# Non-equilibrium thermodynamics

Non-equilibrium thermodynamics is a branch of thermodynamics concerned with studying time-dependent thermodynamic systems, irreversible transformations and open systems. Non-equilibrium thermodynamics is most successful in the study of stationary states, where there are nonzero forces, flows and entropy production, but no time variation.

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## 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 quantities per unit volume (as densities) and assuming that intensive quantities have locally defined values; this means that all thermodynamic variables can be represented by fields. Differences or gradients of intensive parameters are called thermodynamic forces, and they cause flows 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 Ei. Each extensive variable has a conjugate intensive variable called a thermodynamic force:

[itex] I_{i} := \partial{S}/\partial{E_{i}} \mbox{,} \! [itex]

so that

dS = Σi Ii dEi.

Each of the extensive variables Ei is assumed to be conserved. This means that the following continuity equations hold:

[itex] \partial{E_{i}}/\partial{t} + \nabla \cdot \mathbf{J}_{i} = 0 \mbox{,} \! [itex]

where Ji is the flux density of Ei.

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

## Entropy production, the second law, and the Onsager relations

The time-variation of the entropy is then equal to

[itex] \partial{S}/\partial{t} = -\sum_{i} I_{i}\, \nabla \cdot \mathbf{J}_{i} = -\nabla \cdot \sum_{i} I_{i}\mathbf{J}_{i} + \sum_{i} \nabla{I_{i}} \cdot \mathbf{J}_{i} \mbox{.} \! [itex]

Here, Σi IiJi is a reversible entropy flow (resulting in entropy thansfer through the boundaries of the system) and ΣiIi · Ji is the rate of entropy production in the bulk.

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

ΣiIi · Ji ≥ 0.

Otherwise, it would be possible to set up a configuration of thermodynamic forces and flows resulting in a decrease of entropy in an isolated system. This condition restricts what flows are possible in the pressence of given thermodynamic forces, without applying external work.

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:

Ji = Σj LijIj.

The second law of thermodynamics requires that the matrix L be positive definite. Statistical mechanics considerations involving microscopic reversibility of dynamics imply that the matrix L is symmetric. This fact is called the Onsager reciprocal relations.

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