DYNAMICAL SYSTEMS THEORY AND STATISTICAL MECHANICS

The study of nonlinear dynamics associated with the motion of atoms or molecules inside matter is a new topic of nonequilibrium statistical mechanics. At this microscopic level, the nonlinear dynamics may induce chaotic behavior over the time scale of the collisions between the particles. Indeed, the defocusing character of the collisions generates sensitivity to initial conditions and, hence, dynamical randomness. For instance, a mole of air at room temperature and pressure would require a data accumulation rate of about 10³⁴ bits per second in order to follow the gas evolution on the microscopic scale.

Relationships have been established between the characteristic quantities of chaos and the transport coefficients, in particular, thanks to the escape-rate theory. These relationships are based on the observation that the microscopic chaos induces fractal structures in the phase space of out-of-equilibrium systems and in the modes of exponential relaxation toward the thermodynamic equilibrium. These fractal structures are the manifestation of a subtle order in the deterministic chaotic motion of interacting particles.

Currently, the relationships between chaos and transport and the properties of stationary nonequilibrium states are being investigated in different systems such as diffusive and reactive Lorentz gases and multibaker maps, hard-ball fluids, as well as Brownian motion systems.

Selected publications:

P. Gaspard and G. Nicolis, "Transport Properties, Lyapunov Exponents, and Entropy per Unit Time", Phys. Rev. Lett. 65, 1693-1696 (1990).

S. Tasaki and P. Gaspard, "Fick's Law and Fractality of Nonequilibrium Stationary States in a Reversible Multibaker Map", J. Stat. Phys. 81 ,935-987 (1995).

P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge UK, 1998).

F. Bonetto, D. Daems and J. L. Lebowitz, "Properties of the Stationary States in the Thermostatted Lorentz Gas I: The One Particle System", J. Stat. Phys. 101 , 17-34 (2000).

R. Klages, K. Rateitschak and G. Nicolis, "Thermostatting by Deterministic Scattering: Construction of Nonequilibrium Steady States", Phys. Rev. Lett. 84 , 4268-4271 (2000).

S. McNamara and M. Mareschal, "Origin of the Hydrodynamic Lyapunov modes", Phys. Rev. E 64, 051103 (2001).

P. Gaspard, I. Claus, T. Gilbert and J. R. Dorfman, "The Fractality of the Hydrodynamic Modes of Diffusion", Phys. Rev. Lett. 86 , 1506-1509 (2001).

Chaotic diffusion of a charged particle moving according to Newton's equations in a lattice of screened Coulomb potentials: (a) Pair of trajectories starting from the same position with slightly different velocities, showing the sensitivity to initial conditions at the energy E = 3. (b) Cumulative functions of the nonequilibrium steady states corresponding to gradients of concentration across the lattice for the motion at the energies E = 2, 3, 4, 5. (c) Zoom on these cumulative functions, revealing their self-similarity.