Fundamental Statistical Mechanics

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6.2 Individual Ergodic Theorem In 1931 Birkhoff proved a very important theorem which allows us to make some of the ideas of Boltzmann more precise. Although Birkhoff's theorem is still a long way from what is needed in Boltzmann's picture it does define in a useful way the dynamical properties that a system must possess in order to have an equilibrium state. This theorem is concerned with the properties of individual trajectories in phase space. Suppose we are considering a mechanical system with an invariant measure and suppose that we consider some phase function B defined on the constant energy surface, satisfying the condition that ∫ dµ B(Γ)
EXAMPLE: Consider the map given by φ (x) = x + α mod(1) If α is a rational number, nm where n and m are integers, then the mapping repeats the initial point after m iterations. A trajectory starting from an initial point will be periodic and will not spend equal time in regions of equal (Lebesgue) measure, and hence the trajectory is not dense on the unit interval. In this case the system is not ergodic. But if α is irrational no trajectory is periodic and the system is ergodic! To see this consider the time average of a phase function on a trajectory starting at x 1 B (x) = lim N →∞ N N −1 ∑ n= 0 B(φ n (x)) 1 = lim N→∞ N N−1 ∑ n=0 B(x + nα) Consider the Fourier series for B(x): B(x) = a j e 2πijx ∞ ∑ B(x + nα) = a j e j =−∞ 2πijx+ 2πijnα Substituting into the time-average gives 1 B (x) = lim N →∞ N 1 = lim N →∞ N N −1 ∞ ∑ ∑ n= 0 j =−∞ ∞ ∑ j=−∞ 2π ijx+ 2πijnα a je a j e 2πijx e 2πijnα N −1 ∑ n= 0 <strong>Statistical</strong> <strong>Mechanics</strong> Page 38 ∞ ∑ j =−∞ 1 = a0 + lim N →∞ N ∞ ∑ j =−∞ j ≠0 a j e 2πijx 1− e 2πijNα 2 πijα 1 − e The terms with n ≥ 1 don't survive the limit N →∞, thus B (x) = a 0 for α irrational. In this case the time average is a constant - independent of the starting point x and the system is ergodic, where the ensemble average is the average over the circle with the usual Lebesgue measure, 1 ∫ B (x) = dxB(x) = B(x) mc 0
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Fundamental Statistical Mechanics

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