1.2 Model systems

**STADIUM BILLIARD**

This model, introduced by Bunimovich and Sinai, will serve us to demonstrate the existence of**chaos**even in quite small systems. Chaos, it must be understood, is a fundamental prerequisite for the application of statistical rules. It is a basic property of chaotic systems that they will acquire each one of a certain set of ``states'' with equal probability, i.e. with equal relative frequency. It is this ``equal a priory probability'' of states which we need to proceed into the heart of Statistical Mechanics.The stadium billiard is defined as follows. Let a ``light ray'' or ``mass point'' move about in a two-dimensional container with reflecting or ideally elastic walls. The boundaries have both flat and semicircular parts, which gives rise to an efficient mixing of the flight directions upon reflection. Such a system is chaotic, meaning that the motional degrees of freedom exhibit a uniform probability distribution. Since we have the ``phase points'' describing the momentary motion are distributed evenly over the periphery of a circle. The model is easily extended into three dimensions; the velocity then has three components, energy conservation keeps the phase points on the surface of a sphere, and equal a priory probability (or chaos) means that the distribution of points on this ``energy surface'' is homogeneous - nowhere denser or thinner than elsewhere.

Applet Stadium: Start**Simulation: Stadium Billiard in 2 Dimensions**

- See the trajectory of the particle (ray); note the frequency histograms for flight direction and -velocity .

- ``Chaos'' is demonstrated by simultaneously starting a large number of trajectories with nearly identical initial directions: fast emergence of equidistribution on the circle .

[Code: Stadium]

Applet VarSinai: Start **Simulation: Sinai Billiard**

- Ideal one particle gas in a box having randomizers on its walls

- See the trajectory of the particle (ray); note the frequency histograms for flight direction and -velocity

[Code: VarSinai]

**CLASSICAL IDEAL GAS**

particles are confined to a volume . There are no mutual interactions between molecules - except that they may exchange energy and momentum in some unspecified but conservative manner. The theoretical treatment of such a system is particularly simple; nevertheless the results are applicable, with some caution, to gases at low densities. Note that air at normal conditions may be regarded an almost ideal gas.The momentary state of a classical system of point particles is completely determined by the specification of all positions and velocities . The energy contained in the system is entirely kinetic, and in an isolated system with ideally elastic walls remains constant.

**IDEAL QUANTUM GAS**

Again, particles are enclosed in a volume . However, the various states of the system are now to be specified not by the positions and velocities but according to the rules of quantum mechanics. Considering first a single particle in a one-dimensional box of length . The solutions of Schroedinger's equation

(1.19)

In two dimensions - the box being quadratic with side length - we have for the energies

(1.20) If there are particles in the box we have, in three dimensions,

where is the quantum number of particle .

**Note:**In writing the sum 1.21 we only assumed that each of the particles is in a certain state . We have not considered yet if any combination of single particle states is indeed permitted, or if certain might exclude each other (Pauli priciple for fermions.)

**HARD SPHERES, HARD DISCS**

Again we assume that such particles are confined to a volume . However, the finite-size objects may now collide with each other, and at each encounter will exchange energy and momentum according to the laws of elastic collisions. A particle bouncing back from a wall will only invert the respective velocity component.At very low densities such a model system will of course resemble a classical ideal gas. However, since there is now a - albeit simplified - mechanism for the transfer of momentum and energy the model is a suitable reference system for

**kinetic theory**which is concerned with the transport of mechanical quantities. The relevant results will be applicable as first approximations also to moderately dense gases and fluids.In addition, the HS model has a special significance for the simulation of fluids.

Applet Harddisks: Start

Applet Hspheres: Start

**Simulation 1.4: hard discs in a 2D box with periodic boundaries.**[Code: Hdiskspbc, UNDER CONSTRUCTION]

**LENNARD-JONES MOLECULES**

This model fluid is defined by the interaction potential (see figure)

where (potential well depth) and (contact distance) are substance specific parameters.In place of the hard collisions we have now a continuous repulsive interaction at small distances; in addition there is a weak attractive force at intermediate pair distances. The model is fairly realistic; the interaction between two rare gas atoms is well approximated by equ. 1.22.

**Figure 1.7:**Lennard-Jones potential with (well depth) and (contact distance where ). Many real fluids consisting of atoms or almost isotropic molecules are well described by the LJ potential.In simulation one often uses the so-called ``periodic boundary conditions'' instead of reflecting vessel walls: a particle leaving the (quadratic, cubic, ...) cell on the right is then fed in with identical velocity from the left boundary, etc. This guarantees that particle number, energy and total momentum are conserved, and that each particle is at all times surrounded by other particles instead of having a wall nearby. The situation of a molecule well within a macroscopic sample is better approximated in this way.

Applet LJones: Start

**HARMONIC CRYSTAL**

The basic model for a solid is a regular configuration of atoms or ions that are bound to their nearest neighbors by a suitably modelled pair potential. Whatever the functional form of this potential, it may be approximated, for small excursions of any one particle from its equilibrium position, by a harmonic potential. Let denote the equilibrium (minimal potential) distance between two neighbouring particles; the equilibrium position of atom in a one-dimensional lattice is then given by . Defining the*displacements*of the atoms from their lattice points by we have for the energy of the lattice

where is a force constant and is the atomic mass.The generalization of 1.23 to and dimensions is straightforward.

The further treatment of this model is simplified by the approximate assumption that each particle is moving independently from all others in its own oscillator potential: . In going to and dimensions one introduces the further simplifying assumption that this private oscillator potential is isotropically ``smeared out'': . The model thus defined is known as the ``Einstein model'' of solids.

**MODELS FOR COMPLEX MOLECULES**

Most real substances consist of more complex units than isotropic atoms or Lennard-Jones type particles. There may be several interaction centers per particle, containing electrical charges, dipoles or multipoles, and the units may be joined by rigid or flexible bonds. Some of these models are still amenable to a theoretical treatment, but more often than not the methods of numerical simulation - Monte Carlo or molecular dynamics - must be invoked.

**SPIN LATTICES**

Magnetically or electrically polarizable solids are often described by models in which ``spins'' with the discrete permitted values are located at the vertices of a lattice.If the spins have no mutual interaction the discrete states of such a system are easy to enumerate. This is why such models are often used to demonstrate of statistical-mechanical - or actually, combinatorical - relations (Reif: Berkeley Lectures). The energy of the system is then defined by , where is an external field.

Of more physical significance are those models in which the spins interact with each other, For example, the parallel alignment of two neighboring spins may be energetically favored over the antiparallel configuration (

**Ising**model). Monte Carlo simulation experiments on systems of this type have contributed much to our understanding of the properties of ferromagnetic and -electric substances.

2005-01-25