Deterministic transport: from normal to anomalous diffusion (original) (raw)

From Deterministic Chaos to Anomalous Diffusion

2008

Abstract: This is an easy-to-read introduction to foundations of deterministic chaos, deterministic diffusion and anomalous diffusion. The first part introduces to deterministic chaos in one-dimensional maps in form of Ljapunov exponents and dynamical entropies. The second part outlines the concept of deterministic diffusion. Then the escape rate formalism for deterministic diffusion, which expresses the diffusion coefficient in terms of the above two chaos quantities, is worked out for a simple map.

Microscopic chaos and transport in thermostated dynamical systems

2003

A fundamental challenge is to understand nonequilibrium statistical mechanics starting from microscopic chaos in the equations of motion of a many-particle system. In this review we summarize recent theoretical advances along these lines. Particularly, we are concerned with nonequilibrium situations created by external electric fields and by temperature or velocity gradients. These constraints pump energy into a system, hence there must be some thermal reservoir that prevents the system from heating up. About twenty years ago a deterministic and time-reversible modeling of such thermal reservoirs was proposed in form of Gaussian and Nose-Hoover thermostats. This approach yielded simple relations between fundamental quantities of nonequilibrium statistical mechanics and of dynamical systems theory. The main theme of our review is to critically assess the universality of these results. As a vehicle of demonstration we employ the driven periodic Lorentz gas, which is a toy model for the classical dynamics of an electron in a metal under application of an electric field. Applying different types of thermal reservoirs to this system we compare the resulting nonequilibrium steady states with each other. Along the same lines we discuss an interacting many-particle system under shear and heat. Finally, we outline an unexpected relationship between deterministic thermostats and active Brownian particles modeling biophysical cell motility.

Deterministic patterns of noise and the control of chaos

Contemporary Physics, 2002

Real systems in physics, chemistry and biology are always subject to¯uctuations that change qualitatively the systems' dynamics. In particular, rare large¯uctuations are responsible for the nucleation at phase transitions, mutations in DNA sequences, protein transport in cells and failure of electronic devices. In many cases of practical interest systems are away from thermal equilibrium, and understanding the¯uctuations in such systems is one of the fundamental problems of statistical physics that has challenged researchers for decades. Recent progress in the solution of this problem is closely related to the emerging understanding of patterns of deterministic trajectories underlying non-equilibriu m¯uctuations. These trajectories correspond to the Hamilton equations of motion written for the asymptotic solution of the Fokker ± Planck equation and were often thought of as a mere mathematical abstraction. The possibility of quantitative experiments could not be entertained until the appropriate statistical quantity (prehistory probability distribution) had been introduced. In this paper it is shown how such trajectories can be measured experimentally in a number of systems and how the knowledge of these trajectories can be used to solve long standing problems in the theory of¯uctuations and in the control theory.

Chaos, molecular fluctuations, and the correspondence limit

Physical Review A, 1990

Chaos is characterized by sensitive dependence on initial conditions. Trajectories determined by coupled, ordinary differential equations show sensitive dependence when their associated Liapunov exponent is positive. The Liapunov exponent is positive if the Jacobi matrix associated with the coupled differential equations has an eigenvalue with a positive real part, on the average, as the Jacobi matrix evolves along the trajectory. For macrovariable equations, there are also fluctuation

Transport properties of chaotic and non-chaotic many particle systems

Journal of Statistical Mechanics: Theory and Experiment, 2007

Two deterministic models for Brownian motion are investigated by numerical simulations and kinetic theory arguments. The first model consists of a heavy hard disk immersed in a rarefied gas of smaller and lighter hard disks acting as a thermal bath. The second is the same except for the shape of the particles, which is now square. The basic difference in these two systems lies in the interaction: hardcore elastic collisions make the dynamics of the disks chaotic whereas that of squares is not. Remarkably, this difference does not reflect on the transport properties of the two systems: simulations show that the diffusion coefficients, velocity correlations and response functions of the heavy impurity are in agreement with kinetic theory for both the chaotic and non-chaotic model. The relaxation to equilibrium instead is very sensitive to the kind of interactions. These observations are used to think back and discuss some issues connected to chaos, statistical mechanics and diffusion.