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Generating Functional for the BBGKY Hierarchy and the N-Identical-Body Problem

  • Stephen F. Gull
  • Anthony J. M. Garrett
Conference paper
  • 285 Downloads
Part of the Fundamental Theories of Physics book series (FTPH, volume 70)

Abstract

We formulate the N-body problem for identical particles by identifying a generating functional for the reduced n-particle probability density functions, nN. The evolution equation for the generating functional follows by substituting the corresponding representation of the Liouville density P N , called the de Finetti representation, into Liouville’s equation. We illustrate this process in model problems, which indicate that there is a difficulty when the flux of the generator depends on its properties away from the constraint plane of fixed normalisation. The evolution equation for the generating functional then depends crucially on N, the actual number of particles, which in real problems is seldom known exactly. For fixed transition rates, and for bosons satisfying detailed balance, the difficulty does not arise, and the de Finetti representation provides useful insights. For the general case of classical statistical mechanics, the difficulty forces us to conclude that the de Finetti generator is not a useful alternative to the BBGKY hierarchy.

Keywords

Phase Space Evolution Equation Detailed Balance Closure Scheme Classical Statistical Mechanic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Balescu, R.: 1975, Equilibrium and Non-equilibrium Statistical Mechanics, Wiley, New York, USA. Chapters 3 and 14.Google Scholar
  2. Clemmow, P.C. & Dougherty, J.P.: 1969, Electrodynamics of Particles and Plasmas, Addison-Wesley, Reading, Mass., USA.Google Scholar
  3. Inguva, R., Smith, C.R., Huber, T.M. & Erickson, G.: 1987. ‘Variational Method for Classical Fluids’, in: C.R. Smith & G.J. Erickson (eds), Maximum Entropy and Bayesian Spectral Analysis and Estimation Problems, Reidel, Dordrecht, Netherlands. pp 295 – 304.Google Scholar
  4. Jaynes, E.T.: 1986, ‘Some Applications and Extensions of the de Finetti Representation Theorem’, in: P. Goel & A. Zellner (eds), Bayesian Inference and Decision Techniques, Studies in Bayesian Econometrics 6, Kluwer, Dordrecht, Netherlands. pp 31 – 42.Google Scholar
  5. Karkheck, J.: 1989. ‘Kinetic Theory and Ensembles of Maximum Entropy’, in: J. Skilling (ed.), Maximum Entropy and Bayesian Methods, Cambridge, England, 1988, Kluwer, Dordrecht, Netherlands. pp 491 – 496.Google Scholar
  6. Lynden-Bell, D.: 1967. ‘Statistical Mechanics of Violent Relaxation in Stellar Systems’, Mon. Not. Roy. Astron. Soc., 136, 101 – 121.Google Scholar
  7. Rodriguez, C.C.: 1989. ‘The Metrics Induced by the Kullback Number’, in: J. Skilling (ed.), Maximum Entropy and Bayesian Methods, Cambridge, England, 1988, Kluwer, Dordrecht, Netherlands. pp 415 – 422.Google Scholar
  8. Saslaw, W.C.: 1985, Gravitational Physics of Stellar and Galactic Systems, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
  9. Skilling, J.: 1989. ‘Classic Maximum Entropy’, in: J. Skilling (ed. ), Maximum Entropy and Bayesian Methods, Cambridge, England, 1988, Kluwer, Dordrecht, Netherlands. pp 45 – 52.Google Scholar
  10. Skilling, J. & Sibisi, S.: 1995. ‘Your title, please…’, in: J. Skilling & Sibisi, S. (eds), Maximum Entropy and Bayesian Methods, Cambridge, England, 1994, Kluwer, Dordrecht, Netherlands. pp xxx-xxx.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Stephen F. Gull
    • 1
  • Anthony J. M. Garrett
    • 1
  1. 1.Cavendish LaboratoryMullard Radio Astronomy ObservatoryCambridgeUK

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