Microscopic Dynamics of Transport Processes

We apply the powerful tools of dynamical systems theory to the study of the microscopic dynamics of many-body systems representing molecular liquids and gases, both at equilibrium and in steady nonequilibrium states. Computer simulations are used to compute the complete set of Lyapunov.exponents, which describe the exponential growth, and decay, of infinitesimal perturbations in the high-dimensional phase space. Positive exponents are a measure for dynamical chaos and are necessary for the classical axioms of statistical mechanics to hold. The whole project is subdivided into five subprojects: Stationary heat flow: In 1987 we demonstrated the existence of fractal attractors for the phase-space probability density of stationary nonequilibrium systems. This result is consistent with a negative sum of all Lyapunov exponents, and is a fingerprint of the Second Law of thermodynamics. It explains the macroscopic irreversibility of such systems in spite of the time-reversible nature of their equations of motion. The standard proofs involve the use of mathematical thermostats to control the temperature. These thermostats, however, do not occur in nature. Taking heat conduction in a two-dimensional lattice system as the model, we intend to show that these thermostats are a convenient, but ultimately irrelevant, tool, and that the fractality of the probability densities constitutes a general property of stationary transport processes. The rate of phase-volume shrinkage is determined by the rate of irreversible entropy production. Lyapunov instability of a planar gas of rough hard disks: Based on our earlier work on this subject, we intend to study how, the coupling between qualitatively-different molecular degrees of freedom, rotation and translation, affects the chaotic properties of simple models of molecular fluids. Heat conduction in a modified planar Lorentz gas: Recently, the validity of Fick`s law of heat conduction close to equilibrium was mathematically demonstrated. Far away from equilibrium, the temperature profile is a power law. Because of these properties this model is well suited for our intended study of the entropy production and of the fractal nature of the probability density in phase space. Lyapunov instability and particle diffusion in narrow pores: If the diameter of a microscopic channel is gradually reduced such that particles become unable to pass, one observes discontinuities of the pressure, the collision frequency, and of the Lyapunov exponents as a function of channel width. They are reminiscent of phase transitions, which, however, they are not. We intend to study these properties for various particle interactions. Stability by thermodynamic instability, and the problem of negative specific heat: The stability of stars such as our sun is threatened by two gigantic instabilities, the gravothermal effect (negative heat capacity) and thermonuclear reactions. By mutual compensation these effects may lead to exceptional stability for billions of years. We intend to investigate this problem by devising simplified models, for which analytical results may be obtained, and by complementing them with computer simulations.

We apply the powerful tools of dynamical systems theory to the study of the microscopic dynamics of many-body systems representing molecular liquids and gases, both at equilibrium and in steady nonequilibrium states. Computer simulations are used to compute the complete set of Lyapunov.exponents, which describe the exponential growth, and decay, of infinitesimal perturbations in the high-dimensional phase space. Positive exponents are a measure for dynamical chaos and are necessary for the classical axioms of statistical mechanics to hold. The whole project is subdivided into five subprojects: Stationary heat flow: In 1987 we demonstrated the existence of fractal attractors for the phase-space probability density of stationary nonequilibrium systems. This result is consistent with a negative sum of all Lyapunov exponents, and is a fingerprint of the Second Law of thermodynamics. It explains the macroscopic irreversibility of such systems in spite of the time-reversible nature of their equations of motion. The standard proofs involve the use of mathematical thermostats to control the temperature. These thermostats, however, do not occur in nature. Taking heat conduction in a two-dimensional lattice system as the model, we intend to show that these thermostats are a convenient, but ultimately irrelevant, tool, and that the fractality of the probability densities constitutes a general property of stationary transport processes. The rate of phase-volume shrinkage is determined by the rate of irreversible entropy production. Lyapunov instability of a planar gas of rough hard disks: Based on our earlier work on this subject, we intend to study how, the coupling between qualitatively-different molecular degrees of freedom, rotation and translation, affects the chaotic properties of simple models of molecular fluids. Heat conduction in a modified planar Lorentz gas: Recently, the validity of Fick`s law of heat conduction close to equilibrium was mathematically demonstrated. Far away from equilibrium, the temperature profile is a power law. Because of these properties this model is well suited for our intended study of the entropy production and of the fractal nature of the probability density in phase space. Lyapunov instability and particle diffusion in narrow pores: If the diameter of a microscopic channel is gradually reduced such that particles become unable to pass, one observes discontinuities of the pressure, the collision frequency, and of the Lyapunov exponents as a function of channel width. They are reminiscent of phase transitions, which, however, they are not. We intend to study these properties for various particle interactions. Stability by thermodynamic instability, and the problem of negative specific heat: The stability of stars such as our sun is threatened by two gigantic instabilities, the gravothermal effect (negative heat capacity) and thermonuclear reactions. By mutual compensation these effects may lead to exceptional stability for billions of years. We intend to investigate this problem by devising simplified models, for which analytical results may be obtained, and by complementing them with computer simulations.