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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
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.
Adhikari, S.K., and R.D. Amado: 1971, ‘Low-Temperature Behavior of the Quantum Cluster Co-efficients’Phys. Rev. Letters 27 485.
Amdur, I., and E.A. Mason: 1958, ‘Properties of Gases at Very High Temperatures’,Phys. Fluids 1, 370.
Arf, C., K. Imre, and E. Ozizmir: 1965, ‘On the Algebraic Structure of the Cluster Expansion inStatistical Mechanics’, J.Math. Phys. 6, 1179.
Barker, J.A., P.J. Leonard, and A. Pompe: 1966, ‘Fifth Virial Coefficients’,J. Chem. Phys. 44, 4206.
Barker, J.A., and J.J. Monaghan: 1962a, ‘Virial Coefficients for Square-Well Potentials’,J. Chem. Phys. 36, 2558.
Barker, J.A., and J.J. Monaghan: 1962b, ‘Fourth Virial Coefficients for the 12–6 Potential’, J.Chem. Phys. 36 2564.
Barker, J.A., and J.J. Monaghan: 1966, ‘Comment on a Letter of Katsura’,J. Chem. Phys. 45, 3482.
Baumgartl, B.J.: 1967, ‘Second and Third Virial Coefficients of a Quantum Gas from Two-ParticleScattering Amplitude’,Z. Phys. 198 148.
Beth, E., and G.E. Uhlenbeck: 1937, ‘The Quantum Theory of the Non-Ideal Gas.II. Behavior atLow Temperatures’,Physic a 4 915.
Boltzmann, L.: 1899, ‘Über die Zustandgleichung v.d. Waals’,Versl. Acad. Wet. Amsterdam 7, 477.
Boyd, M., S. Larsen, and J.E. Kilpatrick: 1966, ‘Exchange and Direct Second Virial Coefficientsfor Hard Spheres’,J. Chem Phys. 45 499.
Bruch, L.W.: 1967, ‘Second Virial Coefficient for the Exponential Repulsive Potential’,Phys. Fluids 10, 2531.
Bruch, L.W.: 1973a, ‘High Temperature Limit of the Exchange Third Virial Coefficient for HardSpheres’,Prog. Theor. Phys. 50, 1537.
Bruch, L.W.: 1973a, ‘High Temperature Limit of the Exchange Third Virial Coefficient for HardSpheres’,Prog. Theor. Phys. 50, 1835.
Brydges, D.C.: 1986, ‘Convergence of Mayer Expansions’,J. Stat. Phys. 42 425.
Brydges, D.C., and P. Federbush: 1978, ‘A New Form of the Mayer Expansion in Statistical Mechanics’,J. Math. Phys. 19 2064.
Clausius, R.: 1870, ‘Über einen auf die Warme andwendbaren mechanischen Satz’,Ann. d. Phys. 141 124.
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.
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.
Dashen, R., S.-K. Ma, and H.J. Bernstein: 1969, ‘S-Matrix Formulation of Statistical Mechanics’,Phys. Rev. 187, 345.
de Boer, J.: 1940, ‘Contribution to the Theory of Compressed Gases’,Ph.D thesis Amsterdam Univ. (unpublished).
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.
de Boer, J., and A. Michels: 1939, ‘Quantum-Mechanical Calculation of the Second Virial- Coefficient of Helium at Low Temperatures’,Physica 6, 409.
de Boer, J., J. van Kranendonk, and K. Compaan: 1950, ‘The Equation of State of Gaseous He3’,Physic a 16, 545.
De Witt, H.E.: 1962, ‘Analytic Properties of the Quantum Corrections to the Second Virial Coefficient’,J. Math. Phys. 3 1003.
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.
Dymond, J.H., and E.B. Smith: 1980,The Virial Coefficients of Pure Gases and Mixtures Clarendon Press, Oxford.
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.
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.
Fosdick, L.D., and H.F. Jordan: 1966, ‘Path Integral Calculation of the Two-Particle Slater Sum for He4’,Phys. Rev. 143 58.
Fré, P.: 1977, ‘The 5-Matrix Formulation of the Cluster Expansion in Statistical Mechanics’,Fort, d. Phys. 25, 579.
