Skip to main content

Interacting Particles, I: Classical and Quantum Clustering

  • Chapter
Foundations of Statistical Mechanics

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 19))

  • 588 Accesses

Abstract

For all the various models considered up to this point the basic interactions between and among the constituent particles were essentially ignored, and for good reasons. First of all, the resulting simplification in the many-body problem permitted a detailed explication of the major physical features of these systems, and led to the development of some powerful mathematical techniques which will always be useful. Secondly, the resulting physical descriptions in terms of free-particle models are often quite viable in themselves and provide significant insight into the systems and processes they model. This latter situation arises because the particle-particle interactions are actually irrelevant to those macroscopic properties we choose to measure in these cases.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Abramowitz, M., and I. A. Stegun: 1964,Handbook of Mathematical Functions, AMS 55, Natl. Bur. of Stds., Washington.

    MATH  Google Scholar 

  • Adhikari, S.K., and R.D. Amado: 1971, ‘Low-Temperature Behavior of the Quantum Cluster Co-efficients’Phys. Rev. Letters 27 485.

    Article  ADS  Google Scholar 

  • Amdur, I., and E.A. Mason: 1958, ‘Properties of Gases at Very High Temperatures’,Phys. Fluids 1, 370.

    Article  MathSciNet  ADS  Google Scholar 

  • Arf, C., K. Imre, and E. Ozizmir: 1965, ‘On the Algebraic Structure of the Cluster Expansion inStatistical Mechanics’, J.Math. Phys. 6, 1179.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Barker, J.A., P.J. Leonard, and A. Pompe: 1966, ‘Fifth Virial Coefficients’,J. Chem. Phys. 44, 4206.

    Article  ADS  Google Scholar 

  • Barker, J.A., and J.J. Monaghan: 1962a, ‘Virial Coefficients for Square-Well Potentials’,J. Chem. Phys. 36, 2558.

    Article  ADS  Google Scholar 

  • Barker, J.A., and J.J. Monaghan: 1962b, ‘Fourth Virial Coefficients for the 12–6 Potential’, J.Chem. Phys. 36 2564.

    Article  ADS  Google Scholar 

  • Barker, J.A., and J.J. Monaghan: 1966, ‘Comment on a Letter of Katsura’,J. Chem. Phys. 45, 3482.

    Article  ADS  Google Scholar 

  • Baumgartl, B.J.: 1967, ‘Second and Third Virial Coefficients of a Quantum Gas from Two-ParticleScattering Amplitude’,Z. Phys. 198 148.

    Article  ADS  Google Scholar 

  • Beth, E., and G.E. Uhlenbeck: 1937, ‘The Quantum Theory of the Non-Ideal Gas.II. Behavior atLow Temperatures’,Physic a 4 915.

    ADS  Google Scholar 

  • Boltzmann, L.: 1899, ‘Über die Zustandgleichung v.d. Waals’,Versl. Acad. Wet. Amsterdam 7, 477.

    Google Scholar 

  • Boyd, M., S. Larsen, and J.E. Kilpatrick: 1966, ‘Exchange and Direct Second Virial Coefficientsfor Hard Spheres’,J. Chem Phys. 45 499.

    Article  ADS  Google Scholar 

  • Bruch, L.W.: 1967, ‘Second Virial Coefficient for the Exponential Repulsive Potential’,Phys. Fluids 10, 2531.

    Article  ADS  Google Scholar 

  • Bruch, L.W.: 1973a, ‘High Temperature Limit of the Exchange Third Virial Coefficient for HardSpheres’,Prog. Theor. Phys. 50, 1537.

    Article  ADS  Google Scholar 

  • Bruch, L.W.: 1973a, ‘High Temperature Limit of the Exchange Third Virial Coefficient for HardSpheres’,Prog. Theor. Phys. 50, 1835.

    Article  ADS  Google Scholar 

  • Brydges, D.C.: 1986, ‘Convergence of Mayer Expansions’,J. Stat. Phys. 42 425.

    Article  MathSciNet  ADS  Google Scholar 

  • Brydges, D.C., and P. Federbush: 1978, ‘A New Form of the Mayer Expansion in Statistical Mechanics’,J. Math. Phys. 19 2064.

