The Canonical Distribution Function L04
The Canonical Distribution
•Idea: Describe a system “1” in equilibrium with a heat bath “2”, with which it exchanges
energy but not matter. The total system 1+2 is isolated.
•Large-system approximation: We only deal with very large systems with short-range
interactions (with respect to the system size). Then, if both subsystems 1 and 2 are large,
most of the variables in each part are statistically independent of those in the other part,
and the distribution function must be a product
ρ(q, p) = ρ1(q1, p1)ρ2(q2, p2)
of probability distributions for the two sets of variables (q1, p1) and (q2, p2).
•Form of the distribution function: Taking the log of both sides,
ln ρ(q, p) = ln ρ1(q1, p1) + lnρ2(q2, p2),
which means that ln(p, q) must be an additive constant of the motion (recall that ρis a
constant of the motion by Liouville’s theorem). A known result is that the only possibility
is that it be a linear combination of the ones related to spacetime and internal symmetries,
(E,~p,~
L,Q, ...). Restricting our attention to energy, we get
ln ρ(q, p) = α−β H(q, p),for some constants αand β .
The constant αis fixed by renormalization, so we get
ρ(q, p) = 1
Zc
e−βH (q,p),with Zc:= ZΓ
dqdpe−βH (q,p).
•Note: If the number of states available is a rapidly growing function of the energy, in
the limit of high energies ρbecomes a sharply peaked function.
