Lecture 3 : Phase transitions, first part (Thursday 6 Feb 2003)
I Thermodynamic limit theorem for the entropy:
Thermodynamic limit theorem: for suitable hamiltonians
(i) the following limit exists if N has a terminating binary expansion
(although it may be - infinity for some E,N,V, e.g. for a hard spehre system
it is - infinty if N/V is less than the close-packing density):
S(E,N,V):= lim_{m -> infty} 8^{-m}k log [W(8^m E,8^m N,8^m V)/(8^m N)!)]
where W(.,.,.) is the structure function for a cube of volume 8^m V,
i.e. the phase space volume in which a system of 8^m N particles in such
a cube has energy <= 8^m E
(ii) in the interior of the domain where S(E,N,V) is not - infinity it
is continuous, a homogeneous function of degree 1, concave in all three arguments.
We call it the thermodynamic entropy.
An example of a "suitable Hamiltonian" is the one for a system of hard spheres
with an pair interaction u(r) that is attractive (u(r) <= 0))
and decays fast enough at large r for the integral of u(|x|) over all space to
converge.
II. The partition function and the free energy
From the def'n of Z, for a given V and N we have
Z(beta) := (1/N!)int_Omega exp( - beta H(omega) d omega
= (1/N!)int exp(-beta E) W'(E) dE = beta int exp(-beta E) W(E) dE
\sim max_E exp (- beta E + S(E)/k)
by Laplace's method, which means using the maximum of the integrand as an
estimate of the integral
From the formula just above, for any positive number T we have as m -> infty
8^{-m} log Z(1/kT, 8^m N, 8^m V) -> max_E ( - E/kT + S(E,N,V)/k
= - F(T,N,V)/kT
where F(T,N,V) := min_E (E - T S(E,N,V))
F is called the Helmoltz free energy. It is homogeneous and concave
in N,V of degree 1
Inversion formula S(E) = max_T (E-F)/T (graph)
At the extremum, F = E - TS and T = dE/dS, E = d(F/T)/d(1/T) (i.e. S= - dF/dT)
III Phase transitions
These occur when there is some singularity in the thermodynamic functions,
for example when water boils, the density or volume depends
discontinuously on the temperature at constant pressure (draw (P,T) diagram).
Likewise, it depends discontinuously on the pressure at constant temperature.
(draw an isotherm in the (P,V/N) diagram).
In terms of the free energy, P = - dF(T,V)/dV
(since TdS = dE + P dV, dF = SdT - PdV)
so this phase transition corresponds to a flat part on the (F,V/N) isotherm.
IV An example: hard rods in 1 dimension
For hard rods of length b in a box of length L, the Hamiltonian is
H = sum_1^N p_i^2/2m + sum_{i b
+ infty otherwise
Partition function is the same as for N non-interacting particles in
a box of length L - Nb, i.e.
Z(1/kT) = (1/N!) (L-Nb)^N (2 pi m kT)^{N/2}
giving - F/NkT = lim_{N -> infty}(1/N)log Z
= log (L - Nb) - log N + 0.5 log T + const
eqn of state P = - dF/dL = NkT/(L - Nb) (graph)
The 3-D analogue (hard spheres of diameter b ) is
P = NkT/(V - Nv_0) where v_0 := (pi b^3)/6 = vol. of sphere of diameter b
(Van der Waals circa 1900)
A better approx is (from the Percus-Yevick approximation)
PV/NkT \approx (1 + eta + eta^2 +...)/(1-eta)^3
where eta = v_0 N/V
(reference Croxton theory of liquids p 73 if I can't do better)
VdW gives rhs as 1/(1 - 4 eta) which is correct to the first order in eta.
There is a lot of theory about how to calculate such things, both as
a power series in the density (the Mayer expansion) and by means of integral
equations for the pair correlation functin.
V Phase transitions: the mean field approximation
Consider again the hard rods system but add in an attractive interaction,
i.e. hange the potential to u = u_{hs} + u_{lr}
where u_{lr} is a weak long-range attractive (meaning u(r) <= 0)
potential such that its integral converges,
int_a^{infty} u_{lr}(r) = - a
Long-range means that the range is large compared to
the average inter-particle spacing L/N.
If the molecules were uniformly distributed in space with density N/L,
then the total interaction of each particle with the particles to its
right would be approximately
(N/L) int_a^{+infty} u_{lr}(r) dr = - aN/L
(the 'mean field') and so the total interaction energy would be
- aN^2/L. Adding this to the terms we already know gives
"F" = - NkT[ log (L - bN) - log N + 0.5 log T + const] - aN^2/L
and the approximate equation of state is now
"P" = - dF/dL = NkT/(L - Nb) ] - aN^2/L^2
van der Waals (circa 1900) suggested using the corresponding
formula for three dimensions, i.e. (P + akT (N/V)^2 ) (V - Nb') = NkT
where b' = volume of a sphere of diameter b.
Graph illustrates that "P" can be a non-monotonic function of V/N,
in which case "F" is not concave in L (or V).
Physical reason for this is that the assumed uniform distribution
in space is wrong; there would in fact be large fluctuations
of the local value of N/V away from its mean value.
To get a convex F, we could take the convex envelope of "F".
Kac Uhlenbeck and Hemmer, J math Phys 4 (1963) 216-228
showed that the convex envelope recipe (written out below)
is exactly correct in the limit gamma -> 0 for 1-D hard rods with
u_{lr} = - a gamma exp(- gamma r)
The generalization to two-body attractive potentials of the form
u(r) = + infty if 0 < r < b
- gamma^D u_0(gamma r) if r > b
in D dimensions, is
F(N,V) = convex envelope of [F_{a = 0}(N,V) - a N^2/V]
where a = - 0.5 int_{R^D} u_0(r) d^D r.
The convex envelope is calculated for variations of N at fixed V
(J L Lebowitz and OP, J Math Phys 7 (1966) 98-113).
Appendix : sketch of proof of the thermodynamic entropy theorem
(to be added later)