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Onsager reciprocal relations

In thermodynamics, the Onsager reciprocal relations express the equality of certain relations between flows and forces in thermodynamic systems out of equilibrium, but where a notion of local equilibrium exists.

As an example, it is observed that temperature differences in a system lead to heat flows from the warmer to the colder parts of the system. Similarly, pressure differences will lead to matter flow from high-pressure to low-pressure regions. It was observed experimentally that when both pressure and temperature vary, pressure differences can cause heat flow and temperature differences can cause matter flow. Even more surprisingly, the heat flow per unit of pressure difference and the density (matter) flow per unit of temperature difference are equal. This was shown to be necessary by Lars Onsager using 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 thermodynamic potential is internal energy. In a 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

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