Entropy Production

Transport Forces & Irreversible Fluxes

Graduate Physics: Transport Phenomena

The Universal Wind-Down

Every real process is irreversible. In equilibrium thermodynamics, we focus on the endpoints (\(\Delta S\)). In non-equilibrium systems, we care about the rate.

"How do we quantify the exact rate at which the universe 'winds down' during a simple process like heat conduction?"

Local Equilibrium Hypothesis

We assume that for a small volume element \(dV\), thermodynamic state functions remain valid locally.

\(s = s(u, v, c_i)\) holds locally.
Gibbs relation: \(T ds = du + P dv - \sum \mu_i dc_i\)

Validity

Mean free path \(\ll\) Characteristic gradient length scale
Relaxation time \(\ll\) Observation time

Conservation & Continuity

The change in a local quantity is due to flux divergence and internal production.

\(\frac{\partial \rho a}{\partial t} = -\nabla \cdot \mathbf{J}_a + \sigma_a\)

Mass/Energy

\(\sigma = 0\)

Entropy

\(\sigma_s \ge 0\)

Flux (\(\mathbf{J}\))

Transport rate

The Entropy Production Rate

By combining the Gibbs relation with continuity equations for \(u\) and \(n_i\), we find:

\(\sigma_s = \sum_i \mathbf{J}_i \cdot \mathbf{X}_i \ge 0\)

Fluxes (\(\mathbf{J}_i\))

Heat flow, Diffusion, Electric current, Chemical reaction rate.

Conjugate Forces (\(\mathbf{X}_i\))

\(\nabla(1/T)\), \(-\nabla(\mu/T)\), \(\mathbf{E}/T\), Affinity \(A/T\).

Linear Phenomenological Laws

Near equilibrium, fluxes are linear functions of the forces:

\(\mathbf{J}_i = \sum_j L_{ij} \mathbf{X}_j\)

Pure Effects

Fourier's Law, Fick's Law, Ohm's Law (Diagonal terms \(L_{ii}\)).

Cross Effects

Thermo-diffusion (Soret), Thermo-electricity (Peltier/Seebeck) (\(L_{ij}, i \neq j\)).

Summary

1

Non-equilibrium systems are characterized by fluxes and forces.

2

Entropy production rate \(\sigma_s\) is the bilinear product of fluxes and forces.

3

Linear response assumes \(\mathbf{J} \propto \mathbf{X}\) near equilibrium.

Next Lesson: Onsager Reciprocal Relations

Transport Phenomena and Non-Equilibrium Systems

Entropy Production and Transport Forces

Onsager Reciprocal Relations

Linear Response Theory

Jarzynski Equality and Fluctuation Theorems

Stochastic Thermodynamics and Brownian Motion

Entropy Ledger Slides

Flux and Force Workshop Worksheet

Entropy Production Notes Teacher Guide

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Thermoelectric Coupling Worksheet

Coupled Transport Key Answer Key

Fluctuation Response Slides

Kubo Formula Exercises Worksheet

Linear Response Reference Teacher Guide

Far From Equilibrium Slides

Jarzynski Work Analysis Activity

Work Distribution Guide Teacher Resource

Brownian Ledger Slides

Stochastic Motor Design Activity

Stochastic Thermodynamics Cheat Sheet

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