Skip to main content
Log in

On the Unification of Geometric and Random Structures through Torsion Fields: Brownian Motions, Viscous and Magneto-fluid-dynamics

Foundations of Physics Aims and scope Submit manuscript

Abstract

We present the unification of Riemann–Cartan–Weyl (RCW) space-time geometries and random generalized Brownian motions. These are metric compatible connections (albeit the metric can be trivially euclidean) which have a propagating trace-torsion 1-form, whose metric conjugate describes the average motion interaction term. Thus, the universality of torsion fields is proved through the universality of Brownian motions. We extend this approach to give a random symplectic theory on phase-space. We present as a case study of this approach, the invariant Navier–Stokes equations for viscous fluids, and the kinematic dynamo equation of magnetohydrodynamics. We give analytical random representations for these equations. We discuss briefly the relation between them and the Reynolds approach to turbulence. We discuss the role of the Cartan classical development method and the random extension of it as the method to generate these generalized Brownian motions, as well as the key to construct finite-dimensional almost everywhere smooth approximations of the random representations of these equations, the random symplectic theory, and the random Poincaré–Cartan invariants associated to it. We discuss the role of autoparallels of the RCW connections as providing polygonal smooth almost everywhere realizations of the random representations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland/Kodansha, Amsterdam/Tokyo, 1981).

  2. P. Malliavin (1978) Géométrie Différentielle Stochastique Les Presses University Montreal

    Google Scholar 

  3. D. L. Rapoport (2002) Rep. Math. Phys. 49 IssueID1 1–27 Occurrence Handle1004.60060 Occurrence Handle2003e:58055

    MATH  MathSciNet  Google Scholar 

  4. D. L. Rapoport, In Proceedings, Third Inter. Conf. on Dynamical Systems and Differential Equations, Kennesaw, May 2000, Discrete and Cont. Dyn. Sys. special issue, S. Hu (ed.), 2000.

  5. D. L. Rapoport (2002) Rep. Math. Phys. 50 IssueID2 211–250 Occurrence Handle1032.58020 Occurrence Handle2003i:58070

    MATH  MathSciNet  Google Scholar 

  6. D. L. Rapoport (2003) Rand. Oper. Stoch. Eqs. 11 IssueID2 109–136 Occurrence Handle1051.60045 Occurrence Handle2004g:76043

    MATH  MathSciNet  Google Scholar 

  7. D. L. Rapoport, Rand. Oper. Stoch. Eqs. 11(2), 359–380 (2003); ibid. in Instabilities and Nonequilibrium Structures vol. VII & VIII, O. Descalzi, J. Martinez and E. Tirapegui (eds.), 313–332, Complex Phenomenae and Nonlinear Systems Series (Kluwer, Dordrecht, 2004).

  8. K. Kunita (1994) Stochastic Flows and Stochastic Differential Equations Cambridge Univ Press Cambridge

    Google Scholar 

  9. P. Baxendale K. D. Elworthy (1983) Z.Wahrschein.verw.Gebiete 65 245–267

    Google Scholar 

  10. V. I. Arnold B. A. Khesin (1999) Topological Methods in Hydrodynamics Springer New York

    Google Scholar 

  11. J. M. Bismut (1982) Mécanique Analytique Springer Berlin

    Google Scholar 

  12. K. D. Elworthy, In Diffusion Processes and Related Problems in Analysis, M. Pinsky et al. (eds.), vol. II, (Birkhauser, Basel, 1992).

  13. K.D. Elworthy X. M Li (1994) J. Func. Anal. 125 252–286 Occurrence Handle10.1006/jfan.1994.1124 Occurrence Handle95j:60087

    Article  MathSciNet  Google Scholar 

  14. V. I. Arnold, Mathematical Methods in Classical Mechanics (Springer, 1982).

  15. M. Taylor, Partial Differential Equations, vols. I & III (Springer,1995).

  16. M. Ghill and S. Childress, Stretch, Twist and Fold (Springer, 1995).

  17. D. Ebin J. Marsden (1971) Ann. Math. 92 102–163 Occurrence Handle42 #6865

    MathSciNet  Google Scholar 

  18. H. K. Moffatt R. L. Ricca (1992) Procds. Roy. London. 439 411–429 Occurrence Handle1992RSPSA.439..411M Occurrence Handle94b:53006

    ADS  MathSciNet  Google Scholar 

  19. H. Marmanis (1998) Phys. Fluids. 10 IssueID6 1028 Occurrence Handle10.1063/1.869762 Occurrence Handle99f:76070a

    Article  MathSciNet  Google Scholar 

  20. A. Einstein, Annalen Phys. 17 (1905); J. Perrin, C. R. Acad. Sci. Paris 147, 475 and 530 (1908); ibid. Ann. de Chim. et de Phys. 18, 1 (1909); M. Smoluchowski, Ann. der Phys. 21, 756 (1906).

  21. D. Rapoport W. Rodrigues Q. Souza Particlede J. Vaz (1994) Alg. Groups. Geom. 14 27

    Google Scholar 

  22. J. Wenzelburger (1998) J. Geom. Phys. 24 334–352 Occurrence Handle10.1016/S0393-0440(97)00016-8 Occurrence Handle0977.74003 Occurrence Handle99e:73088

    Article  MATH  MathSciNet  Google Scholar 

  23. H. Kleinert (1989) The Gauge Theory of Defects, vols. I and II World Scientific Singapore

    Google Scholar 

  24. D Rapoport (1995) NoChapterTitle Y Aizawa (Eds) Dynamical Systems and Chaos Proceedings(Tokyo, 1994) World Scientific Singapore 73–77

    Google Scholar 

  25. D. Rapoport, in Instabilities and Non-Equilibrium Structures, vol. VI, 359–370, Proceedings (Valparaíso, Chile, December 1995), E. Tirapegui and W. Zeller (eds.), Series in Nonlinear Phenomenae and Complex Systems (Kluwer, Boston, 2000); ibid. in Instabilities and Nonequilibrium Structures, vol. IX O. Descalzi, J. Martinez and S. Rica (eds.), 259–270, Nonlinear Phenomenae and Complex Systems (Kluwer, 2004).

  26. R. W. Sharpe, Differential Geometry, Cartan’s Generalization of the Erlangen Program (Springer, 2000).

  27. D. Rapoport, in Group XXI, Physical Applications and Mathematical Aspects of Algebras, Groups and Geometries, vol. I, Proceedings (Clausthal, 1996), H.D. Doebner et al. (eds.), (World Scientific, Singapore, 1997); ibid. 8(1), 129–146, (1998); ibid. in Gravitation, The Space-Time Structure, Proceedings of the Eighth Latinoamerican Symposium in Relativity and Gravitation, Aguas de Lindoia, Brasil , 220–229, W. Rodrigues and P.~Letelier (eds.) (World Scientific, Singapore, 1995); ibid. Proceedings of the IXth Marcel Grossman Meeting in Relativity, Gravitation and Field Theories, Rome, June 2000, R. Ruffini et al. (eds.) (World Scientific, Singapore, 2003).

  28. V. de Sabbata, and C. Sivaram, Spin and Torsion in Gravitation (World Scientific, Singapore,1991); F. Hehl, J. Dermott McCrea, E. Mielke, and Y. Ne’eman, Phys. Rep. 258, 1–157, 1995.

  29. Z. Schuss, Theory and Applications of Stochastic Differential Equations (Academic Press, Wiley, New York, 1990); C. Gardiner, Handbook of Stochastic Processes,and its applications to Physics, Chemistry and the Natural Sciences, Springer Series in Synergetics (Springer,New York/Berlin, 1991); R. Mazo, Brownian Motion, Fluctuations, Dynamics and its Applications (Clarendon Press, Oxford, 2002); N.G. .Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981); H. Risken, The Fokker–Planck Equation: Methods of Solutions and Applications, 2nd edn. (Springer, New York, 1989).

  30. G. Rham Particlede (1984) Differentiable Manifolds Springer New York

    Google Scholar 

  31. V. I. Arnold (1966) Ann.Inst. Fourier 16 319–361 Occurrence Handle0148.45301

    MATH  Google Scholar 

  32. Yu. Gliklikh, Global Analysis in Mathematical Physics, Applied Mathematics Sciences 122, (Springer, Berlin, 1997); ibid. Ordinary and Stochastic Differential Geometry as a Tool for Mathematical-Physics, (Kluwer, Dordrecht, 1996); ibid. Viscous Hydrodynamics through stochastic perturbations of flow of perfect fluids on groups of diffeomorphisms, Proceedings of the Voronezh State University (Russia), 1, 83–91, 2001.

  33. T. Taylor (1993) Stochastic and Stochastic Rep. 43 179–197 Occurrence Handle0786.60093 Occurrence Handle95g:58133

    MATH  MathSciNet  Google Scholar 

  34. D. Rapoport and S. Sternberg, Ann. Phys. 158, 447 (1984); ibid., N. Cimento 80A, 371 (1984); S. Sternberg, Ann. Phys. 162, 85 (1985).

  35. D. Rapoport, in Trends on Partial Differential Equations of Mathematical Physics (Conference in Honor of Prof. V.A.Solonnikov, Obidos, Portugal, June 2003), 225–241, J. F. Rodrigues et al. (eds.), Birkhauser Progress in Nonlinear Differential Equations and Their Applications, vol. 61 (Birkhauser, Boston, 2004.)

  36. D. Rapoport, Int. J. Theor. Phys. 35(10) 2127–2152, (1987); b. 30(1) 1, 1497 (1991); c. 35(2), 287 (1996)

  37. EC. Stueckelberg (1941) Helv Phys Acta. 14 322–588 Occurrence Handle67.0926 Occurrence Handle4,56f

    MATH  MathSciNet  Google Scholar 

  38. L. P. Horwitz and C. Piron, Helv. Phys. Acta66, 694, (1993); L. P. Horwitz and N. Shnerb, Found. Phys.28, 1509 (1998).

  39. J. Fanchi, Parametrized Relativistic Quantum Theory (Kluwer, 1993); M. Trump and W. Schieve, Classical Relativistic Many-Body Dynamics(Kluwer, 1994).

  40. Lesieur (1994) La Turbulence Presses Univ. Grenoble

    Google Scholar 

  41. R. Durrett (1984) Martingales and Analysis Wordsworth Belmont (California)

    Google Scholar 

  42. O. Oron and L. Horwitz, Found. Phys., IARD (2004, special issue); ibid. Phys. Letts. A 280, 265 (2001); ibid. Found. Phys. 31 951 (2001) and 33 1323 (2003) and 33 1177 (2003); ibid. in Progress in General Relativity and Quantum Cosmology Research, V. Dvoeglazov (ed.), (Nova Science, Hauppage, 2004); L. Horwitz, N. Katz and O. Oron, to appear in Discrete Dyn. Nature Soc. 2004).

  43. Sh. Sklarz and L. Horwitz, Relativistic Mechanics and Continuous Media, to appear.

  44. D. Hestenes G. Sobczyck (1984) Clifford Calculus to Geometric Calculus D Reidel Dordrecht

    Google Scholar 

  45. I. Prigogine (1995) From Being to Becoming Freeman New York

    Google Scholar 

  46. H. Weyl, Space, Time and Matter (Dover (reprinted), New York, 1952).

  47. K.D. Elworthy Y. Yan ParticleLe M. Li Xue (2000) On the Geometry of Diffusion Operators and Stochastic Flows Springer Berlin

    Google Scholar 

  48. A. Bohm and M. Gadella, Dirac Kets, Gamow Vectors and Gel’fand and Triplets, Lecture Notes in Physics 348, (Springer, Berlin, 1989); A. Jaffe and J. Glimm, Quantum Physics: The Functional Integral point of view, (Springer, 1982); P. Cartier and Cecile de Witt Morette, J. Math. Phys. 41(6) 4154 (2000); D. Rapoport and M. Tilli, Hadronic J. 10, 25–34 (1987).

  49. P. Malliavin,Stochastic Analysis, (Springer Verlag, 1999); L. Schwartz, Second order Differential Geometry, Semimartingales and Stochastic Differential Equations on a Smooth Manifold, Lecture Notes in Mathematics 921 (Springer, 1978); P. Meyer, Géometrie Stochastique sans Larmes, in Séminaire des Probabilites XVI, supplement, Lecture Notes in Mathematics 921 (Springer, Berlin, 165–207 (1982).

  50. D. Stroock S.V.S. Varadhan (1984) Multidimensional Diffusion Processes Springer Berlin

    Google Scholar 

  51. E. B. Dynkin, Soviet Math. Dokl. 9(2), 532–535, (1968); Ya. I. Belopolskaya and Yu.L. Dalecki, Stochastic Processes and Differential Geometry (Kluwer, Dordrecht, 1989); S. A. Molchanov, Russian Math. Surveys30, 1–63 (1975).

  52. E. Nelson, Quantum Fluctuations, (Princeton University Press, New Jersey, 1985). Stochastic Processes in Classical and Quantum Systems, Lecture Notes in Physics 262, (Springer, 1986); F. Guerra and P. Ruggiero, Lett. Nuovo Cimento 23, 529 (1978); F. Guerra and P. Ruggiero, Phys. Rev. Letts. 31, 1022 (1973); D. Dohrn and F. Guera, Lett. Nuovo Cimento 22, 121 (1978); E. Aldovandri, D. Dohrn and F. Guerra, J. Math. Phys. 31 (3), 639 (1990).

  53. L. M. Morato N.C. Petroni (2000) J. Phys. A . 33 5833–5848 Occurrence Handle2000JPhA...33.5833C Occurrence Handle2001g:81034

    ADS  MathSciNet  Google Scholar 

  54. D. Hestenes, J. Math. Phys. 14, 893–905 (1973); 15, 1768–1777 (1974); 15, 1778–1786 (1974); 16, 556–572 (1975); Found. Phys.15, 63–87 (1985).

  55. T. Hida, Brownian Motion (Springer Verlag, 1980).

  56. H. Kleinert (1991) Path integrals in Quantum Mechanics, Statistics and Polymer Physics World Scientific Singapore

    Google Scholar 

  57. H. Grabert P. Hanggi P. Talkner (1979) Phys Rev A. 19 IssueID6 2440–2445 Occurrence Handle10.1103/PhysRevA.19.2440 Occurrence Handle1979PhRvA..19.2440G Occurrence Handle80d:81005

    Article  ADS  MathSciNet  Google Scholar 

  58. K. L. Chung and J. P. Zambrini, Introduction to Random Time and Quantum Randomness, 2nd edn. (World Scientific, 2003); A. B. Cruzeiro, Wu Liming and J.~P.~Zambrini, in Stochastic Analysis and Mathematical Physics, ANESTOC ’98 (Santiago, Chile), R. Rebolledo (ed.), (World Scientific, Singapore, 2004).

  59. S. Pope, Turbulent Flows (Cambridge University Press, 2000).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Diego L. Rapoport.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rapoport, D.L. On the Unification of Geometric and Random Structures through Torsion Fields: Brownian Motions, Viscous and Magneto-fluid-dynamics. Found Phys 35, 1205–1244 (2005). https://doi.org/10.1007/s10701-005-6407-y

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-005-6407-y

Keywords

Navigation