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10. The Method of Cluster Expansions

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

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10. The Method of Cluster Expansions

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  1. 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

  2. Cluster expansions = Series expansion to handle inter-particle interactions Applicability : Low density gases Poineers : Mayer : Classical statistics. Kahn-Uhlenbeck, Lee-Yang : Quantum statistics

  3. 10.1. Cluster Expansion for a Classical Gas Central forces : Partition function : 

  4. where Configuration integral Non-interacting system ( uij = 0 ) :  Let L-J potential 

  5. Graphic Expansion All possible pairings 8-particle graphs : = = factorized = =

  6. l - Cluster Each N-particle integral is represented by an N-particlegraph. Graphs of the same topology but different labellings are counted as distinct.  An l-cluster graph is a connectedl-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 :  = =

  7. 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

  8. Examples  V(r1) = volume of gas using r1 as origin. 

  9. 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.

  10.  where    

  11. Z,Z, F, P, n    

  12. 10.2. Virial Expansion of the Equation of State Virial expansion for gases : Invert gives  Mathematica

  13. 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.

  14. 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 :

  15. a2 For :   Blare also called the virial coefficients

  16. 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 )

  17. B2  where Reduced Lennard-Jones potential

  18. Hard Sphere Gas Molecules = Hard spheres  Step potential :  D = diameter of spheres  D D Mathematica

  19. Mathematica   See Pathria, p.314 for values of a4, a5, a6& P. Approximate analytic form of the equation of state for fluids ( ) :

  20. 10.4. General Remarks on Cluster Expansions Cluster Expansion :

  21. Coefficients of Zjkin ( ... )lsum to 0.  Classical ideal gas :  ( ... )l ~ sum of all possible l-clusters are independent of V ( ... )l  V Rushbrooke :

  22. Semi-Invariants Constraint (l) : Semi-Invariants Inversion :

  23. Proof of   inversion      QED

  24. A theorem due to Lagrange : Solution x(z) to eq. is where  

  25. constraint (j1) :  Inversion due to Mayer : constraint (l1) :

  26. 10.5. Exact Treatment of the Second Virial Coefficient u(r) = 0    where Total Reduced Let 

  27.   

  28. Let spectrum of interacting system consist of a discrete (bounded states) part & a continuum (travelling states) part with DOS g(). 

  29. Unbounded states ( n> 0 )  where    l= phase shift

  30. 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 

  31. For the purpose of counting states ( to get g() ), we discretize the spectrum by setting for some .    For a given l ,   k l mis 2l+1 fold degenerate e/omeans l in sum iseven/odd for boson/fermion   For u= 0 : 

  32.  Boson Fermion From § 7.1 &§ 8.1 :

  33. b2(0) From § 5.5 :  same as before Alternatively, using the statistical potential from § 5.5

  34. Hard Sphere Gas In region where u= 0,   Mathematica

  35. No bound states for hard sphere gas.   Mathematica

  36. 10.6. Cluster Expansion for a Quantum Mechanical System

  37. 10.7. Correlations & Scattering

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