Loss of temporal homogeneity and symmetry in statistical systems: Deterministic versus stochastic dynamics

Symmetry

Volumn 2, Issue 3, 2010, Pages 1201-1249

  Gujrati, Purushottam D ^a,b  

^a UNIVERSITY OF AKRON   (United States)

^b The University of Akron   (United States)

Author keywords

Deterministic and stochastic dynamics; Irreversibility and recurrence paradoxes; Molecular chaos assumption; Phase space confinement; Poincar recurrence; Second law of thermodynamics; Temporal and ensemble averages; Temporal asymmetry and inhomogeneity

Indexed keywords

EID: 79551644509     PISSN: None     EISSN: 20738994     Source Type: Journal    
DOI: 10.3390/sym2031201     Document Type: Review

References (74)

1. Noether, E. Invariante Variationsprobleme. Nachr. Konig. Gesell. Wiss. Gottingen Math.-Phys. Kl. Heft 1918, 2, 235-257; English translation in Transp. Theory Statist. Phys. 1971, 1, 186-207.
2. Wigner, E.P. Symmetries and Reflections; Indiana University Press: Bloomington, IN, USA, 1967.
3. When we consider many particles, it is convenient to introduce the concept of a phase space in which a point represents the collections of particles' coordinates and momenta. Thus, each point in the phase space represents a state of the system. A microstate of the system is represented not by a point, but by a volume element h3N, where h is Planck's constant; see Section 5 for more details.
4. Zeh, H.-D. The Physical Basis of the Direction of Time; Springe-Verlag: Berlin, Germany, 1989.
5. Poincaré, H. Mathematics and Science: Last Essays; Dover Publications Inc.: New York, NY, USA, 1963.
6. A system with unique trajectories requiring an invertible one-to-one mapping (7) is what we call a deterministic system in this work. A Hamiltonian system is deterministic in this sense.
7. We are not considering weak interactions where this symmetry is not exact
8. de Hemptinne, X. Non-Equilibrium Statistical Thermodynamics; World Scientific: Singapore, 1992.
9. Chandrasekhar, S. Stochastic Problems in Physics and Astronomy. Rev. Mod. Phys. 1943, 15, 1-89.
10. Ehrenfest, P.; Ehrenfest, T. The Conceptual Foundations of the Statistical Approach in Mechanics, translated by Moravcsik, M.; Cornell University Press: Ithaca, NY, USA, 1959.
11. Daub, E.E. Probability and thermodynamics: The reduction of the second law. ISIS 1969, 60, 318-330.
12. Davies, P.C.W. The Physics of Time Asymmetry; University of California Press: Berkeley, CA, USA, 1977.
13. Hawking, S.W. Arrow of time in cosmology. Phys. Rev. D 1985, 32, 2489-2495.
14. Coveney, P.V. The second law of thermodynamics: entropy, irreversibility and dynamics. Nature 1988, 333, 409-415.
15. Zak, M. Irreversibility in Thermodynamics. Int. J. Theor. Phys. 1996, 35, 347-382.
16. Price, H. Time's Arrow and Archimedes' Point: New Directions for the Physics of Time; Oxford University Press: New York, NY, USA, 1996.
17. Uffink, J. Irreversibility and the Second Law of Thermodynamics. In Entropy; Greven, A., Keller, G., Warnecke, G., Eds.; Princeton University Press: Princeton, NJ, USA, 2003; pp. 121-146.
18. Price, H. The Thermodynamic Arrow: Puzzles and Pseudo-puzzles. arXiv Phys. 2004, 0402040.
19. The reader should pause to guess about our motivation to italicize container. During the process of expansion or at any other time, the gas molecules are always experiencing the walls of the container. Later, we will see that the presence of walls becomes a central concept for breaking the temporal symmetry. Their presence gives rise to boundary conditions for the collisions of gas particles with the walls. These collisions are not described by potentials that are part of the Hamiltonian of the system, and destroy the temporal symmetry, just like the presence of walls destroys the homogeneity of space.
20. Prigogine, I.; Grecos, A.; George, C. On the relation of dynamics to statistical mechanics. Celes. Mech. 1977, 16, 489-507.
21. Maxwell, J.C. Theory of Heat; Longmans, Green & Co: London, UK, 1871; J. Chem. Soc. London 1875, 28, 493-508.
22. Boltzmann, L. Über die mechanische Bedeutung des Zweiten Hauptsatzes der Wärmegleichgewicht. Wien. Ber. 1877, 76, 373-435.
23. Boltzmann, L. Lectures on Gas Theory; University Of California Press: Berkeley, CA, USA, 1964.
24. Clausius, R. Über die Wärmeleitung gasförmiger Körper. Ann. Physik 1862, 115, 1-57.
25. Loschmidt, J. Sitznngsberichte der Akademie der Wissenschaften. Wien. Ber. 1876, 73, 128-142.
26. Kröning, A. Grundziuge einer Theorie der Gase. Ann. Physik 1856, 99, 315-322.
27. Burbury, S.H. Boltzmann's minimum fuction. Nature 1895, 52, 104-105.
28. von Smoluchowski, M. Experimetell nachweisbare der ublichen Thermodynamik widersprechende Molekularphanomene. Physik Z 1912, 13, 1069.
29. Terr Haar, D.; Green, C.D. The statistical aspect of Bolzmann's H-Theorem. Proc. Phys. Soc. A 1953, 66, 153-159.
30. ter Haar, D. Foundations of Statistical Mechanics. Rev. Mod. Phys. 1953, 7, 289-338.
31. Sklar, L. Physics and Chance; Cambridge University Press: Cambridge, UK, 1993.
32. Lebowitz, J.L. Statistical mechanics: A selective review of two central issues. Rev. Mod. Phys. 1999, 71, S346-S357.
33. We discuss this important ansatz later in Section 6.1 in more detail, where we find that the ansatz is not fulfilled in a deterministic dynamics. We suggest that one needs a stochastic dynamics for the ansatz to be satisfied.
34. Poincarè, H. Sur le probl'eme des trois corps et leséquations de la dynamique. Acta Math. 1890, 13, 1-270; see also Chandrasekhar, S. [9].
35. Zermelo, E. On a Theorem of Dynamics and the Mechanical Theory of Heat. Ann. Physik 1896, 57, 485-494;
36. Zermelo, E. On the Mechanical explanation of Irreversible Processes. Wied. Ann. 1897, 60, 392-398.
37. Huang, K. Statistical Mechanics, 2nd ed.; John Wiley and Sons: New York, NY, USA, 1987.
38. Gujrati, P.D. Poincare Recurrence, Zermelo's Second Law Paradox, and Probabilistic Origin in Statistical Mechanics. arXiv 2008, 0803.0983.
39. Hoover, W.G. Computational Statistical Mechanics; Elsevier: Amsterdam, The Netherlands, 1991.
40. Boltzmann, L. Reply to Zermelo's Remarks on the Theory of Heat. Ann. Physik 1896, 57, 773-784;
41. Boltzmann, L. On Zermelo's Paper: On the Mechanical Explanation of Irreversible Processes. Ann. Physik 1897, 60, 392-398.
42. Jäckle, J. On the glass transition and the residual entropy of glasses. Philos. Mag. B 1981, 44, 533-545
43. Jäckle, J. Residual entropy in glasses and spin glasses. Physica B 1984, 127, 79-86.
44. Palmer, R.G. Broken Ergodicity. Adv. Phys. 1982, 31, 669-735.
45. Landau, L.D.; Lifshitz, E.M. Statistical Physics, 3rd ed.; Pergamon Press: Oxford, UK, 1986.
46. Planck, M. The Theory of Heat Radiation. In The history of modern physics 1800-1950; American Institute of Physics: New York, NY, USA, 1988; Volume 11.
47. Lewis, G.N. The entropy of radiation. Proc. Nat. Acad. Sci. 1927, 13, 307-313.
48. Lanyi, G. Thermal Equilibrium Between Radiation and Matter. Found. Phys. 2003, 33, 511-528.
49. Whether the entire universe satisfies the second law is an unsettled problem at present. To verify it requires making measurement of some sort on different parts of an ever-expanding universe at the same instant. It is not clear whether it is possible to send signals to distant receding parts of our expanding universe to be able to make this measurement; most of these parts are probably causally disconnected from us. The idea of an isolated system is based on an exterior from which it is isolated. To test the isolation, we need to perform some sort of test from outside the isolated system. We need to know if we live in a universe or a multiverse. Also, is there a physical boundary to our universe isolating it from outside? By physical, we mean it to be composed of matter and energy. What is outside this boundary, and how can we test or know what is outside, while remaining inside the isolated universe? If there is a physical boundary, does it contain all the matter and energy within it or is there energy outside it? Are dark matter and dark energy confined within this boundary or do they also exist outside it? If it is vacuum outside, does it have any vacuum energy, which is then absorbed by the expanding universe? At present, we do not know answers to these questions. It is highly likely that there is no physical boundary to the universe that we can detect. Everything that we observe is causally connected to us and lies within the universe. Therefore, we cannot see its boundary, which is causally disconnected from us. For all practical purposes, the universe appears to be "unbounded" to us. The only sensible thing we can speak of is a part (within the causally connected observable universe) of the universe, finite in extent within this "unbounded" universe. The surrounding medium of the observable universe and the 3K radiation generate stochasticity and ensure that the observable universe satisfies the second law. In our opinion, causally disconnected parts of the universe have no bearing on the second law. Therefore, we will not worry about this issue here.
