By George F Babits
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Example text
Each such specification determines a macrostate, the set of all phase points whose particles Boltzmann’s Approach to Statistical Mechanics 43 are distributed in the manner specified. Different specifications yield different macrostates and the set of all such macrostates defines a partition of our phase space ΩE , into macrostates. Γ (X) is then the macrostate – the element of this partition – to which X belongs. All points in Γ (X) have, in particular, similar spatial particle densities (and similar bulk-velocity profiles).
Many different phase points correspond to each of the snapshots in Fig. 1. There is nothing particularly special about any specific equilibrium phase point – a phase point corresponding to the snapshot on the right, a system in equilibrium. The dynamics prefers a given equilibrium phase point neither more nor less than it prefers any other given phase point, even a specific far-from-equilibrium phase point, corresponding say to the leftmost snapshot. There are, however, for a system at a given energy E, far more equilibrium phase points than nonequilibrium phase points, overwhelming more, in fact, than the totality of nonequilibrium phase points at that energy – corresponding to all possible ways the system can fail to be in equilibrium, and described, for example, by the various density distributions perceptibly different from the uniform one.
Gross, N. W. Lewis (The New York Academy of Sciences, New York 1996) pp. 131–175 3. W. Thomson: Proceedings of the Royal Society of Edinburgh 3, 325 (1874); reprinted in [4] 4. G. Brush: Kinetic Theory (Pergamon, Oxford 1966) 5. E. Schr¨ odinger: What is Life? : The Physical Aspect of the Living Cell with Mind and Matter & Autobiographical Sketches (Cambridge University Press, Cambridge 1992) 6. E. Schr¨ odinger: What is Life? and Other Scientific Essays (Doubleday Anchor Books, New York 1965) section 6 7.