Combinatorial origins of the canonical ensemble
CCombinatorial origins of the canonical ensemble
Kornelia Ufniarz E-mail: [email protected]
Grzegorz Siudem E-mail: [email protected] Faculty of Physics, Warsaw University of Technology,Koszykowa 75, 00-662 Warsaw, Poland
Abstract.
The Darwin-Fowler method in combination with the steepest descentapproach is a common tool in the asymptotic description of many models arisingfrom statistical physics. In this work, we focus rather on the non-asymptotic behaviorof the Darwin-Fowler procedure. By using a combinatorial approach based on Bellpolynomials, we solve it exactly. Due to that approach, we also show relationships oftypical models with combinatorial Lah and Stirling numbers.
Keywords : Classical statistical mechanics, equilibrium and non-equilibrium
1. Introduction
Proposed by Darwin and Fowler [1, 2] method of steepest descent is a typical approachfor the derivation of canonical (Gibbs) ensembles (see [3, 4, 5]) or other asymptoticproblems on the border on combinatorics and statistical physics [6]. The method baseson the properties of the complex integrals and allows one to efficiently calculate thedesired limit of the typical combinatorial problems. Thought, typically Darwin-Fowlerapproach is concerned with the asymptotics, in this article, we focus mainly on the non-asymptotic case. With the introduced combinatorial approach based on Bell polynomialswe solve exactly the Darwin-Fowler problem purely combinatorically without integralrepresentation.Bell Polynomials have been applied recently to the wide range of problems ofstatistical physics e.g. in the description of gas of clusters [7, 8], partition functionfor ideal gases [9], series expansion for quantum partition functions [10], general latticemodels description [11] and Ising model [12]. With this work, we complete this listwith the application to the canonical ensemble, which additionally reveals unexpectedcombinatorial origins of this fundamental to statistical mechanics distribution. Withthe introduced approach we show that for the typical degeneracies (i.e. constant one a r X i v : . [ m a t h - ph ] A ug ombinatorial origins of the canonical ensemble
2. Darwin-Fowler method
Let us consider an ensemble consisting of the total of K systems with possible energies E = { ε , ε , . . . } with degeneracies Ω = { ω , ω , . . . } , where the energy level ε (cid:96) hasdegeneracy equal to ω (cid:96) . Furthermore, we assume that there exists a quantum of theenergy ε , which without loss of generality means that ε (cid:96) = ε(cid:96) . The above allows one toformulate the following condition for { n (cid:96) } i.e. the numbers of systems with the energiesfrom E (cid:40)(cid:80) ∞ (cid:96) =1 n (cid:96) = K , (cid:80) ∞ (cid:96) =1 ε (cid:96) n (cid:96) = ε N . = ⇒ (cid:40)(cid:80) ∞ (cid:96) =1 n (cid:96) = K , (cid:80) ∞ (cid:96) =1 (cid:96)n (cid:96) = N . (1)In the Darwin-Fowler method we examine the probability distribution over the energylevels P N , K ( (cid:96) ) = n (cid:63) N , K ( (cid:96) ) K , (2)where n (cid:63) N , K ( (cid:96) ) is the number of the systems at (cid:96) -th energy level for the averageconfiguration of ensemble consisting of K systems and having total energy ε N . Weassume that every single configuration occurs with equal probability (with respect tothe degeneracies) thus n (cid:63) N , K = (cid:80) { m r } N , K m (cid:96) W N , K ( { m r } ) (cid:80) { m r } N , K W N , K ( { m r } ) , where W N , K ( { m (cid:96) } ) = K ! (cid:89) (cid:96) ω m (cid:96) (cid:96) m (cid:96) ! , (3)and the summation is taken over all possible configurations satisfying Eq. (1) and factor W N , K ( { m (cid:96) } ) counts the number of possible realisations of the configuration { m (cid:96) } . Withthe introduced notion one can realize that for the function Γ N , K defined asΓ N , K ( ω , ω , . . . ) = K ! (cid:88) { m r } N , K (cid:18) ω m m ! · ω m m ! · · · (cid:19) , (4)one can express Eq. (3) for the number n (cid:63) N , K in the following way n (cid:63) N , K = ω (cid:96) ∂ ln Γ( u , u , . . . ) ∂u (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u r = ω r , r =1 , , ... . (5) ombinatorial origins of the canonical ensemble P N , K distribution can bereduced to determining the value of the function Γ N , K and its derivative. Typically,in the Darwin-Fowler method, this is done by using the steepest descend method (seeAppendix A). Instead of this approximate approach, we solve the problem exactly, usingBell polynomials, which we introduce in the next section.
