Lagrangian formalism and classical statistical ensemble

Abstract

We present a formulation of classical statistical mechanics based on a Lagrangian description on the tangent bundle. In this approach, a Wick rotation from real time to imaginary time is employed as a technical device that facilitates the construction of a Hamiltonian structure expressed in velocity variables. The resulting dynamics preserves a natural measure induced by the associated symplectic form on the tangent bundle. This measure-preserving property enables the consistent definition of classical statistical ensembles directly in terms of Lagrangian variables.

Figures (3)

• Figure 1: Hamiltonian and Lagrangian approaches in the classical and quantum statistical mechanics.
• Figure 2: Area under the time evolution on the tangent bundle.
• Figure 3: Ensemble in the energy range $E\rightarrow E+\delta E$ on the tangent bundle.