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1 Introduction

From the master equation we can derive a Fokker-Planck equation by means of a second order Taylor approximation. The Fokker-Planck equation is a linear partial differential equation of second order so that, not least thanks to the analogy to the Schrödinger equation (250), there exist many solution methods for it [250], there exist many solution methods for it [27, 83, 84, 86, 95, 96, 241, 242, 259]. In contrast to the master equation the Fokker-Planck equation takes into account only the first two jump moments. The mean value and covariance equations, however, agree with those of the master equation.

From the Fokker-Planck equation, by factorizing the pair distribution function, we can gain ‘Boltzmann-Fokker-Planck equations’ which are also derivable from the Boltzmann-like equations by means of a second order Taylor approximation. An application of these new equations is discussed in Chap. 11. Since the Boltzmann-Fokker-Planck equations depend non-linearly on the probability distributions their solution is complicated. However, there exists a recursive solution strategy that is related to Hartree’s self-consistent field method for quantum mechanical many-particle systems [114] (which is applied to the evaluation of atomic and molecular orbitals).

2 Derivation

The continuous master equation (3.5b) can, by writing X′ in the first term as X-Y and in the second term as X+Y, be transformed into

$$\frac{d}{dt} P(X,t) = \int d^M Y \Big[w[Y|X-Y;t] P(X-Y\!,t) - w[Y|X;t]P(X\!,t) \Big] \qquad$$
((6.1a))

with the notation

$$w[Y|X;t] := w(X+Y | X,t) \, .$$
((6.1b))

For the right-hand side of (6.1a) a Taylor approximation [270] can be carried out (cf. [100], pp. 550ff.) which, when applied to the master equation, is denoted as the Kramers-Moyal expansion [84, 160, 193]. Introducing the lth jump moments

$$m_{l_1\dots l_M}(X\!,t) := \int d^M Y \; Y_1^{l_1}\cdot \dots \cdot Y_M^{l_M} w[Y|X;t]$$
((6.2a))

with

$$l := l_1 + \dots + l_M$$
((6.2b))

we obtain

$$\begin{aligned} \!\!\!& & \frac{\partial}{\partial t} P(X,t)\\ \!\!\!&=& \int \! d^M Y \bigg\{ \bigg[ \sum_{l=0}^\infty \!\! \sum_{\scriptstyle l_1,\dots,l_M \atop \scriptstyle (l_1+\dots + l_M = l)} \!\!\!\! \frac{(-Y_1)^{l_1}\dots (-Y_M)^{l_M}}{l_1!\dots l_M!}\ \!\!\!& & \times\, \frac{\partial^{l_1}}{\partial X_1^{l_1}}\dots \frac{\partial^{l_M}}{\partial X_M^{l_M}} \Big(w[Y| X;t] P(X,t)\Big) \bigg] - w[Y|X;t]P(X,t) \bigg\}\\ \!\!\!& = & \sum_{l=1}^\infty \frac{(-1)^l}{l!} \sum_{\scriptstyle l_1,\dots,l_M \atop \scriptstyle (l_1+\dots + l_M = l)} \frac{l!}{l_1!\dots l_M!} \frac{\partial^{l_1}}{\partial X_1^{l_1}}\dots \frac{\partial^{l_M}}{\partial X_M^{l_M}} \Big[ m_{l_1\dots l_M}(X,t) P(X,t) \Big] \, .\end{aligned}$$
((6.2c))

If series (6.2) is convergent, (6.2) and (6.1) are completely equivalent formulations. In particular, the transition rates \(w[Y|X;t]\) of the master equation can be uniquely reconstructed from the corresponding jump moments (6.2a): \(w[Y|X;t]\) is the inverse Fourier transform

$$w[Y|X;t] = \frac{1}{(2\pi)^M} \int d^M \omega \, \exp ( -\mathrm{i} \omega\cdot Y) \varTheta (\omega,X;t)$$
((6.3a))

of the charakteristic function

$$\varTheta(\omega,X;t) := \int d^M Y \, \exp (\mathrm{i} \omega \cdot Y) w[Y|X;t] \, .$$
((6.3b))

In addition, because of

$$m_{l_1\dots l_M}(X,t) = \left. \frac{1}{\mathrm{i}^{l_1 + \dots + l_M}} \frac{\partial^{l_1}}{\partial \omega_1^{l_1}} \dots \frac{\partial^{l_M}}{\partial \omega_M^{l_M}} \varTheta(\omega,X;t) \right|_{\omega_1=\dots=\omega_M=0} \, ,$$
((6.3c))

the Taylor representation of the characteristic function is determined by the jump moments \(m_{l_1\dots l_M}\) (cf. [241], pp. 16ff.):

$$\varTheta(\omega,X;t) = \sum_{l=0}^\infty \sum_{\scriptstyle l_1,\dots,l_M \atop \scriptstyle (l_1+\dots + l_M = l)} m_{l_1\dots l_M}(X,t) \frac{(\mathrm{i}\omega_1)^{l_1}}{l_1!} \cdot \dots \cdot \frac{(\mathrm{i}\omega_M)^{l_M}}{l_M!} \, .$$
((6.3d))

In order to analytically and numerically cope with the Kramers-Moyal expansion (6.2), it is desirable to approximate this by l terms of the infinite series. Since \(P(X,t)\) has the meaning of a probability density, our demand is

$$P(X,t) \stackrel{!}{\ge} 0 \qquad \forall \, X,t \, .$$
((6.4))

