Consider N identical particles of mass m that interact through two-body interactions, vij, with Hamiltonian

H=12m∑ipi2+∑j>ivij ,  (i,j=1,2,⋯,N)

(6.1)

where vij≡v(ri−rj) denotes the potential energy associated with the interaction between particles at positions r_i, r_j, and ∑j>i indicates a sum over N2 pairs of particles.¹ For central forces, vij depends only on the magnitude of the distance between particles, vij=v(|ri−rj|), which we assume for simplicity. The methods developed here can be extended to quantum systems, but the analysis becomes more complicated; we won’t consider interacting quantum gases.²

For our purposes, the precise nature of the interactions underlying the potential energy function v(r) is not important as long as there is a long-range attractive component together with a short-range repulsive force. To be definite, we mention the Lennard-Jones potential (shown in Fig. 6.1) for the interaction between closed-shell atoms, which has the parameterized form,

v(r)=4ϵσr12−σr6 ,

(6.2)

where ϵ is the depth of the potential well, and σ is the distance at which the potential is zero. The r−6 term describes an attractive interaction between neutral molecules that arises from the energy of interaction between fluctuating multipoles of the molecular charge distributions.³ The r−12 term models the repulsive force at short distances that arises from the Pauli exclusion effect of overlapping electronic orbitals. There’s no science behind the r−12 form; it’s analytically convenient, and it provides a good approximation of the interactions between atoms. For the noble gases, ϵ ranges from 0.003 eV for Ne to 0.02 eV for Xe [18, p398]. The parameter σ is approximately 0.3 nm.

To embark on the statistical-mechanical-road, we have in the canonical ensemble⁴

Z(T,V,N)=1N!∫dΓNe−β∑ipi2/2m+∑i>jvij≡1N!λT3NQ(T,V,N) ,

(6.3)

where λ_T occurs from integrating the momentum variables and Q is the configuration integral, the part of the partition function associated with the potential energy of particles,

Q(T,V,N)=∫dNrexp⁡(−β∑i>jvij)=∫dNr∏i>je−βvij≡∫dNr∏i>j1+fij ,

(6.4)

where

fij=f(rij)≡e−βv(rij)−1

(6.5)

is the Mayer function,⁵ shown in Fig. 6.2. In the absence of interactions, fij=0, Q=VN, and we recover Eq. (5.1). With interactions, f(r) is bounded between −1 and e−βVmin−1, where V_min is the minimum value of the interaction potential; f(r) is small for inter-particle separations in excess of the effective range of the potential. Mayer functions allow us to circumvent problems associated with potential functions that diverge⁶ as r→0. At sufficiently high temperatures, |f(r)|≪1, which provides a way of approximately treating the non-ideal gas.

By expanding out the product in Eq. (6.4), a 3N-fold integration is converted into a sum of lower-dimensional integrals known as cluster integrals. The product expands into a sum of terms each involving products of Mayer functions, from zero to all N2=N(N−1)/2 Mayer functions,

∏j>i(1+fij)=1+∑pairs‘fij+∑triples‘[fijfjk+fijfjkfik]+∑quadruples‘[fijfkl+⋯]+⋯ ,

(6.6)

where pairs, triples, quadruples, etc., refer to configurations of two, three, four, and so on, particles known as clusters.⁷ The primes on the summation signs indicate restrictions so that we never encounter terms such as fmm (particles don’t interact with themselves) or f12f12; interactions are between distinct pairs of particles, counted only once. For N=3 particles, Eq. (6.6) generates 232=23=8 terms:

1+f121+f131+f23=1+f12+f13+f23+f12f13+f12f23+f13f23+f12f13f23 .

As N increases, the number of terms rises rapidly. For N=4, there are 242=26=64 terms generated by the product; 64 integrals that contribute to Eq. (6.4). Fortunately, many of them are the same; our task is to learn how to characterize and count the different types of integrals. For N=5, there are 1024 terms—we better figure out how to systematically count the relevant contributions to the partition function if we have any hope of treating macroscopic values of N.

A productive strategy is to draw a picture, or diagram, representing each term in Eq. (6.6). Figure 6.3 shows two circles with letters in them denoting particles at positions r_i and r_j. The circles represent particles, and the line between them represents the Mayer function fij. Note that the physical distance between i and j is taken into account through the value of the Mayer function; the line in Fig. 6.3 indicates an interaction between particles, regardless of the distance between them. If one were to imagine a numbered circle for each of the N particles of the system, with a line drawn between circles i and j for every occurrence of fij in the terms of Eq. (6.6), every term would be represented by a diagram.⁸ Let the drawing of diagrams begin!