    Article  MathSciNet  ADS  Google Scholar 

  • Clausius, R.: 1870, ‘Über einen auf die Warme andwendbaren mechanischen Satz’,Ann. d. Phys. 141 124.

    ADS  Google Scholar 

  • D’Arruda, J. J.: 1973, ‘High-Temperature Quantum Corrections to the Second Virial Coefficient for a Hard-Core- Plus-Attractive-Well-Potential Model’,Phys. Rev. A 7 820.

    Article  ADS  Google Scholar 

  • D’Arruda, J.J., and R.N. Hill: 1970, ‘Quantum Corrections to the Second Virial Coefficient with an Application to the Hard-Core-Plus-Square-Well Potential at High Temperatures’,Phys. Rev A, 1. 1791.

    Article  ADS  Google Scholar 

  • Dashen, R., S.-K. Ma, and H.J. Bernstein: 1969, ‘S-Matrix Formulation of Statistical Mechanics’,Phys. Rev. 187, 345.

    Article  ADS  MATH  Google Scholar 

  • de Boer, J.: 1940, ‘Contribution to the Theory of Compressed Gases’,Ph.D thesis Amsterdam Univ. (unpublished).

    Google Scholar 

  • de Boer, J., and A. Michels: 1938, ‘Contribution to the Quantum-Mechanical Theory of the Equation of State and the Law of Corresponding States. Determination of the Law of Force of Helium’,Physica5, 945.

    Article  ADS  MATH  Google Scholar 

  • de Boer, J., and A. Michels: 1939, ‘Quantum-Mechanical Calculation of the Second Virial- Coefficient of Helium at Low Temperatures’,Physica 6, 409.

    Article  ADS  MATH  Google Scholar 

  • de Boer, J., J. van Kranendonk, and K. Compaan: 1950, ‘The Equation of State of Gaseous He3’,Physic a 16, 545.

    ADS  MATH  Google Scholar 

  • De Witt, H.E.: 1962, ‘Analytic Properties of the Quantum Corrections to the Second Virial Coefficient’,J. Math. Phys. 3 1003.

    Article  ADS  Google Scholar 

  • Douslin, D.R., R.H. Harrison, R.T. Moore, and J.P. McCullough: 1964, ‘P-V-T Relations for Methane’,J. Chem. Engng. Data 9, 358.

    Article  Google Scholar 

  • Dymond, J.H., and E.B. Smith: 1980,The Virial Coefficients of Pure Gases and Mixtures Clarendon Press, Oxford.

    Google Scholar 

  • Edwards, J.C.: 1970, ‘Perturbation-Theory Approach to the First Quantum Correction to the Square-Well High-Temperature Classical Second Virial Coefficient’,Phys. Rev. A 2, 1599.

    Article  ADS  Google Scholar 

  • Feinberg, M.J., A.G. DeRocco: 1964, ‘Intermolecular Forces: The Triangle Well and Some Comparisons with the Square Well and Lennard-Jones’,J. Chem. Phys. 41 3439.

    Article  ADS  Google Scholar 

  • Fosdick, L.D., and H.F. Jordan: 1966, ‘Path Integral Calculation of the Two-Particle Slater Sum for He4’,Phys. Rev. 143 58.

    Article  ADS  Google Scholar 

  • Fré, P.: 1977, ‘The 5-Matrix Formulation of the Cluster Expansion in Statistical Mechanics’,Fort, d. Phys. 25, 579.

    Article  Google Scholar 

  • Frisch, H.L., and E. Helfand: 1960, ‘Conditions Imposed by Gross Properties on the Intermolecular Potential’,J. Chem. Phys. 32 269.

    Article  MathSciNet  ADS  Google Scholar 

  • Gibson, W.G.: 1970, ‘Quantum-Mechanical Second Virial Coefficient at High Temperatures’,Phys. Rev. A2 996.

    ADS  Google Scholar 

  • Gibson, W.G.: 1972a, ‘Quantum Corrections to the Equation of State for Nonanalytic Potentials’,Phys. Rev. A5, 862.

    ADS  Google Scholar 

  • Gibson, W.G.: 1972b, ‘Low-Temperature Expansion of the Third-Cluster Coefficient of a Quantum Gas’,Phys. Rev.A6, 2469.

