Non-equilibrium Thermodynamics

Non-equilibrium thermodynamics is a branch of Thermodynamics concerned with studying Time -dependent Thermodynamic System s, irreversible transformations and open systems. Non-equilibrium thermodynamics, as contrasted with Equilibrium Thermodynamics , is most successful in the study of Stationary State s, where there are nonzero forces, flows and Entropy production, but no time variation.

BASIC CONCEPTS

The basic Thermodynamic Potential in equilibrium thermodynamics is, depending on the conditions, the Internal Energy (''U'') or a variation such as Enthalpy (''H'' = ''U'' + ''P'V''), Helmholtz Free Energy (''F'' = ''U'' - ''T'S'') or Gibbs Free Energy (''G'' = ''U'' + ''P''''V'' - ''T''''S'').
However, in non-equilibrium thermodynamics it is Entropy (''S'') that takes center stage.
Irreversible transformations are characterized by net entropy production.

Non-equilibrium thermodynamics applies to situations where the system under study is not in Thermodynamic Equilibrium but can be broken into subsystems which are sufficiently small to be in equilibrium, while still being large enough that thermodynamics is applicable to them. This hypothesis is known as ''local equilibrium''. In some cases, there will be a discrete collection of systems interacting with each other through a discrete collection of channels. Continuous systems are studied by measuring Extensive Quantiti es per unit volume (as Densiti es) and assuming that Intensive Quantiti es have locally defined values; this means that all thermodynamic variables can be represented by Field s.
Differences or Gradient s of intensive parameters are called ''thermodynamic forces'', and they cause Flow s of the extensive variables.

When an open system is allowed to reach a stationary state, it organizes itself so as to minimize total entropy production. This principle, emphasized by Ilya Prigogine among others, allows one to formulate stationary-state nonequilibrium thermodynamics using Variational Principle s.
Another powerful tool is provided by the Onsager Reciprocal Relations , which assert a certain symmetry between the response of two different flows to each other's thermodynamic forces.

FLOWS AND FORCES

Definitions and the continuity equations

Suppose that entropy ''S'' is given as a function of a collection of extensive variables ''E''_''i''. Each extensive variable has a conjugate intensive variable:

:

I_{i} := \partial{S}/\partial{E_{i}} \mbox{,} \!

such that

: d''S'' = Σ_''i'' ''I''_''i'' d''E''_''i''.

The gradients of the conjugate intensive variables are known as Thermodynamic Forces

:

\mathbf{F}_{i} = -
abla{I_{i}} \! .

These forces cause fluxes of the associated extensive quantities.
Each of the extensive variables ''E''_''i'' is assumed to be conserved. This means that the following continuity equations hold:

:

\partial{E_{i}}/\partial{t} + 
abla \cdot \mathbf{J}_{i} = 0 \mbox{,} \!

where J_''i'' is the Flux Density of ''E''_''i''.

It is possible to add ''source terms'' to the right-hand side if necessary.

Examples

The fundamental relation of thermodynamics

:

dS=rac{1}{T}du+rac{p}{T}dV+\Sigma_{i=1}^krac{\mu_i}{T}dc_i

expresses the change in Entropy ''dS'' of a system as a function of the intensive quantities Temperature ''T'', Pressure ''p'' and ''i''th Chemical Potential

\mu_i

and of the differentials of the extensive quantities Energy ''u'', Volume ''V'' and ''i''th particle Concentration

dc_i

.

Using ''u'', ''V'' and

dc_i

as our basis of the extensive quantities, we then see that the corresponding thermodynamic forces are the gradients of ''1/T'', ''p/T'' and

\mu_i/T

respectively.

ENTROPY PRODUCTION, THE SECOND LAW, AND THE ONSAGER RELATIONS

The time-variation of the entropy is then equal to

:

\partial{S}/\partial{t} = -\sum_{i} I_{i}\, 
abla \cdot \mathbf{J}_{i} = -
abla \cdot \sum_{i} I_{i}\mathbf{J}_{i} + \sum_{i} 
abla{I_{i}} \cdot \mathbf{J}_{i} \mbox{.} \!

Here, Σ_''i'' ''I''_''i''J_''i'' is a reversible entropy flow (resulting in entropy transfer through the boundaries of the system) and Σ_''i'' ∇''I''_''i'' · J_''i'' is the rate of entropy production in the bulk.

In this context, the Second Law Of Thermodynamics can be stated as requiring that the rate of entropy production be nonnegative, that is,

: Σ_''i'' ∇''I''_''i'' · J_''i'' ≥ 0.

Otherwise, it would be possible to set up a configuration of thermodynamic forces and flows resulting in a decrease of entropy in an isolated system. This condition restricts what flows are possible in the presence of given thermodynamic forces, without applying external work.

In the regime where both the flows are small and the thermodynamic forces vary slowly, there will be a Linear Relation between them, parametrized by a Matrix of coefficients conventionally denoted ''L'':

: J_''i'' = Σ_''j'' ''L''_''i''''j'' ∇''I''_''j''.

The second law of thermodynamics requires that the matrix ''L'' be Positive Definite . Statistical Mechanics considerations involving microscopic reversibility of dynamics imply that the matrix ''L'' is Symmetric . This fact is called the '' Onsager Reciprocal Relations ''.

STATIONARY STATES AND THE PRINCIPLE OF MINIMAL ENTROPY PRODUCTION

''Still needed.''

FURTHER READING

• "Into the Cool" By Dorion Sagan, on nonequilibrium thermodynamics and Evolutionary Theory . Interesting and very readable for the layman.

	CATEGORIES ABOUT NON-EQUILIBRIUM THERMODYNAMICS

	non-equilibrium thermodynamics

	fundamental physics conceptsnon-equilibrium thermodynamics

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	thermodynamics

	dynamical systems

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Non-equilibrium Thermodynamics