In general, a system is described by two types of variables. External
variables
whose values are fixed at
by the exterior and internal
variables
that are free to fluctuate, only their mean being fixed to
. Problem to solve is thus the following:
Problem:
Find distribution probability
over the states
of the
considered system
that maximizes the entropy
![{\displaystyle S=-k_{B}\sum P_{l}\ln(P_{l})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2695088336880df9e87ce2433334b6a5ca5142b6)
and that verifies following constraints:
Entropy functional maximization is done using Lagrange multipliers
technique. Result is:
![{\displaystyle P_{l}={\frac {1}{Z}}e^{-\lambda _{1}X_{l}^{1}-\lambda _{2}X_{l}^{2}-...}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38904c49b060f22a17e8d61f24d6d4c8ec551af0)
where function
, called partition function,
\index{partition function}
is
defined by:
![{\displaystyle Z=\sum _{(l)}e^{-\lambda _{1}X_{l}^{1}-\lambda _{2}X_{l}^{2}-...}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/618e539354d594d44fa90e045ee47eed542745b5)
Numbers
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:
![{\displaystyle \beta ={\frac {1}{k_{B}T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95a4a13fa8841e9f6c2b657c4ccccbf3002ee229)
where
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
where
is
called the chemical potential.
Relations on means[1] that:
![{\displaystyle S/k=\sum \lambda _{i}{\bar {X^{i}}}+\ln(Z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a045650d3ad65ff54cf87d0e8f81fd65938775b)
This relation that binds
to
is called a {\bf Legendre transform}.\index{Legendre transformation}
is function of the
's and
's,
is a function of
the
's and
's.
- ↑ They are used to determine Lagrange multipliers
from associated means
} can be written as:
![{\displaystyle -{\frac {\partial }{\partial \lambda _{i}}}\ln Z(y,\lambda _{1},\lambda _{2},...)={\bar {X_{i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5c186d0f59ec3682bb31dd8b4ecbbdb865b02d)
It is useful to define a function
by:
![{\displaystyle L=\ln Z(y,\lambda _{1},\lambda _{2},...)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c689ac5e66cb21a5edb8267aaa80ba054ac44b5)
It can be shown\footnote{
By definition
![{\displaystyle S=-k_{B}\sum P_{l}\ln(P_{l})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2695088336880df9e87ce2433334b6a5ca5142b6)
thus
![{\displaystyle S/k=-\sum P_{l}\ln({\frac {1}{Z}}e^{-\lambda _{1}X_{l}^{1}-\lambda _{2}X_{l}^{2}-...})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9baf3e98351ec9a1e533f1494503668f26d1c43)
![{\displaystyle S/k=1\ln Z+\lambda _{1}{\bar {X_{l}^{1}}}+\lambda _{2}{\bar {X_{l}^{2}}}+...}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eca258ae55d7a06c6d3aa74e46751b63d5503f45)