10.8 The Partition Function for Ensembles
459
Here, N refers to the average number of particles across the ensemble and μ
its conjugate thermodynamic variable called the chemical potential. The quantity
H = E + pV is the enthalpy, which you may recall changes by the amount of heat
into the system under constant pressure.
For systems with a large number of particles, predicted thermodynamic behavior
(i.e. at equilibrium) described by any of the grand canonical ensembles usually is
the same as the canonical ensemble predictions. As we will see, this follows from
the fact that for large N , fluctuations in extensive macroscopic variables are usually
very small, the exception being first-order phase transitions. However, a particular
ensemble may be easier to handle than another choice when applied to a system that
permits heat, work, or particles to be exchanged with the environment.
10.8.4 Equilibrium Averages with Gibbs Ensemble
In deriving ensemble averages for systems at or near equilibrium, we will start with
a quantum description, partly because of its generality (after all, it contains classical
behavior), partly because of the simplicity of notation (which arises because of the
discrete nature of state labels), and partly because the classical assumption of equal
a priori probabilities in phase space simply derives from the assumption of equal a
priori probabilities for each distinct microscopic quantum state.
We will often be dealing with systems with a large number N of particles, perhaps of the order of 10 23 . In that case, a Gibbs ensemble can be well approximated
by taking just one representative system for each possible microstate, i.e N ≈ N ,
the average across the ensemble of the particle number.
Let the N members of a Gibbs ensemble be labeled by the full set of quantum
numbers for the observables of that system. We map these quantum numbers into a
single index k. Each member of the ensemble will have an index k and be in a state
ψ k , with energy E k . Let the probability for the occurrence of such a state be called
P k . As described in Sect. 10.7, the entropy in the ensemble is given by
S = −k B
k
P k ln P k
(10.87)
in both the quantum and the classical setting.
The Canonical Ensemble
Now suppose that the Gibbs ensemble of N systems is isolated from the environment, but each member of the ensemble is allowed to exchange heat energy with
other members. Let N E be the energy of the full ensemble. As the full ensemble is
isolated, N E is fixed, making the average energy E in a system also fixed.
459
Here, N refers to the average number of particles across the ensemble and μ
its conjugate thermodynamic variable called the chemical potential. The quantity
H = E + pV is the enthalpy, which you may recall changes by the amount of heat
into the system under constant pressure.
For systems with a large number of particles, predicted thermodynamic behavior
(i.e. at equilibrium) described by any of the grand canonical ensembles usually is
the same as the canonical ensemble predictions. As we will see, this follows from
the fact that for large N , fluctuations in extensive macroscopic variables are usually
very small, the exception being first-order phase transitions. However, a particular
ensemble may be easier to handle than another choice when applied to a system that
permits heat, work, or particles to be exchanged with the environment.
10.8.4 Equilibrium Averages with Gibbs Ensemble
In deriving ensemble averages for systems at or near equilibrium, we will start with
a quantum description, partly because of its generality (after all, it contains classical
behavior), partly because of the simplicity of notation (which arises because of the
discrete nature of state labels), and partly because the classical assumption of equal
a priori probabilities in phase space simply derives from the assumption of equal a
priori probabilities for each distinct microscopic quantum state.
We will often be dealing with systems with a large number N of particles, perhaps of the order of 10 23 . In that case, a Gibbs ensemble can be well approximated
by taking just one representative system for each possible microstate, i.e N ≈ N ,
the average across the ensemble of the particle number.
Let the N members of a Gibbs ensemble be labeled by the full set of quantum
numbers for the observables of that system. We map these quantum numbers into a
single index k. Each member of the ensemble will have an index k and be in a state
ψ k , with energy E k . Let the probability for the occurrence of such a state be called
P k . As described in Sect. 10.7, the entropy in the ensemble is given by
S = −k B
k
P k ln P k
(10.87)
in both the quantum and the classical setting.
The Canonical Ensemble
Now suppose that the Gibbs ensemble of N systems is isolated from the environment, but each member of the ensemble is allowed to exchange heat energy with
other members. Let N E be the energy of the full ensemble. As the full ensemble is
isolated, N E is fixed, making the average energy E in a system also fixed.