If heat is transferred from system A to system A′ (the surroundings) and the temperature of A′ remains unchanged, then system A′ is called a heat reservoir.
Let system A be in its i-th state with
energy Eᵢ. The combined system A*, consisting of A and A′, is completely
isolated. Therefore, its total energy E* remains constant.
Hence, when the energy of system A is Eᵢ,
the energy of the reservoir A′ is
E′ = E* − Eᵢ. (1)
According to the principle of a priori
probabilities, all accessible microscopic states of an isolated system are
equally probable. Therefore, the probability Pᵢ of finding system A in its
specific i-th state is directly proportional to the number of accessible states
of the reservoir:
Pᵢ ∝ Ω′(E′).
Thus,
Pᵢ = C′Ω′(E′),
or, using Eq. (1),
Pᵢ = C′Ω′(E* − Eᵢ). (2)
where C′ is a proportionality constant
independent of the i-th state.
Since system A is very small compared with
the heat reservoir A′,
Eᵢ ≪ E*,
and therefore
E′ ≈ E*.
The logarithm of Ω′(E* − Eᵢ) can now be
expanded in a Taylor series about E*:
ln Ω′(E’) = ln Ω′(E*) − (∂ ln
Ω′/∂E′) Eᵢ + ⋯
Since A′ acts as a heat reservoir, its
temperature remains constant and the higher-order terms can be neglected.
Hence,
ln Ω′(E* − Eᵢ) ≈ ln Ω′(E*) − (∂ ln
Ω′/∂E′) Eᵢ.
From the definition of β,
β = ∂ ln Ω′/∂E′ = 1/(kT)
where T is the constant temperature of the
heat reservoir and k is the Boltzmann constant.
Therefore,
ln Ω′(E* − Eᵢ) = ln Ω′(E*) − βEᵢ.
Taking the exponential,
Ω′(E* − Eᵢ) = Ω′(E*)
(3)
Substituting Eq. (3) into Eq. (2),
Pᵢ = C′Ω′(E*)
Putting
C = C′Ω′(E*),
we obtain
Pᵢ = C
(4)
This is an important result of statistical
mechanics. It gives the probability of finding a system maintained at constant
temperature T in its i-th state of energy Eᵢ. This is called the probability
distribution function or canonical distribution.
The quantity
is called the Boltzmann factor.
Canonical
Ensemble
An ensemble of systems that are in thermal
contact with a heat reservoir maintained at a constant temperature T, and are
in thermal equilibrium, is called a canonical ensemble when the distribution of
states is given by
Pᵢ = C
Normalization
of the Probability Distribution
The proportionality constant C in Eq. (4)
can be determined using the normalization condition. The total probability of
finding the system in any one of its accessible states must be unity:
Σ Pᵢ = 1.
Using Eq. (4),
Σ C
= 1.
Therefore,
C Σ
= 1,
and hence
C = 1 / Σ
Thus,
Pᵢ = / Σ
(5)
Using this probability distribution
function, various parameters of a system in thermal contact with a reservoir at
constant temperature T can be determined.
The probability of finding a system in the
i-th state of energy Eᵢ, when the system is in thermal equilibrium with a heat
reservoir at constant temperature T, is
Pᵢ = / Z
(1)
where
Z = Σ
(2)
The quantity Z is the PARTITION
FUNCTION. It is the sum of the Boltzmann factors over all accessible states
of the system.
Thus,
Z = Σ ,
β = 1/(kT).
The partition function is one of the most
important quantities in statistical mechanics because many thermodynamic
properties of a system can be derived from Z.