User:Miguel~enwiki/Drafts

This page is intended to contain work in progress on existing wikipedia articles where I feel it would be inappropriate to carry out incremental updates on the article pages. Miguel (talk) 13:41, 4 July 2010 (UTC)

= Onsager reciprocal relations =

Thermodynamic potentials
Entropy can be expressed as a function $$S(E_1,\ldots,E_n)$$ of a collection of conserved extensive quantities whose conjugate thermodynamic potentials are then defined as the intensive quantities $$I_k\colon = {\partial S\over\partial E_k}$$. That is, entropy variations can be expressed as

$$d S = \sum_k I_k dE_k$$

In practice, such an expression will be typically derived from an energy balance equation. For instance, for a fluid system

$$d U = T d S - p d V$$

where $$U$$ is the internal energy and $$V$$ is the volume (both extensive), $$T$$ is the temperature and $$p$$ is the pressure (both intensive), and $$TdS$$ expresses heat transfer into the system while $$p dV$$ is the work done by the system. Writing

$$d S = {1\over T}d U + {p\over T}d V$$

shows that $$1/T$$ is the thermodynamic potential conjugate to internal energy, and $$p\over T$$ is the thermodynamic potential conjugate to volume.

Now assume that all extensive quantities are taken per unit volume, resulting in densities which we denote by lower-case letters such as $$s$$ for the entropy density, $$u$$ for the energy density, and so on. Then,

$$d s = \sum_k I_k ds_k$$

flows and generalised forces
Following Prigogine one can associate to each extensive quantity $$E_k$$ a velocity or flux $$\mathbf{J}_k$$. Local conservation is expressed by the continuity equation

$$\partial_t e_i + \nabla\cdot\mathbf{J_i} = 0$$

Also, to each thermodynamic potential one associates an affinity or generalised force $$\mathbf{X}_k\colon = \nabla I_k$$. The rate of change of the entropy density is, then,

$$\partial_t s = \sum_k I_k \partial_t e_k = - \sum_k I_k \nabla \cdot\mathbf{J}_k = \underbrace{- \nabla\cdot\sum_k I_k \mathbf{J}_k}_{\hbox{boundary term}}  + \underbrace{\sum_k (\nabla I_k) \cdot \mathbf{J}_k}_{\hbox{bulk term}}.$$

local entropy production
Defining an entropy current

$$\mathbf{J}^{(S)} \colon = \sum_k I_k \mathbf{J}_k$$

we have that entropy is locally conserved except for a bulk entropy production term

$$\partial_t s + \nabla\cdot\mathbf{J}^{(S)} = \sum_k \mathbf{X}_k \cdot \mathbf{J}_k.$$

phenomenological coefficients
Assuming deviations from equilibrium are small, then both the forces and fluxes will be small and, moreover, one can assume fluxes depend linearly on the forces via phenomenological coefficients $$L_{ij}$$:

$$\mathbf{J}_i \approx \sum_i L_{ij} \mathbf{X}_{j}$$

Entropy production in the bulk is, then,

$$\partial_t s + \nabla\cdot\mathbf{J}^{(S)} \approx \sum_{ij}L_{ij}\mathbf{X}_i\cdot \mathbf{X}_j$$

The second law of thermodynamics implies that $$(L_{ij})$$ must be a positive-definite matrix. Onsager's reciprocal relations assure as that the skew-symmetric part of this matrix, which has no effect on bulk entropy production, in fact vanishes and $$(L_{ij})$$ is, in fact, symmetric $$L_{ij} = L_{ji}$$