Sep 9, 2013 - law of thermodynamics via the fundamental equation,. dE = T dS - P dV + ÂµdN, that ... classroom has led to better retention and underst...

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arXiv:1309.2017v1 [cond-mat.stat-mech] 9 Sep 2013

1

Department of Chemistry, University of South Florida 4202 E. Fowler Ave., Tampa, FL 33620

A pedagogical approach for deriving the statistical mechanical partition function, in a manner that emphasizes the key role of entropy in connecting the microscopic states to thermodynamics, is introduced. The connections between the combinatoric formula S = k ln W applied to the Gibbs P construction, the Gibbs entropy, S = −k pi ln pi , and the microcanonical entropy expression i

S = k ln Ω are clarified. The condition for microcanonical equilibrium, and the associated role of the entropy in the thermodynamic potential is shown to arise naturally from the postulate of equal a priori states. The derivation of the canonical partition function follows simply by invoking the Gibbs ensemble construction at constant temperature and using the first and second law of thermodynamics (via the fundamental equation dE = T dS − P dV + µdN ) that incorporate the conditions 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 given ensemble emerges naturally from the current approach. Connections to generalized ensemble theory also arise and are presented in this context.

I.

INTRODUCTION

In deriving the partition function for a desired ensemble, the most common approach is to maximize an entropy function with constraints appropriate to the thermodynamic condition. While equivalent to the approach proposed below, such a method (called the traditional approach hereafter) does not make clear to students the explicit role of the assumption of equal a priori states and the corresponding role of the entropy in the thermodynamic potential for the microcanonical ensemble. Indeed, S = k ln Ω is often taken as a postulate[1] and its connection to the statistical formula S = k ln W (appearing on Boltzmann’s tombstone) is not obvious. Further, in the traditional approach, the role of the entropy in understanding equilibrium in non-isolated, open ensembles can be confusing. We note in passing that concerns over the rigor of the method of most probable distribution prompted Darwin and Fowler to develop a derivation of the partition function based upon complex analysis.[2] Also, P infrequently stressed is the Gibbs entropy, S = −k pi ln pi , where pi is the probability of finding a sysi

tem in a given state, which can be invoked for any equilibrium ensemble and associated state probabilities.[3] It is a direct consequence of the statistical entropy formula, S = k ln W , in conjunction with the Gibbs construction of an ensemble that contains a large number of macroscopic subsystems, each consistent with the desired thermodynamic variables; W gives the number of possible realizations within the Gibbs construction for the ensemble under consideration. The Gibbs entropy also permits the derivation of the connection between the characteristic

∗ Present

address: Lawrence Livermore National Laboratory 7000 East Avenue., Livermore, CA 94550

thermodynamic function and the partition function for a given ensemble without further appeal to thermodynamic expressions, as is required in the traditional approach. In the present approach, first, the connections between the statistical formula S = k ln W , the Gibbs entropy, P S = −k pi ln pi , and the microcanonical entropy exi

pression S = k ln Ω are clarified. The condition for microcanonical equilibrium, and the associated role of the entropy in the thermodynamic potential then arises from the postulate of equal a priori states. The derivation of the canonical partition function follows by invoking the Gibbs construction and the first and second law of thermodynamics via the fundamental equation, dE = T dS − P dV + µdN , that incorporates the conditions of conservation of energy and composition without the needs for explicit constraints. The role of the temperature (coming from the constraint of total energy and an appeal to appropriate thermodynamic relationships in the traditional approach) is immediately apparent and also introduced via the fundamental equation. The need for explicit maximization of any function is thus also avoided. Legendre transforming a particular thermodynamic function to include desired thermodynamic control variables for an ensemble of interest and invoking equilibrium leads to the corresponding partition function. Using the resulting probabilities in the Gibbs entropy expression directly connects the partition function to the thermodynamic potential. The central role of the entropy in establishing equilibrium for a given ensemble emerges naturally from the current approach. Connections to generalized ensemble theory also arise and are presented in this context. The present approach is novel in providing clarity as to the roles played by the different formulas and physical quantities of interest. Further, it makes explicit the assumptions inherent in deriving the partition function for an ensemble and provides its direct connection to the relevant thermodynamic potential in a systematic fashion.

