Non-equilibrium thermodynamics

Basic concepts

The basic thermodynamic potential in equilibrium thermodynamics is, depending on the conditions, the internal energy (U) or a variation such as enthalpy (H = U + PV), Helmholz free energy (F = U - TS) or Gibbs free energy (G = U + PV - TS). However, in non-equilibrium thermodynamics it is entropy (S) that takes center stage. Irreversible transformations are characterized by net entropy production.

Non-equilibrium thermodynamics applies to situations where the system under study is not in thermodynamic equilibrium but can be broken into subsystems which are sufficiently small to be in equilibrium, while still being large enough that thermodynamics is applicable to them. This hypothesis is known as local equilibrium. In some cases, there will be a discrete collection of systems interacting with each other through a discrete collection of channels. Continuous systems are studied by measuring extensive quantitieses per unit volume (as densities) and assuming that intensive quantitieses have locally defined values; this means that all thermodynamic variables can be represented by fieldss. Differences or gradients of intensive parameters are called thermodynamic forces, and they cause flowss of the extensive variables.

When an open system is allowed to reach a stationary state, it organizes itself so as to minimize total entropy production. This principle, emphasized by Ilya Prigogine among others, allows one to formulate stationary-state nonequilibrium thermodynamics using variational principles. Another powerful tool is provided by the Onsager reciprocal relations, which assert a certain symmetry between the response of two different flows to each other's thermodynamic forces.

Flows and forces

Suppose that entropy S is given as a function of a collection of extensive variables E_i. Each extensive variable has a conjugate intensive variable called a thermodynamic force:

dS = Σ_i I_i dE_i.

It is possible to add source terms to the right-hand side if necessary. \r

Entropy production, the second law, and the Onsager relations

The time-variation of the entropy is then equal to

In this context, the second law of thermodynamics can be stated as requiring that the rate of entropy production be nonnegative, that is,

Σ_i ∇I_i · J_i ≥ 0.

In the regime where both the flows are small and the thermodynamic forces vary slowly, there will be a linear relation between them, parametrized by a matrix of coefficients conventionally denoted L:

J_i = Σ_j L_ij ∇I_j.

Stationary states and the principle of minimal entropy production

Still needed. \r

Applications

Still needed.