Chaos in systems with many degrees of freedom

Abstract

In this thesis I discuss some of the chaotic properties specific to systems of many particles and other systems with many degrees of freedom. A dynamical system is called chaotic if a typical infinite perturbation of initial conditions grows exponentially with time. The chaoticity of a system is characterised by the Lyapunov exponents, which indicate the possible rates at which an infinitesimal perturbation of initial conditions may grow or decrease. A system has as many Lyapunov exponents as it's phase space has dimensions, and so a system with many degrees of freedom has many Lyapunov exponents. A system is chaotic if it has at least one positive exponent. The sum of the positive Lyapunov exponents equals the maximal rate of information increase of the system and is referred to as the Kolmogorov-Sinai entropy. The dynamical properties of a system are thought to have bearing on the non-equilibrium behaviour.

This thesis contains calculations of Lyapunov exponents of three different systems; namely, systems consisting of many, freely moving hard disks and hard spheres are discussed, as well as the Lorentz gas, which is a system consisting of fixed spherical scatterers with a point particle moving between and colliding elastically with them, and a similar system in which the scatterers are cylindrical.

Chapter 3 contains calculations of the smallest positive and negative exponents of hard disks. These are known from simulations to have interesting behaviour, if the system is large enough. They are explained as belonging to Goldstone modes associated with the symmetries of the tangent space, and the Lyapunov exponents are calculated. This allows their calculation by formulating a generalised Boltzmann equation and solving this perturbatively. There is a slight discrepancy between the predicted values and simulation results, which may be attributed to the neglect of ring-collision terms in the Boltzmann equation.

In chapter 4 the Kolmogorov-Sinai entropy is calculated for the same system. It is known to be proportional to the collision frequency multiplied by a factor that equals the logarithm of the density plus a constant. Previous calculations of this constant have produced unsatisfactory results. I discuss the cause of this and how to calculate the constant correctly.

A system of freely moving hard particles can be described as a point particle in a high-dimensional space, colliding with cylindrical scatterers with very specific positions and orientations. Because this is very similar to the Lorentz gas, where the scatterers are hyperspheres, in chapter 5 I examine the Lyapunov exponents of the Lorentz gas in an arbitrary number of dimensions. The full spectrum is calculated analytically and the similarities and differences between the high-dimensional Lorentz gas and systems of hard disks or spheres are discussed.

In chapter 6 I describe numerical calculations of the Lyapunov exponents of systems consisting of isotropically oriented, homogeneously distributed, cylindrical scatterers. Because of the similarity in the shape of the scatterers, the Lyapunov spectrum of this system is much more similar to the spectrum of hard disks than the spectrum of the Lorentz gas. Using the techniques developed in chapters 4 and 5, I estimate the smallest positive and negative Lyapunov exponents of hard disks and hard spheres which are not associated with Goldstone modes.