2 This approach also makes deriving the partition function for a given ensemble a simplified, straight-forward process, even for more challenging examples such as the isothermal-isobaric ensemble. Using this approach in the classroom has led to better retention and understanding of the foundations of statistical mechanics and an ability for students to confidently apply the machinery to problems that arise in their subsequent work.

II.

THE GIBBS ENTROPY AND THE MICROCANONICAL ENSEMBLE

We begin by introducing the concept of an ensemble of replicas that describe the molecular states corresponding to a given macrostate; this picture is referred to as the “Gibbs construction” herein, due to it’s original introduction by Gibbs[4, 5] who addressed many of the subtleties inherent[1] in the formulation of statistical mechanics. Consider a collection of macroscopic molecular “subsystems” of N molecules within a volume V , each of which is part of the larger Gibbs construction, the totality of which is known as the “system”. No other constraints have yet been imposed, i.e. the system’s macrostate is otherwise unspecified. It is desirable to define the microscopic statistics of this system as thoroughly as possible and then apply any other constraints at the end. Let the total number of subsystems in our collection be known as Ω. Then let ωi , the occupation number, denote the number of subsystems from this collection that are in the same thermodynamic state. These occupations will thus take on a large value P in the thermodynamic limit and they obey a sum rule, ωi = Ω. Note, technically i

the energy is course-grained, i.e. specified to within a small but otherwise arbitrary range (these arguments are presented in detail elsewhere[1, 2]) and the results are insensitive to this choice. First, consider the following combinatoric formula: Se = k ln W {ω} = k ln

Ω! ω1 !ω2 !...

(1)

W {ω} is the number of ways in which the set of occupations {ω} may be arranged consistent with the given macrostate. First, it is to be shown that when evaluated at fixed energy, this quantity Se may be identified with the thermodynamic entropy of the ensemble of systems at equilibrium, with each systems entropy given by S = SΩe . Note, the expression necessarily involves the logarithm of the combinatoric expression to make the entropy an extensive property; for two independent systems the possible number of arrangements is the product of those for the individual systems, S = k ln {W1 W2 } = k ln {W1 } + k ln {W1 } = S1 + S2 . Next the connection between the combinatoric formula Se = k ln W and the Gibbs entropy is presented; the details of this have been given elsewhere.[6] Applying the Sterling approximation[7] to the factorial function gives

the entropy: ( Se = k

Ω ln Ω − Ω −

X

ωi ln ωi +

i

X i

ωi

) (2)

Further substituting ωi = Ωpi where pi = ωi /Ω is identified as the probability of a particular state and using the property of the natural log gives a system’s entropy as:[6] X pi ln pi (3) S = −k i

This is the Gibbs entropy in an as yet unspecified ensemble with its associated probabilities; the Gibbs entropy is an entirely general definition that, for any equilibrium ensemble, specifies the relationship between the partition function and the associated characteristic thermodynamic function. Now, specializing to a set of microcanonical subsystems, and invoking the equilibrium principle of equal a priori states, i.e. pi = 1/Ω, gives the well known result: S = −k

Ω X 1 1 ln = k ln Ω Ω Ω i

(4)

It is also simple and useful to show that the Gibbs entropy, and thus the thermodynamic entropy, is maximized microcanonically[8] by the state-independent probabilities p = pi = 1/Ω. Proceeding, taking the derivative of Equation 3 and setting it to zero as ! X ∂ (5) pi ln pi = 0 −k ∂pj i gives pj = 1/e, a constant value independent of the summation index. Thus, normalizing the probabilities as, Ω P pi = 1 immediately yields pi = 1/Ω. i

Thus, for an isolated system, the assumption of equal a priori states leads to a probability pi that is independent of index, i.e. every subsystem has energy E by construction. Further, the characteristic maximum entropy in the microcanonical equilibrium ensemble also follows. Then applying the Gibbs entropy expression leads to the identification of the thermodynamic entropy as the characteristic function of the microcanonical ensemble and gives its relationship to the N, V, E partition function, Ω(E), which can also be interpreted as the density of states[1] at that energy. III. A SIMPLIFIED DERIVATION OF THE CANONICAL PARTITION FUNCTION

