V.B The Cluster Expansion

Transcription

1 V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f( q ) = exp ( βv( q )), which is obtained by summing over a possibe number of bonds between two points on a cumuant graph. The resuting series is organized in powers of the density N/V, and is most suitabe for obtaining a viria expansion, which expresses the deviations from the idea gas equation of state in a power series k B T = N V + B (T) N V + B (T) ( N V ) +. (V.4) The temperature dependent parameters, B i (T), are known as the viria coefficients and originate from the inter-partice interactions. Our initia goa is to compute these coefficients from first principes. To iustrate a different method of expansion, we sha perform computations in the grand canonica ensembe. With a macro-state M (T, µ, V ), the grand partition function is given by where Q(µ, T, V ) = e βµn Z(N, T, V ) = N! ( ) e βµ N S N, (V.5) N S N = d q i ( + f ij ), (V.6) i= i<j λ and f ij = f( q i q j ). The N(N )/ terms in S N can now be ordered in powers of f ij as N S N = d q i f ij + f ij f k +. (V.7) + i<j i= i<j,k< An efficient method for organizing the perturbation series is to represent the various contributions diagrammaticay. In particuar we sha appy the foowing conventions: (a) Draw N dots abeed by i =,, N to represent the coordinates q through q N, N. (b) Each term in eq.(v.7) corresponds to a product of f ij, represented by drawing ines connecting i and j for each f ij. For exampe, the graph, N, 0

2 represents the integra ( d q ) ( d q d q f ) ( ) ( d q 4 d q 5 d q 6 f 45 f 56 d q N ). As the above exampe indicates, the vaue of each graph is the product of the contributions from its inked custers. Since these custers are more fundamenta, we reformuate the sum in terms of them by defining a quantity b, equa to the sum over a -partice inked custers (one-partice irreducibe or not). For exampe b = = d q = V, (V.8) and b = = d q d q f( q q ). (V.9) There are four diagrams contributing to b, eading to b = d q d q d q f( q q )f( q q ) + f( q q )f( q q ) + f( q q )f( q q ) + f( q q )f( q q )f( q q ). (V.0) A given N-partice graph can be decomposed to n -custers, n -custers,, n - custers, etc. Hence, S N = {n } b n W({n }), (V.) where the restricted sum is over a distinct divisions of N points into a set of custers {n }, such that n = N. The coefficients W({n }) are the number of ways of assigning N partice abes to groups of n -custers. For exampe, the divisions of partices into a -custer and a -custer are,, and. A above graphs have n = and n =, and contribute a factor of b b to S ; thus W(, ) =. In genera, W({n }) is the number of distinct ways of grouping the abes,..., N into bins of n -custers. It can be obtained from the tota number of permutations, N!, after dividing by the number of equivaent assignments. Within each bin of n partices, equivaent assignments are obtained by: (i) permuting the abes in each subgroup in! 0

3 ways, for a tota of (!) n permutations; and (ii) the n! rearrangements of the n subgroups. Hence, W({n }) = N! n. (V.)!(!) n (We can indeed check that W(, ) =!/(!)(!) = as obtained above.) Using the above vaue of W, the expression for S N in eq.(v.) can be evauated. However, the restriction of the sum to configurations such that n = N compicates the evauation. Fortunatey, this restriction disappears in the expression for the grand partition function in eq.(v.6), Q = ( ) e βµ N N! N! λ {n } n!(!) n b n. The restriction in the second sum is now removed by noting that {n }. Therefore, (V.) {n } δ n,n = Q = ( ) e βµ n b n λ n!(!) n = ( e βµ b n! λ! {n } {n } = (e ) βµ n b n! λ = (e ) βµ b exp! λ! {n } ( ) e βµ b = exp λ.! = ) n (V.4) The above resut has the simpe geometrica interpretation that the sum over a graphs, connected or not, equas the exponentia of the sum over connected graphs. This is a quite genera resut that is aso reated to the graphica connection between moments and cumuants discussed in sec.ii.b. The grand potentia is now obtained from n Q = βg = V kt = = ( e βµ λ ) b!. (V.5) In eq.(v.5), the extensivity condition is used to get G = E TS µn = V. Thus the terms on the right hand side of the above equation must aso be proportiona to the voume V. This can be expicity verified by noting that in evauating each b there is an 04

4 integra over the center of mass coordinate that expores the whoe voume. For exampe, b = d q d q f( q q ) = V d q f( q ). Quite generay, we can set im b = V b, V (V.6) and the pressure is now obtained from kt = = ( e βµ λ ) b!. (V.7) The inked custer theorem ensures G V, since if any non-inked custer had appeared in n Q, it woud have contributed a higher power of V. Athough an expansion for the gas pressure, eq.(v.7) is quite different from eq.(v.4) in that it invoves powers of e βµ rather than the density n = N/V. This difference can be removed by soving for the density in terms of the chemica potentia, using N = n Q (βµ) = ( ) e βµ V b λ!. (V.8) = The equation of state can be obtained by eiminating the fugacity x = e βµ /λ, between the equations n = using the foowing steps: = x ( )! b, and (a) Sove for x(n) from ( b = d q/v = ) kt = = x! b, (V.9) x = n b x b x. (V.0) The perturbative soution at each order is obtained by substituting the soution at the previous order in eq.(v.0), x = n + O(n ) x = n b n + O(n ) x = n b (n b n) b n + O(n 4 ) = n b n + ( b b )n + O(n 4 ). (V.) 05

5 (b) Substitute the perturbative resut for x(n) into eq.(v.9), yieding β = x + b x + b 6 x + = n b n + ( b b )n + b n b n + b 6 n + = n b n + ( b b )n + O(n 4 ). The fina resut is in the form of the viria expansion of eq.(v.4), (V.) β = n + B (T)n. = The first term in the series reproduces the idea gas resut. The next two corrections are B = b ( ) = d q e βv( q ), (V.) and B = b b ( ( ) ) = d q e βv( q ) d q d q f( q )f( q ) + d q d q f( q )f( q )f( q q ) = d q d q f( q )f( q )f( q q ). (V.4) The above exampe demonstrates the canceation of the one partice reducibe custer that appears in b. Whie a custers (reducibe or not) appear in the sum for b, as demonstrated in the previous section, ony the one partice irreducibe ones can appear in an expansion in powers of density. The fina expression for the th viria coefficient is ( ) B (T) = d,! (V.5) where d is defined as the sum over a one partice irreducibe custers of points. Note that in terms of d, the partition function can be organized as nz = nz 0 + V = n! d, (V.6) reproducing the above viria expansion from β = nz/ V. 06