We’ll do that in short order, but let’s first consider what we do with diagrams, a process known as “calculating the diagram.” Starting with two-particle diagrams (Fig. 6.3), for each term associated with ∑pairs′fij in Eq. (6.6), there is a corresponding contribution to Eq. (6.4):

∫dNrfij=VN−2∫dridrjf(|ri−rj|)≡VN−22b2(T)V .

(6.7)

The integrations in Eq. (6.7) over the spatial coordinates not associated with particles i and j leave us with VN−2 on the right side. The cluster integral b2(T) is, by definition,

b2(T)≡12V∫dridrjf(rij)=12∫drf(r)=12∫0∞4πr2f(r)dr .

(6.8)

The second equality in Eq. (6.8) is a step we’ll take frequently. Define a new coordinate system centered at the position specified by r_i. With rj−ri≡rji as a new integration variable, we’re free to integrate over r_i. We’ll denote this step as dridrj→dridrji. The quantity b₂ probes the effective range of the two-body interaction at a given temperature (the second moment of the Mayer function) and has the dimension of (length)³. The cluster expansion method works best when the volume per particle V/N≡1/n is large relative to the volume of interaction, i.e., when nb2≪1. The identity of particles is lost in Eq. (6.8). Thus, the N2=N(N−1)/2 terms in ∑pairs′fij all contribute the same value to the configuration integral. Through first order in Eq. (6.6), we have

Q(N,T,V)=VN1+N(N−1)b2(T)V+⋯ .

The partition function (number of states available to the system) is modified (relative to noninteracting particles) by pairs of particles that bring with them an effective volume of interaction, 2b2.

The next term in Eq. (6.6), a sum over three-particle clusters, involves products of two and three Mayer functions. Figure 6.4 shows the diagrams associated with three distinct particles (ijk) joined by two or three lines. For their contribution to Q(T,V,N),

∫dNrfikfkj+fikfij+fijfjk+fijfjkfik≡VN−3(3!Vb3(T)) ,

(6.9)

where, by definition,

b3(T)≡13!V3∫dridrjdkkfikfkj+∫dridrjdrkfikfkjfji→13!V3∫drkdrikdrjkf(rik)f(rjk)+∫drjdrijdrkjf(rik)f(rkj)f(rij)=13!3∫drf(r)2+∫drijdrkjf(rik)f(rkj)f(rij)=2b22(T)+13!∫drijdrkjf(rik)f(rkj)f(rij) .

(6.10)

The third line of Eq. (6.10) follows because rik=rij−rkj (and thus the integral is completely determined by rij and rkj; see Exercise 6.2), and we’ve used Eq. (6.8) in the final line. The factor of 3! is included in the definition to take into account permutations of i, j, k, and thus b₃ is independent of how we label the vertices of the diagram.⁹ The factor of 3 inside the square brackets comes from the equivalence of the three diagrams in Fig. 6.4 under cyclic permutation, i→j→k→i. The quantity b₃ has the dimension of (volume)2.

With Eq. (6.10), we’ve evaluated (formally) the contribution to Q(N,T,V) of a given set of three particles (ijk) that are coupled through pairwise interactions. How many ways can we choose triples? Clearly, N3=N(N−1)(N−2)/3!. Through second order in Eq. (6.6), we have for the configuration integral

Q(T,V,N)=VN1+N(N−1)b2(T)V+N(N−1)(N−2)b3(T)V2+⋯ .

6.1.1 Disconnected diagrams and the linked-cluster theorem

It might seem we’ve discerned the pattern now and we could start generalizing. With the next term in Eq. (6.6) (“quadruples”), we encounter a qualitatively new type of diagram. Figure 6.5 shows the diagram associated with the product of two Mayer functions with four distinct indices. The contribution of this diagram to the configuration integral is

∫dNrfjiflk=VN−4∫dridrjdrkdrlf(rji)f(rlk)→VN−4∫dridrjidrkdrlkf(rji)f(rlk)=VN−2(∫drf(r))2=4VN−2b22 ,

(6.11)

where we’ve used Eq. (6.8). What is the multiplicity of this diagram? There are N2×N−22×12=18N(N−1)(N−2)(N−3) distinct ways the cluster in Fig. 6.5 can be synthesized out of N particles. Figure 6.6 shows the three equivalent diagrams for N=4 particles coupled through two pairwise interactions. Including the cluster integral Eq. (6.11) together with its multiplicity,

Q(T,V,N)=VN(1+N(N−1)b2(T)V+N(N−1)(N−2)b3(T)V2+12N(N−1)(N−2)(N−3)b22(T)V2+⋯) .

