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
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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.
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