According to the Pawula theorem [216, 241] this condition is only fulfilled for \(l \le 2\). In the second order approximation (l = 2), (6.2) takes on the form of the Fokker-Planck equation [84]:

$$\begin{array}{rcl} \frac{\partial}{\partial t} P(X,t) &=& - \sum_{I} \frac{\partial}{\partial X_I} \Big[m_I(X,t)P(X,t)\Big]\\ &&+\; \frac{1}{2} \sum_{I,J} \frac{\partial}{\partial X_I} \frac{\partial}{\partial X_J} \Big[m_{IJ}(X,t)P(X,t)\Big] \end{array}$$
((6.5))

where

$$m_I := m_{l_1\dots l_M} \mathrm{with} l_I = 1, \, l_J = 0 \hbox{ for } J\ne I$$
((6.6a))

and

$$m_{\!I\!J} := \left\{ \begin{array}{ll} m_{l_1\dots l_M} \hbox{with} l_I = 1 = l_J, \, l_K = 0 \hbox{ for } K\ne I,J & \hbox{if } J\ne I \\ m_{l_1\dots l_M} \hbox{with} l_I = 2, \, l_K = 0 \hbox{ for } K\ne I & \hbox{if } J=I. \end{array}\right.$$
((6.6b))

The configurational Fokker-Planck equation corresponding to the configurational master equation (5.50) can be obtained analogously. It reads

$$\begin{array}{rcl} \frac{\partial}{\partial t} P(n,t) &=& - \sum_{a,x} \frac{\partial}{\partial n_{x}^a} \Big[m_{x}^a(n,t)P(n,t)\Big]\\ &&+\; \frac{1}{2} \sum_{a,x}\sum_{b,x'} \frac{\partial}{\partial n_{x}^a}\frac{\partial}{\partial n_{x'}^b} \Big[ m_{x}^a{}_{x'}^b(n,t)P(n,t) \Big] \end{array}$$
((6.7))

if n is formally treated as a continuous quantity. Combining the subscripts x and superscripts a in a vectorial subscript

$$I := (a,x)^{\mathrm{tr}}$$
((6.8))

and replacing \(n_{x}^a \equiv n_I\) by X I as well as n by X, (6.7) has the same form as (6.5) with \(M = AS\). In the following, notation (6.5) will therefore also be used for the configurational Fokker-Planck equation (6.7).

For the second order Taylor approximation to be applicable, the expansion parameters Y I have, on average, to be small compared to the system size

$$\varOmega := \max_I \max_{X',X} | X'_I - X_I | \, .$$
((6.9))

The decisive quantity for this is the smallness parameter δ defined by

$$\begin{aligned} \delta^2 &:= & \max_I \max_{X} \frac{m_{II}(X,t)} {\varOmega^2 \int d^M Y' \, w[Y'|X;t] } \\ &=& \max_I \max_{X} \int d^M Y \left( \frac{Y_I}{\varOmega} \right)^2 \frac{w[Y|X;t]}{\int d^M Y' \, w[Y'|X;t] } \end{aligned}$$
((6.10))

since it is a measure for the relative average jump distance. Transforming master equation (6.1) into scaled co-ordinates

$$Z := \varepsilon X \qquad \hbox{with} \qquad \varepsilon := \frac{1}{\varOmega} \, ,$$
((6.11))

the corresponding Kramers-Moyal expansion takes on the form

$$\begin{aligned} \frac{\partial}{\partial t} P(Z,t) &=& - \sum_{l=1}^\infty \frac{(-\varepsilon)^{l-1}}{l!} \sum_{\scriptstyle l_1,\dots,l_M \atop \scriptstyle (l_1+\dots + l_M = l)} \frac{l!}{l_1!\dots l_M!}\\ && \times \frac{\partial^{l_1}}{\partial Z_1^{l_1}}\dots \frac{\partial^{l_M}}{\partial Z_M^{l_M}} \Big[M_{l_1\dots l_M}(Z,t) P(Z,t)\Big] \qquad \end{aligned}$$
((6.12a))

with the scaled jump moments

$$M_{l_1\dots l_M}(Z,t) := \varepsilon \, m_{l_1\dots l_M}\left(\frac{Z}{\varepsilon},t\right) \, .$$
((6.12b))

Therefore, with regard to the order of magnitude, we find

$$\begin{aligned} & & \varepsilon^{l-1} | M_{l_1\dots l_M}(Z,t)| \; = \; \frac{1}{\varOmega^l} | m_{l_1 \dots l_M}(Z/\varepsilon,t)| \\ &\le & \int d^M Y \left| \left( \frac{Y_1}{\varOmega} \right)^{l_1} \! \cdot \ldots \cdot \left( \frac{Y_M}{\varOmega} \right)^{l_M} \right| \frac{w[Y|X;t]} {\int d^M Y' \, w[Y'|X;t]} \int d^M Y' \, w[Y'|X;t] \\ & \sim & \delta^l \int d^M Y' \, w[Y'|X;t] \, .\end{aligned}$$
((6.13))

According to this, in the case \(\delta \ll 1\) the scaled jump moments become smaller and smaller for increasing values of \(l= l_1 + \dots + l_M\). Consequently, they are most often neglected for \(l > 2\). However, we must not neglect the second jump moments if we do not want to suppress the effect of the fluctuations. Hence as a suitable approximation we obtain the Fokker-Planck equation

$$\begin{aligned} \frac{\partial}{\partial t} P(Z,t) = &- \! \sum_{I} \frac{\partial}{\partial Z_I} \Big[ k_I(Z,t)P(Z,t) \Big] \\ &+ \frac{\varepsilon}{2} \sum_{I,J} \frac{\partial}{\partial Z_I} \frac{\partial}{\partial Z_J} \Big[q_{\!I\!J}(Z,t)P(Z,t)\Big] \end{aligned}$$
((6.14a))

with the scaled first jump moments

$$k_I(Z,t) := M_I(Z,t) := \varepsilon m_I\left(\frac{Z} {\varepsilon},t \right)$$
((6.14b))

and the scaled second jump moments

$$q_{\!I\!J}(Z,t) := M_{\!I\!J}(Z,t) := \varepsilon m_{IJ}\left(\frac{Z}{\varepsilon},t \right) \, .$$
((6.14c))

3 Properties

In the following we will briefly treat the properties of Fokker-Planck equation (6.14).

3.1 The Continuity Equation

Equation (6.14) can be written in the form of the continuity equation

$$\frac{\partial}{\partial t} P(Z,t) = - \nabla j(Z,t) \, .$$
((6.15a))

Here

$$j_I(Z,t) := k_I(Z,t)P(Z,t) - \frac{\varepsilon}{2} \sum_J \frac{\partial}{\partial Z_J} \Big[ q_{IJ}(Z,t)P(Z,t) \Big]$$
((6.15b))

is the component I of the probability current density j.