Frisch, H.L., and E. Helfand: 1960, ‘Conditions Imposed by Gross Properties on the Intermolecular Potential’,J. Chem. Phys. 32 269.
Gibson, W.G.: 1970, ‘Quantum-Mechanical Second Virial Coefficient at High Temperatures’,Phys. Rev. A2 996.
Gibson, W.G.: 1972a, ‘Quantum Corrections to the Equation of State for Nonanalytic Potentials’,Phys. Rev. A5, 862.
Gibson, W.G.: 1972b, ‘Low-Temperature Expansion of the Third-Cluster Coefficient of a Quantum Gas’,Phys. Rev.A6, 2469.
Gibson, W.G.: 1973, ‘Quantum Corrections to the Virial Coefficients for Potentials with Hard Cores’,Phys. Rev. A7 822.
Goldberger, M.L., and A.N. Adams: 1952, ‘The Configurational Distribution Function in Quantum- Statistical Mechanics’,J. Chem. Phys. 20, 240.
Graben, H.W., and R.D. Present: 1964, ‘Third Virial Coefficient for the Sutherland (∞,v) Potential’, Rev.Mod. Phys. 36, 1025.
Gropper, L.: 1937, ‘Connection Between the Second Virial Coefficient and the Phases of Collision Theory’,Phys. Rev. 51 1108.
Guggenheim, E.A.: 1957,Thermodynamics North-Holland, Amsterdam.
Handelsman, R.A., andJ.B. Keller: 1966, ‘Quantum-Mechanical Second Virial Coefficient of a Hard-Sphere Gas at High Temperature’,Phys. Rev. 148 94.
Happel, H.: 1906, ‘Zur Theorie und Prufung der Zustandgleichung’,Ann. d. Phys. 21, 342.
Hemmer, P.C.: 1968, ‘The Hard Core Quantum Gas at High Temperatures’,Phys. Letters 27A 377.
Hemmer, P.C., andK.J. Mork: 1967, ‘Quantum-Mechanical Second Virial Coefficient of a Hard- Sphere Gas at High Temperatures’,Phys. Rev. 158 114.
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.
Hill, R.N.: 1968, ‘Quantum Corrections to the Second Virial Coefficient at High Temperatures’, J.Math. Phys. 9, 1534.
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.
Hirschfelder, J.O., C.F. Curtiss, and R.B. Bird: 1954,Molecular Theory of Gases and Liquids Wiley, New York.
Holborn, L., and J. Otto: 1925:, ‘ÜCber die Isothermen einiger Gases zwischen +400 ° und -183°Z. Phys. 33, 1.
Jancovici, B.: 1969a, ‘Quantum-Mechanical Equation of State of a Hard-Sphere Gas at High Temperatures’,Phys. Rev. 178, 295.
Jancovici, B.: 1969b, ‘Quantum-Mechanical Equation of State of a Hard-Sphere Gas at High Temperatures. II’,Phys. Rev. 184 119.
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.
Jeans, J.: 1925,Dynamical Theory of Gases Cambridge Univ. Press, Cambridge.
Kahn, B.: 1938, ‘On the Theory of the Equation of State’,Ph.D. thesis, Utrecht. [North-Holland, Amsterdam, 1938.]
Kahn, B., and G.E. Uhlenbeck: 1938, ‘On the Theory of Condensation’,Physic a 5, 399.
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.
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.
Keesom, W.H.: 1942,Helium Elsevier, Amsterdam.
Kihara, T.: 1943., ‘The Equation of State of Gases and the Critical State (in Japanese’,Nippon- Sugaku-Buturigakukaisi 17 11.
Kihara, T.: 1948, ‘Determination of Intermolecular Forces from the Equation of State of Gases’, J.Phys. Soc. (Japan) 3, 265.
Kihara, T.: 1951a, ‘Determination of the Intermolecular Forces from the Equation of State of Gases’, J.Phys. Soc. (Japan) 6, 184.
Kihara, T.: 1951b, ‘The Second Virial Coefficient of Non-Spherical Molecules’,J. Phys. Soc. (Japan) 6, 289.
Kihara, T.: 1953a, ‘On Isihara-Hayashida’s Theory of the Second Virial Coefficient for Rigid, Convex Molecules’,J. Phys. Soc. (Japan) 8, 686.
