204
4 Mean Values and Thermodynamics
V , N) =
2πm
h 2
3N
2 V N
N!
E
3N
2 −1
(3N/2)
,
so that the canonical partition function for an N-particle system is likewise obtained
as a Laplace transform,
Z(β, V , N) =
∞
0
(E, V , N)e
−βE dE ,
(4.4.2)
of (E, V , N). An appropriate definition of the inverse Laplace transform correctly
gives V , N) from Z(β, V , N). Moreover, this interrelationship between
V , N) and Z(β, V , N) explains why the microcanonical and canonical
ensembles yield identical final results for the thermodynamic functions of state, as
they ultimately contain the same information about the system being treated.
If, in expression (4.2.2a) for the grand partition function (with γ = βμ), we
approximate the sum over N by an integration over N, we see that
V , μ) =
∞
0
Z(β, V , N)e
−βμN dN
(4.4.3)
gives the grand partition function as a Laplace transform of the canonical partition
function Z(β, V , N). In a similar fashion, we see from expression (4.3.5a) that by
approximating the summation over the volumes V i of the ensemble members by an
integration over continuous volume V , the isobaric–isothermal partition function
, P , N) can be obtained as a different Laplace transform of the canonical
partition function Z(β, V , N), namely,
, P , N) =
∞
0
Z(T , V , N)e
−βP V dV .
(4.4.4)
4.5 Problems for this Chapter
1. Show that the expression for the single-particle internal energy given in
Example 4.1 for an ideal gas in a (weak) gravitational field reduces to u(T ) =
3
2 k B T in the zero gravity g → 0 limit. How does u(T ) behave for very low
temperatures or in a strong gravitational field?
2. Obtain an expression giving the standard deviation σ E in terms of the heat
capacity per particle at constant volume, c V = C V /N, and show that σ E /U ,
with U ≡ E the thermodynamic internal energy, behaves as N
−
1
2 . What
conclusions can thus be drawn from this result with respect to the equivalence or
nonequivalence of results obtained for a macroscopic (thermodynamic) system
using the microcanonical and canonical ensembles?
4 Mean Values and Thermodynamics
V , N) =
2πm
h 2
3N
2 V N
N!
E
3N
2 −1
(3N/2)
,
so that the canonical partition function for an N-particle system is likewise obtained
as a Laplace transform,
Z(β, V , N) =
∞
0
(E, V , N)e
−βE dE ,
(4.4.2)
of (E, V , N). An appropriate definition of the inverse Laplace transform correctly
gives V , N) from Z(β, V , N). Moreover, this interrelationship between
V , N) and Z(β, V , N) explains why the microcanonical and canonical
ensembles yield identical final results for the thermodynamic functions of state, as
they ultimately contain the same information about the system being treated.
If, in expression (4.2.2a) for the grand partition function (with γ = βμ), we
approximate the sum over N by an integration over N, we see that
V , μ) =
∞
0
Z(β, V , N)e
−βμN dN
(4.4.3)
gives the grand partition function as a Laplace transform of the canonical partition
function Z(β, V , N). In a similar fashion, we see from expression (4.3.5a) that by
approximating the summation over the volumes V i of the ensemble members by an
integration over continuous volume V , the isobaric–isothermal partition function
, P , N) can be obtained as a different Laplace transform of the canonical
partition function Z(β, V , N), namely,
, P , N) =
∞
0
Z(T , V , N)e
−βP V dV .
(4.4.4)
4.5 Problems for this Chapter
1. Show that the expression for the single-particle internal energy given in
Example 4.1 for an ideal gas in a (weak) gravitational field reduces to u(T ) =
3
2 k B T in the zero gravity g → 0 limit. How does u(T ) behave for very low
temperatures or in a strong gravitational field?
2. Obtain an expression giving the standard deviation σ E in terms of the heat
capacity per particle at constant volume, c V = C V /N, and show that σ E /U ,
with U ≡ E the thermodynamic internal energy, behaves as N
−
1
2 . What
conclusions can thus be drawn from this result with respect to the equivalence or
nonequivalence of results obtained for a macroscopic (thermodynamic) system
using the microcanonical and canonical ensembles?