10. Interacting matter The grand potential is now Ω= k TVω(z,T ) 10.1. Classical real gas − B We take into account the mutual interactions of atoms and (molecules) p Ω The Hamiltonian operator is = = ω(z,T ) kBT −V N p2 N ∂ω(z,T ) H(N) = i + v(r ), r = r r . ρ = = z . 2m ij ij | i − j | V ∂z i=1 i<j X X Eliminating z we can write the equation of state as For example, for noble gases a good approximation of the p = k T ϕ(ρ,T ). interaction potential is the Lennard-Jones 6–12 -potential B Expanding ϕ as the power series of ρ we end up with the σ 12 σ 6 v(r)=4ǫ . virial expansion. r − r Ursell-Mayer graphs V ( r ) Let’s write Q = dr r e−βv(rij ) N 1 · · · N i<j Z Y r 0 s r 6 = dr r (1 + f ), e 1 / r 1 · · · N ij Z i<j We evaluate the partition sums in the classical phase Y where space. The canonical partition function is f = f(r )= e−βv(rij ) 1 ij ij − −βH(N) ZN (T, V )= Z(T,V,N) = Tr N e is Mayer’s function. 1 V ( r ) dΓ e−βH . classical−→ N! limit, Maxwell- Z f ( r ) Boltzman Because the momentum variables appear only r quadratically, they can be integrated and we obtain - 1 1 1 The function f is bounded everywhere and it has the ZN = dp1 dpN dr1 drN same range as the potential v. In general f is small N! h3N · · · · · · · · · × Z Z (compared to 1), when rij is in the relevant range. 2 0 N pi If v(r) 0, then f 0 and QN = QN = V i.e. we end exp β + v(rij ) ≡ ≡ − 2m up with the Maxwell-Boltzmann ideal gas. We write QN i i<j X X as the power series of Mayer functions fij : 1 1 = Q (T, V ), N! λ3N N T QN = dr1 rN 1+ fij + fij fkl · · · " where Z X(ij) (ijX)<(kl) h2 1/2 λ = T fij fklfmn + . 2πmkBT · · · (ij)<(kl)<(mn) # is the thermal wave length and X Here −β v(rij ) Q (T, V )= dr dr e i<j . 1 N 1 · · · N (ij) pair 1 i < j N, N(N 1) terms Z P ⇔ ≤ ≤ 2 − The grand canonical partition function is (ij) < (kl) (ij) = (kl) and only one ⇔ 6 of terms (ij)(kl), (kl)(ij) selected, Z(T,V,µ) = eβµN Z (T, V ) N 1 1 1 N [ N(N 1)][ N(N 1) 1] terms X N 2 2 − 2 − − 1 z . = 3 QN (T, V ), . N! λT XN Graphs are build from the elements when z = eβµ is the fugacity. We define an intensive function i dr • ⇔ i 1 Z ω(z,T )= ln Z(T,V,µ). i j f . V ⇔ ij 1 Permutations of particles do not change the values of the Denote z graphs, for example ξ = 3 . λT dr1dr2dr3f12f23 = dr1dr2dr3f13f23, The grand canonical partition function is now Z Z Z Z Z Z 1 N as we can see by exchanging the integration variables r Z(T,V,µ) = ξ QN 2 N! and r3. XN qµl 3 3 = l = νl νl!(l!) × 1 2 1 2 {νl} l X Y Example: 1 ξN N!δ N, lν Q 2 = N! l N l ! Q 2 = + X νl X q lνl Q 3 = + + + l l = ν ξ νl!(l!) l + + + + {ν } l Xl Y P 1 q νl = + 3 * + 3 * + = l ξl We classify the graphs: νl! l! {Xνl} Yl h i ∞ ν q in connected (coupled) graphs or a clusters one can 1 ql l l l ξl • = ξ = e l! . get from every black dot ( ) to every black dot νl! l! • l νl=0 l following a chain of lines ( ). Y X h i Y We end up with the cumulant expansion of the grand in unconnected (uncoupled) graphs there are parts canonical partition sum: • that are not connected by a line ( ). ∞ ∞ It is easy to see that an unconnected graph can be 1 N 1 l Z(T,V,µ)= ξ QN = exp ξ ql . factorized as a product of its connected parts. N! l! N=0 "l=0 # The sum of graphs of l connected points is called an X X l-cluster. graphs = exp[ connected graphs] We define q so that it is the sum of all l-clusters, e.g. l PVirial expansionP In the cumulant expansion every ql is proportional to the q = dr V 1 = volume V . We define the cluster integral bl depending Z only on temperature so that q = dr dr f 2 1 2 12 1 1 Z Z bl = ql l! V q3 = dr1dr2dr3(3f12f13 + f12f23f13). 