98-526 Andrea Carati, Luigi Galgani

Planck's Formula in Classical Mechanics (52K, plain Tex) Jul 22, 98

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Abstract. We consider the model studied by Poincar\'e in connection with Planck's law, when he proved the necessity of quantization, namely a system of $N$ independent identical oscillators, each of which interacts through smooth collisions with a gas particle (mimicing a heat reservoir), according to the laws of classical mechanics. We prove that the expected energy distribution of the oscillators obeys Planck's formula, i.e. Planck's law with an action characteristic of the system in place of Planck's constant. This is obtained by combining two ingredients, namely: the conception of Jeans who, following a perspective introduced by Boltzmann, was thinking of Planck's formula as describing a situation of quasi equilibrium very far from equilibrium, and Einstein's conception of the thermodynamic role of the energy fluctuations, which is at the basis of nonequilibrium thermodynamics. In turn, the energy fluctuations are estimated by the most advanced mathematical results presently available for the energy exchanges in elementary collision processes.

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