Phase space and partition function 137A process in whichN,V, andTchange by small amounts dN, dV, and dTleads
to a change dAin the Helmholtz free energy of
dA=(
∂A
∂N
)
V,TdN+(
∂A
∂V
)
N,TdV+(
∂A
∂T
)
N,VdT (4.2.4)via the chain rule. However, sinceA=E−TS, the change inAcan also be expressed
as
dA= dE−SdT−TdS=TdS−PdV+μdN−SdT−TdS=−PdV+μdN−SdT, (4.2.5)where the second line follows from the first law of thermodynamics. By comparing the
last line of eqn. (4.2.5) with eqn. (4.2.4), we see that the thermodynamic variables
obtained from the partial derivatives ofAare:
μ=(
∂A
∂N
)
V,T, P=−
(
∂A
∂V
)
N,T, S=−
(
∂A
∂T
)
N,V. (4.2.6)
These relations define the basic thermodynamics of the canonical ensemble. We must
now establish the link between these thermodynamic relations and the microscopic
description of the system in terms of its HamiltonianH(x).
4.3 The canonical phase space distribution and partition function
In the canonical ensemble, we assume that a system can only exchange heat with
its surroundings. As was done in Section 3.2, we consider two systems in thermal
contact. Let us denote the physical system as “system 1” and the surroundings as
“system 2” (see Fig. 4.1). System 1 is assumed to containN 1 particles in a volume
V 1 , while system 2 containsN 2 particles in a volumeV 2. In addition, system 1 has an
energyE 1 , and system 2 has an energyE 2 , such that the total energyE=E 1 +E 2.
System 2 is taken to be much larger than system 1 so thatN 2 ≫N 1 ,V 2 ≫V 1 ,
E 2 ≫E 1. System 2 is often referred to as athermal reservoir, which can exchange
energy with system 1 without changing its energy appreciably. The thermodynamic
“universe,” composed of system 1 + system 2, is treated within the microcanonical
ensemble. Thus, the total HamiltonianH(x) of the universe is expressed as a sum of
contributions,H 1 (x 1 ) +H 2 (x 2 ) of system 1 and system 2, where x 1 is the phase space
vector of system 1, and x 2 is the phase space vector of system 2.
As was argued in Section 3.4, if we simply solved Hamilton’s equations forthe
total HamiltonianH(x) =H 1 (x 1 ) +H 2 (x 2 ),H 1 (x 1 ) andH 2 (x 2 ) would be separately
conserved because the Hamiltonian is separable. However, the microcanonical distri-
bution, which is proportional toδ(H(x)−E) allows us to consider all possible energies
E 1 andE 2 for whichE 1 +E 2 =Ewithout explicitly requiring a potential coupling