3.2 Normalization

In the following we will occasionally require G auss’ divergence theorem which facilitates the transformation of a volume integral over the divergence \(\nabla j(Z,t)\) of a vector field \(j(Z,t)\) into a surface integral (cf. [246], pp. 1048ff.):

$$\int\limits_V d^M Z \, \nabla j(Z,t) = \int\limits_{\partial V} dO(Z) \cdot j(Z,t) \, .$$
((6.16))

Here \(\partial V\) means the surface of volume V. \(dO(Z)\) denotes vectors of magnitude dO which are perpendicular to the differential surface elements of \(\partial V\) of area dO and point to the outside. Thus we integrate over the component

$$\frac{dO}{dO} \cdot j(Z,t)$$
((6.17))

of \(j(Z,t)\) that is perpendicular to the surface.

Gauss’ divergence theorem and continuity equation (6.15) imply

$$\begin{aligned} \frac{\partial}{\partial t} \int\limits_V d^M Z \, P(Z,t) &=& - \int\limits_V d^M Z \, \nabla\!j(Z,t)\\ &=& - \int\limits_{\partial V} dO \cdot j(Z,t)\\ &=& 0 \end{aligned}$$
((6.18))

if the probability current j vanishes on the surface \(\partial V\) of the system. Hence, for the possibility of normalization

$$\int\limits_V d^M ZP(Z,t) \stackrel{!}{=} 1$$
((6.19))

we must demand the boundary condition

$$j(Z,t)\Big|_{Z\in\partial V} \stackrel{!}{=} 0 \, .$$
((6.20))

In order to guarantee this we will assume that the relations

$$\frac{\partial}{\partial Z_J} P(Z,t) \Big|_{Z\in\partial V} = 0$$
((6.21a))

and

$$P(Z,t)\Big|_{Z\in \partial V} = 0$$
((6.21b))

are valid, i.e. that the probability distribution \(P(Z,t)\) decreases fast enough in the neighbourhood of the surface \(\partial V\) and vanishes on it. This is most often the case.

3.3 The Liouville Representation

The Fokker-Planck equation can be represented in the form

$$\frac{\partial}{\partial t} P(Z,t) = \mathcal{L}(Z,t) P(Z,t)$$
((6.22a))

with the Liouville operator

$$\mathcal{L}(Z,t) := - \sum_{I} \frac{\partial}{\partial Z_I} k_I(Z,t) + \frac{\varepsilon}{2} \sum_{I,J} \frac{\partial}{\partial Z_I} \frac{\partial}{\partial Z_J} q_{IJ}(Z,t) \, .$$
((6.22b))

Since, due to (6.3), every Fokker-Planck equation is equivalent to a master equation (6.1), many properties of the Fokker-Planck equation directly correspond to those of the master equation (cf. Sect. 3.3). Therefore, only a short review of these will be given in the following.

3.4 Non-negativity

If condition \(P(Z,t_0) \ge 0\) is fulfilled for all states Z at an initial time t 0, we have

$$P(Z,t) \ge 0$$
((6.23))

for all later times \(t>t_0\).

3.5 Eigenvalues

The eigenvalues λ of the Liouville operator have the property that their real part is non-positive:

$$\hbox{Re}(\lambda) \le 0$$
((6.24))

(cf. [241], pp. 104f., 143ff.). Since the Fokker-Planck equation possesses the eigenvalue \(\lambda = 0\), the existence of a stationary solution \(P^0(Z)\) is always guaranteed.

3.6 Convergence to the Stationary Solution

For the entropy-like Liapunov function

$$K(t) := \int d^M Z \, P(Z,t) \ln \frac{P(Z,t)}{P'(Z,t)}$$
((6.25))

(where \(P(Z,t)\) and \(P'(Z,t)\) are two solutions of the Fokker-Planck equation (6.14) for different initial distributions) one can show

$$K(t) \ge 0 \, ,$$
((6.26a))
$$\frac{\partial K(t)}{\partial t} \le 0$$
((6.26b))

in the case of time-independent jump moments. If \(P'(Z,t) := P^0(Z)\) is a stationary solution and if \(q_{IJ}(Z) > 0\) and \(P^0(Z) > 0\) holds for almost every Z (i.e. with probability 1), every solution \(P(Z,t)\) of the Fokker-Planck equation will converge to \(P^0(Z)\) in the course of time and the stationary solution \(P^0(Z)\) is unique. (Cf. [84], pp. 61ff.)

4 Solution Methods

4.1 Stationary Solution

The stationary solution \(P^0(Z)\) of the Fokker-Planck equation in several dimensions can be explicitly found only under certain circumstances. First of all, the jump moments \(k_I(Z,t)\) and \(q_{IJ}(Z,t)\) must be time-independent. Furthermore, it will be assumed that in the stationary case not only the divergence \(\nabla j(Z)\) of the probability current \(j(Z)\) vanishes but also the probability current itself:

$$j(Z) \equiv \mathbf{0} \, .$$
((6.27))

Then (6.27) can, by means of the product rule (cf. p. xxviii), be brought into the form

$$\frac{\varepsilon}{2} \sum_J q_{IJ}(Z) \frac{\partial P^0(Z)} {\partial Z_J} = P^0(Z) \left[ k_I(Z) - \frac{\varepsilon}{2} \sum_J \frac{\partial}{\partial Z_J} q_{IJ}(Z) \right] \, .$$
((6.28))

If the matrix \(\underline{q}(Z) \equiv (q_{I\!J}(Z))\) has the inverse \(\underline{q}^{-1}(Z) \equiv \Big(q_{I\!J}^{-1}(Z)\Big)\) we obtain

$$\begin{aligned}\frac{\partial}{\partial Z_I} [\ln P^0(Z)] &=& \frac{\partial P^0(Z)/\partial Z_I}{P^0(Z)}\\ &=& \sum_K q_{IK}^{-1}(Z) \left[ \frac{2}{\varepsilon}k_K(Z) - \sum_J \frac{\partial}{\partial Z_J} q_{KJ}(Z) \right]\\ &=:& f_I(Z) \, .\end{aligned}$$
((6.29))