Specializing the Gibbs construction from the previous section to include temperature, we have a collection of

3 subsystems all possessing the same N, V, T . This can be thought of by placing the subsystems in contact with a large heat bath of temperature T .[1, 8] We now imagine that each subsystem (after having achieved equilibrium with the heat bath by definition) is to be insulated and the energy of the ith subsystem is measured as Ei , and for which there is also an associated macroscopic entropy Si . Of great importance, we also note that the thermodynamic energy Ei is exactly equal to the microscopic configurational energy of the subsystem upon insulation. Furthermore, the details and/or rates involved in the insulation process are irrelevant for an equilibrium ensemble. Using the earlier result, the entropy for a collection of subsystems with a specified energy Ei is Si = k ln Ω(Ei ), where Ω is the number of subsystems with energy Ei in the ensemble. Consider the ratio of the density at energies Ei+1 > Ei : 1

1 e k Si Ω(Ei ) = e− k (Si+1 −Si ) = 1S Ω(Ei+1 ) e k i+1

(6)

The fundamental equation of thermodynamics[9] is now invoked:

Substituting Equation 13 into Equation 6 gives: 1 Ω(Ei ) eβEi = e− k (Si+1 −Si ) = βEi+1 Ω(Ei+1 ) e e−βEi /Q pi = −βEi+1 /Q pi+1 e

where β = 1/kT and pi =

∂E = E − ST ∂S dA = dE − T dS − SdT

A = LT {E} = E − S

(8) (9)

where the condition for canonical equilibrium is that dA = 0 and N, V, T are constant, giving: 0 = dE − T dS 1 dS = dE T

(10) (11)

Integrating between two state points gives: Zi+1 Zi+1 1 dE dS = T

(12)

1 Si+1 − Si = (Ei+1 − Ei ) T

(13)

i

(i.e. the probabil-

ity of choosing the ith state from the entire ensemble at equilibrium the heat bath). The normalization P with e−βEi , may be readily recognized as the factor, Q = i

canonical partition function. Most importantly, we note that the insulation procedure applied to each subsystem has allowed us to identify the macroscopic energy (and entropy) of that microcanonical system, with the microscopic energy of the molecular configuration present at the time of insulation. We can now proceed to use the Gibbs entropy expression, Equation 3, substituting the canonical expression for pi to obtain: X e−βEi e−βEi S = −k ln Q Q i

(7)

The canonical ensemble is given by a state with well defined thermodynamic variables, N, V, T . So the energy function, E(S, V, N ) is Legendre transformed to a new thermodynamic function, the Helmholtz free energy, A(T, V, N ) via:

(15)

i

S=k dE(S, V, N ) = T dS − P dV + µdN

P1/Ωi 1/Ωi

(14)

X Ei e−βEi ln Q X −βEi + kβ e Q i Q i

T S = kT ln Q + hEiN V T

(16)

Above, hEiN V T represents the canonical average energy that is identified with the thermodynamic energy, E.[1, 5] Thus, the relationship A = E − T S = −kT ln Q is obtained directly from the Gibbs entropy. Note, the Gibbs entropy is defined for any set of probabilities and, as was shown above, is simply a consequence of the combinatoric formula S = k ln W interpreted in the context of the Gibbs construction. Thus, an entropy can be associated even with nonequilibrium probabilities. However, in that case, the entropy does not play the role of being the constrained maximized quantity that it does at equilibrium and its utility, in such circumstances, is unclear. Further note, the role of temperature is introduced via the fundamental equation without further appeal to thermodynamic relationships. This emphasizes the role of temperature as the system is in contact with a heat bath – different energy ranges are now accessible with canonical probabilities. The ability of a diathermal system to exchange energy with its surroundings also clarifies how the concepts of work and entropy make sense for an open system and provides their relationship to the temperature.

i

Note, the constraints of fixed temperature, particle number and volume have been explicitly enforced by using the fundamental equation, Equation 7 and dropping differential terms that are fixed canonically.

IV.

