BoltzDistLect14ME501F2011

# BoltzDistLect14ME501F2011 - Purdue University ME 501...

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Patterned Border Template 1 Purdue University School of Mechanical Engineering ME 501: Statistical Thermodynamics Lecture 14: Most Probable Distributions, The Dilute Limit, The Boltzmann Distribution Prof. Robert P. Lucht Room 2204, Mechanical Engineering Building School of Mechanical Engineering Purdue University West Lafayette, Indiana [email protected] , 765-494-5623 (Phone) October 3, 2011

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Patterned Border Template 2 Purdue University School of Mechanical Engineering Lecture Topics • The most probable distribution for the different statistical models. • The Lagrange multipliers. • The Boltzmann distribution.
Patterned Border Template 3 Purdue University School of Mechanical Engineering The Most Probable Distribution Determination of the most probable distribution over energy levels is equivalent to determination of the most probable macrostate. For the most probable macrostate, () ( l n ) 0 mm dW d W But W m is a function of the number distribution { N j } over energy levels, For Maxwell-Boltzmann statistics,  012 ln ln , , , ..... (ln ) 0 m j j j W Wf N N N d W d N N  , ln ln ln j mMB j j j g WN N N N  , ln( ) ln ln ln j mCMB j j j j j j jj j g g N N N N N N  

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Patterned Border Template 4 Purdue University School of Mechanical Engineering The Most Probable Distribution Differentiating with respect to the level populations N j we obtain For Fermi-Dirac statistics, 2 ln ln ln ln jj j j mMB mCMB j j j j j gN g g WW N NN N g N N      (ln ) (ln ) ln ln mMB mCMB j j j j gg d W d W dN dN dN  (ln ) (ln ) ln 0 ( 1) j mMB mCMB j j j g dW d N E N , ln ln ln mFD j j j WN g Ng  
Patterned Border Template 5 Purdue University School of Mechanical Engineering The Most Probable Distribution Differentiating with respect to the level populations N j we obtain For Fermi-Dirac statistics we obtain,  2 ln 1 ln jj j j j j mFD j j j j j j j gN N g N g W Ng NN g N g N g       2 ln ln j j j mFD j j j j j j N g g W g N N g N g ln ln ln j j mFD j j

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BoltzDistLect14ME501F2011 - Purdue University ME 501...

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