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Reynolds’ dream?

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Abstract

For many non-linear phenomena, it is necessary to solve infinite systems of equations for correlation functions with a wide range of parameters. This paper can be seen as a first step in addressing this problem. Correlation functions related to the ϕ3 and to the ϕ4 field theories are described by means of generating Fock space vectors constructed with the help of Cuntz algebra. The equations obtained are easily transformed and general solutions are constructed. Various expansions of these solutions are developed using the explicitly constructed right inverse operators related to linear and non-linear parts of the theory. Based on the idea of information loss and using the language of classical mechanics, a solution to the closure problem for correlation functions is proposed. The method described in this paper can be used to obtain approximated correlation functions with strong and weak non-linearity.

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References

  1. Accardi, L., Aref'eva, I.Ya., Volovich, I.V.: Non-Equilibrium Quantum Field Theory and Entangled Commutation Relations. ArXiv:hep-th/9905035 v1 (5 May 1999)

  2. Achenbach, J.E.: Wave Propagation in Elastic Solids. North-Holland, Amsterdam London (1973)

    MATH  Google Scholar 

  3. Anselmi, D.: A New Perspective on the Philosophical Implications of Quantum Field Theory. Internet (1998)

  4. Berry, H., Perez, D.G., Temam, O.: Chaos in Computer Performance. ArXiv:nlin.AO/0506030 v1 (13 June 2005)

  5. Bożejko, M., Speicher, R.: An Example of a Generalized Brownian Motion. Preprint of University of Wroclaw, Nr. 599 (November 1990)

  6. Dirac, P.A.M.: The Principles of Quantum Mechanics. Belfer Graduate School of Science, Yeshiva University, New York (1964)

    Google Scholar 

  7. Dreyer, W., Kunik, M.: Cold, Thermal and Oscillator Closure of the Atomic Chain. Preprint ISSN 0946-8633, No. 489. Berlin (1999)

  8. Dunca, M., Epshteyn, Y.: On the Adams-Stolz Deconvolution LES Models. Internet; Google (2004)

  9. Esquivel-Avila, J.A.: The dynamics of a nonlinear wave equation. J. Math. Anal. Appl. 279, 135–150 (2003). Available at: www.MATHEMATICSweb.org

    Article  MATH  Google Scholar 

  10. Faith, C.: Algebra: Rings, Modules and Categories I. Springer-Verlag, Berlin, New York (1970)

    Google Scholar 

  11. Fried, E.: An Elementary Introduction to the Abstract Algebra. MIR Publishers, Moscow (1979) (in Russian)

    Google Scholar 

  12. Frost, W., Moulden, T.H. (eds.): Handbook of Turbulence, vol. 1. Plenum Press, New York, London (1977)

    Google Scholar 

  13. Gad-el-hak, M.: Fluid mechanics from the beginning to the third millennium. Int. J. Eng. Ed. 14(3), 177–185 (1998)

    Google Scholar 

  14. Gantmacher, F.R.: Lectures on Analytical Mechanics. Science Publishers, Moscow (1966) (in Russian)

    Google Scholar 

  15. Goldstein, H.: Classical Mechanics. Science Publishers, Moscow (1975) (in Russian)

    MATH  Google Scholar 

  16. Greenberg, O.W.: Theories of violation of statistics. ArXiv:hep-th/0007054 v2 (13 July 2000)

  17. Hańćkowiak, J.: Quasilocal terms in the functional formulation of the S-matrix theory. Acta Phys. Polon. 33, 711 (1968)

    Google Scholar 

  18. Hańćkowiak, J.: In search of physical solutions in quantum and statistical field theory. J. Math. Phys. 33, 1132–1140 (1992a)

    Article  Google Scholar 

  19. Hańćkowiak, J.: In search of physical solutions in quantum field theory. Fortschr. Phys. 40(6), 593–614 (1992b)

    Article  Google Scholar 

  20. Hańćkowiak, J.: The constraint analysis of coupled equations for n-point functions. Rep. Math. Phys. 33(1/2), (1993)

  21. Harville, E.A.: Matrix Algebra from a Statistician's Perspective. Springer, Berlin Heidelberg, New York (2000)

    Google Scholar 

  22. Kempf, A.: Unsharp Degrees of Freedom and the Generating of Symmetries. ArXiv:hep-th/9907160 v1 (19 July 1999)

  23. Kolokoltsov, V.N., Maslov, V.P.: Idempotent Analysis and Applications. Kluwer, Dordrecht (1997)

    MATH  Google Scholar 

  24. Kozyrev, S.V.: The Space of Free Coherent States is Isomorphic to the Space of Distributions on P-Adic Numbers. ArXiv:q-alg/9706020 v1 (17 June 1997)

  25. Kraichnan, R.H., Lewis, R.M.: A space-time functional formalism for turbulence. Comm. Comm. Pure Appl. Math. 15, 397–411 (1962)