GRAND CANONICAL PARTITION FUNCTION

Using the Gibbs construction from the previously derived canonical ensemble, the constraint that all subsystems possess identical N can now be relaxed. We now

4 The ratio of microstates then becomes: 1 Ω(Nj , Ei ) = e− k (Si+1,j+1 −Si,j ) Ω(Nj+1 , Ei+1 ) 1

= e− kT (Ei+1 −Ei −µNj+1 +µNj ) e−βµNj eβEi = −βµNj+1 βEi+1 e e and since pj,i =

P 1/Ω P j,i 1/Ωj,i j

i

pj,i pj+1,i+1 FIG. 1: Gibbs construction for the canonical ensemble. The subsystems of identical N, V are in thermal equilibrium with a large bath at temperature T .

=

eβµNj e−βEi /Ξ eβµNj+1 e−βEi+1 /Ξ

X j

Ω(Nj , Ei ) 1 = e− k (Si+1,j+1 −Si,j ) Ω(Nj+1 , Ei+1 )

(17)

As before, the fundamental equation of thermodynamics can be relied upon to relate the change in entropy to the other variables of our ensemble. In this case, the probabilities that will generate the macrostate corresponding to constant µ, V, T are desired. After doing so, the entropy via the Boltzmann law is used to determine the microscopic states. Legendre transforming the canonical thermodynamic equation to substitute µ for N : ∂A ∂N = A − µN dJ = dA − N dµ − µdN = dE − T dS − SdT − N dµ − µdN

(25)

where the normalization factor Ξ is the grand canonical partition function: Ξ(µ, V, T ) =

consider the ratio of subsystem, having chosen a particular system from the energy level Ei with Nj molecules:

(24)

eβµNj

X

e−βEi

(26)

i

It may be noted that in this case the constraint of constant N was merely relaxed, Legendre transformed to the corresponding macrostate, and the partition function then followed quite naturally and simply. The resulting probabilities can now be substituted into the Gibbs entropy and the relationship between the thermodynamic potential and partition function is thus directly established.

J = LT {A} = A − N

(18) (19)

where at equilibrium dJ = 0 and with constant µ, V, T : 0 = dE − T dS − µdN 1 dS(µ, V, T ) = (dE − µdN ) T

(20) (21)

FIG. 2: Gibbs construction for the grand canonical ensemble. µ has been Legendre transformed to replace N as the macroscopic constant, and so the set of subsystems includes those of differing N values.

Upon integrating, the difference equation for the entropy is found: i+1 j+1 Z Z 1 dN (22) dE − µ dS = T

i+1,j+1 Z i,j

Si+1,j+1 − Si,j

i

V.

ISOTHERMAL-ISOBARIC PARTITION FUNCTION

j

1 = [Ei+1 − Ei − µ (Nj+1 − Nj )] (23) T

Having successfully derived the grand canonical partition function by relaxing the constant N constraint, it

5 can now be shown that the isothermal-isobaric ensemble is generated by starting with constant N V T and relaxing the condition of constant V . Consider the number of subsystems with both volume Vi and energy Ej : 1 Ω(Vj , Ei ) = e− k (Si+1,j+1 −Si,j ) Ω(Vj+1 , Ei+1 )

(27)

Legendre transforming our desired variables to a new characteristic function G, and then applying the condition of equilibrium dG = 0 and our constant differential terms: ∂A = A + PV ∂V dG = dA + P dV + V dP = 0 = dE − T dS − SdT + P dV + V dP 1 → dS = (dE + P dV ) T G=A−V

(28) (29) (30)

FIG. 3: Gibbs construction for the isothermal-isobaric ensemble. P has been Legendre transformed to replace V as the macroscopic constant, and so the set of subsystems includes those of differing V values.

and so the ratio of observable subsystems becomes: In the continuum limit, it can be shown that the canonical partition function Q (written as a sum over energy levels) transforms as:

1 Ω(Vj , Ei ) = e− k (Si+1,j+1 −Si,j ) Ω(Vj+1 , Ei+1 ) 1

= e− kT [Ei+1 −Ei +P (Vi+1 −Vi )] eβEi eβP Vj = βEi+1 βP Vj+1 e e and since pj,i =

(31)

X

e−βEi Ω(Ei )

i

1/Ωj,i PP 1/Ωj,i j

Q(N, V, β) = →

i

Z

∞

dE e−βE Ω(N, V, E)

(34)

