Nonequilibrium statistical description of anomalous diffusion

In this paper, from the unifying viewpoint we will cover our recent work on the nonequilibrium statistical description of anomalous diffusion and application of this theory to explaining late experiment. We will study the motion of a particle under the influence of a random force modeled as Gaussian colored noise with arbitrary correlation and with/without external field. In the very general case, the generalized Langevin equation is presented. We obtain the variances of displacement, velocity and cross variance between displacement and velocity, their asymptotic and crossover behavior. The exact equations for the joint and marginal probability density functions, and their solutions are obtained. Finally the anomalous diffusion is described in the framework of nonequilibrium statistical mechanics. The experimental results (Skjeltorp et al., Phys. Rev. E 58 (1998) 4229) can well be explained by our theory presented in this paper.

[1]  Wang,et al.  Generalized Langevin equations: Anomalous diffusion and probability distributions. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  Wang Long-time-correlation effects and biased anomalous diffusion. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[3]  K. Seki,et al.  Brownian motion of spins revisited , 1998 .

[4]  J. Masoliver,et al.  Linear oscillators driven by Gaussian colored noise: crossovers and probability distributions , 1996 .

[5]  Statistical-mechanical theory of Brownian motion -translational motion in an equilibrium fluid , 1978 .

[6]  Free inertial processes driven by Gaussian noise: Probability distributions, anomalous diffusion, and fractal behavior. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  D. Ramkrishna,et al.  ANOMALOUS DIFFUSION: A DYNAMIC PERSPECTIVE , 1990 .

[8]  Balescu Anomalous transport in turbulent plasmas and continuous time random walks. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  Pawula Rf Dichotomous-noise-driven oscillators. , 1987 .

[10]  A. Aharony Anomalous Diffusion on Percolating Clusters , 1983 .

[11]  B. Alder,et al.  Decay of the Velocity Autocorrelation Function , 1970 .

[12]  L. Reichl Translational Brownian motion in a fluid with internal degrees of freedom , 1981 .

[13]  Pawula Rf Approximating distributions from moments , 1987 .

[14]  LONG-TIME CORRELATION-EFFECTS AND FRACTAL BROWNIAN-MOTION , 1990 .

[15]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[16]  Porr,et al.  Harmonic oscillators driven by colored noise: Crossovers, resonances, and spectra. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  M. Shlesinger,et al.  Beyond Brownian motion , 1996 .

[18]  B. Alder,et al.  Velocity Autocorrelations for Hard Spheres , 1967 .