Fundamental Statistical Mechanics

6. Boltzmann's Ergodic Hypothesis
6.1 Approach to Equilibrium
We now begin to try to solve what might well be called the Fundamental Problem of Statistical
Mechanics: Mechanical systems - isolated ones, at least - are time reversible and recurrent.
However, we observe that large isolated systems often reach a state of thermodynamic
equilibrium. How do we explain our observations in such a way that there are no contradictions
with the laws of mechanics? Boltzmann proposed a resolution along the following lines:
a) Equilibrium statistical mechanics can be formulated in terms of microcanonical ensemble
averages, by using the invariant measure dµ = dS gradH . The ensemble average of a phase
variable B(Γ), is
B(Γ) mc =
The density of states is
∫
Ω(E) = ∫ dµ
H =E
dµB(Γ)
H = E
dµ
H = E
∫ (6.1)
and the thermodynamic entropy is given by S = k B lnΩ(E).
that
Statistical Mechanics Page 35
(6.2)
As an aside, the microcanonical ensemble phase space density is ρ(Γ) = δ(H − E), so
B(Γ) mc =
∫
dΓB(Γ)δ (H − E)
∫ dΓδ(H − E)
(6.3)
If in the laboratory, we were to make very precise measurements of the quantity B(Γ t ), where Γt is the location of the phase point of the system at time t (where for example B(Γ t ) is the force
per unit area on a piston) the values would show wild fluctuations about a more slowly varying
"mean" quantity (as molecules collide with the piston). We might identify the thermodynamic
value with the time average
1
B = B(Γ) t = lim
T →∞ T B(Γ t )dt ∫
T
0
(6.4)