Nonextensive diffusion as nonlinear response

– The porous media equation has been proposed as a phenomenological “nonextensive” generalization of classical diffusion. Here, we show that a very similar equation can be derived, in a systematic manner, for a classical fluid by assuming nonlinear response, i.e. that the diffusive flux depends on gradients of a power of the concentration. The present equation distinguishes from the porous media equation in that it describes generalized classical diffusion, i.e. with r/ √ Dt scaling, but with a generalized Einstein relation, and with power-law probability distributions typical of nonextensive statistical mechanics. One of the characteristic features of the nonextensive thermodynamics introduced by Tsallis is the appearance of non-exponential distribution functions with power-law tails [1,2]. There has been considerable interest recently in the question of how such non-exponential distributions might arise from first-principle considerations [3–6]. In this paper, we show that it is possible to obtain, in a generic manner, power-law distributions for the diffusion of a tracer particle in a liquid by assuming a simple generalization of the usual linear response arguments. In classical transport theory, the probability to find the diffuser at point −→r at time t given that it starts at point −→r 0 at time 0 obeys the advection-diffusion equation ∂ ∂t P (−→r , t;−→r 0, 0) = ∂ ∂−→r · −→u (−→r , t)P (−→r , t;−→r 0, 0) + ∂ ∂−→r · ←→ D · ∂ ∂−→r P (−→r , t;−→r 0, 0) , (1) where −→u (−→r , t) is the drift and ←→ D the diffusion tensor, see e.g. [7]. Note that in general, we could, instead of probabilities, speak of the concentration of a diffusing species or the density of a single-component system undergoing self-diffusion just as well. There are currently two widely explored generalizations of this classical diffusion equation. The first involves a fractional time derivative in the diffusion term [8]. This fractional diffusion equation can be derived, e.g., by considering the behavior of a diffuser on a lattice with the waiting time between hops being governed by a power-law distribution (e.g., a Pareto distribution), a type of Levy flight [8]. A second generalization of the classical diffusion equation sometimes used to model anomalous diffusion is the so-called “porous media equation” ∂ ∂t P (−→r , t;− →r 0, 0) = ∂ ∂−→r · −→u (−→r , t)P (−→r , t;−→r 0, 0) + ∂ ∂−→r · ←→ D · ∂ ∂−→r P (−→r , t;−→r 0, 0) . (2)