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Gibbs Entropy

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ã°¡n style="font:7.0pt "Times New Roman""> Probabilities

ã°¡n style="font:7.0pt "Times New Roman""> Averaging

ã°¡n style="font:7.0pt "Times New Roman""> Entropy

ã°¡n style="font:7.0pt "Times New Roman""> Moles & Gas Constant

ã°¡n style="font:7.0pt "Times New Roman""> Thermodynamics

ã°¡n style="font:7.0pt "Times New Roman""> Partition Function

ã°¡n style="font:7.0pt "Times New Roman""> Gibbs Entropy

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Here we express entropy in terms of probabilities so that we can treat states that are not equally probable. This is usually derived from the Boltzmann entropy and the result is the Gibbs entropy form,

(1)

Where the sum is over all states. This is often the most practical form as it enables us to derive probability distributions like the canonical distribution and the familiar normal distribution among others.

Maximum Entropy

The second law of thermodynamics requires that entropy increases for a closed system until equilibrium is attained. We then require that entropy is maximized while various physical requirements are met. For example, a system may be constrained to satisfy the normalization condition and to have some average energy.

Maximizing S with such constraint equations is done by writing a constraint equation in the form (for instance,). Then construct a new function from the original S and the constraint g:

Here we introduced an undetermined quantity l known as a Lagrange undetermined multiplier. Now is maximized with respect to each using the relations

(1)

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An exercise can illustrate the bare bones of the process:

ã°¡n style='font:7.0pt "Times New Roman"'> A coin has only two states, up U or down D. The only requirement we know is that the sum of the unknown probabilities must equal 1, Write the normalization condition as and construct using:

Finally, apply the equations (1) and solve them with the constraint to find the probabilities.

[ans. ]

The result is hardly surprising because no physical information like average energy was included to bias the results. If a constraint on energy was also included as it should be for up and down states of an electron, we would include a second l and constraint so becomes

Equations (1) are applied as in our exercise, but the effort may not be suitable for a quick read on a lazy afternoon. The results have the form , so to agree with classical thermodynamics we must identify ,

According to information theory, maximum entropy (the MaxEnt principle) is the foundation of statistical mechanics. We simply assert that canonical probabilities result from maximizing missing information (entropy) for systems where there is a known set of accessible energies and whose probabilities satisfy the necessity normalization condition.

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