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3. The Canonical Ensemble. Equilibrium between a System & a Heat Reservoir A System in the Canonical Ensemble Physical Significance of Various Statistical Quantities in the Canonical Ensemble Alternative Expressions for the Partition Function The Classical Systems
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3. The Canonical Ensemble Equilibrium between a System & a Heat Reservoir A System in the Canonical Ensemble Physical Significance of Various Statistical Quantities in the Canonical Ensemble Alternative Expressions for the Partition Function The Classical Systems Energy Fluctuations in the Canonical Ensemble: Correspondence with the Microcanonical Ensemble Two Theorems: the “Equipartition” & the “Virial” A System of Harmonic Oscillators The Statistics of Paramagnetism Thermodynamics of Magnetic Systems: Negative Temperatures
Reasons for dropping the microcanonical ensemble: • Mathematical: Counting states of given E is difficult. • Physical: Experiments are seldom done at fixed E. Canonical ensemble : System at constant T through contact with a heat reservoir. • Let r be the label of the microstates of the system. • Probablity Pr( Er) can be calculated in 2 ways: • Pr # of compatible states in reservoir. • Pr~ distribution of states in energy sharing ensemble.
3.1. Equilibrium between a System & a Heat Reservoir Isolated composite system A(0) = ( System of interest A ) + ( Heat reservoir A ) Heat reservoir : , T = const. Let r be the label of the microstates of A. with Probability of A in state r is
3.2. A System in the Canonical Ensemble Consider an ensemble of N identical systems sharing a total energy E. Let nr = number of systems having energy Er ( r = 0,1,2,... ). = average energy per system Number of distinct configurations for a given E is Equal a priori probabilities (X) means sum includes only terms that satisfy constraint on X. { nr* } =most probable distribution
Method of Most Probable Values To maximize lnW subjected to constraints is equivalent to minimize, without constraint , are Lagrange multipliers
with Let and set Same as sec 3.1 E.g.
Method of Mean Values ~ means “depend on {r} ”. Let Thus Constraints: Note: r in { nr } is a dummy variable that runs from 0 to , including s.
Method of Steepest Descent ( Saddle Point ) is difficult to evaluate due to the energy constraint. Its asymptotic value ( N ) can be evaluate by the MSD. Define the generating function Binomial theorem U removes the energy constraint. where
NU = integers = coefficient of zN U in power expansion of . This is the case if all Er , except the ground state E0= 0, are integer multiples of a basic unit. analytic for |z| < R C : |z| < R ( For { r~ 1 }, sharp min at z = x0 ) Let
N >>1 Fo z real, has sharp min at x0 max along ( i y )-axis For z complex : x0 is a saddle point of .
MSD: On C, integrand has sharp max near x0 . Gaussian dies quickly
C.f. so that With { r = 1 } :
Fluctuations where
3.3. Physical Significance of Various Statistical Quantities in the Canonical Ensemble
3.6. Energy Fluctuations in the Canonical Ensemble: Correspondence with the Microcanonical Ensemble
3.10. Thermodynamics of Magnetic Systems: Negative Temperatures