    ADS  Google Scholar 

  • Gibson, W.G.: 1973, ‘Quantum Corrections to the Virial Coefficients for Potentials with Hard Cores’,Phys. Rev. A7 822.

    ADS  Google Scholar 

  • Goldberger, M.L., and A.N. Adams: 1952, ‘The Configurational Distribution Function in Quantum- Statistical Mechanics’,J. Chem. Phys. 20, 240.

    Article  MathSciNet  ADS  Google Scholar 

  • Graben, H.W., and R.D. Present: 1964, ‘Third Virial Coefficient for the Sutherland (∞,v) Potential’, Rev.Mod. Phys. 36, 1025.

    Article  ADS  Google Scholar 

  • Gropper, L.: 1937, ‘Connection Between the Second Virial Coefficient and the Phases of Collision Theory’,Phys. Rev. 51 1108.

    Article  ADS  Google Scholar 

  • Guggenheim, E.A.: 1957,Thermodynamics North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Handelsman, R.A., andJ.B. Keller: 1966, ‘Quantum-Mechanical Second Virial Coefficient of a Hard-Sphere Gas at High Temperature’,Phys. Rev. 148 94.

    Article  ADS  Google Scholar 

  • Happel, H.: 1906, ‘Zur Theorie und Prufung der Zustandgleichung’,Ann. d. Phys. 21, 342.

    Article  ADS  Google Scholar 

  • Hemmer, P.C.: 1968, ‘The Hard Core Quantum Gas at High Temperatures’,Phys. Letters 27A 377.

    ADS  Google Scholar 

  • Hemmer, P.C., andK.J. Mork: 1967, ‘Quantum-Mechanical Second Virial Coefficient of a Hard- Sphere Gas at High Temperatures’,Phys. Rev. 158 114.

    Article  ADS  Google Scholar 

  • Henderson, D., and S.G. Davison: 1965, ‘Quantum Corrections to the Equation of State for a Steep Repulsive Potential’,Proc. Natl. Acad. Sci. (U.S.A.) 54, 21.

    Article  MathSciNet  ADS  Google Scholar 

  • Hill, R.N.: 1968, ‘Quantum Corrections to the Second Virial Coefficient at High Temperatures’, J.Math. Phys. 9, 1534.

    Article  ADS  Google Scholar 

  • Hill, R.N.: 1974, ‘High-Temperature Exchange Third Virial Coefficient for Hard Spheres via an Asymptotic Method for Path Integrals’ J.Stat. Phys. 11 207.

    Article  ADS  Google Scholar 

  • Hirschfelder, J.O., C.F. Curtiss, and R.B. Bird: 1954,Molecular Theory of Gases and Liquids Wiley, New York.

    MATH  Google Scholar 

  • Holborn, L., and J. Otto: 1925:, ‘ÜCber die Isothermen einiger Gases zwischen +400 ° und -183°Z. Phys. 33, 1.

    Article  ADS  Google Scholar 

  • Jancovici, B.: 1969a, ‘Quantum-Mechanical Equation of State of a Hard-Sphere Gas at High Temperatures’,Phys. Rev. 178, 295.

    Article  ADS  Google Scholar 

  • Jancovici, B.: 1969b, ‘Quantum-Mechanical Equation of State of a Hard-Sphere Gas at High Temperatures. II’,Phys. Rev. 184 119.

    Article  ADS  Google Scholar 

  • Jancovici, B., and S.P. Merkuriev: 1975, ‘Quantum- Mechanical Third Virial Coefficient of a Hard- Sphere Gas at High Temperatures’,Phys. Rev. A 12 2610.

    Article  ADS  Google Scholar 

  • Jeans, J.: 1925,Dynamical Theory of Gases Cambridge Univ. Press, Cambridge.

    MATH  Google Scholar 

  • Kahn, B.: 1938, ‘On the Theory of the Equation of State’,Ph.D. thesis, Utrecht. [North-Holland, Amsterdam, 1938.]

    Google Scholar 

  • Kahn, B., and G.E. Uhlenbeck: 1938, ‘On the Theory of Condensation’,Physic a 5, 399.

    ADS  Google Scholar 

  • Kammerlingh Onnes, H.: 1902, ‘Expression of the Equation of State of Gases and Liquids by Means of Series’,Proc. Kon. Ned. Akad. Wet. (Amsterdam) 4 125.

