The greatest difficulty with classical statistical mechanics may be the fact that some degrees of freedom do not seem to attain the energy expected from the equipartition principle. This is typical of the vibrational modes in molecules and of the high frequencies in the electromagnetic radiation (problem of the ultraviolet cutoff). Boltzmann1 had the intuition that an explanation might be provided if each degree of freedom were to have a characteristic relaxation time until equilibrium, and that such a time should greatly increase with frequency; he spoke in terms of relaxation times of days or centuries. This possibility was seriously considered by Rayleigh and Jeans2, but they could not produce a clear classical mechanism to explain the freezing of the high frequency degrees of freedom; so, the equilibrium concept of the ultraviolet cutoff provided by quantum mechanics was accepted, and Boltzmann's hypothesis was forgotten. Here we point out that a general framework for understanding the ultraviolet cutoff in Boltzmann's dynamical sense in a classical context seems to be provided by Nekhoroshev's3 theorem on Arnold diffusion.

Boltzmann's ultraviolet cutoff and Nekhoroshev's theorem on Arnold diffusion / G. Benettin, L. Galgani, A. Giorgilli. - In: NATURE. - ISSN 0028-0836. - 311:5985(1984), pp. 444-446. [10.1038/311444a0]

Boltzmann's ultraviolet cutoff and Nekhoroshev's theorem on Arnold diffusion

L. Galgani
Secondo
;
A. Giorgilli
Ultimo
1984

Abstract

The greatest difficulty with classical statistical mechanics may be the fact that some degrees of freedom do not seem to attain the energy expected from the equipartition principle. This is typical of the vibrational modes in molecules and of the high frequencies in the electromagnetic radiation (problem of the ultraviolet cutoff). Boltzmann1 had the intuition that an explanation might be provided if each degree of freedom were to have a characteristic relaxation time until equilibrium, and that such a time should greatly increase with frequency; he spoke in terms of relaxation times of days or centuries. This possibility was seriously considered by Rayleigh and Jeans2, but they could not produce a clear classical mechanism to explain the freezing of the high frequency degrees of freedom; so, the equilibrium concept of the ultraviolet cutoff provided by quantum mechanics was accepted, and Boltzmann's hypothesis was forgotten. Here we point out that a general framework for understanding the ultraviolet cutoff in Boltzmann's dynamical sense in a classical context seems to be provided by Nekhoroshev's3 theorem on Arnold diffusion.
Multidisciplinary
Settore FIS/02 - Fisica Teorica, Modelli e Metodi Matematici
Settore MAT/07 - Fisica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/243914
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