Reduced Hamiltonian descriptions

Henry E. Kandrup

doi:10.1103/PhysRevD.50.2425

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Reduced Hamiltonian descriptions

Henry E. Kandrup

Department of Astronomy and Department of Physics and Institute for Fundamental Theory, University of Florida, Gainesville, Florida 32611

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Phys. Rev. D 50, 2425 – Published 15 August, 1994

DOI: https://doi.org/10.1103/PhysRevD.50.2425

Abstract

Suppose that the evolution of some large, isolated system is Hamiltonian, but that one is interested only in the evolution of a smaller piece of the system, i.e., a subsystem. Alternatively, suppose that one is interested in the evolution of two halves of a single Hamiltonian system, but that one is not interested in the details of their mutual interaction. It is well known that, in either case, if the degrees of freedom of the two pieces are not completely decoupled, an exact reduced description of the individual components cannot be Hamiltonian. It is, however, shown here that, if one only allows for the average effects of each piece on the other, neglecting detailed correlations between the two components in a generalized self-consistent field approximation, one is always led to an approximate reduced description which is Hamiltonian.

References (17)

Cf. A. O. Caldeira and A. J. Leggett, Physica A 121, 587 (1983); Ann. Phys. (N.Y.) 149, 374 (1983) B. L. Hu and H. E. Kandrup, Phys. Rev. D 35, 1776 (1987); H. E. Kandrup, ibid. 39, 2253 (1989).
Cf. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer, Berlin, 1992) for a discussion of the special features associated with Hamiltonian systems. The significance of a non-Hamiltonian evolution has been discussed in the context of quantum field theory by, e.g., B. L. Hu, J. P. Paz and Y. Zhang, Phys. Rev. D 45, 2843 (1992); Regular and Chaotic Dynamics (Springer, Berlin, 1992) for a discussion of the special features associated with Hamiltonian systems. The significance of a non-Hamiltonian evolution has been discussed in the context of quantum field theory by, e.g., ibid. 47, 1576 (1993); S. Habib and R. Laflamme, ibid. 42, 4056 (1990); J. P. Paz and S. Sinha, ibid. 44, 1038 (1991), and numerous references contained therein.
Cf. L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (Pergamon, Oxford, 1981); R. K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972).
Cf. R. K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972).
Cf. the recent review by M. Hillary, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984). A systematic approach to the definition of Wigner functions in a curved spacetime has been presented by E. A. Calzetta, S. Habib and B. L. Hu, Phys. Rev. D 37, 2901 (1988).
This system was originally formulated in galactic dynamics by J. H. Jeans, Mon. Not. R. Astron. Soc. 76, 71 (1915). It was introduced in plasma physics by A. Vlasov, Zh. Eksp. Teor. Fiz. 8, 291 (1938).
Cf. the independent oscillator model of G. W. Ford, M. Kac and P. Mazur, J. Math. Phys. 6, 504 (1965), as generalized by S. Habib and H. E. Kandrup, Phys. Rev. D 46, 5303 (1992), as well as Calderia and Leggett [1].
The particular implementation of projection operators used here is based on an approach due to C. R. Willis and R. H. Picard, Phys. Rev. A 9, 1343 (1974), as adapted to the case of a general interacting subsystem and environment by H. E. Kandrup, Phys. Rev. D 39, 2253 (1989).
L. Boltzmann, Wien. Ber. 66, 275 (1872).
Cf. K. Ogorodnikov, Dynamics of Stellar Systems (Macmillan, New York, 1965); J. Binney and S. Tremaine, Galactic Dynamics (Princeton University, Princeton, 1987).
See, e.g., V. Guillemin, Symplectic Techniques in Physics (Cambridge University, Cambridge, England 1984) or J. E. Marsden, Lectures on Mechanics (Cambridge University, Cambridge, England, 1992) for a detailed mathematical discussion of this general approach to Hamiltonian dynamics. A more terse introduction is provided in V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer, Berlin, 1989). Another, more elementary, introduction is provided by A. Dragt, in Physics of High Energy Particle Accelerators, edited by R. A. Carrigan, F. R. Huson, and M. Month, AIP Conf. Proc. No. 87 (AIP, New York, 1982).
P. J. Morrison, Phys. Lett. 80A, 383 (1980). These brackets also arise in other contexts as well, including the case of two-dimensional perfect fluids. Cf. P. J. Morrison, in Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, edited by M. Tagor and Y. M. Treve, AIP Conf. Proc. No. 99, (AIP, New York, 1982).
Cf. D. Lynden-Bell and N. Sanitt, Mon. Not. R. Astron. Soc. 143, 167 (1969).
H. E, Kandrup, Astrophys. J. 351, 104 (1990).
W. Heitler, The Quantum Theory of Radiation (Clarendon, Oxford, 1954).
N. Rostoker and M. N. Rosenbluth, Phys. Fluids 3, 245 (1960).
J. E. Marsden and A. Weinstein, Physica D 4, 394 (1982).