    ADS  Google Scholar 

  • Keesom, W.H.: 1912, ‘On the Deduction from Boltzmann’s Entropy Principle of the Second Virial coefficient for Material particles (in the limit Rigid Spheres of Central Symmetry) Which Exert Central Forces Upon Each Other and for Rigid Spheres of Central Symmetry Containing an Electric Doublet at their Centre’,Comm. Phys. Lab. Leiden, Suppl. 24B 32.

    Google Scholar 

  • Keesom, W.H.: 1942,Helium Elsevier, Amsterdam.

    Google Scholar 

  • Kihara, T.: 1943., ‘The Equation of State of Gases and the Critical State (in Japanese’,Nippon- Sugaku-Buturigakukaisi 17 11.

    Google Scholar 

  • Kihara, T.: 1948, ‘Determination of Intermolecular Forces from the Equation of State of Gases’, J.Phys. Soc. (Japan) 3, 265.

    Article  ADS  Google Scholar 

  • Kihara, T.: 1951a, ‘Determination of the Intermolecular Forces from the Equation of State of Gases’, J.Phys. Soc. (Japan) 6, 184.

    Article  MathSciNet  ADS  Google Scholar 

  • Kihara, T.: 1951b, ‘The Second Virial Coefficient of Non-Spherical Molecules’,J. Phys. Soc. (Japan) 6, 289.

    Article  MathSciNet  ADS  Google Scholar 

  • Kihara, T.: 1953a, ‘On Isihara-Hayashida’s Theory of the Second Virial Coefficient for Rigid, Convex Molecules’,J. Phys. Soc. (Japan) 8, 686.

    Article  ADS  Google Scholar 

  • Kihara, T.: 1953b, ‘Virial Coefficients and Models of Molecules in Gases’,Rev. Mod. Phys. 25, 831.

    Article  ADS  MATH  Google Scholar 

  • Kihara, T., and T. Hikita: 1953, in FourthSymposium on Combustion Williams and Wilkins Co., Baltimore, p.458.

    Google Scholar 

  • Kihara, T., Y. Midzuno, and T. Shizume: 1955, ‘Virial Coefficients and Intermolecular Potential of Helium’,J. Phys. Soc. (Japan) 10 249.

    Article  ADS  Google Scholar 

  • Kilpatrick, J.E., W.E. Keller, E.F. Hammel, and N. Metropolis: 1954, ‘Second Virial Coefficients of He3 and He4’,Phys. Rev. 94 1103.

    Article  ADS  MATH  Google Scholar 

  • Kilpatrick, J.E., W.E. Keller, and E.F. Hammel: 1955, ‘Second Virial Coefficients of Helium from the Exp-Six Potential’,Phys. Rev. 97 9.

    Article  ADS  Google Scholar 

  • Kilpatrick, J.E., and D.I. Ford: 1969, ‘The Virial Equation of State: Its Inversion and Other Manipulations’,Am. J. Phys. 37, 881.

    Article  ADS  Google Scholar 

  • Kirkwood, J.G.: 1933, ‘Quantum Statistics of Almost Classical Assemblies’,Phys. Rev. 44 31. Kramers, H.A.: 1944, ‘Leiden Lectures’.

    Article  ADS  MATH  Google Scholar 

  • Larsen, S.Y.: 1963, ‘Quantum Mechanical Calculation of the Third Virial Coefficient of He4’,Phys. Rev. 130 1426.

    Article  ADS  MATH  Google Scholar 

  • Larsen, S.Y., J. Kilpatrick, E. Lieb, and H. Jordan: 1965, ‘Suppression at High Temperature of Effects due to Statistics in the Second Virial Coefficient of a Real Gas’,Phys. Rev. 140 A129.

    Article  ADS  Google Scholar 

  • Larsen, S.Y., and P.L. Mascheroni: 1970, ‘Quantum-Mechanical Third Virial Coefficient and Three- Body Phase Shifts’,Phys. Rev. A 2 1018.

    Article  ADS  Google Scholar 

  • Lebowitz, J.L., and O. Penrose: 1964, ‘Convergence of Virial Expansions’,J. Math. Phys. 5, 841.

