Temporal Diffusion: From Microscopic Dynamics to Generalised Fokker–Planck and Fractional Equations

Abstract

The temporal Fokker–Planck equation (Boon et al. in J Stat Phys 3/4: 527, 2003) or propagation–dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. We present two generalizations of the temporal Fokker–Planck equation for the first passage distribution function $f_j(r,t)$ of a particle moving on a substrate with time delays $\tau _j$. Both generalizations follow from the first visit recurrence relation. In the first case, the time delays depend on the local concentration, that is the time delay probability $P_j$ is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, $P_j \propto f_j^{\nu - 1}$, the generalized Fokker–Planck equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the situation where the time delays are distributed according to a power law, $P_j \propto \tau _j^{-1-\alpha }$ (with $0< \alpha < 2$), in which case we obtain a fractional propagation-dispersion equation which is the temporal analog of the fractional spatial diffusion equation (with space and time interchanged). The analysis shows how certain microscopic mechanisms can lead to non-Gaussian distributions and non-classical scaling exponents.

Appendices

Appendix 1: Expansion of the Recurrence Relation

We first consider the expansion of $F_{j}^{(\nu )}\left[ f\right]$ (for simplicity in the notation we shall omit the upper index ${(\nu )}$ which will be reintroduced when necessary):

$$\begin{aligned}&F_{j}\left[ f\right] = \left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f} \nonumber \\- & {} \delta t\left\{ j\frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{ \partial x}+\sum _{l=1}^{N}l\frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial y_{l}}\right\} _{f} \times \left( \frac{\partial f\left( r, t\right) }{\partial t} \right) +\ldots \qquad \end{aligned}$$

(36)

where the notation $\left\{ \ldots \right\} _{f}$ means that all the variables $x,y_{1},\ldots$ are to be set equal to the $f\left( r,t\right)$’s in the r.h.s. of Eq. (5). Using this expansion, the generalized recurrence relation (6) becomes

$$\begin{aligned}&\delta r\frac{\partial f\left( r,t\right) }{\partial r} + \frac{1}{2}\left( \delta r\right) ^{2}\frac{\partial ^{2}f\left( r,t\right) }{\partial r^{2}}+\ldots = \nonumber \\&\quad - \delta t\sum _{j=1}^{N}jp_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f}\frac{\partial f\left( r,t\right) }{ \partial t} \nonumber \\&\quad + \frac{1}{2}\left( \delta t\right) ^{2}\sum _{j=1}^{N}jp_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f}\frac{\partial ^{2}f\left( r,t\right) }{\partial t^{2}} \nonumber \\&\quad + \left( \delta t\right) ^{2}\sum _{j=1}^{N}jp_{j}\left\{ j\frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial x}+\sum _{l=1}^{N}l\frac{ \partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial y_{l}}\right\} _{f}\left( \frac{\partial f\left( r,t\right) }{\partial t}\right) ^{2} +\ldots \nonumber \\ \end{aligned}$$

(37)

We now perform a multiscale expansion with

$$\begin{aligned} \frac{\partial }{\partial r} \rightarrow \epsilon \frac{\partial }{\partial {r_1}} \,+\,\epsilon ^2 \frac{\partial }{\partial {r_2}} \;\;\; ;\;\;\; \frac{\partial }{\partial t} \rightarrow \epsilon \frac{\partial }{\partial {t_1}} \,+\,\epsilon ^2 \frac{\partial }{\partial {t_2}}, \end{aligned}$$

(38)

and $f\,=\,f^{(0)}\,+\,\epsilon f^{(1)}\,+\,\,\mathcal{O}(\epsilon ^2)$, where $f^{(0)}$ is the distribution function in the absence of dispersion. To first order, we obtain

$$\begin{aligned} \mathcal{O}(\epsilon ^{1}) : \;\;\;\;\;\; \delta r\frac{\partial f^{(0)}\left( r,t\right) }{\partial r_{1}}=-\delta t \sum _{j=1}^{N}jp_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f^{(0)}}\frac{\partial f^{(0)}\left( r,t\right) }{\partial t_{1}}, \end{aligned}$$

