Gibbs Measures
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A. Gibbs Distribution Suppose a physical system has possible states 1, . . . , n and the energies of these states are E1 , . . . , En . Suppose that this system is put in contact with a much larger “heat source” which is at temperature T . Energy is thereby allowed to pass between the original system and the heat source, and the temperature T of the heat source remains constant as it is so much larger than our system. As the energy of our system is not fixed any of the states could occur. It is a physical fact derived in statistical mechanics that the probability pj that state j occurs is given by the Gibbs distribution e−βEj pj = n −βE , i i=1 e 1 and k is a physical constant. where β = kT We shall not attempt the physical justification for the Gibbs distribution, but we will state a mathematical fact closely connected to the physical reasoning.
1.1. Lemma. Let real numbers a1 , . . . , an be given. Then the quantity F (p1 , . . . , pn ) =
n
−pi log pi +
i=1
n
pi ai
i=1
n has maximum value log i=1 eai as (p1 , . . . , pn ) ranges over the simplex {(p1 , . . . , pn ) : pi ≥ 0, p1 + · · · + pn = 1} and that maximum is assumed only by −1 eai . pj = eaj i
4
1 Gibbs Measures
n This is proved by calculus. The quantity H(p1 , . . . , pn ) = i=1 −pi log pi is called the entropy of the distribution (p1 , . . . , pn ) (note: n ϕ(x) = −x log x is continuous on [0, 1] if we set ϕ(0) = 0.) The term i=1 pi ai is of course the average value of the function a(i) = ai . In the statistical mechanics case ai = −βEi , entropy is denoted S and average energy E. The Gibbs distribution then maximizes S − βE = S −
1 E, kT
or equivalently minimizes E − kT S. This is called the free energy. The principle that “nature minimizes entropy” applies when energy is fixed, but when energy is not fixed “nature minimizes free energy.” We will now look at a simple infinite system, the one-dimensional lattice. Here one has for each integer a physical system with possible states 1, 2, . . . , n. A configuration of the system consists of assigning an xi ∈ {1, . . . , n} for each i:
·
·
x−2
·
x−1
·
x0
x1
·
·
x2
·
x3
·
·
·
Thus a configuration is a point x = {xi }+∞ i=−∞ ∈
{1, . . . , n} = Σn . Z
We now make assumptions about energy: (1) associated with the occurrence of a state k is a contribution Φ0 (k) to the total energy of the system independent of which position it occurs at; (2) if state k1 occurs in place i1 , and k2 in i2 , then the potential energy due to their interaction Φ∗2 (i1 , i2 , k1 , k2 ) depends only on their relative position, i.e., there is a function Φ2 : Z × {1, . . . , n} × {1, . . . , n} → R so that Φ∗2 (i1 , i2 , k1 , k2 ) = Φ2 (i1 − i2 ; k1 , k2 ) (also: Φ2 (j; k1 , k2 ) = Φ2 (−j; k2 , k1 )). (3) all energy is due to contributions of the form (1) and (2). Under these hypotheses the energy contribution due to x0 being in the 0th place is 1 φ∗ (x) = Φ0 (x0 ) + Φ2 (j; xj , x0 ). 2 j=0
(We “give” each of x0 and xk half the energy due to their interaction). We now assume that Φ2 j = supk
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