Since the left-hand side of (6.29) is a gradient, (6.29) can only be fulfilled on the integrability conditions

$$\frac{\partial f_I}{\partial Z_J} \stackrel{!}{=} \frac{\partial f_J}{\partial Z_I} \qquad \forall \, I, J \, .$$
((6.30))

However, if these are satisfied, the stationary solution can be written in the form

$$P^0(Z) = P^0(Z_0) \hbox{e}^{-\phi(Z)}$$
((6.31a))

with the potential

$$\phi(Z) := - \int\limits_{Z_0}^{Z} d Z' \cdot f(Z')$$
((6.31b))

(cf. [84], pp. 106f.). This implies

$$f_I(Z) = - \frac{\partial}{\partial Z_I} \phi(Z) \, .$$
((6.32))

In the one-dimensional case integrability condition (6.30) is automatically fulfilled. The Fokker-Planck equation then possesses the stationary solution

$$P^0(Z) = P^0(Z_0) \frac{q(Z_0)}{q(Z)} \exp\left( \frac{2}{\varepsilon} \int\limits_{Z_0}^Z dZ' \, \frac{k(Z')}{q(Z')} \right) \, .$$
((6.33))

4.2 Path Integral Solution

For the time-dependent solution of the Fokker-Planck equation a path integral representation exists which is related to the quantum mechanical path integral solution of the Schrödinger equation [73, 75]. This reads

$$P(Z,t) = \lim_{n\rightarrow \infty} \int d^M Z_{n-1} \dots \int d^M Z_0 \left[ \prod_{i=1}^{n} P(Z_{i},t_{i}| Z_{i-1},t_{i-1})\right] P(Z_0,t_0)$$
((6.34a))

with

$$Z_n \equiv Z, \qquad t_i := t_0 + i \varDelta t, \qquad \varDelta t := \frac{t-t_0}{n} \, ,$$
((6.34b))

and the conditional probability

$$\begin{aligned}&& P(Z_{i+1},t_{i+1}|Z_{i},t_{i}) := \frac{1}{\sqrt{ (2\pi)_{ }^M \varepsilon \varDelta t |\underline{q} |}}\\ && \times\exp\left(-\frac{1}{2\varepsilon \varDelta t} \Big[ Z_{i+1} - Z_{i} - k(Z_{i},t_{i})\Big]^{\mathrm{tr}} \underline{q}^{-1} (Z_{i},t_{i})\Big[ Z_{i+1} - Z_{i} - k(Z_{i},t_{i}) \Big]\right)\\ &&\end{aligned}$$
((6.34c))

(cf. [241], pp. 73ff.). Here \(\underline{q}^{-1}\) denotes the inverse of \(\underline{q} \equiv ( q_{IJ} )\) and \(|\underline{q}|\) its determinant.

4.3 Interrelation with the Schrödinger Equation

Since quantum mechanics provides many solution methods for the Schrödinger equation, it is sometimes useful that the Fokker-Planck equation can, on certain conditions, be transformed into a Schrödinger equation.

Let us assume the existence of a stationary solution of the form (6.31) which fulfils integrability conditions (6.30) and makes the probability current (6.30) vanish. Then, like in Sect. 3.4.1, the Fokker-Planck equation

$$\frac{\partial}{\partial t} P(Z,t) = \mathcal{L}(Z) P(Z,t)$$
((6.35))

(cf. (6.22)) can, by means of the transformation

$$\psi(Z,t):= \frac{P(Z,t)}{\sqrt{P_0(Z)}} = \hbox{e}^{\phi(Z)/2} P(Z,t) \, ,$$
((6.36a))

be converted into the equation

$$\frac{\partial}{\partial t} \psi(Z,t) = \mathcal{H}\psi(Z,t)$$
((6.36b))

with the Hermit ian (self-adjoint) operator

$$\mathcal{H} := \hbox{e}^{\phi/2} \mathcal{L} \hbox{e}^{-\phi/2} \, .$$
((6.36c))

\(\mathcal{H}\) can be written in the form

$$\mathcal{H} = \frac{\partial}{\partial Z_I} \varepsilon q_{I\!J}^{ } \frac{\partial}{\partial Z_J} - V(Z)$$
((6.36d))

with the potential

$$\begin{array}{rcl} V(Z) &:=& \displaystyle \hbox{e}^{\phi/2} \left( \frac{\partial}{\partial Z_I} \varepsilon q_{I\!J}^{ } \frac{\partial}{\partial Z_J} \hbox{e}^{-\phi/2} \right) \\ &=& \displaystyle \frac{\varepsilon}{4} q_{I\!J} \frac{\partial \phi}{\partial Z_I} \frac{\partial \phi}{\partial Z_J} - \frac{\varepsilon}{2} \left( \frac{\partial}{\partial Z_I} q_{I\!J}^{ } \frac{\partial \phi}{\partial Z_J}\right) \end{array}$$
((6.36e))

(cf. [241], pp. 139ff., and [242]). Now, after formally setting

$$\tau := \frac{t}{\mathrm{i}\hbar} = - \frac{\mathrm{i}}{\hbar} t$$
((6.37a))

and

$$|\psi(\tau)\rangle := \psi(Z,\mathrm{i}\hbar \tau) \, ,$$
((6.37b))

(6.36) takes on the form

$$\mathrm{i}\hbar \frac{\partial}{\partial \tau} |\psi(\tau)\rangle = \mathcal{H} |\psi(\tau) \rangle$$
((6.37c))

of a Schrödinger equation [250] with the ‘mass tensor’

$$\frac{\hbar^2}{2\varepsilon} q_{IJ}^{-1} \, .$$
((6.38))

