Fundamental Statistical Mechanics

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6. Boltzmann's Ergodic Hypothesis 6.1 Approach to Equilibrium We now begin to try to solve what might well be called the Fundamental Problem of Statistical Mechanics: Mechanical systems - isolated ones, at least - are time reversible and recurrent. However, we observe that large isolated systems often reach a state of thermodynamic equilibrium. How do we explain our observations in such a way that there are no contradictions with the laws of mechanics? Boltzmann proposed a resolution along the following lines: a) Equilibrium statistical mechanics can be formulated in terms of microcanonical ensemble averages, by using the invariant measure dµ = dS gradH . The ensemble average of a phase variable B(Γ), is B(Γ) mc = The density of states is ∫ Ω(E) = ∫ dµ H =E dµB(Γ) H = E dµ H = E ∫ (6.1) and the thermodynamic entropy is given by S = k B lnΩ(E). that Statistical Mechanics Page 35 (6.2) As an aside, the microcanonical ensemble phase space density is ρ(Γ) = δ(H − E), so B(Γ) mc = ∫ dΓB(Γ)δ (H − E) ∫ dΓδ(H − E) (6.3) If in the laboratory, we were to make very precise measurements of the quantity B(Γ t ), where Γt is the location of the phase point of the system at time t (where for example B(Γ t ) is the force per unit area on a piston) the values would show wild fluctuations about a more slowly varying "mean" quantity (as molecules collide with the piston). We might identify the thermodynamic value with the time average 1 B = B(Γ) t = lim T →∞ T B(Γ t )dt ∫ T 0 (6.4)
Boltzmann realised that one could identify the microcanonical ensemble average 6.1 with the infinite time average 6.4, that is B = B(Γ) mc Statistical Mechanics Page 36 (6.5) if he made the hypothesis that the trajectory of a typical point on the constant-energy surface (except for a set of points of zero measure) spends equal time in regions of equal measure. This hypothesis was called the ergodic hypothesis by Boltzmann and it is of central interest for the foundations of statistical mechanics. i To see how this hypothesis works, subdivide the constant energy surface into a fine grid. The average of B in each grid region i is B i . Then T 1 T B(Γ t )dt ∫ ≈ 0 ∑ i τ i T B i where τi T is the fraction of the time that the trajectories spend in region i between t = 0 and T . Using the ergodic hypothesis, we can write so that τ i T = µ i µ(E) B = ∑ i µ i µ(E) Bi = B(Γ) mc Thus equilibrium statistical mechanics could be justified, for isolated systems, if we could prove that the ergodic hypothesis is correct for a large class of physical systems.
Page 1: Fundamental Statistical Mechanics R
Page 5 and 6: EXAMPLE: Consider the map given by
Page 7 and 8: A metrically decomposable system is
Page 9 and 10: Define a reduced distribution funct
Page 11: 10. Lyapunov exponents for a Map 11

Fundamental Statistical Mechanics

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