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)

Thermodynamic potentials

Entropy can be expressed as a function ${\displaystyle S(E_{1},\ldots ,E_{n})}$ of a collection of conserved extensive quantities whose conjugate thermodynamic potentials are then defined as the intensive quantities ${\displaystyle I_{k}\colon ={\partial S \over \partial E_{k}}}$. That is, entropy variations can be expressed as

${\displaystyle dS=\sum _{k}I_{k}dE_{k}}$

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

${\displaystyle dU=TdS-pdV}$

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

${\displaystyle dS={1 \over T}dU+{p \over T}dV}$

shows that ${\displaystyle 1/T}$ is the thermodynamic potential conjugate to internal energy, and ${\displaystyle p \over T}$ 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 ${\displaystyle s}$ for the entropy density, ${\displaystyle u}$ for the energy density, and so on. Then,

${\displaystyle ds=\sum _{k}I_{k}ds_{k}}$

flows and generalised forces

Following Prigogine^[1] one can associate to each extensive quantity ${\displaystyle E_{k}}$ a velocity or flux ${\displaystyle \mathbf {J} _{k}}$. Local conservation is expressed by the continuity equation

${\displaystyle \partial _{t}e_{i}+\nabla \cdot \mathbf {J_{i}} =0}$

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

${\displaystyle \partial _{t}s=\sum _{k}I_{k}\partial _{t}e_{k}=-\sum _{k}I_{k}\nabla \cdot \mathbf {J} _{k}=\underbrace {-\nabla \cdot \sum _{k}I_{k}\mathbf {J} _{k}} _{\hbox{boundary term}}+\underbrace {\sum _{k}(\nabla I_{k})\cdot \mathbf {J} _{k}} _{\hbox{bulk term}}.}$

local entropy production

Defining an entropy current

${\displaystyle \mathbf {J} ^{(S)}\colon =\sum _{k}I_{k}\mathbf {J} _{k}}$

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

${\displaystyle \partial _{t}s+\nabla \cdot \mathbf {J} ^{(S)}=\sum _{k}\mathbf {X} _{k}\cdot \mathbf {J} _{k}.}$

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 ${\displaystyle L_{ij}}$:

${\displaystyle \mathbf {J} _{i}\approx \sum _{i}L_{ij}\mathbf {X} _{j}}$

Entropy production in the bulk is, then,

${\displaystyle \partial _{t}s+\nabla \cdot \mathbf {J} ^{(S)}\approx \sum _{ij}L_{ij}\mathbf {X} _{i}\cdot \mathbf {X} _{j}}$

The second law of thermodynamics implies that ${\displaystyle (L_{ij})}$ 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 ${\displaystyle (L_{ij})}$ is, in fact, symmetric ${\displaystyle L_{ij}=L_{ji}}$

proof of Onsager reciprocity

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