AlternativeDerivation of thePartitionFunction ...

1 [ ] 9 Sep 2013 Alternative derivation of the Partition Function for Generalized EnsemblesJonathan L. Belof and Brian Space11 Department of Chemistry, University of South Florida4202 E. Fowler Ave., Tampa, FL 33620A pedagogical approach for deriving the statistical mechanical partition function, in a mannerthat emphasizes the key role of entropy in connecting the microscopic states to thermodynamics, isintroduced. The connections between the combinatoric formulaS=klnWapplied to the Gibbsconstruction, the Gibbs entropy,S= k ipilnpi, and the microcanonical entropy expressionS=kln are clarified.

2 The condition for microcanonical equilibrium, and the associated roleof the entropy in the thermodynamic potential is shown to arise naturally from the postulate ofequala prioristates. The derivation of the canonical partition functionfollows simply by invokingthe Gibbs ensemble construction at constant temperature and using the first and second law ofthermodynamics (viathe fundamental equationdE=T dS PdV+ dN) that incorporate theconditions of conservation of energy and composition without the needs for explicit constraints;other ensemble follow easily. The central role of the entropy in establishing equilibrium for a givenensemble emerges naturally from the current approach.

3 Connections to generalized ensemble theoryalso arise and are presented in this INTRODUCTIONIn deriving the partition function for a desired ensem-ble, the most common approach is to maximize an en-tropy function with constraints appropriate to the ther-modynamic condition. While equivalent to the approachproposed below, such a method (called the traditionalapproach hereafter) does not make clear to students theexplicit role of the assumption of equala prioristatesand the corresponding role of the entropy in the ther-modynamic potential for the microcanonical ,S=kln is often taken as a postulate[1] and itsconnection to the statistical formulaS=klnW(appear-ing on Boltzmann s tombstone) is not obvious.

4 Further,in the traditional approach, the role of the entropy in un-derstanding equilibrium in non-isolated, open ensemblescan be confusing. We note in passing that concerns overthe rigor of the method of most probable distributionprompted Darwin and Fowler to develop a derivation ofthe partition function based upon complex analysis.[2]Also, infrequently stressed is the Gibbs entropy,S= kPipilnpi, wherepiis the probability of finding a sys-tem in a given state, which can be invoked for any equi-librium ensemble and associated state probabilities.[3] Itis a direct consequence of the statistical entropy formula,S=klnW, in conjunction with the Gibbs constructionof an ensemble that contains a large number of macro-scopic subsystems, each consistent with the desired ther-modynamic variables;Wgives the number of possible re-alizations within the Gibbs construction for the ensembleunder consideration.

5 The Gibbs entropy also permits thederivation of the connection between the characteristic Present address:Lawrence Livermore National Laboratory7000 East Avenue., Livermore, CA 94550thermodynamic function and the partition function for agiven ensemble without further appeal to thermodynamicexpressions, as is required in the traditional the present approach, first, the connections betweenthe statistical formulaS=klnW, the Gibbs entropy,S= kPipilnpi, and the microcanonical entropy ex-pressionS=kln are clarified. The condition formicrocanonical equilibrium, and the associated role ofthe entropy in the thermodynamic potential then arisesfrom the postulate of equala prioristates.

6 The deriva-tion of the canonical partition function follows by in-voking the Gibbs construction and the first and secondlaw of thermodynamicsviathe fundamental equation,dE=T dS P dV+ dN, that incorporates the con-ditions of conservation of energy and composition with-out the needs for explicit constraints. The role of thetemperature (coming from the constraint of total energyand an appeal to appropriate thermodynamic relation-ships in the traditional approach) is immediately appar-ent and also introducedviathe fundamental need for explicit maximization of any function is thusalso avoided.

7 Legendre transforming a particular ther-modynamic function to include desired thermodynamiccontrol variables for an ensemble of interest and invok-ing equilibrium leads to the corresponding partition func-tion. Using the resulting probabilities in the Gibbs en-tropy expression directly connects the partition functionto the thermodynamic potential. The central role of theentropy in establishing equilibrium for a given ensembleemerges naturally from the current approach. Connec-tions to generalized ensemble theory also arise and arepresented in this present approach is novel in providing clarity asto the roles played by the different formulas and physicalquantities of interest.

8 Further, it makes explicit the as-sumptions inherent in deriving the partition function foran ensemble and provides its direct connection to the rel-evant thermodynamic potential in a systematic approach also makes deriving the partition func-tion for a given ensemble a simplified, straight-forwardprocess, even for more challenging examples such as theisothermal-isobaric ensemble. Using this approach in theclassroom has led to better retention and understandingof the foundations of statistical mechanics and an abil-ity for students to confidently apply the machinery toproblems that arise in their subsequent THE GIBBS ENTROPY AND THEMICROCANONICAL ENSEMBLEWe begin by introducing the concept of an ensemble ofreplicas that describe the molecular states correspondingto a given macrostate.

9 This picture is referred to as the Gibbs construction herein, due to it s original introduc-tion by Gibbs[4, 5] who addressed many of the subtletiesinherent[1] in the formulation of statistical a collection of macroscopic molecular subsys-tems ofNmolecules within a volumeV, each of whichis part of the larger Gibbs construction, the totality ofwhich is known as the system . No other constraintshave yet been imposed, system s macrostate isotherwise unspecified. It is desirable to define the micro-scopic statistics of this system as thoroughly as possibleand then apply any other constraints at the the total number of subsystems in our collection beknown as.

10 Then let i, the occupation number, denotethe number of subsystems from this collection that are inthe same thermodynamic state. These occupations willthus take on a large value in the thermodynamic limitand they obey a sum rule,Pi i= . Note, technicallythe energy is course-grained, to within asmall but otherwise arbitrary range (these arguments arepresented in detail elsewhere[1, 2]) and the results areinsensitive to this , consider the following combinatoric formula:Se=klnW{ }=kln ! 1! 2!..(1)W{ }is the number of ways in which the set of occu-pations{ }may be arranged consistent with the givenmacrostate.