Noether’s theorem in statistical mechanics

Ergodic theorem, ergodic theory, and statistical mechanics.

This perspective highlights the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundam...

Shannon’s Statistical Inference and the H-theorem in Quantum Statistical Mechanics

A formulation of quantum statistical mechanics, which incoporates the statistical inference of Shannon as its integral part, is discussed. Our basic idea is to distinguish the dynamical entropy of von Neumann in terms of the density matrix ρ̂(t), H = −kTrρ̂ ln ρ̂, and the statistical amount of uncertainty of Shannon, S = −k ∑ n pn ln pn, with pn = (ρ̂)nn in the representation where the total energy...

Justification in Statistical Mechanics

According to a standard view of the second law of thermodynamics, our belief in the second law can be justified by pointing out that low entropy macrostates are less probable than high entropy macrostates, and then noting that a system in an improbable state will tend to evolve toward a more probable state. I would like to argue that this justification of the second law of thermodynamics is fun...

Statistical Mechanics

Interventionism in Statistical Mechanics

I defend the idea that the fact that no system is entirely isolated (“Interventionism”) can be used to explain the successful use of the microcanonical distribution in statistical mechanics. The argument turns on claims about what is needed for an adequate explanation of this fact: I argue in particular that various competing explanations do not meet reasonable conditions of adequacy, and that ...