1 1 Z Z Z = dr dr (1 + f ) . One can show that l! V 1 · · · l ij Z Z i<j Y connected graphs νl N! ql QN = δ N, lνl ν . Now (l!) l νl! ∞ l ! l l 1 {Xνl} X Y ω(z,T )= ln Z(T,V,µ)= ξlb (T ). V l Here Q l=1 X ν tells the number of l-clusters. For the density we get • l ∞ ∞ ∞ ∞ ∂ω l = ρ = z = lξ bl(T ). • · · · ∂z ν =0 ν =0 ν =0 l=1 {Xνl} X1 X2 X3 X We solve ξ as the power series of the density ρ, substitute it into the expression of ω and collect equal powers of ρ δ N, lνl restricts the number of the integration • ! together and end up with the virial expansion Xl variables (black dots ) to N. 2 3 • p = kB Tω(z,T )= kBT [ρ + B2(T )ρ + B3(T )ρ + ]. N! · · · tells how many ways there are to pick the νl The virial coefficients Bn(T ) are now functions of the • l(l!) clusters from the set of N points. cluster integrals bl(T ) l n , e.g. Q { | ≤ } 1 −βv(r) Everyone of the possible νl! permutations of the B2(T ) = b2(T )= dr 1 e • l-clusters gives an identical contribution and must be − 2 − 2 Z h i counted only once. That’s why the divisor νl! in the B3(T ) = 4b2 2b3 νl − ql 3 factor . B4(T ) = 20b + 18b2b3 3b4. νl! − 2 − 2 the function f(r ) deviates from zero only in the • iN Second virial coefficient range of the interaction. We assume that the term f(r )f(r ) can deviate from zero only if • iN jN the interaction has a hard core, i.e. the interaction is particles i and j are within the range of N. • strongly repulsive when r <σ. ∼ the probability that the particle j is in the range of • the particle i is 1 . on the average, the interaction is attractive when ∝ V • r >σ, but the temperature is so high that βv(r) 1 if the particle j is in the range of the particle i, then there.∼ ≪ • the integral over the variable r is 1, since the N ∝ Now function f has a short range. 0, when r <σ e−βv(r) ≈ 1 βv(r), when r∼>σ, We see, that − ∼ N−1 N−1 and 1 N 2 ∞ drN f(riN )f(rjN ) = . 2 −βv(r) ∝ V O V B2 = 2π dr r 1 e Z i<j i<j − X X Z0 σ h ∞ i In the low density limit we get 2π dr r2 +2π dr r2βv(r) ≈ 0 σ N−1 Z Z −β viN a drN e i = V + (N 1) dr f(r) = b . − − T Z P Z N 2 Here + O V 2π 3 b = σ V + (N 1) dr f(r). 3 ≈ − ∞ Z a = 2π dr r2v(r) > 0. − Iterating we have Zσ N−2 With these approximations we end up with the van der −β vij Q dr dr e i<j Waals equation of state. For hard spheres the virial N ≈ 1 · · · N−2 × coefficients can be calculated exactly. Denoting Z Z P V + (N 2) dr f(r) 2π − × b = σ3 Z 0 3 V + (N 1) dr f(r) we get − Z n Bn N−1 N n 2 b0 V 1+ ρ dr f(r) , 5 2 ≈ ···≈ N 3 8 b0 n=0 Z 3 Y 4 0.287 b0 4 where ρ = N/V . Now 5 0.110 b0 5 6 0.039 b0 j N j x 2 lim 1+ x 1+ − x 1+ x→0 N N ≈ 2 Low density gas so We write N−1 nx x N xN N−1 2 −β vij lim 1+ 1+ e , QN = dr1 drN−1e i<j x→0 N ≈ 2 ≈ n=0 · · · × Y h i h i Z Z N−1 P −β v(riN ) i.e. in the low density limit drN e i . 1 Nρ dr f(r) (0) 1 Nρ dr f(r) Z P Q V N e 2 = Q e 2 . With the help of Mayer’s functions this is N ≈ N R R N−1 N−1 (0) N Here QN = V is QN for the ideal gas. Since the exp β viN = [1 + fiN ] canonical partition sum was − " i # i X Y N−1 1 ZN = 3N QN , = 1+ f(riN ) N!λT i X the energy can be written like N−1 + f(riN )f(rjN )+ . · · · F (T,V,N) = kBT ln ZN i<j − X N 2k T = F (T,V,N) B dr f(r), Choose randomly two particles, i and j say.