Published February 17, 2021 | Version 1
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# Lecture "Introduction to Nonequilibrium Thermodynamics - Onsager's variational principle"

## Creators

• 1. WWU Münster, Institut für Theoretische Physik

## Description

The data deposit contains an introductory  video lecture by Uwe Thiele, WWU Münster, Institut für Theoretische Physik on "Introduction to Nonequilibrium Thermodynamics - Onsager’s variational principle" together with the slides in pdf format and this info file.

The lecture gives a basic introduction to phenomenological nonequilibrium thermodynamics with the aim to show how typical kinetic equations in gradient dynamics form (that are e.g., employed in Soft Matter science) are derived employing Onsager’s variational principle.

The lecture should be suitable for advanced Bachelor students, Master students and beginning PhD students of the natural sciences and other interested people. It is a stand-alone lecture on the derivation of equations like the Allen-Cahn and the Cahn-Hilliard equation, but was given in the context of a lecture course "Introduction to the Theory of Phase Transitions" (ITPT). Assumed is some knowledge of equilibrium thermodynamics (e.g., from some Bachelor Thermodynamics course or the first part of the ITPT lecture course) and basic variational calculus (e.g., as used when deriving Newton's law from the Lagrangian in classical mechanics).

We start with a brief recapitulation of some elements of equilibrium thermodynamics, discuss how the increase in entropy in a nonequilibrium process close to equilibrium can be described for a system characterized by a single state variable. There, the usual linear relation between thermodynamic force and thermodynamic flux/rate is introduced. After considering two examples (elastically bound mass point in viscous medium, heat exchange between two reservoirs) Onsager's reciprocity relations are introduced for systems with several state variables.

Then the Onsager variational principle is discussed - introducing the Rayleighian, and showing that kinetic equations are obtained by minimising the Rayleighian with respect to the rates of change of state variables. The procedure is then used to derive the diffusion equation, and generic gradient dynamics equations for conserved and nonconserved scalar order parameter fields, first individually and then in the case of a mixed dynamics. After generalisation to several coupled scalar order parameter fields the lecture concludes with a summary and outlook.

The two archetypical cases of nonconserved and conserved gradient dynamics, i.e., Allen-Cahn-type and Cahn-Hilliard-type dynamics, are considered
in the two follow up lectures "Allen-Cahn-type phase-transition dynamics" (see https://dx.doi.org/10.5281/zenodo.4545328) and "Cahn-Hilliard-type phase-transition dynamics" (see https://dx.doi.org/10.5281/zenodo.4545361), respectively.

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