(39)

and to second order

$$\begin{aligned}&\mathcal{O}(\epsilon ^{2}) : \;\;\;\;\;\; \delta r\frac{\partial f^{(1)}\left( r,t\right) }{\partial r_{1}}+ \delta r\frac{\partial f^{(0)}\left( r,t\right) }{\partial r_{2}}+\frac{1}{2}\left( \delta r\right) ^{2}\frac{\partial ^{2}f^{(0)}\left( r,t\right) }{\partial r_{1}^{2}} \nonumber \\&\quad = -\delta t\sum _{j=1}^{N} j p_{j}\left\{ \frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial x}+\sum _{l=1}^{N} \frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial y_{l}}\right\} _{f^{(0)}} f^{(1)}\left( r,t\right) \frac{\partial f^{(0)}\left( r,t\right) }{\partial t_{1}} \nonumber \\&\qquad - \delta t\sum _{j=1}^{N}jp_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f} \left( \frac{\partial f^{(1)}\left( r,t\right) }{\partial t_{1}} +\frac{\partial f^{(0)}\left( r,t\right) }{\partial t_{2}} \right) \nonumber \\&\qquad + \frac{1}{2}\left( \delta t\right) ^{2}\sum _{j=1}^{N}j^{2}p_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f}\frac{\partial ^{2}f^{(0)}\left( r,t\right) }{\partial t_{1}^{2}} \nonumber \\&\qquad + \left( \delta t\right) ^{2}\sum _{j=1}^{N}jp_{j}\left\{ j\frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial x}+\sum _{l=1}^{N}l\frac{ \partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial y_{l}}\right\} _{f^{(0)}}\left( \frac{\partial f^{(0)}\left( r,t\right) }{\partial t_{1}} \right) ^{2}.\nonumber \\ \end{aligned}$$

(40)

From the normalization condition (with (7) where $\frac{\partial f\left( r,t\right) }{\partial t}$ is unconstrained)

$$\begin{aligned} 1= & {} \sum _{j=1}^{N}p_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f^{(0)}} \,, \nonumber \\ 0= & {} \sum _{j=1}^{N}p_{j}\left\{ j\frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial x}+\sum _{l=1}^{N}l\frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial y_{l}}\right\} _{f^{(0)}}. \end{aligned}$$

(41)

It is easy, for instance, to check that these relations are indeed verified in the case of the power law (4). Differentiating (39) with respect to $r_1$ and reinserting (39) in the result, we obtain

$$\begin{aligned}&\left( \delta r\right) ^{2}\frac{\partial ^{2}f^{(0)}\left( r,t\right) }{\partial r_{1}^{2}} = -\delta r\delta t\sum _{j=1}^{N}jp_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f^{(0)}}\frac{\partial ^{2}f^{(0)}\left( r,t\right) }{\partial t_{1}\partial r_{1}} \nonumber \\&\quad = \left( \delta t\right) ^{2}\sum _{j=1}^{N}jp_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f^{(0)}}\frac{\partial }{\partial t_{1}} \left\{ \sum _{l=1}^{N}lp_{l}F_{l}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f^{(0)}}\frac{\partial f^{(0)}\left( r,t\right) }{\partial t_{1}} \nonumber \\&\quad = \left( \delta t\right) ^{2}\sum _{j=1}^{N}jp_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f^{(0)}} \times \nonumber \\&\quad \quad \times \sum _{l=1}^{N}lp_{l} \left\{ \frac{\partial F_{l}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial x}+\sum _{k=1}^{N}\frac{\partial F_{l}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial y_{k}}\right\} _{f^{(0)}} \left( \frac{\partial f^{(0)}\left( r,t\right) }{\partial t_{1}}\right) ^{2} \nonumber \\&\quad \quad + \left( \delta t\right) ^{2}\left( \sum _{j=1}^{N}jp_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f^{(0)}}\right) ^{2}\frac{\partial ^{2}f^{(0)}\left( r,t\right) }{\partial t_{1}^{2}}. \end{aligned}$$

(42)