5 Mean Value and Covariance Equations

First, let us define the mean value of a function in the same way as in (3.168). Then, by multiplying the Fokker-Planck equation (6.14) by Z K and \(Z_K Z_L\) and integrating over Z we obtain

$$\frac{d\langle Z_K\rangle}{dt} = \langle k_K(Z,t)\rangle$$
((6.39))

and

$$\frac{d \langle Z_L Z_K \rangle}{dt} = \langle Z_K k_L(Z,t) \rangle + \langle Z_L k_K(Z,t) \rangle + \varepsilon \langle q_{KL}(Z,t) \rangle \, .$$
((6.40))

Here we applied the relations

$$\sum_I Z_K \frac{\partial}{\partial Z_I} k_I = \sum_I \frac{\partial}{\partial Z_I} \Big( Z_K k_I \Big) - k_K \, ,$$
((6.41a))
$$\begin{aligned} \sum_{I,J} Z_K \frac{\partial}{\partial Z_I} \frac{\partial}{\partial Z_J} q_{IJ} &=& \sum_{I,J} \frac{\partial}{\partial Z_I} \frac{\partial}{\partial Z_J} \Big( Z_K q_{IJ} \Big)\\ &&- \sum_I \frac{\partial}{\partial Z_I} \Big( q_{KI} + q_{IK} \Big) \, , \end{aligned}$$
((6.41b))
$$\sum_I Z_K Z_L \frac{\partial}{\partial Z_I}k_I = \sum_I \frac{\partial}{\partial Z_I} \Big(Z_K Z_L k_I\Big) - \Big( Z_K k_L + Z_L k_K \Big) \, ,$$
((6.41c))
$$\begin{aligned} \sum_{I,J} Z_K Z_L \frac{\partial}{\partial Z_I} \frac{\partial}{\partial Z_J} q_{IJ} &=& \sum_{I,J} \frac{\partial}{\partial Z_I} \frac{\partial}{\partial Z_J} \Big( Z_K Z_L q_{IJ} \Big) + \Big( q_{KL} + q_{LK} \Big)\\ &&- \sum_I \frac{\partial}{\partial Z_I} \Big[ Z_L \Big(q_{IK} + q_{KI}\Big) + Z_K \Big(q_{IL} + q_{LI}\Big) \Big] \, . \qquad\\ \end{aligned}$$
((6.41d))

Apart from this, we used Gauss’ divergence theorem and neglected the surface terms because of the assumed vanishing of \(P(Z,t)\) on the surface \(\partial V\).

Finally, for the covariances

$$\sigma'_{KL}(t) := \langle Z_K Z_L \rangle - \langle Z_K \rangle \langle Z_L \rangle$$
((6.42))

we find the equations

$$\frac{d\sigma'_{KL}}{dt} = \Big\langle \varepsilon q_{KL}(Z,t)\Big\rangle + \Big\langle (Z_K - \langle Z_K\rangle)k_L(Z,t)\Big\rangle + \Big\langle (Z_L - \langle Z_L\rangle)k_K(Z,t)\Big\rangle \,.$$
((6.43))

If the mean value equations (6.39) and the covariance equations (6.43) are subject to an inverse transformation into the co-ordinates \(X = Z/\varepsilon\), they read

$$\frac{d\langle X_I\rangle}{dt} = \langle m_I(X,t)\rangle$$
((6.44))

and

$$\frac{d\sigma_{I\!J}^{ }}{dt} = \Big\langle m_{I\!J}(X,t)\Big\rangle + \Big\langle (X_I - \langle X_I\rangle)m_J(X,t)\Big\rangle + \Big\langle (X_J - \langle X_J\rangle)m_I(X,t)\Big\rangle \, .$$
((6.45))

A comparison with (3.172) and (3.175) shows that these equations completely agree with the mean value and covariance equations of master equation (3.5).

During the evaluation of (6.39) and (6.43) we will, in general, apply the same approximations as in Sect. 3.5 in order to obtain closed equations. By means of a method analogous to the projector formalism of Sect. 3.2.4.2 one can also derive exact closed mean value equations from (6.22) which, however, like (3.72), contain a time integral (i.e. they are non-local in time: cf. [292], pp. 73ff.).

5.1 Interpretation of the Jump Moments

The interpretation of the first and second jump moments follows directly from mean value Equation (6.44) and covariance Equation (6.45). If X is the state of the considered system at the time t this implies

$$P(X',t) = \delta_{X'X} \qquad \hbox{and} \qquad \langle f(X',t) \rangle_t = f(X,t)$$
((6.46))

for an arbitrary function \(f(X,t)\). Consequently, the following holds:

$$\begin{aligned} m_I(X,t) &=& \displaystyle \langle m_I(X'\!,t)\rangle_t\\ &=& \displaystyle \frac{d}{dt} \langle X'_{\!I} \rangle_{t}\\ &=& \displaystyle \lim_{\varDelta t \rightarrow 0} \frac{1}{\varDelta t} \Big( \langle X'_{\!I} \rangle_{t+\varDelta t} - \langle X'_{\!I} \rangle_t \Big)\\ &=& \displaystyle \lim_{\varDelta t \rightarrow 0} \frac{1}{\varDelta t} \langle X'_{\!I} - X_I \rangle_{t+\varDelta t} \end{aligned}$$
((6.47a))