    Article  MathSciNet  ADS  Google Scholar 

  • Lee, T.D., and C.N. Yang: 1959a, ‘Many-Body Problem in Quantum Statistical Mechanics.I. General Formulation’,Phys. Rev. 113 1165.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Lee, T.D., and C.N. Yang: 1959b, ‘Many-Body Problem in Quantum Statistical Mechanics.II. Virial Expansion for Hard-Sphere Gas’,Phys. Rev. 116 25.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Lee, T.D., and C.N. Yang: 1960a, ‘Many-Body Problem in Quantum Statistical Mechanics.III. Zero-Temperature Limit for Dilute Hard Spheres’,Phys. Rev. 117 12.

    Article  MathSciNet  ADS  Google Scholar 

  • Lee, T.D., and C.N. Yang: 1960b, ‘Many-Body Problem in Quantum Statistical Mechanics.IV. Formulation in Terms of Average Occupation Number in Momentum Space’,Phys. Rev 117 22.

    Article  MathSciNet  ADS  Google Scholar 

  • Lee, T.D., and C.N. Yang: 1960c, ‘Many-Body Problem in Quantum Statistical Mechanics.V. Degenerate Phase in Bose-Einstein Condensation’,Phys. Rev. 117 897.

    Article  MathSciNet  ADS  Google Scholar 

  • Lennard-Jones, J.E.: 1924, ‘On the Determination of Molecular Fields.II. From the Equation of State of a Gas’,Proc. Roy. Soc. (London) A106 463.

    ADS  Google Scholar 

  • Lennard-Jones, J.E.: 1931, ‘Cohesion’,Proc. Phys. Soc. (London) A43 461.

    Article  ADS  Google Scholar 

  • Lieb, E.: 1966, ‘Quantum Mechanical Extension of the Lebowitz-Penrose Theorem on the van der Waals Theory’, J.Math. Phys. 7, 1016.

    Article  MathSciNet  ADS  Google Scholar 

  • Lieb, E.: 1967, ‘Calculation of Exchange Second Virial Coefficient of a Hard-Sphere Gas by Path Integrals’, J.Math. Phys. 8, 43.

    Article  ADS  Google Scholar 

  • London, F.: 1954,Superfluids, Vol.11 Wiley, New York, pp. 18–30.

    Google Scholar 

  • Majumdar, R.: 1929, ‘Equation of State’,Bull. Calcutta Math. Soc. 21 107.

    MATH  Google Scholar 

  • Margenau, H., and N.R. Kestner: 1969,Theory of Intermolecular Forces Pergamon, New York.

    Google Scholar 

  • Mascheroni, P.L.: 1970, ‘Low-Temperature Behavior for the Quantum Virial Coefficients’,Phys. Rev. Letters 25, 726.

    Article  ADS  Google Scholar 

  • Mason, E.A., and T.H. Spurling: 1969,The Virial Equation of State Pergamon, Oxford.

    Google Scholar 

  • Mayer, J.E., and M.G. Mayer: 1940,Statistical Mechanics Wiley, New York.

    MATH  Google Scholar 

  • McGinnies, R.T., and L. Jansen: 1956, ‘Validity of the Assumption of Two-Body Interactions in Molecular Physics.I.’,Phys. Rev. 101 1301.

    Article  ADS  Google Scholar 

  • Michels, A., Hub. Wijker, and H.K. Wijker: 1949, ‘Isotherms of Argon Between 0°C and 150 °C and Pressures Up to 2 900 Atmospheres’,Physica , 15 627.

    Article  ADS  Google Scholar 

  • Michels, A., A. Visser, R.J. Lunbeck, and G.J. Wolkers: 1952, ‘Isotherms and Thermodynamic Functions of Methyl Fluoride at Temperatures Between 0 °C and 150 °C and at Pressures Up to 150 Atmospheres’,Physica 18, 114.

    Article  ADS  Google Scholar 

  • Michels, A., J.M. Levelt, and W. de Graff: 1958, ‘Compressibility Isotherms of Argon at Temperatures Between -25 °C and -155 °C, and at Densities Up to 640 Amagat’,Physica 24, 659.

    Article  ADS  Google Scholar 

  • Mie, G.: 1903, ‘Zur kinetischen Theorie der einatomigen Korper’,Ann. d. Phys. 11 657. Mohling, F.: 1963, ‘Quantum Corrections to the Second Virial Coefficient for Helium at HighTemperatures’,Phys. Fluids 6 1097.

