Abstract
A classical gas at equilibrium satisfies the locality conditionif the correlations between local fluctuations at a pair of remote small regions diminish in the thermodynamic limit. The gas satisfies a strong locality conditionif the local fluctuations at any number of remote locations have no (pair, triple, quadruple....) correlations among them in the thermodynamic limit. We prove that locality is equivalent to a certain factorizability condition on the distribution function. The analogous quantum condition fails in the case of a freeBose gas. Next we prove that strong locality is equivalent to the total factorizability of the distribution function, and thus (given Liourille’s theorem) to the Maxwell Boltzmann distribution for an ideal gas.
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M. Planck.Ann. Phys. (Leipzig) 1, 69 122 (1900).
There is a quantum field analogue of this problem. Malament (unpublished) considered a free quantum field in the vacuum state and two detectors located in separate positions in space. (A detector is just a system with two distinct states, “ground state” and “exited state”.) He proved that if one detector jumps to the exited state then the probability that the second detector will also do so increases, despite the spacelike separation between the two events. This type of nonlocalily should be distinguished from the (stronger) notion of J. Bell.Physics 1, 195 200 (1964). On the subject of nonlocalily in quantum field theory, see M. Redhead.Found. Phys. 25, 123 137 (1995), and references therein!
See. for example. Y. S. Chow, and H. Teicher.Probability Theory: Independence, Interchangeability. Martingales (Springer. New York. 1978), p. 186.
The derivations of the distribution from maximum entropy considerations fall in the second category. It can be shown that taking maximum entropy is equivalent to assuming total factorizability of the distribution function.
L. D. Landau, and E. M. Lifshitz,Statistical Physics. Part 1, 3rd edn. (Pergamon, Oxford. 1980). p. 7.
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Dedicated to Professor Max Jammer on the occasion of his eightieth birthday. April 13. 1995.
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Pitowsky, I., Shoresh, N. Locality, factorizability, and the Maxwell Boltzmann distribution. Found Phys 26, 1231–1242 (1996). https://doi.org/10.1007/BF02275627
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DOI: https://doi.org/10.1007/BF02275627