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Cluster Expansion in Gas Physics: Virial Coefficients and Quantum Systems

Explore cluster expansions to evaluate virial coefficients, gas equilibria, and quantum mechanical correlations. Learn about exact treatment of coefficients and correlations in scattering dynamics. Utilize cluster expansion for classical and quantum gases.

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Cluster Expansion in Gas Physics: Virial Coefficients and Quantum Systems

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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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