# Lecture 5 - Lecture 5 Canonical Ensemble Difficult to...

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1 Lecture 5 Canonical Ensemble Difficult to control total energy E ; fixed temperature is more practical. On the technical side—microcanonical ensemble is difficult to use in practice because of difficulty in evaluating the volume of phase space or number of states accessible to the system. Total system can be expressed by a microcanomical ensemble with total Energy t E () tt E “volume” of the phase space accessible to the total system for Energy between and t t t E E E The probability that the total system is an element of volume t d of the appropriate phase space is d C d   if the energy between and t t t E E E =0 otherwise What is the probability that the subsystem is in s d without specifying the condition of the reservoir? volume of the phase space of then reservoir (so that total energy is in about ). s s r d C d EE Our task is to evaluate r --- if we know the subsystem is in s d , how much phase space is accessible to the reservoir? (I will discuss a spin-system example to elaborate) Entropy of Reservoir / r Sk r r r S k n e    Now r t s E E E 

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2 We may take st EE  Expand: () ( ) ( ) ( ) rt r r r t s r t s t SE S E S E E S E E E Then ( ) ( ) 1 exp exp r t r t rs t S E S E E k k E
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Lecture 5 - Lecture 5 Canonical Ensemble Difficult to...

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