CanonicalEnsemblesLect29ME501F2011

# CanonicalEnsemblesLect29ME501F2011 - Purdue University ME...

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Patterned Border Template 1 Purdue University School of Mechanical Engineering ME 501: Statistical Thermodynamics Lecture 29: Statistical Mechanics for Systems of Dependent Particles: The Canonical Ensemble and the Grand Canonical Ensemble Prof. Robert P. Lucht Room 2204, Mechanical Engineering Building School of Mechanical Engineering Purdue University West Lafayette, Indiana [email protected] , 765-494-5623 (Phone) November 28, 2011

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Patterned Border Template 2 Purdue University School of Mechanical Engineering Lecture Topics • Introduction to the canonical ensemble and the canonical partition function. • Determination of ensemble average energy and entropy from the canonical partition function. • Determination of other thermodynamic properties for the canonical partition function. • Evaluation of the canonical partition function for distinguishable and indistinguishable particles. • Evaluation of fluctuations in internal energy from the canonical partition function.
Patterned Border Template 3 Purdue University School of Mechanical Engineering Lecture Topics • Introduction to the grand canonical ensemble and the grand canonical partition function. • Determination of thermodynamic properties from the grand canonical partition function. • The grand canonical partition function for indistinguishable, weakly interacting particles in the dilute limit. • Fluctuations in assembly particle number.

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Patterned Border Template 4 Purdue University School of Mechanical Engineering Introduction to the Canonical Ensemble • The canonical ensemble method is very useful for dealing with collections of interacting particles (dense gases, liquids) and for characterizing equilibrium fluctuations in thermodynamic properties. • The disadvantage of the canonical ensemble is the mathematical and conceptual complexity. Up to now we have been dealing with the microcanonical ensemble in which the number N of particles in the system (assembly), the volume of the system and the internal energy U of the system is fixed. • The canonical ensemble is a collection of systems, where is a very large number. For each system i in the ensemble, the number of particles N and the of the system is fixed, but the internal energy E i is not. The total energy E of all systems in the ensemble is fixed, but the systems can exchange energy with each other.
Patterned Border Template 5 Purdue University School of Mechanical Engineering Introduction to the Canonical Ensemble • Constraints on the canonical ensemble: = ( 1) i i E  = ( 2) ii i E EE

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Patterned Border Template 6 Purdue University School of Mechanical Engineering Introduction to the Canonical Ensemble • Our objective is to determine thermodynamic properties, starting with the determination of the ensemble properties. Each member system in the ensemble may be in any microstate consistent with the specified N , T , and . The state of the ensemble is specified in terms of the distribution of the member systems of the ensemble over the possible
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CanonicalEnsemblesLect29ME501F2011 - Purdue University ME...

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