Introduction to Mathematical Physics/Statistical physics/Entropy maximalization

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In general, a system is described by two types of variables. External variables y^i whose values are fixed at y_j by the exterior and internal variables X^i that are free to fluctuate, only their mean being fixed to \bar{X^i}. Problem to solve is thus the following:

Problem:

Find distribution probability P_l over the states (l) of the considered system that maximizes the entropy


S=-k_B\sum P_l \ln (P_l)

and that verifies following constraints:

\begin{matrix}
\sum X_{l}^iP_l&=&\bar{X^i}\\
\sum P_l&=&1
\end{matrix}

Entropy functional maximization is done using Lagrange multipliers technique. Result is:


P_l=\frac{1}{Z}e^{-\lambda_1 X^{1}_l-\lambda_2 X^{2}_l- ...}

where function Z, called partition function, \index{partition function} is defined by:


Z=\sum_{(l)} e^{-\lambda_1 X^{1}_l-\lambda_2 X^{2}_l- ...}

Numbers \lambda_i are the Lagrange multipliers of the maximization problem considered.

Example:

In the case where energy is free to fluctuate around a fixed average, Lagrange multiplier is:


\beta=\frac{1}{k_BT}

where T is temperature.\index{temperature} We thus have a mathematical definition of temperature.

Example:

In the case where the number of particles is free to fluctuate around a fixed average, associated Lagrange multiplier is noted \beta\mu where \mu is called the chemical potential.

Relations on means[1] that:


S/k=\sum \lambda_i \bar{X^i}+\ln(Z)

This relation that binds L to S is called a {\bf Legendre transform}.\index{Legendre transformation} L is function of the y^i's and \lambda_j's, S is a function of the  y^i's and \bar{X^j}'s.

  1. They are used to determine Lagrange multipliers \lambda_i from associated means \bar{X_i}} can be written as:

    
-\frac{\partial}{\partial \lambda_i}\ln Z(y,\lambda_1,\lambda_2,...) =
\bar{X_i}

    It is useful to define a function L by:

    
L=\ln Z(y,\lambda_1,\lambda_2,...)

    It can be shown\footnote{ By definition

    
S=-k_B\sum P_l \ln (P_l)

    thus

    
S/k=-\sum P_l\ln(\frac{1}{Z}e^{-\lambda_1 X^{1}_l-\lambda_2 X^{2}_l- ...})

    
S/k=1\ln Z+\lambda_1 \bar{X^{1}_l}+\lambda_2 \bar{X^{2}_l}+ ...