10. The Method of Cluster Expansions Cluster Expansion for a Classical Gas Virial Expansion of the Equation of State Evaluation of the Virial Coefficients General Remarks on Cluster Expansions Exact Treatment of the Second Virial Coefficient Cluster Expansion for a Quantum Mechanical System Correlations & Scattering
Cluster expansions = Series expansion to handle inter-particle interactions Applicability : Low density gases Poineers : Mayer : Classical statistics. Kahn-Uhlenbeck, Lee-Yang : Quantum statistics
10.1. Cluster Expansion for a Classical Gas Central forces : Partition function :
where Configuration integral Non-interacting system ( uij = 0 ) : Let L-J potential
Graphic Expansion All possible pairings 8-particle graphs : = = factorized = =
l - Cluster Each N-particle integral is represented by an N-particle graph. Graphs of the same topology but different labellings are counted as distinct. An l-cluster graph is a connected l-particle graph. ( Integral cannot be factorized. ) E.g., 5-cluster : = Integrals represented by l-clusters of the same topology has the same value. All possible 3-clusters : = =
Cluster Integrals Cluster integral : Let = dimension of X. X is dimensionless ru = range of u For a fixed r1 , is indep of V. is indep of size & shape of system
Examples V(r1) = volume of gas using r1 as origin.
ZN Let ml = # of l-cluster graphs for each N-particle graph Let be the sum of all graphs that satisfy # of distinct ways to assign particles into is Let there be pl distinct ways to form an l-cluster, with each giving an integral Il j . Then the sum of all distinct products of ml of these l-clusters is The factor ml ! arises because the order of Il j within each product is immaterial.
where
Z, Z, F, P, n
10.2. Virial Expansion of the Equation of State Virial expansion for gases : Invert gives Mathematica
In general : (see §10.4 for proof ) irreducible cluster integral ( dimensionless ) Irreducible means multiply-connected, i.e., more than one path connecting any two vertices. c.f.
10.3. Evaluation of the Virial Coefficients Lennard-Jones potential : minimum Precise form of repulsive part ( u > 0 ) not important. Can be replaced by impenetrable core ( u = r < r0 ). Precise form of attractive part ( u < 0 ) important : Useful adjustable form :
a2 For : Bl are also called the virial coefficients
van der Waals Equation for for c.f. van der Waals eq. v0 = molecular volume see Prob 1.4 r0 = molecular diameter Condition ( dilute gas )
B2 where Reduced Lennard-Jones potential
Hard Sphere Gas Molecules = Hard spheres Step potential : D = diameter of spheres D D Mathematica
See Pathria, p.314 for values of a4 , a5 , a6 & P. Mathematica See Pathria, p.314 for values of a4 , a5 , a6 & P. Approximate analytic form of the equation of state for fluids ( ) :
10.4. General Remarks on Cluster Expansions
Coefficients of Zjk in ( ... )l sum to 0. Classical ideal gas : ( ... )l ~ sum of all possible l-clusters are independent of V ( ... )l V Rushbrooke :
Semi-Invariants Constraint (l ) : Semi-Invariants Inversion :
Proof of inversion QED
A theorem due to Lagrange : Solution x(z) to eq. is where
constraint (j1) : Inversion due to Mayer : constraint (l1) :
10.5. Exact Treatment of the Second Virial Coefficient u(r) = 0 where Total Reduced Let
Let spectrum of interacting system consist of a discrete (bounded states) part & a continuum (travelling states) part with DOS g().
Unbounded states ( n > 0 ) where l = phase shift
For the purpose of counting states ( to get g() ), we discretize the spectrum by setting for some . For a given l , k l m is 2l+1 fold degenerate e/o means l in sum is even/odd for boson/fermion For u = 0 :
Boson Fermion From § 7.1 & § 8.1 :
b2(0) From § 5.5 : same as before Alternatively, using the statistical potential from § 5.5
Hard Sphere Gas In region where u = 0, Mathematica
No bound states for hard sphere gas. Mathematica