Geometric approach to response theory in non-Hamiltonian systems

Journal of Mathematical Chemistry

Volumn 32, Issue 4, 2002, Pages 339-360

  Ezra, Gregory S ^a  

^a Department of Obstetrics Gynecology   (United States)

Author keywords

Applied differential forms; Non Hamiltonian dynamics; Response theory

Indexed keywords

ARTICLE; GEOMETRY; THEORY; THERMODYNAMICS; TIME;

EID: 0141497178     PISSN: 02599791     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022901505641     Document Type: Article

References (48)

1. R. Abraham and J.E. Marsden, Foundations of Mechanics (Benjamin/Cummings, Reading, MA, 1978).
2. B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge University Press, Cambridge, 1980).
3. R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications (Springer-Verlag, New York, 1988).
4. B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, Modern Geometry - Methods and Applications, Part I. The Geometry of Surfaces, Transformation Groups, and Fields (Springer-Verlag, New York, 1992).
5. T. Frankel, The Geometry of Physics (Cambridge University Press, Cambridge, 1997).
6. V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1978).
7. J.V. José and E.J. Saletan, Classical Dynamics (Cambridge University Press, Cambridge, 1998).
8. J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry (Springer-Verlag, New York, 1999).
9. S.F. Singer, Symmetry in Mechanics (Birkhauser, New York, 2001).
10. J. Liouville, Sur la Théorie de la Variation des constantes arbitraires, J. Math. Pures Appl. 3 (1838) 342-349.
11. S. Guiasu, La Méchanique statistique non conservative, Rev. Roum. Math. Pures Appl. 11 (1966) 541-557.
12. G. Gerlich, Die Verallgemeinerte Liouville-Gleichung, Physica 69 (1973) 458-466.
13. W.-H. Steeb, The Lie derivative, invariance conditions and physical laws, Z. Naturforsch. A 33 (1978) 742-748.
14. W.-H. Steeb, Generalized Liouville equation, entropy and dynamic systems containing limit cycles, Physica A 95 (1979) 181-190.
15. L. Andrey, The rate of entropy change in non-Hamiltonian systems, Phys. Lett. A 111 (1985) 45-46.
16. L. Andrey, Note concerning the paper "The rate of entropy change in non-Hamiltonian systems", Phys. Lett. A 114 (1986) 183-184.
17. J.D. Ramshaw, Remarks on entropy and irreversibility in non-Hamiltonian systems, Phys. Lett. A 116 (1986) 110-114.
18. M.E. Tuckerman, C.J. Mundy and M.L. Klein, Toward a statistical thermodynamics of steady states, Phys. Rev. Lett. 78 (1997) 2042-2045.
19. M.E. Tuckerman, C.J. Mundy and G.J. Martyna, On the classical statistical mechanics of non-Hamiltonian systems, Europhys. Lett. 45 (1999) 149-155.
20. S. Melchionna, Constrained systems and statistical distribution, Phys. Rev. E 61 (2000) 6165-6170.
21. M.E. Tuckerman, Y. Liu, G. Ciccotti and G.J. Martyna, Non-Hamiltonian molecular dynamics: Generalizing Hamiltonian phase space principles to non-Hamiltonian systems, J. Chem. Phys. 115 (2001) 1678-1702.
22. D.J. Evans and G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids (Academic, New York, 1990).
23. W.G. Hoover, Computational Statistical Mechanics (Elsevier, New York, 1991).
24. G.P. Morriss and C.P. Dettmann, Thermostats: Analysis and application, CHAOS 8 (1998) 321-336.
25. J.R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge University Press, Cambridge, 1999).
26. S. Nosé, Constant temperature molecular dynamics methods, Prog. Theor. Phys. Suppl. 103 (1991) 1-46.
27. B.L. Holian, W.G. Hoover and H.A. Posch, Resolution of Loschmidt's paradox: The origin of irreversible behavior in reversible atomic dynamics, Phys. Rev. Lett. 59 (1987) 10-13.
28. B.L. Holian and D.J. Evans, Classical response theory in the Heisenberg picture, J. Chem. Phys. 83 (1985) 3560-3566.
29. B.L. Holian, Classical response theory propagators: An illustrative example, J. Chem. Phys. 84 (1986) 1762-1763.
30. C.J. Mundy, S. Balasubramanian, K. Bagchi, M.E. Tuckerman, G.J. Martyna and M.L. Klein, Nonequilibrium Molecular Dynamics, Reviews in Computational Chemistry, Vol. 14 (Wiley-VCH, New York, 2000) pp. 291-397.
31. P. Reimann, Comment on "Toward a statistical thermodynamics of steady states", Phys. Rev. Lett. 80 (1998) 4104.
32. W.G. Hoover, D.J. Evans, H.A. Posch, B.L. Holian and G.P. Morriss, Comment on "Toward a statistical thermodynamics of steady states", Phys. Rev. Lett. 80 (1998) 4103.
33. M.E. Tuckerman, C.J. Mundy and M.L. Klein, Reply to comment on "Toward a statistical thermodynamics of steady states", Phys. Rev. Lett. 80 (1998) 4105-4106.
34. W.G. Hoover, Liouville's theorems, Gibbs' entropy, and multifractal distributions for nonequilibrium steady states, J. Chem. Phys. 109 (1998) 4164-4170.
35. W.-J. Tzeng and C.-C. Chen, The statistical thermodynamics of steady states, Phys. Lett. A 246 (1998) 52-54.
36. W.G. Hoover, The statistical thermodynamics of steady states, Phys. Lett. A 255 (1999) 37-41.
37. J.D. Ramshaw, Remarks on non-Hamiltonian statistical mechanics, Europhys. Lett. 59 (2002) 319-323.
38. G.S. Ezra, On the statistical mechanics of non-Hamiltonian systems: The generalized Liouville equation, entropy, and time-dependent metrics, Preprint (2002).
39. P. Choquard, Variational principles for thermostatted systems, CHAOS 8 (1998) 350-356.
40. M. Pettini, Geometrical hints for a nonperturbative approach to Hamiltonian dynamics, Phys. Rev. E 47 (1993) 828-850.
41. L. Casetti, M. Pettini and E.G.D. Cohen, Geometric approach to Hamiltonian dynamics and statistical mechanics, Phys. Rep. 337 (2000) 238-341.
42. M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990).
43. W.G. Hoover, Canonical dynamics: Equilibrium phase space distributions, J. Chem. Phys. 31 (1985) 1695-1697.
44. H.A. Posch, W.G. Hoover and F.J. Vesely, Canonical dynamics of the Nosé oscillator: Stability, order and chaos, Phys. Rev. A 33 (1986) 4253-4265.
45. S. Tiwari and R. Ramaswamy, Nosé-Hoover dynamics of a nonintegrable Hamiltonian system, J. Mol. Struct. (Theochem) 361 (1996) 111-116.
46. B.L. Holian, G. Ciccotti, W.G. Hoover, B. Moran and H.A. Posch, Nonlinear-response theory for time-independent fields: Consequences of the fractal nonequilibrium distribution function, Phys. Rev. A 39 (1989) 5414-5421.
47. R.D. Levine and C.E. Wulfman, A unified description of regular and chaotic motion in classical mechanics, Chem. Phys. Lett. 87 (1982) 105-108.
48. A. Lasota and M.C. Mackey, Probabilistic Properties of Deterministic Systems (Cambridge University Press, Cambridge, 1985).