3. Bell Polynomials and their combinatorics
Bell Polynomials (introduced in [13], see also sec. 3.3 in [14]) are inseparably linked withthe famous Fa´a di Bruno’s formula i.e. the generalization of the chain rule (derivationof the composition of two functions) to higher derivatives. However, the formula hasbeen published [15] 30 years before Bell was born, so naturally, it was stated withoutthat notion (see Eq. (6)). We, however, use its version presented in Eq. (7), wherewe introduced Bell polynomials. For the historical background of the Fa´a di Bruno’sformula see [16, 17]. Fa´a di Bruno’s formula (in clasical formulation in Eq. (6) and withthe usage of Bell polynomials B n, k in Eq. (7)) states d n dx n F ( G ( x )) = (cid:88) { m r } n n ! m ! m ! · · · m n ! · F ( m + ··· + m n ) ( G ( x )) · n (cid:89) j =1 (cid:20) G ( j ) ( x ) j ! (cid:21) m j = (6)= n (cid:88) k =1 F ( k ) ( G ( x )) · B n,k (cid:0) G (cid:48) ( x ) , G (cid:48)(cid:48) ( x ) , . . . , G ( n − k +1) ( x ) (cid:1) , (7)where F and G are analytical functions and summation in Eq. (6) is taken over integers m i such 1 · m + 2 · m + 3 · m + · · · + n · m n = n. Bell Polynomials B nk from Eq. (7) can be defined in the two equivalent ways –combinatorically and analytically. Let us recall those two approaches in the followingsubsections. As in the previous section, let us consider two analytic functions F and G such that F (0) = 0, which means that F ( x ) = ∞ (cid:88) n =1 f n x n n ! , G ( x ) = ∞ (cid:88) n =0 g n x n n ! . Then we ask about the series expansion of the composition of both functions, similarlyto the consideration in Fa´a di Bruno’s formula (cf. Eqs. (6, 7)) G ( F ( x )) = ∞ (cid:88) n =0 c n x n n ! . ombinatorial origins of the canonical ensemble B n,k and connect them with the coefficients c n , f n , g n as follows (cid:40) c = g ,c n = (cid:80) nk =1 g k B n, k ( f , . . . , f n − k +1 ) for n > . (8)Combining Eqs. (6) and (8) one obtains the following analytical definition of Bellpolynomials B n,k ( a , a , . . . , a n − k +1 ) = (cid:88) { m r } n,k n ! m ! m ! · · · m n − k +1 ! n (cid:89) j =1 (cid:18) a j j ! (cid:19) m j , (9)where the summation is taken over all non-negative integers { m r } which satisfy (cid:40) · m + 2 · m + 3 · m + · · · + n · m n = n,m + m + m + · · · + m n = k. (10)Let us note the similarity of conditions (1) and (10), which makes Bell polynomials anatural tool for describing Darwin-Fowler’s formalism. Let us also compare definitionof function Γ N , K given in the Eq. (4) with Bell polynomial given by the Eq. (9). Now let us ask seemingly totally different and non-connected to the previous onequestion: what is the number of possible decompositions of a set of n elements into k clusters (subsets)? Additionally, we assume that every cluster of size l has a l ≥ { m r } which satisfy (cid:40) · m + 2 · m + 3 · m + · · · + n · m n = n,m + m + m + · · · + m n = k, where m (cid:96) describes the number of subsets of size (cid:96) . In this situation, the number ofpossible implementations of such a division is equal to n ! m ! m ! · · · m n − k +1 ! n (cid:89) j =1 (cid:18) a j j ! (cid:19) m j , which, summed over all possible partitions { m r } N , K leads to Bell polynomials. Theformal proof of the equivalence between analytic and combinatorial definition of BellPolynomials can be found in [14], however let us describe this fact in the followingexample. Example 1.
