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2. Elements of Ensemble Theory
Phase Space of a Classical System Liouville’s Theorem & Its Consequences The Microcanonical Ensemble Examples Quantum States & the Phase Space
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At equilibrium: Each macrostate represents a huge number of microstates. Observed values ~ time averaged over microstates. Ensemble theory: Ensemble average = Time average
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2.1. Phase Space of a Classical System
Hamiltonian formulism of mechanics: Each dynamical state of a system of N particles is specified by 3N position coordinates { qi }, and 3N momentum coordinates { pi }, i.e., by a point { qi , pi } in the 6N –D phase space of the system. Time evolution of the system is described by a path in phase space satisfying For a conservative system, its phase space trajectory is restricted to the hypersurface For a system in thermal equilibrium with a heat reservoir, its phase space trajectory is restricted to the “hypershell”
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Ensemble of a system: Set of identical copies of a system that includes every distinct state that satisfies the given boundary conditions and gives rise to the given macrostate. In the thermodynamical limit ( N, V , N / V = finite ) , the phase points of the ensemble form a continuum. The ensemble average of a dynamical function f is given by = density function An ensemble is stationary if f is t – indep. Ensemble of a system in equilibrium must be stationary.
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2.2. Liouville’s Theorem & Its Consequences
Liouville’s theorem where the Poisson bracket is defined as : # of phase points are conserved The current density of phase points is a 6N-D vector
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where Hence, the Liouville eq. is just the equation of continuity of the phase points : For a system in equilibrium
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Solution 1: region of acceptible microstates in phase space. Microcanonical ensemble Principle of equal a priori probability : Solution 2: Canonical ensemble Chap 3:
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2.3. The Microcanonical Ensemble
Hypersurface Hypershell f ensemble average of f = time average of f for systems in eqm. = time average of f 2 av are indep = long time average of f over 1 ensemble member = f measured Let 0 fundamental volume of one microstate. Then
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2.4. Examples Microcanonical Classical ideal gas:
Surface area of an n-D sphere (App.C) is
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Sec 1.4 : Volume of state per degree of freedom = h
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Single Free Particle Confined to V
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Simple Harmonic Oscillator
Ellipse
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2.5. Quantum States & the Phase Space
Phase space volume of 1 quantum state may be estimated from the uncertainty principle: Among the first to suggest are Tetrode, Sackur, & Bose. Bose (black-body radiation) : for photons Cf. Rayleigh’s number of modes Caution: Above formulae consider only a single polarization component.
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