Helmholtz energy function (Hermann Ludwig Ferdinand von Helmholtz)
Definition of (for arbeit):
where U is the internal energy, T is the temperature and S is the entropy.
(TS) is a conjugate pair. The differential of this function is
From the second law of thermodynamics one obtains
thus one arrives at
For A(T,V) one has the following total differential
The following equation provides a link between classical thermodynamics and
where is the Boltzmann constant, T is the temperature, and is the canonical ensemble partition function.
- Main article: Ideal gas Helmholtz energy function
A quantum correction can be calculated by making use of the Wigner-Kirkwood expansion of the partition function, resulting in (Eq. 3.5 in ):
where is the mean squared force on any one atom due to all the other atoms.