Invariant Measures in Hamiltonian Systems: The Analytical Foundations of Statistical Physics

TL;DR

This paper develops an invariant measure on Hamiltonian level sets, linking it to statistical physics partition functions and providing a new perspective on Hamiltonian system analysis.

Contribution

It constructs an invariant measure on Hamiltonian level sets that connects microcanonical and canonical ensembles, offering a probabilistic framework for Hamiltonian systems.

Findings

The measure generates the microcanonical partition function used in physics.

The measure can be transformed into the canonical partition function asymptotically.

Provides an alternative approach to Simon's second problem.

Abstract

We construct a measure in the hamiltonian function level sets that is invariant under the hamiltonian flow for short times and flow preserving for arbitrarily long times. This allows a probabilistic approach to the study of hamiltonian systems, in the space of states with fixed energy. We prove that this measure generates the microcanonical partition function employed in physics and show that it can be transformed into the canonical partition function in an asymptotic limit, hence reproducing classical Statistical Physics. We also argue that this gives an alternative solution to Simon's second problem.