Onsager reciprocal relations

As an example, it is observed that statistical mechanics.

Similar "reciprocal relations" occur between different pairs of forces and flows in a variety of physical systems.

The theory developed by Onsager is much more general than this example and capable of treating more than two thermodynamic forces at once.

Example: Fluid system

Thermodynamic potentials, forces and flows

The basic fluid system, the energy density $\ u$ depends on matter density $\ r$ and entropy density $\ s$ in the following way:

$\ du= T ds + m dr$

where $\ T$ is temperature and $\ m$ is a combination of pressure and chemical potential. We can write

$ds = \frac{1}{T} du - \frac{m}{T} dr$ .

The extensive quantities $\ u$ and $\ r$ are conserved and their flows satisfy continuity equations:

$\partial_{t}u + \nabla \cdot \mathbf{J}_{u} = 0 \!$

and

$\partial_{t}r + \nabla \cdot \mathbf{J}_{r} = 0 \!$ ,

where $\partial_{t}$ indicates the partial derivative with respect to time $\ t$ , and $\ \nabla \cdot \!$ indicates the divergence of the flux densities $\ \mathbf{J} \!$ .

The gradients of the conjugate variables (thermodynamics) of $\ u$ and $\ r$ , which are $\frac{1}{T}$ and $-\frac{m}{T}$ , are thermodynamic forces and they cause flows of the corresponding extensive variables. In the absence of matter flows,

$\mathbf{J}_{u} = k\, \nabla\frac{1}{T} \!$ ;

and, in the absence of heat flows,

$\mathbf{J}_{r} = -k'\, \nabla\frac{m}{T} \!$ ,

where $\ \nabla$ now indicates the gradient.

The reciprocity relations

In this example, when there are both heat and matter flows, there are "cross-terms" in the relationship between flows and forces (the proportionality coefficients are customarily denoted by $\ L$ ):

$\mathbf{J}_{u} = L_{uu}\, \nabla\frac{1}{T} - L_{ur}\, \nabla\frac{m}{T} \!$

and

$\mathbf{J}_{r} = L_{ru}\, \nabla\frac{1}{T} - L_{rr}\, \nabla\frac{m}{T} \!$ .

The Onsager reciprocity relations state the equality of the cross-coefficients $\ L_{ur}$ and $\ L_{ru}$ . Proportionality follows from simple dimensional analysis (i.e., both coefficients are measured in the same units of temperature times mass density).

Abstract formulation

Let $\ E_{i}$ be the extensive variables on which entropy $\ S$ depends. In the following analysis, these symbols will refer to densities of these thermodynamic quantities. Then,

$\ dS=\sum_{i}I_{i}dE_{i}$

where

$I_{i} :=\frac{\partial{S}}{\partial{E_{i}}} \!$

defines the intensive quantity $\ I_{i}$ conjugate to the extensive quantity $\ E_{i}$ .

The gradients of the intensive quantities are thermodynamic forces:

$\mathbf{F}_{i} = -\nabla{I_{i}} \!$

and they cause fluxes $\ J_{i}$ of the extensive quantities satisfying continuity equations

$\partial_{t}E_{i} + \nabla \cdot \mathbf{J}_{i} = 0 \!$

The fluxes are proportional to the thermodynamic forces by a matrix of coefficients $\ L_{ij}$

$\mathbf{J}_{i}=\sum_{j}L_{ij}\mathbf{F}_{j} \!$

Then,

$\partial_{t}E_{i} = \nabla \cdot \sum_{j} L_{ij}\, \nabla{I_{j}} \!$

Introducing a susceptibility matrix

$\sigma_{ij} = \frac{\partial{E_{i}}}{\partial{I_{j}}} \!$

we have

$\sum_{j} \sigma_{ij}\, \partial_{t}I_{j} = \nabla \cdot \sum_{j} L_{ij}\, \nabla{I_{j}} \!$

Categories: Non-equilibrium thermodynamics

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Onsager_reciprocal_relations". A list of authors is available in Wikipedia.