Kihara, T.: 1953b, ‘Virial Coefficients and Models of Molecules in Gases’,Rev. Mod. Phys. 25, 831.
Kihara, T., and T. Hikita: 1953, in FourthSymposium on Combustion Williams and Wilkins Co., Baltimore, p.458.
Kihara, T., Y. Midzuno, and T. Shizume: 1955, ‘Virial Coefficients and Intermolecular Potential of Helium’,J. Phys. Soc. (Japan) 10 249.
Kilpatrick, J.E., W.E. Keller, E.F. Hammel, and N. Metropolis: 1954, ‘Second Virial Coefficients of He3 and He4’,Phys. Rev. 94 1103.
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.
Kilpatrick, J.E., and D.I. Ford: 1969, ‘The Virial Equation of State: Its Inversion and Other Manipulations’,Am. J. Phys. 37, 881.
Kirkwood, J.G.: 1933, ‘Quantum Statistics of Almost Classical Assemblies’,Phys. Rev. 44 31. Kramers, H.A.: 1944, ‘Leiden Lectures’.
Larsen, S.Y.: 1963, ‘Quantum Mechanical Calculation of the Third Virial Coefficient of He4’,Phys. Rev. 130 1426.
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.
Larsen, S.Y., and P.L. Mascheroni: 1970, ‘Quantum-Mechanical Third Virial Coefficient and Three- Body Phase Shifts’,Phys. Rev. A 2 1018.
Lebowitz, J.L., and O. Penrose: 1964, ‘Convergence of Virial Expansions’,J. Math. Phys. 5, 841.
Lee, T.D., and C.N. Yang: 1959a, ‘Many-Body Problem in Quantum Statistical Mechanics.I. General Formulation’,Phys. Rev. 113 1165.
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.
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.
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.
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.
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.
Lennard-Jones, J.E.: 1931, ‘Cohesion’,Proc. Phys. Soc. (London) A43 461.
Lieb, E.: 1966, ‘Quantum Mechanical Extension of the Lebowitz-Penrose Theorem on the van der Waals Theory’, J.Math. Phys. 7, 1016.
Lieb, E.: 1967, ‘Calculation of Exchange Second Virial Coefficient of a Hard-Sphere Gas by Path Integrals’, J.Math. Phys. 8, 43.
London, F.: 1954,Superfluids, Vol.11 Wiley, New York, pp. 18–30.
Majumdar, R.: 1929, ‘Equation of State’,Bull. Calcutta Math. Soc. 21 107.
Margenau, H., and N.R. Kestner: 1969,Theory of Intermolecular Forces Pergamon, New York.
Mascheroni, P.L.: 1970, ‘Low-Temperature Behavior for the Quantum Virial Coefficients’,Phys. Rev. Letters 25, 726.
Mason, E.A., and T.H. Spurling: 1969,The Virial Equation of State Pergamon, Oxford.
Mayer, J.E., and M.G. Mayer: 1940,Statistical Mechanics Wiley, New York.
McGinnies, R.T., and L. Jansen: 1956, ‘Validity of the Assumption of Two-Body Interactions in Molecular Physics.I.’,Phys. Rev. 101 1301.
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.
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.
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.
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.
Mohling, F., and W.T. Grandy, Jr.: 1965, ‘Quantum Statistics of Multicomponent Systems’, J.
Math. Phys. 6, 348.
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.
Nagamiya, T.: 1940b, ‘Statistical Mechanics of One-Dimensional Substances, II.’,Proc. Phys.-Math. Soc. Japan 22, 1034.
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.
Nijboer, B.R.A., and F. Fieschi: 1953, ‘On the Radial Distribution Function of a Compressed Gasof Rigid Spheres’,Physica 19 545.
Nilsen, T.S.: 1969a, ‘Quantum-Mechanical Second Virial Coefficient of a Hard-Sphere Gas at HighTemperatures’,Phys. Rev. 51 4675.
Nilsen, T.S.: 1969b, ‘Quantum Corrections to the Square-Well Second Virial Coefficient’,Phys. Rev. 186 262.
Opfer, J.E., K. Luszczynski, and R.E. Norberg: 1965, ‘Nuclear Magnetic Susceptibility of He3Vapor’,Phys. Rev. 140 A100.