    Google Scholar 

  26. Kreimer, D.: On the Hopf Algebra of Perturbative Quantum Field Theories. ArXiv:q-alg/9707029 v4 (15 January 1998)

  27. Landau, L., Lifshic, E.: Hydrodynamics. Science Publishers, Moscow (1988)

    Google Scholar 

  28. Layton, W., Lewandowski, R.: An Exposition on the Time-Averaged Accuracy of Approximate De-convolution, Large Eddy Simulation Models of Turbulence. Internet (2004)

  29. Marcinek,W.:On Generalized Statistics and Interactions. In: Coherent States, Quantization and Gravity, Schlichenmaier, M., Strasburger, A., Twareque Ali, S., Odziejewich, A. (eds.) Publishers of Warsaw University (2001)

  30. Maurin, K.: Analysis, Part 1. PWN-Polish Scientific Publishers, Warsaw and D. Reidel Publishing Company, Dordrecht, Holland/Boston, USA (1976)

    Google Scholar 

  31. Monin, A.S., Jaglom, A.M.: Statistical Hydromechanics, vol. 2. MIR Publishers, Moscow (1967)

    Google Scholar 

  32. Musser, G.: Was Einstein Right? Scientific American (September 2004)

  33. Na, T.Y.: Computational Methods in Engineering Boundary Value Problems. Academic Press, New York, San Francisco (1979)

    MATH  Google Scholar 

  34. Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)

    MATH  Google Scholar 

  35. Nusse, H.E., Yorke, J.A.: Dynamics: Numerical Explorations. Springer-Verlag, New York (1997)

    Google Scholar 

  36. Pang, T.: Computational Physics. Cambridge University Press, Cambridge (1997)

  37. Penrose, R.: Shadows of the Mind. A Search for the Missing Science of Consciousness. Oxford University Press (1994)

  38. Pogorzelec, A.: Initial value problems with ill-determined linear systems with right invertible operators. Demonstrat. Math. 16, 407–420 (1983)

    MATH  Google Scholar 

  39. Polyanin, A.E., Zaitsev, V.F.: Exact Solutions for Ordinary Eifferential Equations. CRC Press, Boca Raton, Tokyo (2000)

    Google Scholar 

  40. Przeworska-Rolewicz, E.: Introduction to Algebraic Analysis. PWN Publishers, Warsaw (1988)

    Google Scholar 

  41. Rizzi, S.A., Muravyov, A.A.: Equivalent Linearization Analysis of Geometrically Nonlinear Random Vibrations Using Commercial Finite Element Codes. NASA/TP-2002-211761 (Internet) (2002)

  42. Roache, P.J.: Computational Fluid Dynamics. Hermosa Publishers, Albuqerque (1976)

    Google Scholar 

  43. Rzewuski, J.: Field Theory. Part II. Iliffe Books LTE., London (1969)

    Google Scholar 

  44. Schulz, S.: Four Lectures on Differential-Algebraic Equations. Humbolt Universität zu Berlin (Internet) (2003)

  45. Socha, L.: Equations for Moments in Stochastical Dynamical Systems. PWN Publishers, Warsaw (1993) (in Polish)

    Google Scholar 

  46. Spindler, K.: Abstract Algebra with Applications, vol. 1. Marcel Dekker, New York, Basel, Hong Kong (1994)

    Google Scholar 

  47. 't Hooft, G.: On Peculiarities and Pit Falls in Path Integrals. ArXiv:hep-th/0208054 v1 (7 August 2002a)

  48. 't Hooft, G.: Determinism Beneath Quantum Mechanics. ArXiv:quant-ph/0212095 v1 (16 December 2002b)

  49. Vasiliev, F.P.: Numerical Methods of Solutions of Extreme Problems. Science Publishers, Moscow (1980) (in Russian)

    Google Scholar 

  50. Vasiliev, A.H.: Functional Methods in Quantum Field Theory and Statistic. Leningrad University, Leningrad (1976)

    Google Scholar 

  51. Weinzierl, S.: Algebraic Algorithms in Perturbative Calculations. ArXiv:hep-th/0305260 v1 (30 May 2003)

  52. Wal, A., Kuźma, M. (eds.): Hidden Symmetry of Physical Structures, Recipe of Weyel. Pedagogical University of Rzeszów (1997)

  53. Weinberg, S.: The Quantum Theory of Fields. The Press Syndicate of the University of Cambridge (1996)

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  1. Department of Mechanical Engineering, University of Zielona Góra. ul. Podgórna 50, PL, 65–246, Zielona Góra, Poland

    Jerzy Hańćkowiak

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  1. Jerzy Hańćkowiak
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Correspondence to Jerzy Hańćkowiak.

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Hańćkowiak, J. Reynolds’ dream?. Nonlinear Dyn 50, 191–211 (2007). https://doi.org/10.1007/s11071-006-9151-8

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