Chapter 4
Mean Values and Thermodynamics
This chapter develops expressions relating thermodynamic functions to ensemble
partition functions. A thermodynamic function that is determined directly from
a given ensemble partition function is termed the characteristic function for that
ensemble: for the canonical ensemble, it is the Helmholtz energy. Expressions
are obtained for each of the traditional thermodynamic functions in terms of
the canonical, grand, and isothermal–isobaric ensemble partition functions and
their partial derivatives. The canonical ensemble is shown to give simpler formal
expressions for U and A, while the isothermal–isobaric ensemble gives simpler
formal expressions for H and G: this correlates with what are referred to in
thermodynamics as the ‘natural’ variables for these state functions. It is established
that the statistical expression giving the entropy in terms of probabilities is identical
for the canonical and grand ensembles.
4.1 Canonical Ensemble: Closed Systems
In order to make the connection between the canonical ensemble and thermodynamics, let us begin with a determination of the mean energy associated with a
single-particle canonical distribution. The mean energy, denoted is defined as
≡
r
p r E r =
r E r e −βE r
r e −βE r
=
r
E r e
−βE r /z(β) ,
(4.1.1)
in which p r is the probability that the particle is found in energy state E r , and
the sum is over all accessible energy states of the single-particle system. If we
now associate the mean value (ensemble average) of E with the per particle
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9_4
169
Mean Values and Thermodynamics
This chapter develops expressions relating thermodynamic functions to ensemble
partition functions. A thermodynamic function that is determined directly from
a given ensemble partition function is termed the characteristic function for that
ensemble: for the canonical ensemble, it is the Helmholtz energy. Expressions
are obtained for each of the traditional thermodynamic functions in terms of
the canonical, grand, and isothermal–isobaric ensemble partition functions and
their partial derivatives. The canonical ensemble is shown to give simpler formal
expressions for U and A, while the isothermal–isobaric ensemble gives simpler
formal expressions for H and G: this correlates with what are referred to in
thermodynamics as the ‘natural’ variables for these state functions. It is established
that the statistical expression giving the entropy in terms of probabilities is identical
for the canonical and grand ensembles.
4.1 Canonical Ensemble: Closed Systems
In order to make the connection between the canonical ensemble and thermodynamics, let us begin with a determination of the mean energy associated with a
single-particle canonical distribution. The mean energy, denoted is defined as
≡
r
p r E r =
r E r e −βE r
r e −βE r
=
r
E r e
−βE r /z(β) ,
(4.1.1)
in which p r is the probability that the particle is found in energy state E r , and
the sum is over all accessible energy states of the single-particle system. If we
now associate the mean value (ensemble average) of E with the per particle
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9_4
169