Lagrangian dynamics in inhomogeneous and thermal environments, An application of the Onsager-Machlup theory I

TL;DR

This paper derives the Onsager-Machlup Lagrangian from the Fokker-Planck equation, explores environmental effects on dynamics, and examines phenomena like friction, thermal balance, and phase transitions in inhomogeneous and thermal settings.

Contribution

It introduces a method to derive the Lagrangian from the Fokker-Planck equation and models complex environmental interactions affecting stochastic dynamics.

Findings

Friction and dissipation emerge naturally from the Fokker-Planck framework.

Time-dependent temperature models reveal self-consistent cooling and heating dynamics.

Dynamical balance can lead to phase transitions under certain conditions.

Abstract

We straight-forwardly derive the Onsager-Machlup Lagrangian from the Fokker-Planck equation and show that friction and dissipation are a natural property of the equation of motion. We develop a method to calculate the local variance $\sigma_{2}\,b(q)^{2}$ and identify this function as a Helmholtz-factor. In both meanings the function $b(q)$ describes properties of the environment. For application, we examine the free fall through a barometric medium and model a blow of wind by a solitonic pulse running through the medium. We treat harmonic oscillators immersed in a thermal bath, finding intuitive as well as counter-intuitive phenomena of friction. By allowing the temperature to be time-dependent, the dynamical process of cooling and heating becomes self-consistently available. We find a state of dynamical balance between system and environment. Last, we show that dynamical balance is related to adiabatic thermodynamic processes. In a special case, dynamical balance can induce a real phase-transition.