Using this result in (40) gives

$$\begin{aligned}&\delta r\frac{\partial f^{(1)}\left( r,t\right) }{\partial r_{1}}+\delta r \frac{\partial f^{(0)}\left( r,t\right) }{\partial r_{2}} \nonumber \\&\quad = -\delta t\sum _{j=1}^{N}jp_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f^{(0)}}\ \left( \frac{\partial f^{(1)}\left( r,t\right) }{\partial t_{1}} + \frac{\partial f^{(0)}\left( r,t\right) }{\partial t_{2}} \right) \nonumber \\&\quad \quad - \delta t\sum _{j=1}^{N} j p_{j}\left\{ \frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial x}+\sum _{l=1}^{N} \frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial y_{l}}\right\} _{f^{(0)}} f^{(1)}\left( r,t\right) \frac{\partial f^{(0)}\left( r,t\right) }{\partial t_{1}} \nonumber \\&\quad \quad + \frac{1}{2}\left( \delta t\right) ^{2}\left( \sum _{j=1}^{N}j^{2}p_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f^{(0)}}-\left( \sum _{j=1}^{N}jp_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f^{(0)}}\right) ^{2}\right) \frac{ \partial ^{2}f^{(0)}\left( r,t\right) }{\partial t_{1}^{2}} \nonumber \\&\quad \quad + \left( \delta t\right) ^{2}\left( \begin{array}{c} \sum _{j=1}^{N}jp_{j}\left\{ j\frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial x}+\sum _{l=1}^{N}l\frac{\partial F_{j}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial y_{l}}\right\} _{f^{(0)}} \nonumber \\ -\sum _{j=1}^{N}jp_{j}\left\{ F_{j}\left( x;y_{1}\ldots ,y_{N}\right) \right\} _{f^{(0)}} \sum _{l=1}^{N}lp_{l}\left\{ \frac{\partial F_{l}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial x}+\sum _{k=1}^{N} \frac{\partial F_{l}\left( x;y_{1}\ldots ,y_{N}\right) }{\partial y_{k}}\right\} _{f^{(0)}} \end{array} \right) \nonumber \\&\quad \quad \times \left( \frac{\partial f^{(0)}\left( r,t\right) }{\partial t_{1}} \right) ^{2}, \end{aligned}$$

(43)

After recombining first and second order terms, resummation yields Eq. (9).

Appendix 2: General Confluent Equation

$$\begin{aligned} \frac{d^{2}w}{dx^{2}}+x\frac{dw}{dx}+\left( 2\nu - 1 \right) w=0 , \end{aligned}$$

(44)

From AS 13.1.35 in [8], the general confluent equation is

$$\begin{aligned} 0= & {} w^{\prime \prime }+\left( \frac{2a}{y}+2f^{\prime }(y)+\frac{bh^{\prime }(y)}{h(y)}-h^{\prime }(y)-\frac{h^{\prime \prime }(y)}{h^{\prime }(y)} \right) w^{\prime } \nonumber \\&\quad + \left( \frac{bh^{\prime }(y)}{h(y)}-h^{\prime }(y)-\frac{h^{\prime \prime }(y)}{h^{\prime }(y)}\right) \left( \frac{a}{y}+f^{\prime }(y)\right) w \nonumber \\&\quad + \left( \frac{a(a-1)}{y^{2}}+\frac{2af^{\prime }(y)}{y}+f^{\prime \prime }(y)+f^{\prime 2}(y)-\frac{c h^{\prime 2}(y)}{h(y)}\right) w. \end{aligned}$$

(45)

This equation and Eq. (44) match provided

$$\begin{aligned} a = 0 \;\;\;;\;\;\;\; b = \frac{1}{2} \;\;\;;\;\;\;\; c = 1 - {\nu } \;\;\;;\;\;\;\; f(y) = h(y)=\frac{y^{2}}{2}, \end{aligned}$$

and the general solution then is

$$\begin{aligned} w\left( y\right) =A\exp \left( -\frac{y^{2}}{2}\right) M\left( {1 - \nu },\frac{1}{2},\frac{y^{2}}{2}\right) +B\exp \left( -\frac{y^{2}}{2}\right) U\left( 1 - {\nu },\frac{1}{2},\frac{y^{2}}{2}\right) , \end{aligned}$$

(46)

or, using AS 13.1.3 in [8], with $y \equiv x$,

$$\begin{aligned} w\left( x\right) = {A}\,\left[ M\left( {1 - {\nu }},\frac{1}{2},\frac{x^{2}}{2}\right) +x \,\frac{B}{A}\, M\left( \frac{3}{2}- {\nu },\frac{3}{2},\frac{x^{2}}{2}\right) \right] \exp \left( -\frac{x^{2}}{2}\right) . \end{aligned}$$

(47)

Note that $w\left( 0\right) = A$ and $w^{\prime }\left( 0\right) =~B$.

