Statistical mechanics, structure of matter Theorem

Prigogine's minimum entropy production principle

Builds on Entropy production Onsager reciprocal relations Positive definite matrix

Statement

Let a system in the linear regime have thermodynamic forces $X = (X_1, \dots, X_n)$ and fluxes $J_i = \sum_{k=1}^{n} L_{ik} X_k,$ where $L = (L_{ik})$ is a constant, symmetric, positive definite matrix. Let its entropy production be $\sigma(X) = \sum_{i} J_i X_i = X^T L X$ . Fix $m$ of the forces, $X_1 = a_1, \dots, X_m = a_m$ , with $1 \le m < n$ , and let the other forces vary. Then $\sigma$ , restricted to these forces, has exactly one minimum point. A point $X$ with $X_i = a_i$ for $i \le m$ is this minimum if and only if $J_{m+1} = \cdots = J_n = 0.$ So the stationary state, in which the unconstrained fluxes vanish, is the state of least entropy production compatible with the constraints.

Remarks

The proof is a direct computation. For $j > m$ , symmetry of $L$ gives $\partial \sigma / \partial X_j = 2 \sum_k L_{jk} X_k = 2 J_j$ . The restriction of $\sigma$ is a quadratic function whose Hessian is twice a principal submatrix of $L$ , which is positive definite, so it has one minimum, where the gradient vanishes. The symmetry $L_{ik} = L_{ki}$ comes from the Onsager reciprocal relations; without it the gradient is not $2J$ and the stationary state need not minimise $\sigma$ . Example: in thermal diffusion with a fixed temperature difference, the steady state with zero mass flux is the minimum of $\sigma$ at that temperature difference.

Going deeper

For a student

The principle holds only in the linear regime with constant coefficients and symmetric $L$ . Since $\sigma$ decreases along the approach to the steady state in this regime, it also acts as a Lyapunov function and gives stability of the steady state. Far from equilibrium, or when the coefficients depend on the state, the principle fails in general.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Entropy concept
Entropy production concept
Local thermodynamic equilibrium concept
Positive definite matrix concept
Symmetric matrix concept
Onsager reciprocal relations statement
Prigogine's minimum entropy production principle statement

Tag	`PJVSCHMM6Z`
URI	`m2://PJVSCHMM6Z`
Kind	statement, theorem
Writing	W0 stub
MSC	82C35, 80A17
Status	This copy of the site has no build data. Run `m2 status PJVSCHMM6Z`.
About	Entropy production
Uses	Entropy production, Onsager reciprocal relations, Positive definite matrix
Changed	3 November 2026