Statistical Mechanics - Physics at Oregon State University

2.1. INTRODUCTION. 25
PS(s) ∝ gR(U0 − ɛs) (2.4)
Therefore the ratio of the probabilities of finding the system in state 1 and 2 is
PS(1)
PS(2) = gR(U0 − ɛ1) 1
k (SR(U0−ɛ1)−SR(U0−ɛ2))
= e B
gR(U0 − ɛ2)
Because the reservoir is very large, the difference between the entropies is
SR(U0 − ɛ1) − SR(U0 − ɛ2) ≈ −(ɛ1 − ɛ2)
∂S
(U0) = −
∂U V,N
ɛ1 − ɛ2
TR
(2.5)
(2.6)
The reservoir and the system are in thermal equilibrium, and have the same
temperature T. We drop the subscript on the temperature because is is the same
for both system and reservoir. We find
or in general
PS(1)
PS(2) = e− ɛ 1 −ɛ 2
k B T (2.7)
ɛs − k PS(s) ∝ e B T (2.8)
This is the well-known Boltzmann factor. The only reference to the reservoir
is via the temperature T, no other details of the reservoir are involved.
Normalization requires P (s) = 1 and hence we define
Z(T ) =
s
ɛs − k e B T (2.9)
PS(s) = 1 ɛs
k e− B T (2.10)
Z
The probability PS(s) depends on the temperature T of the system, and
although there is no explicit reference to the reservoir there is still an implicit
reservoir through T. Specifying T requires contact with an outside reservoir.
Also, the probability only depends on the energy ɛs of the state and not on
any other property of this state. All different states with the same energy have
the same probability. The quantity Z(T ) is called the partition function. We
will see later that for a description of a system at temperature T the partition
function is all we need to know. The probability distribution PS(s) of this form
is called the Boltzmann distribution. The partition function is also a function of
volume and number of particles. The dependence on all extensive state variables,
excluding the entropy, is via the energy of the quantum states. In general, the
energy eigenvalues ɛs are functions of the volume V and the number of particles
N. Note that the quantum states we consider here are N-particle quantum states.
The energy ɛs(V, N) is the energy of a collective state and is not equal to the
energy of a given particle!