(6.12)

It’s not apparent yet, but the term we just added to Eq. (6.12) is bad news.

Let’s think about what we’re trying to do. We seek the partition function for a system of interacting particles. But what do we do with that once we find it? All thermodynamic information can be obtained from the free energy, ln⁡Z (Eq. (4.58) or (4.78)). Write Eq. (6.3), Z(N,T,V)=ZtrQ(N,T,V)/VN, where Z_tr is the partition function for the translational degrees of freedom, Eq. (5.1). Then, ln⁡Z=ln⁡Ztr+ln⁡Q/VN≡ln⁡Ztr+ln⁡1+A, where

A≡[1VN∫dNr∏j>i1+fij]−1 .

Make the assumption (to be verified) that A is small compared with unity. Using Eq. (6.12), we can write A=A1+A2+A3+⋯. Apply the Taylor series,¹⁰

ln⁡(1+A)≈A1+A2+A3+⋯−12(A12+2A1A2+2A1A3+⋯)+13(A13+⋯)+⋯ .

Put in the explicit expressions for A₁, A₂, A₃ from Eq. (6.12), keeping terms proportional to b22:

ln⁡(1+A)=N(N−1)b2V+N(N−1)(N−2)1V22b22+12N(N−1)(N−2)(N−3)b22V2−12N2(N−1)2b22V2+⋯ ,

(6.13)

where we’ve used Eq. (6.10). We know that the free energy is extensive in the thermodynamic limit (see Eq. (4.89)),

lim⁡n=N/V fixedN,V→∞N−1F(N,V,T)=f(n,T) ,

and thus we expect that ln⁡(1+A)~Nf(n,T) as N→∞. Examine Eq. (6.13) for N≫1:

ln⁡(1+A)=N(nb2+2n2b22+12Nn2b22︸Diverges with NTrouble:−12Nn2b22︸Divergent term removedTragedy averted:+⋯) .

(6.14)

The first two terms in Eq. (6.14) are indeed intensive quantities that depend on n and T. The third term, however, which comes from the diagram in Fig. 6.5, is not intensive—it scales with N. That’s the bad news in Eq. (6.12)—there are “too many” disconnected diagrams; their contributions prevent the free energy from possessing a thermodynamic limit. Fortunately, the third term in Eq. (6.14) (that scales with N) is cancelled by the fourth term, i.e., A₃ (from the disconnected diagram) is cancelled by a term in the Taylor series for ln⁡(1+A), 12A12. Is that a coincidence? Do cancellations like that occur at every order? Formulating an answer is problematic—we’ve relied on a Taylor series that’s valid only when the terms in A are small, yet they’re not small (they scale with N), but at the same time they seem to miraculously disappear from the expansion. Something deeper is at work.

The linked-cluster theorem is a fundamental theorem in graph theory, that only connected diagrams contribute to the free energy. Before stating the theorem (which we won’t prove), let’s try to “psyche out” what the issue is with disconnected diagrams. The diagrams in Fig. 6.4 are linked clusters, where each vertex of a diagram is connected to at least one line. The diagram in Fig. 6.5 is a disconnected cluster—there is not a path between any vertices of the graph. Figure 6.7 shows graphs involving N=3,4,5,6 particles interacting by three lines, where we’ve left the vertices unlabeled (free graphs). The first three are linked clusters, the remaining two are disconnected. Suppose one of the particles, k, in the linked clusters of Fig. 6.4 is far removed from particles i and j; in that case, the Mayer function fik or fjk vanishes, implying that the contribution of the diagram to the configuration integral vanishes. When particles are within an interaction distance, there is a distinct type of energy configuration that’s counted in the partition function. As the particles become sufficiently separated, leaving no interaction among them, such contributions vanish. Now consider the disconnected diagrams, such as in Fig. 6.5 or 6.7: One can freely separate the disjoint parts of the diagram (which are not in interaction with each other), placing them anywhere in the system, in which case the interactions represented by the disconnected parts have already been counted in the partition function. The expansion we started with in Eq. (6.6) generates disconnected diagrams, which overcount various configurations. The linked-cluster theorem tells us that only connected diagrams contribute to the free energy (and thus to all thermodynamic information). We need evaluate the partition function taking into account connected diagrams only.

