From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit to anomalous diffusion. We consider here the case of subdiffusive processes, which are semi-Markovian and correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to know fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power-law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time

The Unequal Twins - Probability Distributions Aren't Everything

It is the common lore to assume that knowing the equation for the probability distribution function (PDF) of a stochastic model as a function of time tells the whole picture defining all other characteristics of the model. We show that this is not the case by comparing two exactly solvable models of anomalous diffusion due to geometric constraints: The comb model and the random walk on a random walk (RWRW). We show that though the two models have exactly the same PDFs, they differ in other respects, like their first passage time (FPT) distributions, their autocorrelation functions and their aging properties

Fluctuations, Higher Order Anharmonicities, and Landau Expansion for Barium Titanate

Correct phenomenological description of ferroelectric phase transitions in barium titanate requires accounting for eighth-order terms in the free energy expansion, in addition to the conventional sixth-order contributions. Another unusual feature of BaTiO_3 crystal is that the coefficients B_1 and B_2 of the terms P_x^4 and P_x^2*P_y^2 in the Landau expansion depend on the temperature. It is shown that the temperature dependence of B_1 and B_2 may be caused by thermal fluctuations of the polarization, provided the fourth-order anharmonicity is anomalously small, i. e. the nonlinearity of P^4 type and higher-order ones play comparable roles. Non-singular (non-critical) fluctuation contributions to B_1 and B_2 are calculated in the first approximation in sixth-order and eighth-order anharmonic constants. Both contributions increase with the temperature, which is in agreement with available experimental data. Moreover, the theory makes it possible to estimate, without any additional assumptions, the ratio of fluctuation (temperature dependent) contributions to coefficients B_1 and B_2. Theoretical value of B_1/B_2 appears to be close to that given by experiments.Comment: 5 pages, 1 figur

Irreversible and reversible modes of operation of deterministic ratchets

We discuss a problem of optimization of the energetic efficiency of a simple rocked ratchet. We concentrate on a low-temperature case in which the particle's motion in a ratchet potential is deterministic. We show that the energetic efficiency of a ratchet working adiabatically is bounded from above by a value depending on the form of ratchet potential. The ratchets with strongly asymmetric potentials can achieve ideal efficiency of unity without approaching reversibility. On the other hand we show that for any form of the ratchet potential a set of time-protocols of the outer force exist under which the operation is reversible and the ideal value of efficiency is also achieved. The mode of operation of the ratchet is still quasistatic but not adiabatic. The high values of efficiency can be preserved even under elevated temperatures

Understanding Anomalous Transport in Intermittent Maps: From Continuous Time Random Walks to Fractals

We show that the generalized diffusion coefficient of a subdiffusive intermittent map is a fractal function of control parameters. A modified continuous time random walk theory yields its coarse functional form and correctly describes a dynamical phase transition from normal to anomalous diffusion marked by strong suppression of diffusion. Similarly, the probability density of moving particles is governed by a time-fractional diffusion equation on coarse scales while exhibiting a specific fine structure. Approximations beyond stochastic theory are derived from a generalized Taylor-Green-Kubo formula.Comment: 4 pages, 3 eps figure