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2. Elements of Ensemble Theory 1. Phase Space of a Classical System 2. Liouville’s Theorem & Its Consequences 3. The Microcanonical Ensemble 4. Examples 5. Quantum States & the Phase Space At equilibrium: Each macrostate represents a huge number of microstates. Observed values ~ time averaged over microstates. Ensemble theory: Ensemble average = Time average 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 qi H pi pi H qi 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” i 1,2, ,3N H qi , pi E H qi , pi E E , E E 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 f d 3N q d 3 N p f q, p q, p, t d 3N q d An ensemble is stationary if 3N p q, p, t 0 t = density function f is t – indep. Ensemble of a system in equilibrium must be stationary. Liouville’s Theorem & Its Consequences 2.2. d qi , pi , t H H qi pi dt t q p t q p p q i i i i i i i i d , H dt t f H f H f , H pi qi i qi pi where the Poisson bracket is defined as : # of phase points are conserved Liouville’s theorem d 0 dt The current density of phase points is a 6N-D vector j v v qi , pi j v v qi , pi qi pi q pi j i pi pi i qi i qi where qi pi i q p i i qi pi i q p , H i i H H 0 i qi pi pi qi Hence, the Liouville eq. is just the equation of continuity of the phase points : d j 0 dt t 0 For a system in equilibrium t H H , H 0 q p p q i i i i i H H , H 0 q p p q i i i i i Solution 1: q, p const 0 q, p otherwise region of acceptible microstates in phase space. Principle of equal a priori probability : Solution 2: 1 d 3 N q d 3 N p f q, p Microcanonical ensemble q, p H q, p , H i Chap 3: f d H H H H 0 d H qi pi pi qi H kT q, p exp Canonical ensemble 2.3. const q, p 0 The Microcanonical Ensemble 1. Hypersurface q, p H q, p E H q, p E 2. Hypershell otherwise 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. S k ln k ln 0 Then 0 1 2 2.4. Examples d 3N q d 3N q, p d 3 N q d 3 N p Microcanonical Classical ideal gas: d d 3N 3N q V N p 3N pr2 E 2m 2 r 1 Surface area of an n-D sphere (App.C) is R 2mE d 3N d 3N p 2 mE yr2 m d 3N y r 1 2 n /2 Sn R R n 1 n / 2 1 1 R 2m E 2m E 2 2 2 3 N /2 3 N 1 /2 p 2 mE 3N / 2 3N E 2m E 2m N 2 mE V 3N / 2 3 N /2 E 2 mE V N 3N / 2 3 N /2 Sec 1.4 : E 3N V 2 E h3 N 2 mE 3N / 2 ! 3 N /2 Volume of state per degree of freedom = h 0 h3N Single Free Particle Confined to V 1 P 1 p P2 E 2m p2 2m 1 1 4 3 3 3 d q d p V P 3 3 p P h h 3 d 1 dp p 1 2 V 4 p p g p p 3 h 1 E 1 V 4 3/2 2 mE h3 3 d 1 2 V 3 2m 3/2 1/2 a d h Simple Harmonic Oscillator p2 1 2 H kq 2m 2 q A cos t k m p m A sin t E q2 p2 1 2 2 A m A 1 m 2 A2 2 q2 p2 1 2 2E / m 2mE 2E m 2 2mE 1 E 1 d 1 d h Ellipse E n n 2 1 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 0 h DoF are Tetrode, Sackur, & Bose. p Bose (black-body radiation) : q p h c for photons 2 4 h h 3 d q d p V 4 h V c 3 c c 3 3 2 4 2 Cf. Rayleigh’s number of modes V 3 c 0 h 3 Caution: Above formulae consider only a single polarization component.