 

The precise form of the linked-cluster theorem depends on whether we’re in the classical or quantum realm, Fermi or Bose, but the central idea remains the same. We present a version given by Uhlenbeck and Ford[55, p40]. Consider a quantity F_N that’s a weighted sum over the graphs G_N (connected or disconnected) of N labeled points, FN≡∑GNW(GN), where W(GN), the weight given to a graph, is in our application the product of the multiplicity and the cluster integral associated with that type of graph. The N-particle configuration integral Q(N,T,V) is just such a function as F_N. Define another quantity f_l as a weighted sum over connected graphs, fl≡∑ClW(Cl), where the sum is over the connected graphs C_l (of the set G_N) with labeled points. The theorem states that

1+F(x)=ef(x) ,

(6.15)

where F(x) and f(x) are generating functions ¹¹ of the quantities F_N and f_l:

F(x)≡∑N=1∞FNxNN!    f(x)≡∑l=1∞flxll! .

(6.16)

So far, we’ve considered the case of fixed N, yet the generating functions in Eq. (6.15) apply for an unlimited number of particles. That finds a perfect application in the grand canonical ensemble, which is where we’re heading (see Section 6.1.3). The linked-cluster theorem can be remembered as the equality

 

∑ all diagrams=exp⁡∑ (all connected diagrams) .

6.1.2 Obtaining Z(N,T,V)

We now give a general definition of the cluster integral associated with n-particle diagrams:

bn(T)≡1n!V∑n-particle diagramsall connected∫(∏connected n-particle diagramlk ∈ the set of bonds in aflk)dr1⋯drn .

(6.17)

Equation (6.17) is consistent with Eqs. (6.8) and (6.10) for n=2,3. (By definition, b1=1.) The purpose of the factor of n! is to make the value of b_n independent of how we’ve labeled the n vertices of the diagram (required by the linked-cluster theorem), and the factor of 1/V cancels the factor of V that always occurs in evaluating cluster integrals of connected diagrams—we’re free to take one vertex of the graph and place it anywhere in the volume V. The quantity b_n has the dimension of (volume)n−1: For an n-particle diagram, we integrate over the coordinates of n−1 particles relative to the position of the nth particle. The cluster integral b_n is therefore independent of the volume of the system as long as V is not too small.

There are many ways that a given set of particles can be associated with clusters. Suppose K particles are partitioned into m₂ two-particle clusters, m₃ three-particle clusters, and so on. The integral over dr1⋯drN of this collection of clusters factorizes (because each cluster is connected)

(1!Vb1)m1(2!Vb2)m2⋯(j!Vbj)mj⋯=∏j=1N(j!Vbj)mj ,

(6.18)

where, to systematize certain formulas we’re going to derive, we’ve introduced the unit cluster, b₁, which is not a particle; such terms contribute the factors of VN−K seen in Eqs. (6.7) and (6.9). Associated with any given placement of N particles into a cluster is a constraint,

∑l=1Nlml=N .

(6.19)

Figure 6.8 shows a set of diagrams for N=4 particles in which we show the unit clusters. You should verify that Eq. (6.19) holds for each of the diagrams in Fig. 6.8.

How many distinct ways can N distinguishable particles¹² be partitioned into m₁ unit clusters, m₂ two-particle clusters, ⋯, m_j clusters of j-particles, and so on? The number of ways of dividing N distinguishable objects among labeled boxes so that there is one object in each of m₁ boxes, two objects in each of m₂ boxes, etc., is given by the multinomial coefficient, Eq. (3.12),

N!(1!)m1(2!)m2⋯(j!)mj⋯ .

We don’t want to count as separate, however, configurations that differ by permutations among clusters of the same kind. To prevent overcounting, we have to divide the multinomial coefficient by m1!m2!⋯mj!⋯. The combinatorial factor is therefore

 

N!∏j=1N(j!)mjmj! .

(6.20)

The contribution to the configuration integral of the collection of clusters characterized by the particular set of integers {mj} is therefore the product of the expressions in (6.18) and (6.20):

N!∏jj!Vbjmj(j!)mjmj!=N!∏jVbjmjmj! .

