Lecture II

Another service from Omega

An Introduction to Markov Chain Monte Carlo

*****
Lecture II

Lecture II

Abstract

Statistical Physics (part 2), the original Metropolis Algorithm, Simulated Annealing.

Phase Space

It is convenient to visualize a mechanical system as a point in the 6N dimensional space (q,p) of all the positions and momentums of all the N particles. (**add picture here**)

Due to the complexity of macroscopic systems (N ~ 1024) it was necessary to abandom determinism and use statistics to describe the system. The predictions of statistical physics are expected to hold only on the average.

Instead of the precise innitial conditions (which are unknown), statistical physics describes the system by a probability distribution over phase space, r(q,p) for t = 0. As it will be seen later, Hamilton's equations imply the conservation, at all time, of this innitial distribution. This is the famous, Liouville's theorem. The determination of r is then the first step.

Maxwell-Boltzmann-Gibbs Distribution

Different forms for r are found to be needed depending on the particular data available about the system. We will be concerned only with the so called cannonical distribution. We assume that the system is not isolated but in thermal equilibrium with a heat bath at constant temperature T. Statistically this is equivalent to the assumption that the average energy of the molecules is constant. The novel idea of Boltzmann was to discretize phase-space to find the most likely distribution for r.

Each particle has a definite position and momentum. Subdivide the positions and momentums for each particle into m (6 dimensional) cells of equal size. Assume that these cells are small enough so that the value of the energy within each cell is approximately constant. Let Ej be the energy in the j-th cell. Assume further that the cells, eventhough small, they are still big enough to accomodate lots of particles inside. These are reasonable assumptions justified by the smallness of atomic dimensions ( ~ 10-8 cm), the size of typical N and the smoothness of energy surfaces. This discretization of the phase-space for each molecule into m equal size cells induces a discretization of the phase-space of the system into, mN equal size cells. With the help of this discretization, the state s of the system is specified by,

s {1,2,,m }N
indicating the cell number for each of the N particles. If the particles are assumed to be identical and indistinguishable, then permutations of the molecules with a given cell number have no physical consequences. All it matters is how many molecules end up in each of the cells and not which ones did. Thus, the actual set of distinguishable physical states is much smaller than mN it is,

(N+m-1)!
N! (m-1)!
corresponding to the number of ways of splitting N into the m cells. There are,

G = N!
n1! n2!nm!
ways of throwing the N molecules into the m cells in such a way that n1 of them are in the first cell, n2 in the second cell, etc. If we assume apriori that the molecules have equal chance of ending in any of the cells then the number of ways can be turned into a probability for the state s = (n1,,nm),

P = N!
n1! n2!nm!
×constant
Hence, the most likely distribution of balls among the cells is the one that maximizes this probability subject to whatever is knonw about the system. When the temperature is all we know we maximize P subject to the constraint that the average energy is fixed at kT. Where k is a fenomenological (not fundamental) constant needed to change the units from ergs (units of energy) to degrees (usual units for temperature). It is known as the Boltzmann constant and it is about,

k = 1.380 ×10-14 ergs per degree centigrade
Using the fact that N and the nj are large we can use Stirling's approximation,

logn! ~ n logn - n
to get,

logP
=
NlogN -

j 
( njlognj - nj) + constant
=
-N

j 
pj logpj + constant
where,

pj = nj
N
.
Thus, P is the probability of observing the probability distribution (p1,,pm). A probability of a probability... A prior!

P e-Njpj logpj
Known as an entropic prior, for the quantity in the exponent (sans N) is the famous expression for the entropy of a probability distribution. If we treat the pj as if they were continuous variables we can obtain the most likely apriori distribution by solving,


max
s.t. 
-

j 
pj logpj


j 
pj = 1


j 
Pj Ej = k T


File translated from TEX by TTH, version 2.32.
On 5 Jul 1999, 22:59.