Partition function

2007-05-20T17:54:19Z

84.79.223.157: New page: The '''partition function''' of a system in contact with a thermal bath at temperature <math>T</math> is the normalization constant of the Boltzmann distribution function, and therefor...

The '''partition function''' of a system in contact with a thermal bath
at temperature <math>T</math> is the normalization constant of the [[Boltzmann distribution]]
function, and therefore its expression is given by

:<math>Z(T)=\int \Omega(E)\exp(-E/k_BT)\,dE</math>,

where <math>\Omega(E)</math> is the density of states with energy <math>E</math> and <math>k_B</math>
the [[Boltzmann constant]].

The partition function of a system is related to its [[free energy]] through the formula

:<math>F=-k_BT\log Z.</math>

This connection can be derived from the fact that <math>k_B\log\Omega(E)</math> is the
entropy of a system with total energy <math>E</math>. This is an extensive magnitude in the
sense that, for large systems (i.e. in the [[thermodynamic limit]], when the number of
particles <math>N\to\infty</math>
or the volume <math>V\to\infty</math>), it is proportional to <math>N</math> or <math>V</math>.
In other words, if we assume <math>N</math> large, then

:<math>k_B\log\Omega(E)=Ns(e),</math>

where <math>s(e)</math> is the entropy per particle in the [[thermodynamic limit]], which is
a function of the energy per particle <math>e=E/N</math>. We can
therefore write

:<math>Z(T)=N\int \exp\{N(s(e)-e/T)/k_B\}\,de.</math>

Since <math>N</math> is large, this integral can be performed through [[steepest descent]],
and we obtain

:<math>Z(T)=N\exp\{N(s(e_0)-e_0/k_BT)\}</math>,

where <math>e_0</math> is the value that maximizes the argument in the exponential; in other
words, the solution to

:<math>s'(e_0)=1/T.</math>

This is the thermodynamic formula for the inverse temperature provided <math>e_0</math> is
the mean energy per particle of the system. On the other hand, the argument in the exponential
is

:<math>\frac{1}{k_BT}(TS(E_0)-E_0)=-\frac{F}{k_BT}</math>

the thermodynamic definition of the [[free energy]]. Thus, when <math>N</math> is large,

:<math>F=-k_BT\log Z.</math>

Boltzmann distribution

2007-05-20T17:29:43Z

84.79.223.157:

The '''Maxwell-Boltzmann distribution function''' is a function ''f(E)'' which gives the
probability that a system in contact with a thermal bath at temperature ''T'' has energy
''E''. This distribution is ''classical'' and is used to describe systems with ''identical''
but ''distinguishable'' particles.

:<math>f(E) = \frac{1}{Z} \exp(-E/k_B T)</math>

where the normalization constant ''Z'' is the [[partition function]] of the system.
[[Category: Statistical mechanics]]

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Partition function

Boltzmann distribution