The kinetic theory of gases illustrates how random collisions between hard spheres can lead to macroscopic properties such as pressure and temperature. The spheres perform simple random walks and travel a known ‘average’ distance between collisions. A paper by S. G. Cox and G. J. Ackland (Physical Review Letters 84, 2362–2365; 2000) holds some surprises for the physics of colliding particles. By studying simple elastic collisions between three particles on a ring, the authors discover complex random-walk behaviour in a mathematical space (called a phase space) that tracks momentum exchanges. Although the phase space is a simple hexagonal lattice, the path and rate of exploration of this phase space varies greatly, depending on the mass ratios of the particles. This complexity of behaviour has implications for the question of how underlying microscopic behaviour can be averaged to obtain macroscopic laws. This question has a long history in statistical mechanics based around the concept of ergodicity.
Statistical mechanics measures an ensemble average, which means repeating an experiment many times and averaging the results. Physicists would rather study time series and have developed many methods for analysing them. For example, measuring the voltage as a function of time across a resistance can be studied through its power spectrum or wavelet transform. The mathematical condition allowing one to equate a time-series average with an ensemble average is called ergodicity. The idea is that, with time, a system will pass through almost all of its possible position and momentum states, allowing a time and ensemble average to be equivalent. The ergodic condition was first used in this context by Paul Ehrenfest in 1911 and mathematically refined by Garrett Birkhoff in 1931. Ehrenfest was seeking to explain, at a fundamental level, Ludwig Boltzmann's ideas on how microscopic behaviour underlies macroscopic laws. Ergodicity was never proven to be generally valid, although it remains a central assumption of statistical mechanics. Most dynamical systems of practical interest are not ergodic.
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