Classical markovian kinetic equations: Explicit form and H-theorem

Transport Theory and Statistical Physics

Volumn 28, Issue 4, 1999, Pages 325-348

  Tzanakis, Constantinos ^a   Grecos, Alkis P ^b  

^a UNIVERSITY OF CRETE   (Greece)

^b UNIVERSITÉ LIBRE DE BRUXELLES   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033247678     PISSN: 00411450     EISSN: None     Source Type: Journal    
DOI: 10.1080/00411459908205847     Document Type: Article

References (55)

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6. The litterature is extensive, see e.g. Ref.3 and P. Résibois and M. de Leener, Classical kinetic theory of fluids, Wiley, New York(1977); R. Kubo, M. Toda and N. Hashitsume, Statistical Physics, Vol II, Springer Berlin (1985).
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11. E.B. Davies, Comm. Math. Phys. 39, 91 (1974); E.B. Davies, Math. Annalen 219, 147 (1976);
12. e.g N.G. van Kampen and I. Oppenheim, Physica A138, 231 (1986), section 2 and p.239 eq.(3.5); V. Rudyak and I. Ershov, Physica A219, 351 (1995), eq.(15); J. Piasecki and G. Szamel, Physica A143, 114 (1987), section 3 eq.(29); S.A. Aldeman, J. Chem. Phys. 64, 124 (1976), eq(3.19); M. Tokuyama, Physica A169, 147 (1990), eq.(4.9a); Ref. 3, eq(18.1.18); P. Grigolini, in Noise in nonlinear dynamical systems vol.I, F. Moss and P.V.E. McClintock (eds.), (1988), ch.5, eq(58); T.W. Marshall, Physica A 103, 172 (1980), section 3 and eq(3.15).
13. e.g N.G. van Kampen and I. Oppenheim, Physica A138, 231 (1986), section 2 and p.239 eq.(3.5); V. Rudyak and I. Ershov, Physica A219, 351 (1995), eq.(15); J. Piasecki and G. Szamel, Physica A143, 114 (1987), section 3 eq.(29); S.A. Aldeman, J. Chem. Phys. 64, 124 (1976), eq(3.19); M. Tokuyama, Physica A169, 147 (1990), eq.(4.9a); Ref. 3, eq(18.1.18); P. Grigolini, in Noise in nonlinear dynamical systems vol.I, F. Moss and P.V.E. McClintock (eds.), (1988), ch.5, eq(58); T.W. Marshall, Physica A 103, 172 (1980), section 3 and eq(3.15).
14. e.g N.G. van Kampen and I. Oppenheim, Physica A138, 231 (1986), section 2 and p.239 eq.(3.5); V. Rudyak and I. Ershov, Physica A219, 351 (1995), eq.(15); J. Piasecki and G. Szamel, Physica A143, 114 (1987), section 3 eq.(29); S.A. Aldeman, J. Chem. Phys. 64, 124 (1976), eq(3.19); M. Tokuyama, Physica A169, 147 (1990), eq.(4.9a); Ref. 3, eq(18.1.18); P. Grigolini, in Noise in nonlinear dynamical systems vol.I, F. Moss and P.V.E. McClintock (eds.), (1988), ch.5, eq(58); T.W. Marshall, Physica A 103, 172 (1980), section 3 and eq(3.15).
15. e.g N.G. van Kampen and I. Oppenheim, Physica A138, 231 (1986), section 2 and p.239 eq.(3.5); V. Rudyak and I. Ershov, Physica A219, 351 (1995), eq.(15); J. Piasecki and G. Szamel, Physica A143, 114 (1987), section 3 eq.(29); S.A. Aldeman, J. Chem. Phys. 64, 124 (1976), eq(3.19); M. Tokuyama, Physica A169, 147 (1990), eq.(4.9a); Ref. 3, eq(18.1.18); P. Grigolini, in Noise in nonlinear dynamical systems vol.I, F. Moss and P.V.E. McClintock (eds.), (1988), ch.5, eq(58); T.W. Marshall, Physica A 103, 172 (1980), section 3 and eq(3.15).