(6.21)

Note that we don’t need to indicate the range of the index j—clusters for which mj=0 don’t affect the value of the product.

There will be a contribution to the configuration integral for each set of the numbers {mj},

Q(N,T,V)=N!    ∑             {mj}∑j=1Njmj=N∏j=1N(Vbj)mjmj!,

(6.22)

where ∑{mj} indicates to sum over all conceivable sets of the numbers m_j that are consistent with Eq. (6.19). For the noninteracting system, there are no clusters: m1=N with mj≠1=0, for which Eq. (6.22) reduces to V^N. The partition function for N particles is therefore (see Eq. (6.3)):

Z(N,T,V)=1λT3N    ∑             {mj}∑j=1Njmj=N∏j=1N(Vbj)mjmj!.

(6.23)

Once we have the partition function, we know a lot. Combining Eq. (6.25) with Eq. (4.78),¹⁵

P=kT∂ln⁡ZG∂V|T,μ=kT∑j=1∞bjξj=kTξ+b2ξ2+b3ξ3+⋯N=z∂ln⁡ZG∂z|T,V=V∑j=1∞jbjξj=Vξ+2b2ξ2+3b3ξ3+⋯ .

(6.27)

These formulas are fugacity expansions ( ξ=z/λT3), such as we found for the ideal quantum gases (Section 5.5.3). It’s preferable to express P in the form of a density expansion (density is more easily measured than chemical potential). We can invert the expansion for N to obtain a density expansion of the fugacity. Starting from n=∑j=1∞jbjξj in Eq. (6.27), we find using standard series inversion methods, through third order in n:

eβμλT3=ξ=n−2b2n2+8b22−3b3n3+O(n4) .

(6.28)

Equation (6.28) should be compared with Eq. (5.71). Substituting Eq. (6.28) into the expression for P in Eq. (6.27), we find, through second order,

P=nkT1−b2n+4b22−2b3n2+O(n3) .

(6.29)

Equation (6.29), a density expansion of P, is known as the virial expansion. It reduces to the ideal gas equation of state in the case of no interactions.

The virial expansion was introduced (in 1901¹⁶) as a parameterized equation of state,

P=nkT1+B2(T)n+B3(T)n2+B4(T)n3+⋯ ,

where the quantities Bn(T) are the virial coefficients, which are tabulated for many gases.¹⁷ Virial coefficients are not measured directly; they’re determined from an analysis of PVT data. The most common practice is a least-squares fit of PV values along isotherms as a function of density. Statistical mechanics provides (from Eq. (6.29)) theoretical expressions for the virial coefficients:

B2=−b2    B3=4b22−2b3    B4=−20b23+18b2b3−3b4 ,

(6.30)

where the expression for B₄ is the result of Exercise 6.6. The virial coefficients require inter-particle interactions for their existence. For example, using Eq. (6.8),

B2(T)=−12∫drf(r) .

(6.31)

An equation of state proposed by van der Waals¹⁸ in 1873 takes into account the finite size of atoms as well as their interactions. The volume available to gas atoms is reduced (from the volume V of the container) by the volume occupied by atoms. Van der Waals modified the ideal gas law, to

P=NkTV−Nb ,

where b>0 is an experimentally determined parameter for each type of gas. The greater the number of atoms, the greater is the excluded volume. Van der Waals further reasoned that the pressure would be lowered by attractive interactions between atoms. The decrease in pressure is proportional to the probability that two atoms interact, which, in turn, is proportional to the square of the particle density. In this way, van der Waals proposed the equation of state,

P=NkTV−Nb−an2 ,

where a>0 is another material-specific parameter to be determined from experiment.¹⁹ The van der Waals equation of state is usually written

P+an2(V−Nb)=NkT .

(6.32)

It’s straightforward to show that Eq. (6.32) implies for the second virial coefficient,

B2vdw(T)=b−akT .