0

pi,j pi+1,j+1

=

e−βEi e−βP Vj /∆ e−βEi+1 e−βP Vj+1 /∆

where the normalization factor, X X e−βEi e−βP Vj ∆= j

(32)

(33)

i

is the isothermal-isobaric partition function. Again, the associated probabilities can now be substituted into the Gibbs entropy and the relationship between the thermodynamic potential and partition function is thus directly established. VI.

or, in other words, the canonical partition function is the Laplace transform of the microcanonical partition function Ω. How does it work the other way? Let’s apply the inverse Laplace transform to Q:

CONNECTION WITH GENERALIZED ENSEMBLE THEORY

Based on the previous sections, it may be noted that if a partition function for any macrostate in thermal equilibrium is to be derived, one shall always arrive at an expression that involves an exponential function; this follows as a consequence of the Boltzmann Law. Furthermore, the partition function being the normalization factor to express this as a probability means that one will always have a discrete sum. Therefore, our partition functions will always be some variation on a theme amounting to a “sum of exponentials.”

I 1 Ω(N, V, E) = dβ eβE Q(N, V, β) 2πi γ+i∞ Z Z∞ 1 dβ eβ(E−H) = dΩ 2πi

(35)

γ−i∞

−∞

where the phase space differential form dΩ = (h3N N !)−1 dx1 ...dx3N dp1 ...dp3N . Now, β = σ + iτ and because no singularity is present in the right-half of the complex plane, the contour may be taken vertically through γ = 0. Since Re(β) = 0 along the integration, the substitution β = −iτ can be made:

Ω(N, V, E) =

Z∞

1 dΩ 2π

Z∞

dτ eiτ (E−H)

−∞

−∞

=

Z∞

−∞

dΩ δ(H − E)

(36)

6 where indeed Equation 36 can be identified as the microcanonical partition function. Thus, any constant energy shell ensemble may be Laplace transformed to an ensemble of a new intensive variable. As the partition functions are related to one another through the Laplace transform, this is isomorphic to the thermodynamic potentials (to each of which may be associated a particular partition function) being related through the Legendre transform.[10] The quantum harmonic oscillator is an illustrative example of how the canonical partition function may be transformed to the microcanonical case: I 1 ΩHO = dβ eβE QHO 2πi Z∞ 1 2 iτ ~ω 1 −iτ E e = dτ e 2π 1 − eiτ ~ω

Finally, the similarities between the derivation method demonstrated and the relations known from generalized ensemble theory have been noted. It is our hope that the formulaic approach presented here will be of utility in both research and pedagogy.

−∞

=

1 2π

Z∞

dτ eiτ ( 2 ~ω)−E

−∞

1

X

eiτ ~ωn

n

X 1 −E = δ ~ω n + 2 n

(37)

The reason that the aforementioned “recipe” for generating the partition function (as outlined in Sections III,IV and V) in an arbitrary ensemble works is due to the thermodynamic relations and the Boltzmann law. The underlying mathematical structure that allows this has also been previously formulated as generalized ensemble theory.[11–15] VII.

emphasizes the central role that (maximizing) the Boltzmann entropy plays in connecting the molecular states of the system to the observable thermodynamics. Using this technique in a classroom setting for a beginning graduate class in statistical mechanics has led to systemization and demystification of the derivation for useful ensembles. Also, the role of Legendre transforms to introduce thermodynamic control variables appears naturally and is tied directly to both the derivation of the ensemble and corresponding partition function. Within this formalism, students are clear on how the thermodynamic potential relates to a given ensemble and the role of equal a priori states. Further, relating the partition function to the thermodynamic potential using the Gibbs entropy is straightforward and no further appeal to thermodynamic expressions is required as the relevant thermodynamic connection was included from the start of the derivation.

CONCLUSIONS

An approach is presented for deriving partition functions that is an alternative to more common methods. It

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VIII.

ACKNOWLEDGEMENTS

The authors acknowledge funding from the U.S. Department of Energy, Basic Energy Sciences (Grant No. DE0GG02-07ER46470). Lawrence Livermore National Laboratory is operated by Lawrence Livermore National Security, LLC, for the U.S. Department of Energy, National Nuclear Security Administration under Contract DE-AC52-07NA27344.

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