Physica A 280 (2000) 106–114 www.elsevier.com/locate/physa Planck’s formula and glassy behavior in classical nonequilibrium statistical mechanics A. Carati ∗; 1 , L. Galgani Dipartimento di Matematica, Universit a di Milano, Via Saldini 50, 20133 Milano, Italy Abstract In statistical mechanics it is well known that one has to take into account the role of the relaxation times to equilibrium. We discuss the relevance of this fact for systems of weakly coupled harmonic oscillators in a classical framework, showing how one has a behavior similar to that of glassy systems and how one nds an analogue of Planck’s formula. From this point of view, quantum equilibrium statistical mechanics thus appears as a rst-order approximation in classical nonequilibrium statistical mechanics very far from equilibrium. c  2000 Published by Elsevier Science B.V. All rights reserved. PACS: 05.20.−y; 05.45.−a; 05.70.Ln; 61.43.Fs; 82.20.Mj Keywords: Planck’s law; Relaxation times; Glasses 1. Introduction In thermodynamics one always makes reference to some relaxation time to equilib- rium; for example, for a gas inside a cylinder one commonly considers the displace- ment of a piston, and requires that a suciently long time be elapsed for the gas to have reached a new equilibrium state, and in such a way one considers a sequence of equilibrium states. The main thesis we want to discuss here is that, according to classical mechanics, there exists a high nonuniformity in the relaxation times for the dierent “internal” degrees of freedom, such times increasing exponentially fast with the frequencies of the internal motions, so that one is somehow compelled to take such a high nonuniformity into account. The zeroth-order approximation turns out to be just the one in which such dierences are altogether neglected (i.e., one considers * Corresponding author. Fax: +39-02 7063 0346. E-mail addresses: carati@unimi.it (A. Carati), galgani@unimi.it (L. Galgani) 1 Grant from Fondazione Cariplo per la Ricerca Scientica. 0378-4371/00/$ - see front matter c  2000 Published by Elsevier Science B.V. All rights reserved. PII: S0378-4371(99)00625-1