s} = - \eta(t) \nabla \cdot [ \pmb{g}(\pmb{r}) ]\) is stochastic because of the term \(\eta(t) \sim \mathcal{N}(0, 2\kappa)\).
The solution to the above stochastic differential equation is,
\[ \rho(t) = \exp(\mathcal{L}_{d} t + \int_{0}^{t} \mathcal{L}_{s} \ d{t'}) \rho(0) \]
Ensemble average in the form of MGF
polynomial form of exponential function
The property of the stochastic term \(\eta(t)\) can be evaluated by taking ensemble average to the solution,
\[ \begin{equation} <\rho> = \exp( \mathcal{L}_{d} t ) < \exp(\int_{0}^{t} \mathcal{L}_{s} \ d{t'}) > \rho(0) \label{ensemble} \end{equation} \]
The exponential function \(<\exp(\int_{0}^{t} \mathcal{L}_{1} \ d{t'})>\) has a polynomial form,
\[ <\exp(\int_{0}^{t} \mathcal{L}_{s} \ d{t'})> = \sum_{k=0}^{\infty} \frac{ <(\int_{0}^{t} \mathcal{L}_{s} \ d{t'})^{k}> }{k!} \]
We only need to know \(<\int_{0}^{t} \mathcal{L}_{s} \ d{t'}>\) and \(<(\int_{0}^{t} \mathcal{L}_{s} \ d{t'})^{2}>\), which are,
\[ <\int_{0}^{t} \mathcal{L}_{s} \ d{t'}> = - \int_{0}^{t} <\eta(t')> \nabla \cdot [ \pmb{g}(\pmb{r}) ] \ d{t'} = 0 \]
and,
\[ \begin{aligned} < (\int_{0}^{t} \mathcal{L}_{s} \ d{t'})^{2} > &= \int_{0}^{t} \int_{0}^{t} \nabla \cdot \big[ \pmb{g}(\pmb{r}(t_{1})) \nabla \cdot [ \pmb{g}(\pmb{r}(t_{2})) ] \big] <\eta(t_{1}) \eta(t_{2})> \ d{t_{1}} d{t_{2}} \\ &= 2 \kappa \int_{0}^{t} \int_{0}^{t} \nabla \cdot \big[ \pmb{g}(\pmb{r}(t_{1})) \nabla \cdot [ \pmb{g}(\pmb{r}(t_{2})) ] \big] \delta(t_{1} - t_{2}) \ d{t_{1}} d{t_{2}} \\ &= 2 \kappa \int_{0}^{t} \nabla \cdot \big[ \pmb{g}(\pmb{r}(t_{1})) \nabla \cdot [ \pmb{g}(\pmb{r}(t_{1})) ] \big] \ d{t_{1}} \end{aligned} \]
It should be noticed that the square of operator \(\mathcal{L}^{2} f\) is equivalent to \(\mathcal{L} [ \mathcal{L} f]\).
MGF for Gaussian variables
\(<\exp( \int_{0}^{t} \mathcal{L}_{s} \ d{t'} )>\) is the moment generating function \(M_{X}(\tau)\) of variable \(X = \int_{0}^{t} \mathcal{L}_{s} \ d{t'}\) when \(\tau = 1\). The integral of Gaussian variable \(\eta(t)\) is also Gaussian, thus its moment generating function is,
\[ M_{X}(1) = \exp( \mu + \frac{\sigma^{2}}{2} ) \]
Substituting it into the equation \(\eqref{ensemble}\),
\[ <\rho> = \exp( - \nabla \cdot [\pmb{f}(\pmb{r})] t) \exp( \kappa \int_{0}^{t} \nabla \cdot \big[ \pmb{g}(\pmb{r})\nabla \cdot [ \pmb{g}(\pmb{r}) ] \big] \ d{t'} ) \rho_{0} \]
Partial derivative of time
Then we take the partial derivative of \(<\rho>\) with respect to time,
\[ \begin{equation} \partial_{t} <\rho> = - \nabla \cdot [\pmb{f} <\rho>] + \kappa \nabla \cdot \big[ \pmb{g} \nabla \cdot [\pmb{g} <\rho>] \big] = \mathcal{L} <\rho> \label{FPK} \end{equation} \]
where the operator \(\mathcal{L} = - \nabla \cdot \big[ \pmb{f} - \kappa \pmb{g} \nabla \cdot [\pmb{g}] \big]\).
We only take out the integrand in the exponential factor \(\exp( \kappa \int_{0}^{t} \nabla \cdot \big[ \pmb{g}(\pmb{r}) \nabla \cdot [ \pmb{g}(\pmb{r}) ] \big] \ d{t'})\) instead of deriving \(\pmb{g}\) and \(\pmb{r}\) continuingly. Because the partial derivative with respect to time is the change along with time at a fixed position \(\pmb{r}\). It's the total derivative that all functions need to be derived with respect to time.
The equation \(\eqref{FPK}\) is what we want, namely the Fokker-Planck-Kramers equation.