(6.33)

The van der Waals equation of state provides a fairly successful model of the thermodynamic properties of gases. It doesn’t predict all properties of gases, but it predicts enough of them for us to take the model seriously. It’s the simplest model of an interacting gas we have. Can the phenomenological parameters of the model, (a,b), can be related to the properties of the inter-particle potential? Let’s see if the second virial coefficient as predicted by statistical mechanics, −b2, has the form of that in Eq. (6.33), i.e., can we establish the correspondence

b2↔?−b+akT

for suitably defined quantities (a,b) ? To do that, let’s calculate b₂ for the Lennard-Jones potential. From Eq. (6.8),

12πb2=∫0∞r2e−βv(r)−1dr≈−∫0σr2dr−β∫σ∞v(r)r2dr ,

(6.34)

where, referring to Fig. 6.2, for 0≤r≤σ we’ve taken the repulsive part of the potential as infinite (hard core potential) so that the Mayer function is equal to −1, and for r≥σ we’ve approximated the Mayer function with its high-temperature form, −βv(r). With these approximations, we find

b2=−2π3σ3+16π9ϵkTσ3 .

(6.35)

The correspondence therefore holds: b~σ3 is an excluded volume provided by the short-range repulsive part of the inter-particle potential, and the parameter a~ϵσ3 is an energy-volume associated with the attractive part of the potential. Statistical mechanics validates the assumptions underlying the van der Waals model, illustrating the role of microscopic theories in deriving phenomenological theories. Moreover, as we now show, with some more analysis we can provide a physical interpretation of the types of interactions that give rise to the virial coefficients.

We now derive the virial expansion in another way, one that features additional techniques of diagrammatic analyses.²⁰ Consider B3=2(2b22−b3); Eq. (6.30). Using Eq. (6.10) (for b₃), we see that a cancellation occurs among the terms contributing to B₃, leaving us with one integral:

B3=−13∫drijdrkjf(rik)f(rkj)f(rij) .

(6.36)

The diagram corresponding to this integral is shown in the left part of Fig. 6.9. What about the diagrams that don’t contribute to B₃? Time for another property of diagrams. A graph is irreducible when each vertex is connected by a bond to at least two other vertices, as in the left part of Fig. 6.9.²¹ A reducible graph has certain points, articulation points, where it can be cut into two or more disconnected parts, as in the right part of Fig. 6.9. Graphs can have more than one articulation point; see Fig. 6.10. A linked graph having no articulation points is irreducible. Figure 6.11 shows the three types of diagrams: Unlinked, reducible, and irreducible. As we now show, only irreducible diagrams contribute to the virial coefficients.

Rewrite Eq. (6.3) (as we did in Section 6.1.1), Z(T,V,N)=ZtrQ(T,V,N)/VN, where Z_tr is the partition function for the ideal gas, Eq. (5.1). We now write Q/VN in a new way:

1VNQ(T,V,N)=1VN∫dNre−βV(r1,⋯,rN)≡〈e−βV(r1,⋯,rN)〉0 ,

(6.37)

where V(r1,⋯,rN)≡∑j>iv(rij) is the total potential energy of particles having the instantaneous positions r1,⋯,rN, and we’ve introduced a new average symbol,

〈(⋯)〉0≡1VN∫dNr(⋯) .

Equation (6.37) interprets the configuration integral as the expectation value of e−βV(r1,⋯,rN) with respect to a non-thermal (in fact, geometric) probability distribution²² where the variables r1,⋯,rN have a uniform probability density (1/V)N inside a container of volume V. Using Eq. (4.57), we obtain

−βF−Fideal=ln⁡〈e−βV(r1,⋯,rN)〉0 ,

(6.38)

where Fideal≡−kTln⁡Ztr is the free energy of the ideal gas. The right side of Eq. (6.38) is the contribution to the free energy arising solely from inter-particle interactions.

We encountered just such a quantity in Eq. (3.34), the moment generating function 〈eθx〉=∑n=0∞θn〈xn〉/n!, where, we stress, the average symbols 〈 〉 are associated with a given (not necessarily thermal) probability distribution. The logarithm of 〈eθx〉 defines the cumulant generating function, Eq. (3.59),²³

ln⁡〈eθx〉=∑n=1∞θnn!Cn ,

(6.39)

where each quantity C_n (cumulant) contains combinations²⁴ of the moments 〈xk〉, 1≤k≤n. Explicit expressions for the first few cumulants are listed in Eq. (3.62). To apply Eq. (6.39) to Eq. (6.38), set θ=−β and associate the random variable x with the total potential energy, V=∑j>iv(rij). Thus we have the cumulant expansion of the free energy (with ΔF≡F−Fideal):

−βΔF=∑n=1∞(−β)nn!Cn=−βC1+β22C2−β33!C3+⋯=−β〈V〉0+β22〈V2〉0−〈V〉02−β33!〈V3〉0−3〈V2〉0〈V〉0+2〈V〉03+O(β4) ,

(6.40)

where we’ve used Eq. (3.62) for C1,C2,C3. We examine the first few terms of Eq. (6.40).

