Continuous Time Random Walk

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The Continuous Time Random Walk (CTRW) provides a mathematical framework for the study of transport in heterogenous media. It is much more general than usual transport models based on (local, Markovian) Ordinary Differential Equations, and in particular can handle transport in systems without characteristic scales (such as systems in a state of Self-Organised Criticality or SOC).

Motivation

Interestingly, the absence of local characteristic scales means that effective transport coefficients (the diffusivity etc.) become dependent on the system size, as is indeed suggested by experimental scaling laws for plasma confinement.

In the framework of transport in plasmas, it is believed that the presence of trapping regions (such as turbulent eddies, magnetic islands, internal transport barriers) may lead to sub-diffusion, whereas the occurrence of streamers and profile self-regulation (via turbulence) may lead to super-diffusion. The goal of the CTRW approach is to model the effective transport in the presence of these complex phenomena.

Starting point

CTRW theory [1] starts from the single-particle step distribution function (in one dimension)

$ \xi(\Delta x, \Delta t; x, t) $

giving the probability that a particle, located at position x at time t, takes a step of size Δx after waiting a time Δt > 0. All particles are assumed to be identical.

The Master Equation

By making some suitable additional assumptions regarding the nature of this single-particle step distribution, it is possible to compute the average behaviour of the system in the limit of infinitely many particles, and to deduce an evolution equation for the particle (probability) density. This equation is a (Generalized) Master Equation.

In standard CTRW theory, it is customary to assume that the single particle step distribution is separable, i.e., that Δx is independent from Δt, so that

$ \xi( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \psi(\Delta t; x,t) $

In addition, homogeneity in space and time is assumed (i.e., p and ψ do not depend on x and t). However, recently it was shown that a Master Equation can also be derived in the case that p depends on x and t, while ψ depends on x (but not t). [2] This significant extension of the standard CTRW model has led to the development of a model with very interesting properties from the point of view of plasma transport (see the cited reference and [3]).

The Generalized Master Equation (GME) can be written in the form

$ \frac{\partial n(x,t)}{\partial t} = \int_0^t \left ( \int{K(x-x',t-t',x',t')n(x',t')dx'} - n(x,t')\int{K(x-x',t-t',x',t')dx'}\right )dt' $

where n is the particle (probability) density, and K a kernel of the form

$ K( \Delta x, \Delta t; x,t) = p(\Delta x; x,t+\Delta t) \phi(\Delta t; x) $

The GME is an integro-differential equation, generalizing the usual (partial differential) equations for transport. The particle flux at any point in space depends on the global distribution of the transported particle density field, and on its history (although the history dependence can be eliminated by choosing a Markovian waiting time distribution).

The treatment of boundary conditions in a GME is different from standard differential equations. [4] The final (quasi) steady state of the system is a function of the balance between sources and sinks, rather than of imposed values or gradients at the system boundaries.

Fractional Differential Equations

While the Master Equation corresponds to a CTRW in the limit of many particles, the Fractional Differential Equation (FDE) corresponds to a Master Equation in the fluid limit. [5] The fluid limit is the limit in which only the part of the dynamics that is dominant for large scales and long times is retained, and is useful for understanding the steady state properties of a solution.

To obtain the fractional differential operators, it is necessary to make an assumption regarding the shape of the distributions appearing in the kernel K. Invoking the Generalized Limit Theorem for the sums of random variables, [6] these distributions are taken to be Lévy distributions. While the step distribution can be any Lévy distribution, the waiting time distribution must be positive extremal, since Δt > 0. This choice allows modelling both sub- and super-diffusive transport, and in the appropriate limit, standard ("Fickian") transport is recovered. If nothing else, this serves to show that all of the above constitute generalizations (on various levels) of the usual transport equations.

The main numerical advantage of the FDE approach over the GME is that the FDE allows constructing the final solution in the long-time limit by a single integration, [7] whereas the GME must be iterated in time. The FDE approach can be used fruitfully to model transport in fusion plasmas, i.e., finite-size systems. [8] On the other hand, the FDE approach does not capture some of the (interesting) dynamical behaviour inherent in the GME approach. [9]

References

  1. R. Balescu, Aspects of Anomalous Transport in Plasmas, Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307
  2. B.Ph. van Milligen, R. Sánchez, and B.A. Carreras, Probabilistic finite-size transport models for fusion: anomalous transport and scaling laws, Phys. Plasmas 11, 5 (2004) 2272
  3. B.Ph. van Milligen, B.A. Carreras, and R. Sánchez, Phys. Plasmas 11, 3787 (2004)
  4. B.Ph. van Milligen, I. Calvo, and R. Sánchez, Continuous time random walks in finite domains and general boundary conditions: some formal considerations, J. Phys. A: Math. Theor. 41 (2008) 215004
  5. R. Sánchez, B.A. Carreras, and B.Ph. van Milligen, Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations, Phys. Rev. E 71 (2005) 011111
  6. B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions of Sums of Independent Random Variables, Addison-Wesley, Reading, MA (1954)
  7. V.E. Lynch et al, Numerical methods for the solution of partial differential equations of fractional order, Journal of Computational Physics 192, 2 (2003) 406-421
  8. D. del-Castillo-Negrete, Fractional diffusion models of nonlocal transport, Phys. Plasmas 13 (2006) 082308
  9. B.Ph. van Milligen, B.A. Carreras, V.E. Lynch and R. Sánchez, Pulse propagation in a simple probabilistic transport model, Nucl. Fusion 47 (2007) 189