Let us consider two examples of Bell Polynomials with n = 6 , k = 3 and n = 6 , k = 4 . In both cases the coefficients of the polynomials count the number of aombinatorial origins of the canonical ensemble possible partition of the set into clusters of sizes given by indices of a : B , ( a , a , a , a , a , a ) = 15 (cid:124)(cid:123)(cid:122)(cid:125) = ( ) a a + 60 (cid:124)(cid:123)(cid:122)(cid:125) = ( )( ) a a a + 15 (cid:124)(cid:123)(cid:122)(cid:125) = ( )( ) a ,B , ( a , a , a , a , a , a ) = 20 (cid:124)(cid:123)(cid:122)(cid:125) = ( ) a a + 45 (cid:124)(cid:123)(cid:122)(cid:125) = ( )( ) = a a . In the following section we need the formula for the derivation of Bell polynomials ∂B n, k ( a , a , . . . ) ∂a (cid:96) = (cid:18) n(cid:96) (cid:19) B n − (cid:96), k − ( a , a , . . . ) , (11)which we prove in Appendix B. Let us also recall another important property of BellPolynomials after [14] B n,k ( abx , ab x , ... ) = a k b n B n,k ( x , x , ... ) . (12) The Bell polynomials B n,k discussed in the previous sections are sometimes calledexponential polynomials (see [14]). Let us consider a minor modification of them (cid:98) B n,k ,which for the sake of distinction, are called ordinary Bell polynomials (see Eq. [3o]in [14]). (cid:98) B n,k ( a , a , . . . ) = k ! n ! B n,k (1! a , a , . . . ) = (cid:88) { m r } n,k k ! m ! m ! · · · m n − k +1 ! n (cid:89) j =1 a m j j , (13)As we will see in the following sections, ordinary polynomials are more natural fordescribing the Darwin-Fowler procedure, rather than exponential ones. Bell polynomials also allow one to conveniently express known combinatorial numbersin a compact form. There are several versions of the so-called Bell transform (see[18, 19]), but we will focus on expressing Lah and Stirling numbers by exponential Bellpolynomials (see [14, 20]). As we will see in the following sections, these numbers areclosely related to the distributions in the Darwin-Fowler approach. • Unsigned Lah numbers (see A105278 in [21] and Eq. [3h] in [14]) L ( n, k ) = B n,k (1! , , . . . ) = (cid:18) n − k − (cid:19) n ! k ! . (14) • Unsigned Stirling numbers of the first kind (see A008275 in [21] and Eq. [3i] in[14]) | S ( n, k ) | = B n,k (0! , , , . . . ) . (15) • Stirling numbers of the second kind (see A008277 in [21] and Eq. [3g] in [14]) S ( n, k ) = B n,k (1 , , , . . . ) . (16) ombinatorial origins of the canonical ensemble
4. Bell Polynomial approach to Darwin-Fowler model
Using Bell polynomials (both exponential and ordinary) enables to transform theexpression for Γ N , K given by Eq. (4) as followsΓ( ω , ω , . . . ) = K ! N ! B N , K (1! ω , ω , . . . ) = (cid:98) B N , K ( ω , ω , . . . ) . (17)Thus, the expression for a n (cid:63) N , K (see Eqs. (5) and (13)) which we are looking for canbe further simplified n (cid:63) N , K ( (cid:96) ) = N !( N − (cid:96) )! ω (cid:96) B N − (cid:96), K− (1! ω , ω , · · · ) B N , K (1! ω , ω , · · · ) = K ω (cid:96) (cid:98) B N − (cid:96), K− ( ω , ω , · · · ) (cid:98) B N , K ( ω , ω , · · · ) (cid:124) (cid:123)(cid:122) (cid:125) = P N , K ( (cid:96) ) , (18)which is the exact result for finite K and N . As illustrations of this main result let usconsider specific forms of degeneration in the following sections.