and

$$\begin{aligned} \displaystyle m_{IJ}(X,t) &=& \displaystyle \frac{d}{dt} \sigma_{I\!J}^{ }(t) + 0 + 0 \; = \; \frac{\partial \langle X'_{\!I} X'_{\!J}\rangle_t}{\partial t} - \frac{\partial (\langle X'_{\!I} \rangle_t \langle X'_{\!J} \rangle_t)}{\partial t} \\ &=& \frac{\partial \langle X'_{\!I} X'_{\!J}\rangle_t}{\partial t} - \langle X'_{\!I}\rangle_t \frac{\partial \langle X'_{\!J} \rangle_t}{\partial t} - \langle X'_{\!J}\rangle_t \frac{\partial \langle X'_{\!I} \rangle_t}{\partial t} \\ &=& \displaystyle \lim_{\varDelta t \rightarrow 0} \frac{1}{\varDelta t} \Big( \langle X'_{\!I} X'_{\!J} \rangle_{t+\varDelta t} - X_I X_J \Big)\\ &&- \lim_{\varDelta t \rightarrow 0} X_I \frac{1}{\varDelta t} \Big( \langle X'_{\!J} \rangle_{t+\varDelta t} - X_J \Big)\\ &&- \lim_{\varDelta t \rightarrow 0} X_J \frac{1}{\varDelta t} \Big( \langle X'_{\!I} \rangle_{t+\varDelta t} - X_I \Big)\\ &=& \displaystyle \lim_{\varDelta t \rightarrow 0} \frac{1}{\varDelta t} \Big\langle (X'_{\!I} - X_I)(X'_{\!J} - X_J) \Big\rangle_{t + \varDelta t} \end{aligned}$$
((6.47b))

with

$$\Big\langle f(X'\!,t+\varDelta t) \Big\rangle_{t+\varDelta t} = \sum_{X'} f(X'\!,t+\varDelta t) P(X'\!,t+\varDelta t) \, .$$
((6.48))

That is, m I determines the systematic change of the mean value \(\langle X_I \rangle\) (the drift) and \(m_{IJ}\) the change of the covariance \(\sigma_{IJ}\) (the diffusion, i.e. the broadening of the distribution \(P(X,t)\) by fluctuations). Therefore, we call the first jump moments m I the drift coefficients and the second jump moments \(m_{IJ}\) the diffusion coefficients [151, 241, 292].

6 Boltzmann-Fokker-Planck Equations

In the following we will start with a Fokker-Planck equation for a system that consists of N subsystems α and proceed in a similar way as in Chap. 4. Again we combine the states x α of the N subsystems α in the vectors

$$X := (x_1,\dots,x_\alpha,\dots,x_N)^{\mathrm{tr}} \, ,$$
((6.49a))
$$x_{\alpha} := (x_{\alpha 1},\dots,x_{\alpha i},\dots, x_{\alpha m})^{\mathrm{tr}} \in \gamma_\alpha \, .$$
((6.49b))

Then in (6.5) we have \(I := (\alpha, i)^{\mathrm{tr}}\) and \(M = m \cdot N\).

Now let us define the vectors

$$X^\alpha := (\mathbf{0},\dots,\mathbf{0},x_\alpha,\mathbf{0}, \dots,\mathbf{0})^{\mathrm{tr}}$$
((6.50))

and

$$X^{\alpha_1\dots\alpha_k} := \sum_{i=1}^k X^{\alpha_i} \,.$$
((6.51))

The transition rates \(w[Y|X;t]\) can be written in the form

$$w[Y|X;t] := \left\{ \begin{array}{ll} w_{\alpha_1}[Y^{\alpha_1}|X^{\alpha_1};t] & \hbox{if } Y = Y^{\alpha_1} \\ w_{\alpha_1\alpha_2}[Y^{\alpha_1\alpha_2} |X^{\alpha_1\alpha_2};t] & \hbox{if } Y = Y^{\alpha_1\alpha_2} \\ \vdots & \vdots \\ w_{\alpha_1\dots\alpha_N}[Y^{\alpha_1\dots\alpha_N} |X^{\alpha_1\dots\alpha_N};t] & \hbox{if } Y = Y^{\alpha_1\dots\alpha_N} \\ 0 & \hbox{otherwise} \end{array} \right.$$
((6.52))

in which they are decomposed into contributions

$$w_{\alpha_1\dots\alpha_k}[Y^{\alpha_1\dots\alpha_k}| X^{\alpha_1\dots\alpha_k};t]$$
((6.53))

that describe the interactions between k subsystems. The restriction

$$w_{\alpha_1\dots\alpha_k} \equiv 0 \qquad \hbox{if two subscripts} \alpha_i, \alpha_j \hbox{are identical}$$
((6.54))

guarantees that self-interactions are excluded.

Furthermore, only the terms \(k\le 2\) will be taken into account which is well justified if the transitions of the subsystems are essentially due to spontaneous transitions w α and pair interactions \(w_{\alpha\beta}\). Then, as jump moments of the corresponding Fokker-Planck equation

$$\begin{aligned} \frac{\partial}{\partial t} P(X,t) &=& - \sum_{\alpha, i} \frac{\partial}{\partial x_{\alpha i}} \Big[m_{\alpha i}(X,t) P(X,t)\Big]\\[6pt] &&+\; \frac{1}{2} \sum_{\alpha, i}\sum_{\beta, j} \frac{\partial}{\partial x_{\alpha i}}\frac{\partial} {\partial x_{\beta j}} \Big[m_{\alpha i \beta j}(X,t)P(X,t)\Big] \end{aligned}$$
((6.55a))

result

$$\begin{aligned} m_{\alpha i}(X,t) &:= & \int d^M Y \, y_{\alpha i}w[Y|X;t] \\ &=& \sum_\beta \int d^M Y \, y_{\alpha i}w_\beta[Y^\beta|X^\beta;t] \\ &&+\; \frac{1}{2} \sum_{\beta,\gamma} \int d^M Y \, y_{\alpha i}w_{\beta \gamma} [Y^{\beta\gamma}|X^{\beta \gamma};t] \\ &=& \int d^M Y \, y_{\alpha i}w_\alpha[Y^\alpha|X^\alpha;t] \\ &&+\; \frac{1}{2} \left( \sum_\gamma \int d^M Y \, y_{\alpha i}w_{\alpha \gamma} [Y^{\alpha\gamma}|X^{\alpha \gamma};t] \right.\\ & &+\; \left. \sum_{\beta} \int d^M Y \, y_{\alpha i}w_{\beta \alpha} [Y^{\beta\alpha}|X^{\beta \alpha};t] \right)\\ &=& m_{\alpha i}^\alpha (X^\alpha,t) + \sum_{\beta} m_{\alpha i}^{\alpha \beta}(X^{\alpha\beta},t) \end{aligned}$$
((6.55b))