    Google Scholar 

  • Mohling, F., and W.T. Grandy, Jr.: 1965, ‘Quantum Statistics of Multicomponent Systems’, J.

    Google Scholar 

  • Math. Phys. 6, 348.

    Google Scholar 

  • Muir, T.: 1960, ATreatise on the Theory of Determinants, Dover, New York. Nagamiya, T.: 1940a, ‘Statistical Mechanics of One-Dimensional Substances, I.’,Proc. Phys.-Math. Soc. Japan 22, 705.

    Google Scholar 

  • Nagamiya, T.: 1940b, ‘Statistical Mechanics of One-Dimensional Substances, II.’,Proc. Phys.-Math. Soc. Japan 22, 1034.

    MathSciNet  MATH  Google Scholar 

  • Nijboer, B.R.A., and L. Van Hove: 1952, ‘Radial Distribution Function of a Gas of Hard Spheresand the Superposition Approximation’,Phys. Rev. 85 777.

    Article  ADS  MATH  Google Scholar 

  • Nijboer, B.R.A., and F. Fieschi: 1953, ‘On the Radial Distribution Function of a Compressed Gasof Rigid Spheres’,Physica 19 545.

    Article  ADS  MATH  Google Scholar 

  • Nilsen, T.S.: 1969a, ‘Quantum-Mechanical Second Virial Coefficient of a Hard-Sphere Gas at HighTemperatures’,Phys. Rev. 51 4675.

    Google Scholar 

  • Nilsen, T.S.: 1969b, ‘Quantum Corrections to the Square-Well Second Virial Coefficient’,Phys. Rev. 186 262.

    Article  ADS  Google Scholar 

  • Opfer, J.E., K. Luszczynski, and R.E. Norberg: 1965, ‘Nuclear Magnetic Susceptibility of He3Vapor’,Phys. Rev. 140 A100.

    Article  ADS  Google Scholar 

  • Osborn, T.A., and T.Y. Tsang: 1976, ‘A Quantum Theory of Higher Virial Coefficients’,Ann. Phys. (N.Y.) 101 119.

    Article  MathSciNet  ADS  Google Scholar 

  • Pais, A., and G.E. Uhlenbeck: 1959, ‘On the Quantum Theory of the Third Virial Coefficient’,Phys. Rev. 116 250.

    Article  ADS  MATH  Google Scholar 

  • Penrose, O.: 1963, ‘Convergence of Fugacity Expansions for Fluids and Lattice Gases’, J.Math Phys., 4. 1312.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Ree, F.H., and W.G. Hoover: 1964, ‘Fifth and Sixth Virial Coefficients for Hard Spheres and HardDisks’, J.Chem. Phys. 40, 939.

    Article  MathSciNet  ADS  Google Scholar 

  • Ree, F.H., and W.G. Hoover: 1967, ‘Seventh Virial Coefficients for Hard Spheres and Hard Disks’,J.Chem. Phys. 46, 4181.

    Google Scholar 

  • Reed, T.M., and K.E. Gubbins: 1973,Applied Statistical Mechanics, McGraw-Hill, New York, Chap.4.

    Google Scholar 

  • Reiner, A.S.: 1966, ‘Application of Faddeev Techniques to the Quantum Theory of the Third VirialCoefficient’,Phys. Rev. 151 170.

    Article  MathSciNet  ADS  Google Scholar 

  • Rice, W.E., and J.O. Hirschfelder: 1954, ‘Second Virial Coefficients of Gases Obeying a ModifiedBuckingham (Exp-Six) Potential’,J. Chem. Phys. 22, 187.

    Article  ADS  Google Scholar 

  • Rosen, P.: 1953, ‘The Nonadditivity of the Repulsive Potential of Helium’,J. Chem. Phys. 21, 1007.

    Article  ADS  Google Scholar 

  • Rowlinson, J.S.: 1964a, ‘An Equation of State of Gases at High Temperatures and Densities’,Mol. Phys. 7, 349.

    Article  MathSciNet  ADS  Google Scholar 

  • Rowlinson, J.S.: 1964b, ‘The Statistical Mechanics of Systems with Steep Intermolecular Potentials’,Mol. Phys. 8, 107.