16. e.g N.G. van Kampen and I. Oppenheim, Physica A138, 231 (1986), section 2 and p.239 eq.(3.5); V. Rudyak and I. Ershov, Physica A219, 351 (1995), eq.(15); J. Piasecki and G. Szamel, Physica A143, 114 (1987), section 3 eq.(29); S.A. Aldeman, J. Chem. Phys. 64, 124 (1976), eq(3.19); M. Tokuyama, Physica A169, 147 (1990), eq.(4.9a); Ref. 3, eq(18.1.18); P. Grigolini, in Noise in nonlinear dynamical systems vol.I, F. Moss and P.V.E. McClintock (eds.), (1988), ch.5, eq(58); T.W. Marshall, Physica A 103, 172 (1980), section 3 and eq(3.15).
17. e.g N.G. van Kampen and I. Oppenheim, Physica A138, 231 (1986), section 2 and p.239 eq.(3.5); V. Rudyak and I. Ershov, Physica A219, 351 (1995), eq.(15); J. Piasecki and G. Szamel, Physica A143, 114 (1987), section 3 eq.(29); S.A. Aldeman, J. Chem. Phys. 64, 124 (1976), eq(3.19); M. Tokuyama, Physica A169, 147 (1990), eq.(4.9a); Ref. 3, eq(18.1.18); P. Grigolini, in Noise in nonlinear dynamical systems vol.I, F. Moss and P.V.E. McClintock (eds.), (1988), ch.5, eq(58); T.W. Marshall, Physica A 103, 172 (1980), section 3 and eq(3.15).
18. e.g N.G. van Kampen and I. Oppenheim, Physica A138, 231 (1986), section 2 and p.239 eq.(3.5); V. Rudyak and I. Ershov, Physica A219, 351 (1995), eq.(15); J. Piasecki and G. Szamel, Physica A143, 114 (1987), section 3 eq.(29); S.A. Aldeman, J. Chem. Phys. 64, 124 (1976), eq(3.19); M. Tokuyama, Physica A169, 147 (1990), eq.(4.9a); Ref. 3, eq(18.1.18); P. Grigolini, in Noise in nonlinear dynamical systems vol.I, F. Moss and P.V.E. McClintock (eds.), (1988), ch.5, eq(58); T.W. Marshall, Physica A 103, 172 (1980), section 3 and eq(3.15).
19. A. Dimakis and C. Tzanakis, J Physics A 29, 577 (1996), section 2.
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21. G. Lindblad, Comm. Math. Phys. 48, 119 (1976); V. Gorini, A. Kossakowski and E.C.G. Sudarshan, J. Math. Phys. 17, 821 (1976); V. Gorini, A. Frigerio, M. Veri, A. Kossakowski and E.C.G. Sudarshan, Rep. Math. Physics 13, 149 (1978).
22. G. Lindblad, Comm. Math. Phys. 48, 119 (1976); V. Gorini, A. Kossakowski and E.C.G. Sudarshan, J. Math. Phys. 17, 821 (1976); V. Gorini, A. Frigerio, M. Veri, A. Kossakowski and E.C.G. Sudarshan, Rep. Math. Physics 13, 149 (1978).
23. ∞| < ε, ∀x ∈ X - K: cf. K. Taira, Diffusion processes and Partial Differential Equations, Academic Press, Boston (1988), p.12.
24. M. Reed and B. Simon, Methods of modern mathematical physics I: Functional Analysis, Academic Press, New York (1974), chIV.
25. W. Rudin, Real and Complex analysis, McGraw-Hill, London (1970), chs. 6, 3, 2, 7.
26. K. Yosida, Functional analysis, Springer, Berlin (1971), chs.IX, IV, II.
27. E.B. Dynkin, Markov Processes, Vol I, Springer, Berlin (1965), ch.1.
28. E.B. Davies, One-parameter semigroups, Academic Press, London (1981), chs. 1, 2, 7.
29. J-M. Bony, P. Courrège and P. Priouret, Ann. Inst. Fourier (Grenoble) 18 No2, 369 (1968), théorème V and section I.2.
30. R.F. Pawula, Phys. Rev. 162, 186 (1967). It contains Lemma A.2.1 specialized to the Kramers-Moyal expansion of the linear Boltzmann equation.
31. J.V. Pulé and A. Verbeure, J. Math. Phys. 20, 2286 (1979). Lemma A.2.1 is shown using a different definition of dissipativity, a priori stronger than (3.4).
32. E. Nelson, Dynamical theories of Brownian motion, Princeton University Press, Princeton (1967); its theorem 5.3 outlines the derivation of (4.2) using (4.1).
33. See Ref.18, théorèmes IX, IX', XIV; cf. Ref12, theorem 9.4.1 and K. Sato, T. Ueno, J. Math. Kyoto Univ. 4-3, 529 (1965), Lemma 6.1.
34. That this should follow by an explicit construction of a counter example (provided by Lemma A.2.1) has been pointed to one of us (C.T.) by E.B. Davies (private communication).
35. ij is a nonegative - definite matrix function.
36. a for some M > 0 and ∥ ∥ is the Euclidean norm.
37. The proof is contained in L. Stoica, Local operators and Markov processes, Springer Lecture Notes in Mathematics vol. 816, Berlin (1980), proof of theorem 3.3 in ch.I.
38. R. Courant and D. Hilbert, Methods of mathematical physics, Vol II, Wiley, New York (1962), sections IV.2, IV.7; J.R. Mika and J. Banasiak, Singularly perturbed evolution equations with applications to kinetic theory, World Scientific, Singapore 1995, section 3.7.
39. R. Courant and D. Hilbert, Methods of mathematical physics, Vol II, Wiley, New York (1962), sections IV.2, IV.7; J.R. Mika and J. Banasiak, Singularly perturbed evolution equations with applications to kinetic theory, World Scientific, Singapore 1995, section 3.7.
40. O.A. Oleinik and E.B. Radkevich, Second order equations with nonegative characteristic form, Hogy Nauky, Moskow (1971), (in Russian), theorem 1.8.2 p.86.
41. A.D. Venttsell, Theory of Probability and its applications, 4, 164 (1959), theorem 1; Ref.12, theorem 9.5.1; Ref.18, théorème XIII.
42. Ref.12, theorems 10.1.1, 10.3; K. Taira, On the existence of Feller semigroups with boundary conditions, Memoirs of the AMS 99 No475 (1992), theorems 1, 2; cf. K. Sato et al. (Ref.22), theorem 5.2. For the construction of MS on compact manifolds see Ref.18, théorèmes XVI, XVI', XIX.
43. Ref.12, theorems 10.1.1, 10.3; K. Taira, On the existence of Feller semigroups with boundary conditions, Memoirs of the AMS 99 No475 (1992), theorems 1, 2; cf. K. Sato et al. (Ref.22), theorem 5.2. For the construction of MS on compact manifolds see Ref.18, théorèmes XVI, XVI', XIX.
44. P.R. Halmos, Measure theory, Van Nostrand, New York (1950), sections 36, 55.
45. tA)dμ̃ for the observables A (see e.g. Ref.15 p.392), though this has no physical interpretation.
46. X Φdv/N) ≤ (latin small letter esh) h ○ Φdv/N.
47. →) on a set of measure zero we get the result.
48. n) ≤ a < 1 its existence is ensured as well. However (3.11) is essential in the context of kinetic theory. Nevertheless, even in this case we do not know if this invariant measure is absolutely continuous with respect to the Lebesgue measure (cf.(5.4″)).
49. P.M. Alberti and A. Uhlmann, J. Math. Phys. 22, 2345 (1981), theorem 1.2; P.M. Alberti and A. Uhlmann, Math. Nachrichten 97, 279 (1980), proposition 5.3.
50. P.M. Alberti and A. Uhlmann, J. Math. Phys. 22, 2345 (1981), theorem 1.2; P.M. Alberti and A. Uhlmann, Math. Nachrichten 97, 279 (1980), proposition 5.3.
51. For detailed proofs see J. Lamberti, Stochastic Processes, Springer, New York (1977), sections 7.6, 7.7, and Ref.12, theorems 9.2.1-9.2.3.
52. J.A. van Casteren, Generators of strongly continuous semigroups, Pitnam Publishing Inc., Boston (1985), theorem 2.1.
53. For the proof see Ref.18, théorème V in connection with théorème III and §§0.1.4, I.2.6, I.2.1, I.2.10.
54. Its existence is implied by the paracompactness of X.
55. For their physical interpretation see E. Wong, in Proceeding of symposia in applied Mathematics vol. XVI, AMS, Providence (1964), p.264.