and

$$\begin{aligned} m_{\alpha i \beta j}(X,t) &:=& \int d^M Y \, y_{\alpha i}y_{\beta j}w[Y|X;t] \\ &=& \sum_\beta \int d^M Y \, y_{\alpha i}y_{\beta j} w_\gamma[Y^\gamma|X^\gamma;t]\\ &&+\; \frac{1}{2} \sum_{\gamma,\delta} \int d^M Y \, y_{\alpha i}y_{\beta j} w_{\gamma\delta}[Y^{\gamma\delta}|X^{\gamma\delta};t] \\ &=& \delta_{\alpha \beta} \left[ m_{\alpha i \alpha j}^\alpha (X^{\alpha},t) + \sum_{\gamma}m_{\alpha i \alpha j}^{\alpha\gamma} (X^{\alpha\gamma},t) \right]\\ &&+\; m_{\alpha i \beta j}^{\alpha \beta}(X^{\alpha\beta},t) \, . \qquad \end{aligned}$$
((6.55c))

Here the following notations were introduced:

$$m_{\alpha i}^\gamma (X^\gamma\!,t) :=\int d^M Y \, y_{\alpha i}w_\gamma[Y^\gamma|X^\gamma;t] \, ,$$
((6.56a))
$$m_{\alpha i}^{\gamma\delta}(X^{\gamma\delta}\!,t) := \int d^M Y \, y_{\alpha i}w_{\gamma\delta}[Y^{\gamma\delta} |X^{\gamma\delta};t] \, ,$$
((6.56b))
$$m_{\alpha i \beta j}^\gamma (X^\gamma\!,t) := \int d^M Y \, y_{\alpha i}y_{\beta j} w_\gamma[Y^\gamma|X^\gamma;t] \, ,$$
((6.56c))
$$m_{\alpha i \beta j}^{\gamma\delta}(X^{\gamma\delta}\!,t) := \int d^M Y \, y_{\alpha i}y_{\beta j} w_{\gamma\delta}[Y^{\gamma\delta}|X^{\gamma\delta};t] \, .$$
((6.56d))

Now we integrate Fokker-Planck equation (6.55) in an analogous way as (4.12) over all variables x β with \(\beta \ne \alpha\) and again assume the factorization

$$P_{\alpha\beta}(x_\alpha,x_\beta;t) := P_\alpha(x_\alpha,t)P_\beta(x_\beta,t)$$
((6.57))

of the pair distribution function \(P_{\alpha\beta}(x_\alpha,x_\beta;t)\). Then we obtain, apart from

$$\int d^m x_{\alpha} \, P_{\alpha}(x_\alpha,t) = 1 \, ,$$
((6.58))

the equations

$$\begin{aligned} \frac{\partial }{\partial t}P_\alpha(x_\alpha,t) &=& - \sum_{i} \frac{\partial}{\partial x_{\alpha i}} \Big[K_{\alpha i} (x_\alpha,t) P_\alpha(x_\alpha,t)\Big]\\ &&+\; \frac{1}{2} \sum_{i, j} \frac{\partial}{\partial x_{\alpha i}}\frac{\partial} {\partial x_{\alpha j}} \Big[Q_{\alpha i j}(x_\alpha,t) P_\alpha(x_\alpha,t)\Big] \, . \end{aligned}$$
((6.59a))

The effective drift coefficients

$$K_{\alpha i}(x_\alpha,t) := m_{\alpha i}^{\alpha}(x_\alpha,t) + \sum_{\beta} \int d^m x_\beta \, m_{\alpha i}^{\alpha \beta} (x_\alpha,x_\beta,t)P_\beta(x_\beta,t)$$
((6.59b))

and the effective diffusion coefficients

$$Q_{\alpha i j}(x_\alpha,t) := m_{\alpha i \alpha j}^{\alpha} (x_\alpha,t) + \sum_{\beta} \int d^m x_\beta \, m_{\alpha i \alpha j}^{\alpha \beta} (x_\alpha,x_\beta,t) P_\beta(x_\beta,t)$$
((6.59c))

make use of the conventions

$$m_{..}^\alpha (x_\alpha,t) \equiv m_{..}^\alpha(X^\alpha\!,t)\, ,$$
((6.60a))
$$m_{..}^{\alpha\beta}(x_{\alpha},x_{\beta},t) \equiv m_{..}^{\alpha\beta}(X^{\alpha\beta}\!,t)$$
((6.60b))

and take into account, each with its last term, the interactions between the subsystems β. Moreover, we applied Gauss’ divergence theorem in order to convert the terms

$$\sum_j \int d^m x_\beta \, \frac{\partial}{\partial x_{\beta j}} \left( \frac{\partial}{\partial x_{\alpha i}} \Big[m_{\alpha i \beta j}(X,t)P(X,t)\Big] \right)$$
((6.61))

for \(\beta \ne \alpha\) into vanishing surface integrals (cf. Sect. 6.3.2). Finally, we used the relation

$$m_{..}^{\gamma\delta} \equiv 0 \qquad \hbox{for} \qquad \gamma = \delta$$
((6.62))

which follows from (6.54).

Distinguishing only A different types a of elements consisting of N a subsystems (cf. Sect. 4.3) implies

$$\gamma_\alpha \equiv \gamma_a \hbox{if} \alpha \in a \, ,$$
((6.63a))
$$P_\alpha \equiv P_a \hbox{if} \alpha \in a \, ,$$
((6.63b))
$$\int d^m x \, P_a(x,t) = 1 \, ,$$
((6.63c))
$$m^\alpha_{..} \equiv \widehat{m}^{a}_{..} \hbox{if} \alpha \in a \, ,$$
((6.63d))

and

$$m^{\alpha\beta}_{..} \equiv \left\{ \begin{array}{ll} \widehat{m}^{ab}_{..} &\hbox{if } \alpha \ne \beta, \hbox{ } \alpha \in a, \hbox{ } \beta \in b \\ 0 &\hbox{if } \alpha = \beta. \end{array} \right.$$
((6.63e))