    Article  ADS  Google Scholar 

  • Ruelle, D.: 1969,Statistical Mechanics Benjamin, New York.

    MATH  Google Scholar 

  • Sherwood, A.E., and J.M. Prausnitz: 1964a, ‘Third Virial Coefficient for the Kihara, Exp-6, andSquare-Well Potentials’,J. Chem. Phys. 41 413.

    Article  ADS  Google Scholar 

  • Sherwood, A.E., and J.M. Prausnitz: 1964b, ‘Intermolecular Potential Functions and the Secondand Third Virial Coefficients’,J. Chem. Phys. 41 429.

    Article  ADS  Google Scholar 

  • Sherwood, A.E., and E.A. Mason: 1965, ‘Virial Coefficients for the Exponential Repulsive Potential’,Phys. Fluids 8, 1577.

    Article  ADS  Google Scholar 

  • Sherwood, A.E., A.G. DeRocco, and E.A. Mason: 1966, ‘Nonadditivity of Intermolecular Forces:Effects on the Third Virial Coefficient’,J. Chem. Phys. 44 2984.

    Article  ADS  Google Scholar 

  • Siegert, A.J.F.: 1952, ‘Note on the Configuration Probabilities’,J.Chem. Phys. 20 572.

    Article  ADS  Google Scholar 

  • Smith, R.A.: 1974, ‘Density Expansion of the Magnetic Susceptibility’,Ann. Phys. (N.Y.) 83 245.

    Article  ADS  Google Scholar 

  • Sutherland, W.: 1893, ‘The Viscosity of Gases and Molecular Force’,Phil. Mag. 36 507.

    Google Scholar 

  • Taylor, J.R.: 1972,Scattering Theory Wiley, New York.

    Google Scholar 

  • Theumann, A.: 1970, ‘Quantum Corrections to the Second Virial Coefficient for a Square-WellPotential’,J. Math. Phys. 11 1772.

    Article  ADS  Google Scholar 

  • Uhlenbeck, G.E., and E. Beth: 1936, ‘The Quantum Theory of the Non-Ideal Gas.I. Deviationsfrom the Classical Theory’,Physica 3, 729.

    Article  MATH  Google Scholar 

  • Ursell, H.D.: 1927, ‘The Evaluation of Gibbs‘ Phase Integral for Imperfect Gases’,Proc. Camb. Phil. Soc. 23, 685.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • van der Waals, J.D.: 1873, ‘Over de continuiteit van den gas-en vloeisstoftoestand’,Ph.D thesis Leiden (unpublished).

    Google Scholar 

  • van Laar, J. J.: 1899, ‘Calculation of the Second Correction to the Quantityb of the Equation ofCondition of van der Waals’,Proc. Roy. Acad. Sei. Amsterdam 1, 273.

    Google Scholar 

  • White, D., T. Rubin, P. Camky, and H.L. Johnston: 1960, ‘The Virial Coefficients of Helium from20 to 300 °K’,J. Phys. Chem. 64 1607.

    Article  Google Scholar 

  • Widom, B.: 1954, ‘The Virial Series of the Ideal Bose-Einstein Gas’,Phys. Rev. 96 16.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Wigner, E.P.: 1932, ‘On the Quantum Correction for Thermodynamic Equilibrium’,Phys. Rev. 40 749.

    Article  ADS  MATH  Google Scholar 

  • Yntema, J.L., and W.G. Schneider: 1950a, ‘Compressibility of Gases at High Temperatures.III. The Second Virial coefficient of Helium in the Temperature Range 600 °C to 1 200 °C’,J. Chem. Phys. 18 641.

    Article  ADS  Google Scholar 

  • Yntema, J.L., and W.G. Schneider: 1950b, ‘On the Intermolecular Potential of Helium’,J. Chem. Phys. 18, 646.

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1987 D. Reidel Publishing Company, Dordrecht, Holland

About this chapter

Cite this chapter

Grandy, W.T. (1987). Interacting Particles, I: Classical and Quantum Clustering. In: Foundations of Statistical Mechanics. Fundamental Theories of Physics, vol 19. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3867-0_7

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-3867-0_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8219-8

  • Online ISBN: 978-94-009-3867-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics