Skip to main content
Log in

Stochastic representation of nearly-Gaussian, nonlinear processes

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

The use of polynomial functionals of the white noise process is discussed for the treatment of nonlinear random processes. It is noted that such treatments are useful for nearly-Gaussian processes. Applications of such representations to nonlinear systems and to nonlinear fluid mechanics problems (turbulence) are reviewed.

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.

Similar content being viewed by others

References

  1. J. F. Barrett, “The use of functionals in the analysis of nonlinear physical systems,”J. Electronics and Control 15:567–615 (1963).

    Google Scholar 

  2. G. K. Batchelor,Homogeneous Turbulence, Cambridge University Press (1953).

  3. E. R. Benton, “Solutions illustrating the decay of dissapation layers in Burgers' nonlinear diffusion equation,”Phys. Fluids 10:2113–2119 (1967).

    Google Scholar 

  4. M. Bergman, Ph.D. Thesis, University of Paris (1969), to be published.

  5. S. E. Bodner, “Turbulence theory with a time-varying Wiener-Hermite basis,” (1968), to be published.

  6. Burgers, “Correlation problems in a one-dimensional model of turbulence,”Proc. Roy. Acad. Sc. Netherlands 53:247–260 (1950).

    Google Scholar 

  7. R. H. Cameron and W. T. Martin, “The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals,”Ann. Math. 48:385–392 (1957).

    Google Scholar 

  8. R. H. Cameron, “A family of integrals serving to connect the Wiener and Feynman integrals,”J. Math. and Phys. 39:126–140 (1960).

    Google Scholar 

  9. G. H. Canavan and C. E. Leith, “Lagrangian Wiener-Hermite expansion for turbulence,”Phys. Fluids 11:2759–2761 (1968).

    Google Scholar 

  10. S. Chandrasekhar, “A theory of turbulence,”Proc. Roy. Soc. (London) A229:1–19 (1955).

    Google Scholar 

  11. W. Clever, (1969), private communication.

  12. J. D. Cole, “On a quasi-linear parabolic equation occurring in aerodynamics,”Quart. Appl. Math. 9:225–236 (1951).

    Google Scholar 

  13. M. Doi and T. Imamura, “Wiener-Hermite expansion with time dependent ideal random functional,” (1969), private communication, to be published.

  14. E. P. Gyftopoulos and R. J. Hooper, “Signals for transfer functio mneasurements in nonlinear systems,”J. Nuclear Energy A/B (GB) 18:20–21 (1964).

    Google Scholar 

  15. E. P. Gyftopoulos and R. J. Hooper, AEC Symposium series TID9:343–356 (1967).

    Google Scholar 

  16. G. H. Harris and L. Lapidus, “The identification of non-linear systems,”Industrial and Engineering Chemistry 59:66–81 (1967).

    Google Scholar 

  17. I. Hirschsohn, M. S. Thesis, University of California, San Diego, (1969), to be published.

    Google Scholar 

  18. E. Hopf, “The partial differential equation Ut + UUx=μUxx,”Commun. Pure Appl. Math. 3:201–230 (1950).

    Google Scholar 

  19. T. Imamura, W. C. Meecham, and A. Siegel, “Symbolic calculus of the Wiener-Hermite functionals,”J. Math. Phys. 6:695–706 (1965).

    Google Scholar 

  20. K. Ito and M. Nisio, “On stationary solutions of a stochastic differential equation,”J. Math. Kyoto Univ. 4:1–75 (1964).

    Google Scholar 

  21. D. T. Jeng, R. Foerster, S. Haaland, and W. C. Meecham. “Statistical initial-value problem for Burgers' model equation of turbulence,”Phys. Fluids 9:2114–2120 (1966).

    Google Scholar 

  22. D.-T. Jeng, (1969), private communication, to be published.

  23. W.-H. Kahng and A. Siegel, “Symmetry properties of Cameron-Martin-Wiener kernels,”Phys. Fluids (1969), in press.

  24. R. H. Kraichnan, “Direct-interaction approximation for a system of several interacting simple shear waves,”Phys. Fluids 6:1603–1609 (1963).

    Google Scholar 

  25. S. H. Kyong and E. P. Gyftopoulos, “A direct method for a class of optimal control problems,”IEEE Trans. Automatic Control AC-13:240–245 (1968).

    Google Scholar 

  26. Y. W. Lee and M. Schetzen, “Measurement of the Wiener kernels of a non-linear system by cross-correlation,”Int. J. Control 2:237–254 (1965).

    Google Scholar 

  27. W. C. Meecham and A. Siegel, “Wiener-Hermite expansion in model turbulence at large Reynolds numbers,”Phys. Fluids 7:1178–1190 (1964).

    Google Scholar 

  28. W. C. Meecham and D.-T. Jeng, “Use of the Wiener-Hermite expansion for nearly-normal turbulence,”J. Fluid Mech. 32:225–249 (1968).

    Google Scholar 

  29. W. C. Meecham and M. Y. Su, “Prediction of equilibrium properties for nearly-normal model turbulence,”Phys. Fluids (1969), in press.

  30. D. W. Moomaw, “A study of Burgers' model equation with application to the statistical theory of turbulence,” Ph.D. Thesis, University of Michigan (1962).

  31. J. C. J. Nihoul, “The stochastic transform and the study of homogeneous turbulence,”Physica 31:141–152 (1965).

    Google Scholar 

  32. Y. Ogura, “A consequence of the zero-fourth-cumulant approximation in the decay of isotropic turbulence,”J. Fluid Mech. 16:33 (1963).

    Google Scholar 

  33. S. A. Orszag and L. R. Bissonnette, “Dynamical properties of truncated Wiener-Hermite expansions,”Phys. Fluids 10:2603–2613 (1967).

    Google Scholar 

  34. I. Proudman and W. H. Reid, “On the decay of normally distributed and homogeneous turbulent velocity field,”Phil. Trans. A247:163–189 (1954).

    Google Scholar 

  35. P. G. Saffman, “Application of the Wiener-Hermite expansion to the diffusion of a passive scalar in a homogeneous turbulent flow,” (1969), to be published.

  36. M. Schetzen, “Determination of optimum non-linear systems for generalized error criteria based on the use of gate functions,”IEEE Trans. Information Theory 117–125 (January, 1965).

  37. R. W. Stewart, “Triple velocity correlations in isotropic turbulence,”Proc. Camb. Phil. Soc. 47:146–157 (1951).

    Google Scholar 

  38. J. H. Thomas, “Numerical experiments on a model system form agnetohydrodynamic turbulence,”Phys. Fluids 11:1245–1250 (1968).

    Google Scholar 

  39. M. S. Uberoi, “Correlations involving pressure fluctuations in homogeneous turbulence,” NACA Tech. Note 3116 (1954).

  40. H. L. Van Trees,Synthesis of Optimum Non-Linear Control Systems, MIT Technology Press (1962).

  41. L. Vassilopoulos, “Applications of non-linear statistical theory of ship performance in random seas,”Intl. Shipbuilding Progress 14:54 (1967).

    Google Scholar 

  42. R. W. Warming, “Statistical properties of an exact solution of Burgers' model of turbulence,” Ph.D. Thesis, University of Minnesota (1963).

  43. N. Wiener,Non-Linear Problems in Random Theory, MIT Technology Press (1958).

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Meecham, W.C. Stochastic representation of nearly-Gaussian, nonlinear processes. J Stat Phys 1, 25–40 (1969). https://doi.org/10.1007/BF01007239

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01007239

Key words

Navigation