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Register a new userAn Introduction to Non-diffusive Transport Models
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Alexandre Bovet
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The process of diffusion is the most elementary stochastic transport process. Brownian motion, the representative model of diffusion, played a important role in the advancement of scientific fields such as physics, chemistry, biology and finance. However, in recent decades, non-diffusive transport processes with non-Brownian statistics were observed experimentally in a multitude of scientific fields. Examples include human travel, in-cell dynamics, the motion of bright points on the solar surface, the transport of charge carriers in amorphous semiconductors, the propagation of contaminants in groundwater, the search patterns of foraging animals and the transport of energetic particles in turbulent plasmas. These examples showed that the assumptions of the classical diffusion paradigm, assuming an underlying uncorrelated (Markovian), Gaussian stochastic process, need to be relaxed to describe transport processes exhibiting a non-local character and exhibiting long-range correlations. This article does not aim at presenting a complete review of non-diffusive transport, but rather an introduction for readers not familiar with the topic. For more in depth reviews, we recommend some references in the following. First, we recall the basics of the classical diffusion model and then we present two approaches of possible generalizations of this model: the Continuous-Time-Random-Walk (CTRW) and the fractional Levy motion (fLm).
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We discuss how to construct models of interacting anyons by generalizing quantum spin Hamiltonians to anyonic degrees of freedom. The simplest interactions energetically favor pairs of anyons to fuse into the trivial (identity) channel, similar to the quantum Heisenberg model favoring pairs of spins to form spin singlets. We present an introduction to the theory of anyons and discuss in detail how basis sets and matrix representations of the interaction terms can be obtained, using non-Abelian Fibonacci anyons as example. Besides discussing the golden chain, a one-dimensional system of anyons with nearest neighbor interactions, we also present the derivation of more complicated interaction terms, such as three-anyon interactions in the spirit of the Majumdar-Ghosh spin chain, longer range interactions and two-leg ladders. We also discuss generalizations to anyons with general non-Abelian su(2)_k statistics. The k to infinity limit of the latter yields ordinary SU(2) spin chains.
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After a short excursion from discovery of Brownian motion to the Richardson law of four thirds in turbulent diffusion, the article introduces the L{e}vy flight superdiffusion as a self-similar L{e}vy process. The condition of self-similarity converts the infinitely divisible characteristic function of the L{e}vy process into a stable characteristic function of the L{e}vy motion. The L{e}vy motion generalizes the Brownian motion on the base of the $alpha$-stable distributions theory and fractional order derivatives. The further development of the idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorovs equation for arbitrary Markovian processes. As particular case we obtain the fractional Fokker-Planck equation for L{e}vy flights. Some results concerning stationary probability distributions of L{e}vy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally we discuss results on the same characteristics and barrier crossing problems with L{e}vy flights, recently obtained with different approaches.
We study the behavior of stationary non-equilibrium two-body correlation functions for Diffusive Systems with equilibrium reference states (DSe). A DSe is described at the mesoscopic level by $M$ locally conserved continuum fields that evolve through coupled Langevin equations with white noises. The dynamic is designed such that the system may reach equilibrium states for a set of boundary conditions. In this form, just by changing the equilibrium boundary conditions, we make the system driven to a non-equilibrium stationary state. We decompose the correlations in a known local equilibrium part and another one that contains the non-equilibrium behavior and that we call {it correlations excess} $bar C(x,z)$. We formally derive the differential equations for $bar C$. We define a perturbative expansion around the equilibrium state to solve them order by order. We show that the $bar C$s first-order expansion, $bar C^{(1)}$, is always zero for the unique field case, $M=1$. Moreover $bar C^{(1)}$ is always long-range or zero when $M>1$. Surprisingly we show that their associated fluctuations, the space integrals of $bar C^{(1)}$, are always zero. Therefore, the fluctuations are dominated by the local equilibrium behavior up to second order in the perturbative expansion around the equilibrium. We derive the behaviors of $bar C^{(1)}$ in real space for dimensions $d=1$ and $2$ explicitly, and we apply the analysis to a generic $M=2$ case and, in particular, to a hydrodynamic model where we explicitly compute the two first perturbative orders, $bar C^{(1),(2)}$, and its associated fluctuations.
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