The first cumulant C₁ is the average potential energy:

C1=〈V(r1,⋯,rN)〉0=∑j>i〈v(rij)〉0=∑j>i1VN∫dNrv(rij)→N21V∫drv(r) ,

(6.41)

where we’ve used Eq. (3.66), that the cumulant associated with a sum of independent random variables is the sum of cumulants associated with each variable. We’re interested in the thermodynamic limit. Making use of Eq. (3.44), Nn~N→∞Nn/n!, in Eq. (6.41),

lim⁡N/V=nN,V→∞C1N=12n∫v(r)dr .

(6.42)

Cumulants must be extensive, so that the free energy as calculated with Eq. (6.40) is extensive.²⁵ We associate the integral in Eq. (6.42) with the graph in Fig. 6.12, which is nominally the same as Fig. 6.3, with an important exception—Fig. 6.3 represents the Mayer function fij between particles i,j, whereas Fig. 6.12 represents their direct interaction, vij.

The cumulant C₂ is the fluctuation in potential energy:²⁶

C2=〈V2(r1,⋯,rN)〉0−〈V(r1,⋯,rN)〉02=∑j>i∑l>k〈vijvkl〉0−〈vij〉0〈vkl〉0 ,

(6.43)

where vij≡v(rij). Let’s analyze the structure of the indices in Eq. (6.43), because that’s the key to this method. There are three and only three possibilities in this case:

1. 
   

   No indices in common; unlinked terms. Consider 〈v12v34〉0. Because we’re averaging with respect to a probability distribution in which r₁₂ and r₃₄ can be varied independently,

   
   

   〈v12v34〉0=〈v12〉0〈v34〉0 .

   
   

   (6.44)

   
   

   The averages of products of potential functions with distinct indices factor, and as a result C₂ vanishes identically—a fortunate development: There are 12N2N−22~N→∞N4/8 ways to choose, out of N indices, two sets of pairs having no elements in common. We’ll call terms that scale with N too rapidly to let the free energy have a thermodynamic limit, super extensive. The important point is that every unlinked term arising from 〈vijvkl〉0 (which we’re calling super extensive) is cancelled by a counterpart, 〈vij〉0〈vkl〉0. The diagrams representing 〈vijvkl〉0 for no indices in common are those in Fig. 6.11 (a).

   
   

    

   
   
2. 
   

   One index in common; reducibly-linked terms. Consider 〈v12v23〉0:

   
   

   〈v12v23〉0=1V3∫dr1dr2dr3v12v23→1V2∫dr12dr32v12v23=〈v12〉0〈v23〉0 .

   
   

   (6.45)

   
   

   Because of the averaging procedure 〈 〉0, 〈v12v23〉0 factorizes, implying that C₂ vanishes for these terms. Such terms correspond to reducibly linked diagrams (see Fig. 6.11 (b))—the common index represents an articulation point where the graph can be cut and separated into disconnected parts. Reducible graphs make no contribution to the free energy.

   
   
3. 
   

   Both pairs of indices in common; irreducibly-linked graphs. Consider the case, from Eq. (6.43), where k=i and l=j,

   
   

   C2=∑j>i〈vij2〉0−〈vij〉02 .

   
   

   (6.46)

   
   

   The first term in Eq. (6.46), the average of the square of the inter-particle potential,

   
   

   〈vij2〉0=1V2∫dridrjvij2→1V∫drijvij2≡1V∫drv2(r)~1V ,

   
   

   (6.47)

   
   

   whereas

   
   

   〈vij〉02=1V∫drv(r)2~1V2 .

   
   

   (6.48)

   
   

   We’re assuming the integrals ∫drv(r) and ∫drv2(r) exist. Noting how the terms scale with volume in Eqs. (6.47), (6.48), only the first term survives the thermodynamic limit,

   
   

   lim⁡N/V=nN,V→∞C2N=12n∫v2(r)dr ,

   
   

   (6.49)

   
   

   where we’ve used ∑j>i=N2. Figure 6.13 shows the diagram representing the integral in Eq. (6.49). This is a new kind of diagram where vij2 is represented by two bonds between particles i,j; it has no counterpart in the Mayer cluster expansion (we never see terms like f12f12).