Diffusive Limits of the Master Equation in Inhomogeneous Media

Abstract

Diffusion is the macroscopic manifestation of disordered molecular motion. Mathematically, diffusion equations are partial differential equations describing the fluid-like large-scale dynamics of parcels of molecules. Spatially inhomogeneous systems affect in a position-dependent way the average motion of molecules; thus, diffusion equations have to reflect somehow this fact within their structure. It is known since long that in this case an ambiguity arises: there are several ways of writing down diffusion equations containing space dependence within their parameters. These ways are all potentially valid but not necessarily equivalent, meaning that the different diffusion equations yield different solutions for the same data. The ambiguity can only be resolved at the microscopic level: a model for the stochastic dynamics of the individual molecules must be provided, and a well-defined diffusion equation then arises as the long-wavelength limit of this dynamics. In this work we introduce and employ the integro-differential Master Equation (ME) as a tool for describing the microscopic dynamics. We show that is possible to provide a parameterized version of the ME, in terms of a single numerical parameter ($\alpha$), such that the different expressions for the diffusive fluxes are recovered for different values of $\alpha$. This work aims to fill a gap in the literature, where the ME was shown to deliver just one diffusive limit. In the second part of the paper some numerical computer models are introduced, both to support analytical considerations, and to extend the scope of the ME to more sophisticated scenarios, beyond the simplest $\alpha$-parameterization.

Appendix

The rationale of Ryskin’s result is based upon the Central Limit Theorem (CLT). The total displacement of a particle is approximated as the sum of several uncorrelated jumps. The single jumps are not necessarily identical, but picked up from some statistical distribution. Regardless of the details of the distribution, provided that the variance of the jumps remains finite, the CLT warrants that the total displacement distributes according to a statistical distribution that quickly approaches a Gaussian distribution after even a moderate numbers of steps.

For brevity we will sketch Ryskin’s proof for homogeneous systems only. Its generalization to inhomogeneous systems adds some mathematical labour but does not differ conceptually. In homogeneous systems P depends just from the difference of its arguments: $P = P(x-z)$, as explained in Sect. 3. This allows for a dramatic simplification after the Fourier transform of Eq. (8) is taken:

$$\begin{aligned} {\partial \tilde{n}(k,t) \over \partial t} = - {\tilde{n}(k,t) \over \tau } + {\tilde{n}(k,t) \tilde{P}(k) \over \tau } \end{aligned}$$

(30)

Equation (30) can be solved analytically:

$$\begin{aligned} \tilde{n}(k,t) = \tilde{n}(k,t=0) \times \exp \left( \int _0^t {dt \over \tau } (\tilde{P}(k)-1) \right) = \exp \left( {t \over \tau } (\tilde{P}(k)-1) \right) \end{aligned}$$

(31)

Formally, n(x, t) comes from the inverse transform of (31) and it does not appear analytically computable for generic P. However, we can write

$$\begin{aligned} \tilde{P}(k) = \int dz \, e^{i k z} P(z) = \sum _{m=0}^\infty {(i k)^m \over m!} \int dz \, z^m \, P(z) \end{aligned}$$

(32)

We will be considering just specularly symmetric transitions, as customary: $P(x-z) = P(z-x)$; the moments in Eq. (32) become

$$\begin{aligned} \int dz \, P = 1 = \tilde{P}(k=0) \end{aligned}$$

(33)

$$\begin{aligned} \int dz \, z \, P = 0 \end{aligned}$$

(34)

$$\begin{aligned} \int dz \, z^2 P = \sigma ^2 \end{aligned}$$

(35)

$$\begin{aligned} \int dz \, z^m P = <z^m> \equiv \mu _m , \quad m \ge 3 \end{aligned}$$

(36)

and $\mu _m = 0$ for all odd m’s.

Let us now rewrite Eq. (32) using the trigonometric expression of the exponential:

$$\begin{aligned} \tilde{P}(k)= & {} \int dz \, \cos (k z) P(z) + i \int dz \, \sin (k z) P(z) \\ \nonumber\rightarrow & {} |\tilde{P}(k)|^2 = \left( \int dz \, \cos (k z) P(z) \right) ^2 + \left( \int dz \, \sin (k z) P(z) \right) ^2 \\ \nonumber= & {} <\cos (k z)>^2 + < \sin (k z) >^2 \end{aligned}$$

(37)

This implies that $|\tilde{P}(k)| \le 1$ for generic $k \ne 0$. Furthermore,

$$\begin{aligned} |\tilde{P}(k)| \rightarrow 0, k \rightarrow \infty \end{aligned}$$

(38)

To demonstrate this, let us note that $\sin (k z), \cos (k z)$ are periodic with wavelength $\lambda = 2 \pi /k \rightarrow 0, k \rightarrow \infty$, while P is a smoothly varying function, hence is almost constant over $\lambda$: $P(z) \approx P(z+\lambda ) = P_0$. Therefore

$$\begin{aligned} \int _0^\lambda dz \, \cos (k z) P(z) \approx P_0 \int _0^\lambda dz \, \cos (k z) \approx 0 \end{aligned}$$

(39)

(the same holds for $\sin (k z)$).

We define $m, \varDelta t$ such that $t = j \varDelta t$, with j integer and $\varDelta t \approx O(\tau )$. Equation (31) becomes

$$\begin{aligned}&\exp \left( {t \over \tau } (\tilde{P}(k) -1) \right) = \left[ \exp \left( {\varDelta t \over \tau }(\tilde{P}(k) -1) \right) \right] ^j = \\ \nonumber&\left[ \exp \left( {\varDelta t \over \tau } \left( - {k^2 \sigma ^2 \over 2} + \sum _{m=3}^\infty {(i k)^m \over m!} \mu _m \right) \right) \right] ^j \end{aligned}$$

(40)

In the last line of (41) we have taken advantage of (33)–(36).

Let us define $\xi = j^{1/2} k$. Equation (41) becomes

$$\begin{aligned} {\tilde{n}(k,t) \over \tilde{n}(k,t=0)} = \exp \left[ {\varDelta t \over \tau } \left( -{\xi ^2 \sigma ^2 \over 2} + \sum _{m=3}^\infty {1 \over j^{{m \over 2}-1}} {(i \xi )^m \over m!} \mu _m \right) \right] \end{aligned}$$

(41)

Then, we consider separately the two limits $\xi > 1$ and $\xi \le 1$ (Notice that the boundary between $\xi > 1$ and $\xi \le 1$ is a dynamical one: it varies with time, i.e. with j). The former limit corresponds, for any fixed time, to taking $k \rightarrow \infty$ and therefore the result (38) holds: there is not contribution to the density from features at these wavelengths. Conversely, when $\xi \le 1$ the first term inside the exponent in Eq. (41) dominates over the others hence we can retain just it and, reverting to the original variables

$$ \tilde{n}(k,t) = \tilde{n}(k,t) \times \exp \left[ -{t \over \tau } {(k \sigma )^2 \over 2} \right] $$

which is the propagator of the diffusion equation, with diffusivity = $\sigma ^2/2\tau$. This concludes the proof.