5. Special cases of the degeneracy
Firstly let us assume that there is no degeneracy, i.e. ω (cid:96) = ω = constans, which changesEq. (18) to the following form (see Eq. (12)) n (cid:63) N , K ( (cid:96) ) = N !( N − (cid:96) )! B N − (cid:96), K− (1! , , , . . . ) B N , K (1! , , , . . . ) = N !( N − (cid:96) )! L ( N − (cid:96),
K − L ( N , K ) , where we can spot Lah numbers (see Eq. (14)), which can be further simplified n (cid:63) N , K ( (cid:96) ) = N !( N − (cid:96) )! (cid:0) N − (cid:96) − K− (cid:1) ( N − (cid:96) )!( K− (cid:0) N − K− (cid:1) N ! K ! = K (cid:0) N − (cid:96) − K− (cid:1)(cid:0) N − K− (cid:1) == K ( K −
1) (
N − (cid:96) − N − K )!(
N −
N − K − (cid:96) + 1)! (cid:124) (cid:123)(cid:122) (cid:125) P N , K ( (cid:96) ) . (19)Typically we would look for the asymptotic form of Eq. (19) using the steepest descentmethod, which is described in Appendix A. However, we will use the compact form ofLah numbers and apply the Stirling’s approximation in ♠ and obtain P N , K = ( K −
1) (
N − (cid:96) − N −
N − K )!(
N − K − (cid:96) + 1)! ♠ ≈ ♠ ≈ ( K −
1) (
N − (cid:96) − N − (cid:96) − e − ( N − (cid:96) − ( N − N − e − ( N − ( N − K ) N −K e − ( N −K ) ( N − K − (cid:96) + 1)
N −K− (cid:96) +1 e − ( N −K− (cid:96) +1) == (
K − e ( N − K ) (cid:96) − ( N − (cid:96) − (cid:96) (cid:18) − (cid:96) N − (cid:19) N − (cid:18) (cid:96) − N − K − (cid:96) + 1 (cid:19)
N −K− (cid:96) +1 == e K − N − K (cid:18)
N − KN − (cid:96) − (cid:19) (cid:96) (cid:18) − (cid:96) N − (cid:19) N − (cid:18) (cid:96) − N − K − (cid:96) + 1 (cid:19)
N −K− (cid:96) +1 . ombinatorial origins of the canonical ensemble U of the system is constant even in thelimit of large K i.e. N = U K one gets final form of the energy distribution as P N , K ( (cid:96) ) ≈ e K − U − K (cid:124) (cid:123)(cid:122) (cid:125) K→∞ −−−→ ( U − − (cid:18) ( U − K U K − (cid:96) − (cid:19) (cid:96) (cid:124) (cid:123)(cid:122) (cid:125) K→∞ −−−→ ( U − U ) (cid:96) (cid:18) − (cid:96)U K − (cid:19) U K− (cid:124) (cid:123)(cid:122) (cid:125) K→∞ −−−→ e − (cid:96) ×× (cid:18) (cid:96) − U − K − (cid:96) + 1 (cid:19) ( U − K− (cid:96) +1 (cid:124) (cid:123)(cid:122) (cid:125) K→∞ −−−→ e (cid:96) − , which results in the expected formula for the distribution P ∞ ( (cid:96) ) = e βε(cid:96) Z , (20)where U = (1 − e βε ) − and Z = U − For the one-dimensional harmonic oscillator the weight of energy ε (cid:96) is equal to ω (cid:96) = ( (cid:96) + 1) , which implies the following form of the energy distribution from Eq. (18) P N , K ( (cid:96) ) = ( (cid:96) + 1) (cid:98) B N − (cid:96), K− (2 , , . . . ) (cid:98) B N , K (2 , , . . . ) . (21)Eq. (21) can be further generalized for the D -dimensional harmonic oscillator, becausedegenerations are then given as ω (cid:96) = ( (cid:96) + 1)( (cid:96) + 2) . . . ( (cid:96) + D ) =: ( (cid:96) + 1) ( D ) , where ( (cid:96) + 1) ( D ) denotes the rising factorial for brevity. With such degeneracies one facethe problem of determination the values of Bell polynomials B n,k (2 ( D ) , ( D ) , . . . ), whichcan be done with the following formula B n,k (2 ( D ) , ( D ) , . . . ) = [( D − k n (cid:88) r = k | S ( n, r ) | S ( r, k ) D r , (22)where S , S are the Stirling numbers of the first and second kind respectivelly, see Eqs.(15) and (16). Let us note that Eq. (22) follows from Eq. (8.50) in [20] and allows one toobtain the following final formula for the most probable configuration for D -dimensionalharmonic oscillator n (cid:63) N , K = N !( N − (cid:96) )! ( (cid:96) + 1) ( D ) ( D − (cid:80) N − (cid:96)r = K− | S ( N − (cid:96), r ) | S ( r, K − D r (cid:80) N r = K | S ( N , r ) | S ( r, K ) D r . (23)As we can see, the above Eq. (23) combines the energy distribution for a harmonicoscillator with the Stirling numbers. ombinatorial origins of the canonical ensemble
6. Acknowledgement
We would like to thank Agata Fronczak for stimulating discussions (and pointing outour mistakes at the early stage of the work). GS work has been supported by theNational Science Centre of Poland (Narodowe Centrum Nauki, NCN) under grant no.2015/18/E/ST2/00560.
Appendix A. Steepest descend approach to Darwin-Fowler
As we mentioned previously, typically in Darwin-Fowler approach one is only interestedin the result in the thermodynamical limit
K → ∞ with N = U K . Keeping the abovein mind let us define a generating function for Γ N , K in the following way G K ( z ) = ∞ (cid:88) U =0 z K U Γ K U, K , (A.1)It is easy to see (compare Eq. (4) or more detailed discussion in [4, 5]) that due to themultinomial theorem this generating function simplifies to G K ( z ) = ( ω z ε + ω z ε + . . . ) K = [ g ( z )] K . (A.2)Let us now focus on the specific case and assume (after Huang [4]) that degeneraciesare constant i.e. ω (cid:96) = 1, and ε = 1 which simplifies the function as follows g ( z ) = 1 + z + z + z + · · · = 11 − z . (A.3)From Eq. (A.3) one see that Γ K U, K is the coefficient of z K U in the expansion of G K ( z )in powers of z , hence Γ K U, K = 12 πi (cid:73) [ g ( z )] K z K U +1 dz. (A.4)For real positive z function g ( z ) monotonically increases with a radius of convergence z = R . The function 1 /z K U +1 is a monotically descreasing function of real positive z .Hence the function [ f ( z )] K /z K U +1 has a minimum at z = x for z ∈ [0 , R ]. In addition,functions f ( z ) and 1 /z K U +1 are analytic, therefore the integrand I ( z ) = [ f ( z )] K /z K U +1 is also analytic and satisfies the Cauchy-Riemann equation (cid:18) ∂ ∂x + ∂ ∂y (cid:19) = 0 , hence (cid:18) ∂I∂z (cid:19) z = x = 0 , (cid:18) ∂ I∂x (cid:19) z = x > , (cid:18) ∂ I∂y (cid:19) z = x > , (A.5)where z = xi + y . One can see that x is a saddle point. Thus let us define u ( z ) as I ( z ) = e K u ( z ) , u ( z ) = ln g ( z ) − ( K U + 1) ln z. (A.6) ombinatorial origins of the canonical ensemble ∂ I∂x K→∞ −−−→ ∞ , ∂ I∂y K→∞ −−−→ ∞ . Therefore the saddle point touches an infinitely sharp peak and an infinitely steep valleyin the limits as
