Lecture 1
Provisional outline of the course
1. Fundamentals
2. Thermodynamic limit
3 and 4 phase transitions
5 and 6 non-equilibrium stat. mech. (+ Monte Carlo?)
I'm restricting this course to classical statistical mechanics
I The basis of equlibrium stat. mech.
The two essential ingredients making it different from
mordinary mechanics (a) probability (b) large number of particles
We model some physical system as a mechanical system of N particles
usually enclosed in a box of some kind, say a cube of volume V.
Describe its dynamical state as a point omega = (p_1..q_{nu N})
in the phase space Omega. (More generally Omega can have one component for
each value of N, i.e. Omega = sum_M Omega_N, but I won't do this)
Model the statistics by a measure d mu(omega) = rho(omega) d omega
Use Hamiltonian dynamics, the Ham. equations of motion
d omega / dt = matrix times dH/omega
Liouville's eqn : phase space volume is an invariant of the motion.
That is to say, the Hamiltonian dynamics preserves the Lebesgue measure.
Consequently, (d/dt) rho(omega , t) = 0
II A mathematical model for equilibrium: the ergodic problem
A fact of experience: isolated systems approach equilibrium.
We model equilibrium by an invariant rho , i.e. one that is invariant
under the Hamiltonian dynamics --- or nearly so.
Simplest invariant p.s.d.s are rho = f(H(omega)).
An ergodic system (in SM ) means one for which these are the
only invariant p.s.d.s
The ergodic problem (much mathematical activity) is whether a
particular system is ergodic. Important results:
Sinai Billiards:
Ya G Sinai Russian Math Surveys 25 (1970) 137 proves
ergodicity (also mixing) for billiards on a closed table with
finite concave boundaries and also concave obstacles; for more
complete proof, Gallavotti Springer LN in physics 38 (1975) 36-296.
This is equivalent to the problem of two hard disks
on a torus.
Chernov and Sinai 1987 two billiard balls on a nu-torus
Kramli, Simanyi, Szasz 1991,2 three and four balls on a
nu-torus (nu >= 3 for N=4)
Simanyi 1991,1992 N balls on a nu-torus with N <= nu
Szasz 1993 Physica A 194 (1993) 86-92 reviews the above
The KAM theorem: a system of anharmonically coupled oscillators.
Moser in Stable and Random Motions in Dynamical Systems
(Princeton 1973) gives:
H = 0.5 sum_i p_i^2 + V with V analytic having a min
at q = 0 and the Hessian there has incommensurable eigenvalues,
then, subject to a certain nondegeneracy condition on the
third and fourth order terms in V, the system has a set of
trajectories of non-zero measure which are restricted to a
subset of the energy manifold. Ergo, it is non-ergodic.
Roughly speaking, one can obtain ergodic-type results if the system is
hyperbolic (chaotic) which means that the long-time behaviour is very
sensitive to the initial conditions. A more recent review article is
JR Dorfman, Physics reports, 301 (1998) 151-185 Deterministic chaos and
the foundations of the kinetic theory of gases.
From now on we assume the system to be ergodic.
Physically this is likely to be correct for liquids and gases.
For solids some further justification is required
III The microcanonical ensemble (distribution) and its relatives
These are particular invariant phase-space densities of the form
rho(omega) = f(H(omega))
Step function ensemble (rare): rho is prop to Step(E - H(omega)).
Normalizing factor is the integrated structure function W(E)
Microcanonical ensemble: Phase space density rho is prop.
to delta (H(omega) - E)
Normalizing denominator is the structure function W'(E)
Thickened MCE: rho is prop. to
Step(H(omega) - E) - Step(H(omega) - (E - DeltaE))
Normalizing denominator is W(E) - W(E- Delta E)
Canonical ensemble: rho is prop to exp (- beta H(omega))
Normalizing factor is the partition function
Z(beta) = int exp (- beta H(omega)) d omega.
It is the Laplace transrform of the structure function.
Grand canonical ensemble: refers to the case where
Omega = Sum_N Omega_N
IV Adiabatic changes: the entropy.
Suppose H depends also on some externall controlled parameter x,
for example the position of a piston.
Model the system by a thickened microcanonical ensemble.
Consider an adiabatic change of x , i.e. H = H(omega, x(t))
so slow that equilibrium
is maintained, i.e. the distribution
rho(omega , t) = f(H(omega,t), t).
By considering the step ensemble we see that the
integrated structure function W(E,x)
is an adiabatic invaraiant.