# What is meant by “linear” in non-equilibrium thermodynamics?

I'm trying to learn a bit about non-equilibrium thermodynamics, and am currently reading de Groot and Mazur. In it, there is a quote right in the beginning, talking about the phenomenological equations, such as Fick's law and Fourier's law, being linear:

Non-equilibrium thermodynamics, in its present form, in mainly restricted to the study of such linear phenomena. Very little of a sufficiently general nature is known outside this linear domain. This is not a very serious restriction however, since even in rather extreme physical situations, transport processes, for example, are still described by linear laws.

I can think of several examples that seem nonlinear to me, e.g. materials where the thermal conductivity depends on the temperature, or the magnetic permeability of ferromagnetic materials.

Are these cases simply not covered by this theory (in which case I would consider this quite a serious restriction), or do I misunderstand a) what they mean by "linear" or b) something else?

The so-called "linear irreversible thermodynamics" makes two independent assumptions:

(1) Denote the intensive and extensive parameters by $$Y_k$$ and $$X_k$$, then the static equilibrium Gibbs equation $$dS= \frac{1}{T} (dU + pdV +Y_2dX_2 + Y_3dX_3-...$$ also holds dynamically for the entropy production rate $$\sigma$$. $$\sigma = \frac{1}{T}\big(\dot u + \sum_k Y_k\dot x_k \big)$$ Here $$x_k$$ are the local densities of the extensive variables $$X_k$$, resp., and $$X_0=U, Y_0=1$$. The "dot" signifies substantial time derivative and is important when various conservation laws including that of mass, energy, momentum, etc., applied. The time derivatives $$J_k= \dot x_k$$ are the fluxes while the $$F_k = \frac{Y_k}{T}$$ are the thermodynamic forces that generate these fluxes.

(2) It is assumed that the fluxes and forces are coupled, and the coupling can be described by a set of constitutive relationship between them that is linear: $$J_k = \sum_m \mathcal L_{km} F_{m}$$ where the $$\mathcal L_{km}$$ are the so-called phenomenological coefficients and are supposed to be given and independent of the fluxes/forces but may be locally varying with the space coordinates.

Onsager proved for a large class of materials that the matrix $$\mathcal L$$ is symmetrical $$\mathcal L_{km}=\mathcal L_{mk}$$ from which Prigogine derived that the function defined by the integral $$\mathcal K =\sum_k \int_0^{F_k} J_k dF_k$$ is non-increasing in time $$\frac{d\mathcal K}{dt} \le 0$$ and in steady state $$\mathcal K$$ reaches its minimum. Equally important that the integral is path independent, in other words the differential form $$\sum_k J_k dF_k$$ is integrable and the integral $$\mathcal K$$ is a potential function. Again, this conclusion needs the assumption that the constitutive relationship between the fluxes and the forces be both linear and symmetric.

A differential equation is said to be linear if it can be written in the form:

$$a_n(t)\frac{d^nx}{dt^n}+a_{n-1}(t)\frac{d^{n-1}x}{dt^{n-1}}+\cdots+a_1(t)\frac{dx}{dt}+a_0(t)x=g(t)$$

where each coefficient $$a_n$$ and the non-homogeneous term $$g(t)$$ are functions only of the independent variable $$t$$.

This can be expanded to partial differential equations. The heat equation, for instance, is:

$$\dot{u}=\alpha \nabla^2u\\ \frac{\partial u}{\partial t}=\alpha\left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right)$$

Notice there is a linear combination of the derivatives of $$u$$, so the equation is said to be linear.

• As I understand it, this is not about the balance equations but the constitutive equations, so for example, $B = μ(H) H$ for ferromagnetism. These are not necessarily differential equations. – Psirus Aug 16 '19 at 17:02