Dynamics of Many-Body Systems

Computer simulations of many body systems will be performed to clarify the chaotic properties described by the set of Lyapunov exponents. For systems in thermodynamic equilibrium, the emphasis will be on the Lyapunov modes, and for nonequilibrium systems on the fractal struture of the probability distribution of the instantaneous states. To generate nonequilibrium states, both time reversible and stochastic thermostats will bei used. Furthermore, using a novel particle method, Smoothed Particle Applied Mechanics (SPAM), for the simulation of continuous flows, we will extend these considerations to mesoscopic systems, to bridge the gap between microscopic and macroscopic scales in space and time. The Lyapunov exponents are the rate constants for the exponential growth, or shrinkage, of (infinitesimal) perturbations of a chaotic many body system. We have found that the perturbation associated with the maximum exponent is highly localized in space. For large systems, the smallest positive exponents, however, are represented by delocalized collective wave-like patterns in space, to which we refer as Lyapunov modes. Our discovery provides a new link between dynamical systems theory and the dynamics of fluids and solids. Our study aims to establish the connection between the Lyapunov modes and the familiar hydrodynamic modes. This will require large-scale computer simulations on parallel processors. In particular, we will study the fluid-to-solid phase transition, the sensitivity of the modes to a variation of the interparticle forces, and the extension of these considerations to nonequilibrium systems. The fractal nature of the probability distribution is a manifestation of the Second Law of thermodynamics and is responsible for the observed irreversibility. The models to be studied will include the famous model of Brownian motion, driven out of equilibrium by external forces. Many-body systems with gravitational forces will also be considered, where we concentrate on the influence of the conservation laws on thefinal equilibrium state. We have shown that SPAM, augmented with proper boundary conditions, may be used to study drop condensation and low-Reynolds number laboratory flows with complex geometries. An extension to phase separation for two-component liquid mixtures will be developed.

The first part of the project is concerned with the dynamics of many-body systems resembling fluids or solids. The instability of the particle trajectories with respect to small perturbations of an initial state (Lyapunov instability) was studied by computing the rate constants for the exponential growth, or decay, of such perturbations, the so- called Lyapunov exponents. For equilibrium systems we found that the fastest growing perturbations are strongly localized in physical space. Only a small fraction of all particles contributes to the growth at any instant of time. This is a strong argument in favour of the existence of Lyapunov exponents for macroscopic systems. The perturbations associated with the slowest growth or decay, however, are coherently spread out over the whole space and form wave-like patterns, which we have called Lyapunov modes. They are similar, but not identical, to the classical modes of fluctuating hydrodynamics (e.g. phonons). There exist non-propagating and propagating Lyapunov modes, whose symmetries were clarified. The modes are a consequence of the spontaneous breaking of the translational symmetry of so-called zero modes (with infinite wave length) associated with the vanishing Lyapunov exponents and the conservation laws of the system. For systems in nonequilibrium stationary states we verified our pioneering result of 1987 that the probability distribution for the visited states becomes a fractal object with a dimension strictly smaller than the dimension ascribed to the system in equilibrium. This holds also for systems driven away from equilibrium by stochastic random forces. This important result suggests that the details of the driving are unimportant and that the existence of fractals in state space is physically relevant and not an interesting artefact. The fractals are a fingerprint of the Second Law of thermodynamics and essential for our understanding of macroscopic irreversibility. In the second group of problems we studied systems with a negative heat capacity. This property, well known in astrophysics but unfamiliar to most physicists, means that the temperature of a system is reduced if the energy is increased, and vice versa. It occurs for predominantly gravitating matter (stars and galaxies), but has also been found for nano-particles in the laboratory. We showed that for ordinary matter a negative heat capacity may occur, if the system reaches only a small subset of all possible states with a constant energy or, if the number of particles is small, such as in atomic clusters. A negative specific heat is also responsible for the stability of stars such as the sun. There, it counterbalances a potentially explosive energy input due to thermonuclear reactions. With a simple model we demonstrated how the choice between implosion (gravothermal instability) and explosion (thermonuclear instability) is converted into stable coexistence, which may last for billions of years. Only if one of the competing mechanisms gets weaker than the other, we may expect either a violent supernova implosion or a thermonuclear explosion in the core of the star.