K → ∞ . If we choose the contour of integration to be a circle centeredin z = 0 with radius x the main part of integral comes from the neighrhood of x . Thusto compute the integral one can extend the intergrand around z = x , henceΓ K U, K = 12 πi (cid:73) e K u ( z ) dz ≈ e K u ( x ) π (cid:90) x − x dye − / K u (cid:48)(cid:48) ( x ) y ≈ e K u ( x ) (cid:112) π K u (cid:48)(cid:48) ( x ) . (A.7)From Eqs. (5) and (A.7), the average number n (cid:63) ( (cid:96) ) of systems in the degenerate stateof energy ε l ε(cid:96) is thus given by the expression n (cid:63) ( (cid:96) ) ∝ exp( βε(cid:96) ) , (A.8)which is consistent with the result from Bell polynomial approach (see Eq. (20)). Appendix B. Proof of Eq. 11
We want to prove that the derivative of Bell polynomial follows Eq. (11) i.e. ∂B n, k ( a , a , . . . ) ∂a (cid:96) = (cid:18) n(cid:96) (cid:19) B n − (cid:96), k − ( a , a , . . . ) . Let us start with the left-hand side of the original equation for n > (cid:96) and k > ∂B n, k ( a , a , . . . ) ∂a (cid:96) = (cid:88) { m r } n,k n ! m ! · · · m n − k +1 ! (cid:18) (cid:96) ! (cid:19) m (cid:96) ∂a m (cid:96) (cid:96) ∂a (cid:96) (cid:89) j (cid:54) = (cid:96) (cid:18) a j j ! (cid:19) m j = ( ♠ ) , which can be further transformed as follows due to the elementary differentiation( ♠ ) = (cid:88) { m r } n,k ,m (cid:96) (cid:54) =0 n ! ( (cid:96) !) − m (cid:96) (cid:81) j (cid:54) = (cid:96) (cid:16) a j j ! (cid:17) m j m ! · · · m n − k +1 ! ∂a m (cid:96) (cid:96) ∂a (cid:96) + (cid:88) { m r } n,k ,m (cid:96) =0 n ! ( (cid:96) !) − m (cid:96) (cid:81) j (cid:54) = (cid:96) (cid:16) a j j ! (cid:17) m j m ! · · · m n − k +1 ! ∂a m (cid:96) (cid:96) ∂a (cid:96) (cid:124) (cid:123)(cid:122) (cid:125) =0 == 1 (cid:96) ! (cid:88) { m r } n,k ,m (cid:96) (cid:54) =0 n ! m ! · · · ( m (cid:96) − · · · m n − k +1 ! (cid:16) a (cid:96) (cid:96) ! (cid:17) m (cid:96) − (cid:89) j (cid:54) = (cid:96) (cid:18) a j j ! (cid:19) m j = ( ♣ ) . Let us note that thanks to the condition of the summation given by Eq. (10) one cansimplify the above into the form( ♣ ) = 1 (cid:96) ! (cid:88) { m r } n − (cid:96),k − n ! m ! · · · ( m (cid:96) )! · · · m n − k +1 ! (cid:89) j (cid:18) a j j ! (cid:19) m j = n ! (cid:96) !( n − (cid:96) )! B n − (cid:96), k − ( a , a , . . . ) , which ends the proof. ombinatorial origins of the canonical ensemble References [1] Darwin C G, Fowler R H 1922 XLIV. On the partition of energy
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