Appendix 3: Derivation of the Fractional Temporal Diffusion Equation

The probability for $\hat{t} =T_{\alpha }$ is

$$\begin{aligned} f_{\alpha }(T_{\alpha },N)&= \int _{0}^{\infty } \delta (T_{\alpha }-\frac{1}{N^{1/\alpha }}\sum _{i=1}^{N}t_{i})p(t_{1})\ldots p(t_{N})dt_{1}\ldots dt_{N} \end{aligned}$$

(48)

so that

$$\begin{aligned} \tilde{f}_{\alpha }(\omega ,N) \equiv \int _{-\infty }^{\infty } e^{i \omega T}f_{\alpha }(T_{\alpha }) = \left( \tilde{p}\left( \frac{\omega }{N^{1/\alpha }}\right) \right) ^N , \end{aligned}$$

(49)

where a simple calculation gives

$$\begin{aligned} \tilde{p}(\omega ) = \alpha (-i\omega t_{0})^{\alpha }\Gamma (-\alpha ,-i \omega t_{0}) , \end{aligned}$$

(50)

which has the expansion for small $\omega t_{0}$

$$\begin{aligned} \tilde{p}(\omega ) = \alpha (-i\omega t_{0})^{\alpha }\Gamma (-\alpha )-\alpha \sum _{k=0}^{\infty }\frac{(-i \omega t_{0})^{k}}{k!(k-\alpha )}. \end{aligned}$$

(51)

Thus, taking the inverse Fourier transform and expanding in 1 / N leads to the result

$$\begin{aligned}&f_{\alpha }(T_{\alpha },N) = \int _{-\infty }^{\infty } \left( \tilde{p}\left( \frac{\omega }{N^{1/\alpha }}\right) \right) ^N e^{-i \omega T_{\alpha }} \frac{d\omega }{2\pi } \nonumber \\= & {} \int _{-\infty }^{\infty } \exp \left( -i \omega T_{\alpha }+N \ln \tilde{p}\left( \frac{\omega }{N^{1/\alpha }}\right) \right) \frac{d\omega }{2\pi } \nonumber \\= & {} \int _{-\infty }^{\infty } \exp \left( -i \omega T_{\alpha } +\alpha \left( -i \omega t_{0}\right) ^{\alpha } \Gamma (-\alpha ) + \frac{\alpha }{1-\alpha } (-i \omega t_{0})N^{1-1/\alpha } \right) \nonumber \\&\times \; \exp \left( \mathcal{{O}}(N^{1-2/\alpha })\right) \frac{d\omega }{2\pi }, \end{aligned}$$

(52)

so that the higher order terms are negligible for large N provided $\alpha < 2$. The probability density that the first arrival time to reach position N is $T = N^{1/\alpha }T_{\alpha }$ is therefore

$$\begin{aligned}&f(T,N) = \int _{0}^{\infty } f_{\alpha }(T_{\alpha },N)\delta (T-N^{1/\alpha }T_{\alpha }) dT_{\alpha } \\&\quad = N^{-1/\alpha }\int _{-\infty }^{\infty } \exp \left( -i \omega N^{1-1/\alpha } \left( \frac{T}{N} + \frac{\alpha }{1-\alpha } t_{0}\right) \right) \\&\quad \quad \times \exp \left( \alpha \left( -i \omega t_{0}\right) ^{\alpha } \Gamma (-\alpha ) +\mathcal{{O}}(N^{1-2/\alpha })\right) \frac{d\omega }{2\pi }. \end{aligned}$$

Rescaling the integration variable gives

$$\begin{aligned} f(T,N)= & {} \int _{-\infty }^{\infty } exp\left( i \omega \left( T + \frac{\alpha }{1-\alpha } t_{0} N\right) +\alpha \left( i \omega t_{0}\right) ^{\alpha } \Gamma (-\alpha )N\right) \frac{d\omega }{2\pi } \end{aligned}$$

(53)

which is the Levy-stable distribution with stability parameter $\alpha$. Defining the spatial variable $r \equiv N \delta r$, it is easy to see that this distribution satisfies the fractional temporal diffusion equation

$$\begin{aligned} \frac{\partial }{\partial r}f\left( T;r/\delta r\right) = \left[ \frac{\alpha t_{0}}{(1-\alpha ) \delta r}\frac{\partial }{\partial T}+\frac{{\alpha }t_{0}^{\alpha }}{ \delta r} \Gamma \left( -\alpha \right) \frac{\partial ^{\alpha }}{\partial T^{\alpha }}\right] f_{\alpha }\left( T;r/\delta r\right) . \end{aligned}$$

(54)