Osborn, T.A., and T.Y. Tsang: 1976, ‘A Quantum Theory of Higher Virial Coefficients’,Ann. Phys. (N.Y.) 101 119.
Pais, A., and G.E. Uhlenbeck: 1959, ‘On the Quantum Theory of the Third Virial Coefficient’,Phys. Rev. 116 250.
Penrose, O.: 1963, ‘Convergence of Fugacity Expansions for Fluids and Lattice Gases’, J.Math Phys., 4. 1312.
Ree, F.H., and W.G. Hoover: 1964, ‘Fifth and Sixth Virial Coefficients for Hard Spheres and HardDisks’, J.Chem. Phys. 40, 939.
Ree, F.H., and W.G. Hoover: 1967, ‘Seventh Virial Coefficients for Hard Spheres and Hard Disks’,J.Chem. Phys. 46, 4181.
Reed, T.M., and K.E. Gubbins: 1973,Applied Statistical Mechanics, McGraw-Hill, New York, Chap.4.
Reiner, A.S.: 1966, ‘Application of Faddeev Techniques to the Quantum Theory of the Third VirialCoefficient’,Phys. Rev. 151 170.
Rice, W.E., and J.O. Hirschfelder: 1954, ‘Second Virial Coefficients of Gases Obeying a ModifiedBuckingham (Exp-Six) Potential’,J. Chem. Phys. 22, 187.
Rosen, P.: 1953, ‘The Nonadditivity of the Repulsive Potential of Helium’,J. Chem. Phys. 21, 1007.
Rowlinson, J.S.: 1964a, ‘An Equation of State of Gases at High Temperatures and Densities’,Mol. Phys. 7, 349.
Rowlinson, J.S.: 1964b, ‘The Statistical Mechanics of Systems with Steep Intermolecular Potentials’,Mol. Phys. 8, 107.
Ruelle, D.: 1969,Statistical Mechanics Benjamin, New York.
Sherwood, A.E., and J.M. Prausnitz: 1964a, ‘Third Virial Coefficient for the Kihara, Exp-6, andSquare-Well Potentials’,J. Chem. Phys. 41 413.
Sherwood, A.E., and J.M. Prausnitz: 1964b, ‘Intermolecular Potential Functions and the Secondand Third Virial Coefficients’,J. Chem. Phys. 41 429.
Sherwood, A.E., and E.A. Mason: 1965, ‘Virial Coefficients for the Exponential Repulsive Potential’,Phys. Fluids 8, 1577.
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.
Siegert, A.J.F.: 1952, ‘Note on the Configuration Probabilities’,J.Chem. Phys. 20 572.
Smith, R.A.: 1974, ‘Density Expansion of the Magnetic Susceptibility’,Ann. Phys. (N.Y.) 83 245.
Sutherland, W.: 1893, ‘The Viscosity of Gases and Molecular Force’,Phil. Mag. 36 507.
Taylor, J.R.: 1972,Scattering Theory Wiley, New York.
Theumann, A.: 1970, ‘Quantum Corrections to the Second Virial Coefficient for a Square-WellPotential’,J. Math. Phys. 11 1772.
Uhlenbeck, G.E., and E. Beth: 1936, ‘The Quantum Theory of the Non-Ideal Gas.I. Deviationsfrom the Classical Theory’,Physica 3, 729.
Ursell, H.D.: 1927, ‘The Evaluation of Gibbs‘ Phase Integral for Imperfect Gases’,Proc. Camb. Phil. Soc. 23, 685.
van der Waals, J.D.: 1873, ‘Over de continuiteit van den gas-en vloeisstoftoestand’,Ph.D thesis Leiden (unpublished).
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.
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.
Widom, B.: 1954, ‘The Virial Series of the Ideal Bose-Einstein Gas’,Phys. Rev. 96 16.
Wigner, E.P.: 1932, ‘On the Quantum Correction for Thermodynamic Equilibrium’,Phys. Rev. 40 749.
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.
Yntema, J.L., and W.G. Schneider: 1950b, ‘On the Intermolecular Potential of Helium’,J. Chem. Phys. 18, 646.
Author information
Authors and Affiliations
Rights 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