[c1] Cf. A. O. Caldeira and A. J. Leggett, Physica A 121, 587 (1983); Ann. Phys. (N.Y.) 149, 374 (1983) B. L. Hu and H. E. Kandrup, Phys. Rev. D 35, 1776 (1987); H. E. Kandrup, ibid. 39, 2253 (1989).

[c2] Cf. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer, Berlin, 1992) for a discussion of the special features associated with Hamiltonian systems. The significance of a non-Hamiltonian evolution has been discussed in the context of quantum field theory by, e.g., B. L. Hu, J. P. Paz and Y. Zhang, Phys. Rev. D 45, 2843 (1992); Regular and Chaotic Dynamics (Springer, Berlin, 1992) for a discussion of the special features associated with Hamiltonian systems. The significance of a non-Hamiltonian evolution has been discussed in the context of quantum field theory by, e.g., ibid. 47, 1576 (1993); S. Habib and R. Laflamme, ibid. 42, 4056 (1990); J. P. Paz and S. Sinha, ibid. 44, 1038 (1991), and numerous references contained therein.

[c3] Cf. L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (Pergamon, Oxford, 1981); R. K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972).

[c4] Cf. R. K. Pathria, Statistical Mechanics (Pergamon, Oxford, 1972).

[c5] Cf. the recent review by M. Hillary, R. F. O'Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984). A systematic approach to the definition of Wigner functions in a curved spacetime has been presented by E. A. Calzetta, S. Habib and B. L. Hu, Phys. Rev. D 37, 2901 (1988).

[c6] This system was originally formulated in galactic dynamics by J. H. Jeans, Mon. Not. R. Astron. Soc. 76, 71 (1915). It was introduced in plasma physics by A. Vlasov, Zh. Eksp. Teor. Fiz. 8, 291 (1938).

[c7] Cf. the independent oscillator model of G. W. Ford, M. Kac and P. Mazur, J. Math. Phys. 6, 504 (1965), as generalized by S. Habib and H. E. Kandrup, Phys. Rev. D 46, 5303 (1992), as well as Calderia and Leggett [1].

[c8] The particular implementation of projection operators used here is based on an approach due to C. R. Willis and R. H. Picard, Phys. Rev. A 9, 1343 (1974), as adapted to the case of a general interacting subsystem and environment by H. E. Kandrup, Phys. Rev. D 39, 2253 (1989).

[c9] L. Boltzmann, Wien. Ber. 66, 275 (1872).

[c10] Cf. K. Ogorodnikov, Dynamics of Stellar Systems (Macmillan, New York, 1965); J. Binney and S. Tremaine, Galactic Dynamics (Princeton University, Princeton, 1987).

[c11] See, e.g., V. Guillemin, Symplectic Techniques in Physics (Cambridge University, Cambridge, England 1984) or J. E. Marsden, Lectures on Mechanics (Cambridge University, Cambridge, England, 1992) for a detailed mathematical discussion of this general approach to Hamiltonian dynamics. A more terse introduction is provided in V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer, Berlin, 1989). Another, more elementary, introduction is provided by A. Dragt, in Physics of High Energy Particle Accelerators, edited by R. A. Carrigan, F. R. Huson, and M. Month, AIP Conf. Proc. No. 87 (AIP, New York, 1982).

[c12] P. J. Morrison, Phys. Lett. 80A, 383 (1980). These brackets also arise in other contexts as well, including the case of two-dimensional perfect fluids. Cf. P. J. Morrison, in Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, edited by M. Tagor and Y. M. Treve, AIP Conf. Proc. No. 99, (AIP, New York, 1982).

[c13] Cf. D. Lynden-Bell and N. Sanitt, Mon. Not. R. Astron. Soc. 143, 167 (1969).

[c14] H. E, Kandrup, Astrophys. J. 351, 104 (1990).

[c15] W. Heitler, The Quantum Theory of Radiation (Clarendon, Oxford, 1954).

[c16] N. Rostoker and M. N. Rosenbluth, Phys. Fluids 3, 245 (1960).

[c17] J. E. Marsden and A. Weinstein, Physica D 4, 394 (1982).

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