Andrea Carati, Luigi Galgani
Planck's Formula in Classical Mechanics
(52K, plain Tex)
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.