So we obtain (with respect to the considerably reduced number of variables) equations which are significantly simplified:

$$\begin{array}{rcl} \frac{\partial}{\partial t}P_a(x,t) &=& - \sum_{i} \frac{\partial}{\partial x_{i}} \Big[K_{a i} (x,t) P_a(x,t)\Big] \\ &&+\; \frac{1}{2} \sum_{i, j} \frac{\partial}{\partial x_{i}}\frac{\partial} {\partial x_{j}} \Big[Q_{a i j}(x,t) P_a(x,t)\Big] \end{array}$$
((6.64a))

with the effective drift coefficients

$$\begin{array}{rcl} K_{a i}(x,t) &:=& \displaystyle m_{a i}^{a}(x,t) + \sum_{b} \int d^m y \, m_{a i}^{ab} (x,y,t)P_b(y,t) \\ &=& \displaystyle \int d^m x' \, \varDelta x'_i w^a(x'|x;t) \, , \end{array}$$
((6.64b))

the effective diffusion coefficients

$$\begin{array}{rcl} Q_{a i j}(x,t) &:=& \displaystyle m_{a i a j}^{a} (x,t) + \sum_{b} \int\! d^m y \, m_{a i a j}^{ab} (x,y,t) P_b(y,t) \\ &=& \displaystyle \int d^m x' \, \varDelta x'_i \varDelta x'_j w^a(x'|x;t) \, , \end{array}$$
((6.6.4c))

and

$$m^a_{..} := \widehat{m}^a_{..} \, ,$$
((6.65a))
$$m^{ab}_{..} := \left\{ \begin{array}{ll} N_b\cdot \widehat{m}^{ab}_{..} & \hbox{if } b\ne a \\ (N_a -1)\cdot \widehat{m}^{ab}_{..} & \hbox{if } b=a \,. \end{array} \right.$$
((6.65b))

The result (6.64) could also (and even very easily) have been obtained from the Boltzmann-like equations (4.22) by means of a second order Taylor approximation (cf. Sect. 11.2 and [184]). Thus it is suggested to denote Eqs. (6.64) as the ‘Boltzmann-Fokker-Planck equations’.

6.1 Self-Consistent Solution

Due to factorization assumption (6.57) Eqs. (6.64) are closed equations, i.e. they contain no functions (like, for instance, pair distribution functions \(P_{\alpha\beta}(x_\alpha,x_\beta;t)\)) which cannot be determined by the others. However, since they are non-linear in \(P_a(x,t)\) due to the dependence of the effective drift and diffusion coefficients on \(P_b(y,t)\), we can only apply an approximation method to determine their solution. Having this in mind, we start from suitable probability distributions \(P_a^{(0)}(x,t)\) as e.g. the solutions of the Fokker-Planck equations which neglect pair interactions. These are given by

$$K_{a i}(x,t) := m_{a i}^{a}(x,t)$$
((6.66a))

and

$$Q_{a i j}(x,t) := m_{a i a j}^{a}(x,t) \, .$$
((6.66b))

With \(P_a^{(0)}(x,t)\) we evaluate the effective drift and diffusion coefficients for the next iteration step which gives the next approximation \(P_a^{(1)}(x,t)\) etc. Hence our iteration method has the form

$$\begin{array}{rcl} \displaystyle \frac{\partial}{\partial t} P_a^{(k+1)}(x,t) &=& \displaystyle - \sum_{i} \frac{\partial}{\partial x_{i}} \Big[ K_{a i}^{(k)}(x,t) P_a^{(k+1)}(x,t) \Big]\\ &&+\; \displaystyle \frac{1}{2} \sum_{i, j} \frac{\partial}{\partial x_{i}}\frac{\partial} {\partial x_{j}} \Big[ Q_{a i j}^{(k)}(x,t) P_a^{(k+1)}(x,t) \Big] \end{array}$$
((6.67a))

with

$$\int d^m x \, P_a^{(k+1)}(x,t) = 1 \, ,$$
((6.67b))
$$K_{a i}^{(k)}(x,t) := m_{a i}^{a}(x,t) + \sum_b \int d^mym_{a i}^{ab} (x,y,t)P_b^{(k)}(y,t) \, ,$$
((6.67c))

and

$$Q_{a i j}^{(k)}(x,t) := m_{a i a j}^{a}(x,t) + \sum_b \int d^m y \, m_{a i a j}^{ab}(x,y,t) P_b^{(k)}(y,t) \, .$$
((6.67d))

If the sequences of the functions \(P_a^{(k)}(x,t)\) converge for \(k \rightarrow \infty\), we have

$$P_a(x,t) = \lim_{k\rightarrow \infty} P_a^{(k)}(x,t) \, .$$
((6.68))

The recursive method (6.67) is especially useful for the determination of stationary solutions \(P_a^0(x)\). It formally corresponds to the quantum mechanical self-consistent field method of Hartree (cf. [49, 114]). An analogous recursive solution method can also be formulated for Boltzmann-like equations (4.22).

In the one-dimensional case (m = 1), the recursion formula for stationary solutions \(P_a^0(x)\) of the Boltzmann-Fokker-Planck equations is particularly simple. Here the stationarity condition

$$0 \stackrel{!}{=} - \frac{\partial}{\partial x} \Big[K^{(k)}_{a}(x) P_a^{0(k+1)}(x)\Big] + \frac{1}{2} \frac{\partial^2}{\partial x^2} \Big[Q^{(k)}_{a}(x) P_a^{0(k+1)}(x)\Big]$$
((6.69))

can, due to (6.33), be cast into the form

$$P_a^{0(k+1)}(x) = P_a^{0(k+1)}(x_0) \frac{Q_{a}^{(k)}(x_0)}{Q_{a}^{(k)}(x)} \exp\left( 2 \int\limits_{x_0}^x dx' \, \frac{K_{a}^{(k)}(x')}{Q_{a}^{(k)}(x')} \right)$$
((6.70))

with

$$\int dx \, P_a^{0(k+1)} (x) = 1 \, .$$
((6.71))