50. This is impossible at least due to the presence of the remanent 3 K radiation from the big bang that permeates the entire universe. We will neglect this radiation and other thermal radiation from the walls and other external bodies when we consider a deterministic dynamics. They will become an integral part of the discussion when we deal with stochastic dynamics.
51. A truly isolated system is really an idealization and will not correctly represent a physical system, as noted in the previous footnote. For a correct representation, the description requires a probabilistic approach, which follows from the loss of temporal inhomogeneity; see the discussion leading to (9).
52. Gibbs, J.W. Elementary Principles in Statistical Mechanics; Ox Bow Press: Woodbridge, VA, USA, 1981.
53. Tolman, R.C. The Principles of Statistical Mechanics; Oxford University: London, UK, 1959.
54. Rice, S.A.; Gray, P. The Statistical Mechanics of Simple Liquids; John Wiley & Sons: New York, NY, USA, 1965.
55. Rice, O.K. Statistical Mechanics, Thermodynamics and Kinetics; W.H. Freeman: San Francisco, CA, USA, 1967.
56. Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J. 1948, 27, 379-423; 623-656.
57. Jaynes, E.T. Papers on Probability, Statistics and Statistical Physics. Resenkrantz, R.D., Ed.; Reidel Publishing: Dordrecht, The Netherlands, 1983.
58. Sethna, J.P. Statistical Mechanics: Entropy, Order Parameters and Complexity; Oxford University Press: New York, NY, USA, 2006.
59. Becker, R. Theory of Heat, 2nd Ed.; Leibfried, G., Ed.; Springer-Verlag: New York, NY, USA, 1967.
60. Nemilov, S.V. Thermodynamic and Kinetic Aspects of the Vitreous State; CRC Press: Boca Raton, FL, USA, 1995.
61. Gutzow, I.; Schmelzer, J. The Vitreous State: Thermodynamics, Structure, Rheology and Crystallization; Springer: Berlin, Germany, 1995.
62. Gujrati, P.D. Where is the residual entropy of a glass hiding? arXiv 2009, 0908.1075.
63. The division in cells is to ensure that the number of microstates does not become infinite even for a finite system (finite N, E and V)
64. Even the collisions are deterministic in such a system
65. Kac, M. Some remarks on the use of probability in classical statistical mechanics. Bull. Acad. Roy. Belg. 1956, 42, 356-361.
66. Henin, F. Entropy, dynamics and molecular chaos, Kac's model. Physica 1974, 77, 220-246.
67. Thompson, C.J. Mathematical Statistical Mechanics; Princeton University: Princeton, NJ, USA, 1979.
68. Gujrati, P.D. Irreversibility, Molecular Chaos, and A Simple Proof of the Second Law. arXiv 2008, 0803.1099.
69. Fernando, P. Lack of Molecular Chaos and the Role of Stochasticity in Kac's Ring Model; The University of Akron: Akron, OH, USA, 2009.
70. Indeed, it is found that the entire phase space with 2N microstates is broken into disjoint components, so that the initial microstate in a given component evolves into microstates belonging to this component alone; no microstates from other components occur in the evolution. However, Poincaré's recurrence theorem applies to each component separately.
71. Hoover, W.G. Molecular Dynamics; Springer-verlag: Berlin, Germany, 1986.
72. As the system is no longer isolated because of its interaction with the environment, E, N, V need not remain constant and may fluctuate. However, as long as we are dealing with very weak environmental noise, we can safely treat the system as quasi-isolated in that the widths of their spread can be neglected.
73. Indeed, for a macroscopic system, the probability to come back to a previously generated microstate will be almost negligible
74. Gautam, M. The Role of Walls' Stochastic Forces in Statistical Mechanics: Phenomenon of Time Irreversibility; The University of Akron: Akron, OH, USA, 2009.