20.1 Non-equilibrium thermodynamics

Introduction to Non-Equilibrium Thermodynamics

Non-equilibrium thermodynamics extends classical thermodynamics to systems that have gradients in temperature, pressure, or concentration. While classical thermodynamics describes systems at rest (in equilibrium), most real-world processes are not at equilibrium. Transport phenomena, chemical reactions, biological systems, and materials processing all involve continuous flows of energy or matter driven by these gradients.

Fundamentals

Classical thermodynamics tells you about initial and final states but says nothing about the path between them. Non-equilibrium thermodynamics fills that gap by describing how systems evolve over time.

This framework applies to a wide range of problems:

• Transport phenomena: heat transfer, mass transfer, and fluid dynamics
• Chemical kinetics: reaction rates and combustion processes
• Biological systems: membrane transport, metabolic networks, and biophysics
• Materials science: phase transformations, diffusion, and interfacial phenomena

The central idea is that thermodynamic gradients (differences in temperature, pressure, or chemical potential across a system) act as driving forces that produce fluxes (flows of heat, mass, or momentum). The relationship between forces and fluxes is the core of the theory.

Entropy Production in Non-Equilibrium Processes

In equilibrium thermodynamics, you compare entropy between states. In non-equilibrium thermodynamics, you care about the rate of entropy production, which quantifies how fast irreversibility is accumulating inside the system.

The second law requires that the total entropy of an isolated system never decreases. For non-equilibrium processes specifically, the local rate of entropy production $\sigma$ is always positive:

$\sigma \geq 0$

This rate is calculated as the sum of products of each flux $J_i$ with its corresponding thermodynamic force $X_i$:

$\sigma = \sum_i J_i X_i$

For example, in heat conduction the flux is the heat flow and the force is the temperature gradient. A steeper gradient means faster entropy production and more energy dissipation. This expression is the starting point for deriving the phenomenological transport equations discussed below.

Advanced Concepts in Non-Equilibrium Thermodynamics

Linear vs. Far-from-Equilibrium Regimes

Near equilibrium (linear regime): When a system is only slightly displaced from equilibrium, fluxes are linearly proportional to the thermodynamic forces driving them. This is the regime where familiar transport laws hold:

• Fourier's law for heat conduction: $J_q = -\kappa \nabla T$
• Fick's law for diffusion: $J_m = -D \nabla c$

These linear phenomenological equations work well for many engineering applications and are mathematically tractable.

Far from equilibrium (nonlinear regime): When gradients become large, the linear approximation breaks down. Fluxes may depend on forces in nonlinear ways, and entirely new phenomena can appear. Studying these systems requires more advanced frameworks such as extended irreversible thermodynamics or rational thermodynamics.

One of the most striking features of far-from-equilibrium systems is the emergence of dissipative structures, where organized patterns spontaneously form and are maintained by a continuous flow of energy through the system. Notable examples include:

• Rayleigh-Bénard convection cells: A fluid heated from below organizes into regular hexagonal convection patterns once the temperature gradient exceeds a critical threshold.
• Belousov-Zhabotinsky reaction: A chemical reaction that produces oscillating color changes and traveling wave patterns in solution.
• Turing patterns: Reaction-diffusion systems that generate stable spatial patterns (stripes, spots), proposed as a mechanism for biological morphogenesis.

These phenomena illustrate that driving a system far from equilibrium can produce self-organization rather than simply more disorder.

Onsager Reciprocal Relations

Lars Onsager showed that when multiple transport processes occur simultaneously in a system near equilibrium, they can be coupled in a way that obeys a fundamental symmetry.

The general linear relation between fluxes and forces is:

$J_i = \sum_j L_{ij} X_j$

Here $L_{ij}$ are the phenomenological coefficients that form a matrix relating each flux to every thermodynamic force. Onsager's key result is that this matrix is symmetric:

$L_{ij} = L_{ji}$

This symmetry has a physical consequence: if a gradient in variable $j$ can drive a flux of quantity $i$, then a gradient in variable $i$ will equally drive a flux of quantity $j$. Two well-known cross-coupling effects demonstrate this:

• Soret effect (thermophoresis): A temperature gradient drives mass diffusion, causing species in a mixture to separate.
• Dufour effect: A concentration gradient drives a heat flux, the reciprocal counterpart of the Soret effect.

The Onsager relations are not just theoretical elegance. They constrain the number of independent transport coefficients you need to measure, and they are directly relevant to the design of devices that exploit coupled transport, such as thermoelectric generators (converting heat flow into electrical current) and fuel cells (coupling chemical reactions to ion transport).