Cumulant Analysis of the Statistical Properties of a Deterministically Thermostated Harmonic Oscillator

  • A. N. ArtemovEmail author


Usual approach to investigate the statistical properties of deterministically thermostated systems is to analyze the regime of the system motion. In this work the cumulant analysis is used to study the properties of the stationary probability distribution function of the deterministically thermostated harmonic oscillators. This approach shifts attention from the investigation of the geometrical properties of solutions of the systems to the studying a probabilistic measure. The cumulant apparatus is suitable for studying the correlations of dynamical variables, which allows one to reveal the deviation of the actual probabilistic distribution function from canonical one and to evaluate it. Three different thermostats, namely the Nosé–Hoover, Patra-Bhattacharya and Hoover–Holian ones, were investigated. It is shown that their actual distribution functions are non-canonical because of nonlinear coupling of the oscillators with thermostats. The problem of ergodicity of the deterministically thermostated systems is discussed.


Cumulant analysis Deterministic thermostat Statistical properties 



The author thanks A. Samoletov for interesting and useful discussions.


  1. 1.
    Nose, S.: A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 81, 511–519 (1984)ADSCrossRefGoogle Scholar
  2. 2.
    Hoover, W.G.: Canonical dynamics: equilibrium phase-space distributions. Phys. Rev. A 31, 1695–1697 (1985)ADSCrossRefGoogle Scholar
  3. 3.
    Patra, P.K., Bhattacharya, B.: A deterministic thermostat for controlling temperature using all degrees of freedom. J. Chem. Phys. 140, 064106 (2014)ADSCrossRefGoogle Scholar
  4. 4.
    Hoover, W.G., Holian, B.L.: Kinetic moments method for the canonical ensemble distribution. Phys. Lett. A 211, 253–257 (1996)ADSCrossRefGoogle Scholar
  5. 5.
    Kusnezov, D., Bulgac, A., Bauer, W., Kusnezov, D.: Canonical ensembles from Chaos. Ann. Phys. 204, 155–185 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Martyna, G.J., Klein, M.L., Tuckerman, M.: Nose–Hoover chains: the canonical ensemble via continuous dynamics. J. Chem. Phys. 97, 2635–2643 (1992)ADSCrossRefGoogle Scholar
  7. 7.
    Watanabe, H., Kobayashi, H.: Ergodicity of a thermostat family of the Nose–Hoover type. Phys. Rev. E 75, 040102(R) (2007)ADSCrossRefGoogle Scholar
  8. 8.
    Patra, P.K., Bhattacharya, B.: An ergodic configurational thermostat using selective control of higher order temperatures J. Chem. Phys. 142, 194103-1-8 (2015)Google Scholar
  9. 9.
    Samoletov, A., Vasiev, V.: Dynamic principle for ensemble control tools. J. Chem. Phys. 147, 204106 (2017)ADSCrossRefGoogle Scholar
  10. 10.
    Posch, H.A., Hoover, W.G., Vesely, F.J.: Canonical dynamics of the Nose oscillator: stability, order, and chaos. Phys. Rev. A 33, 4253–4265 (1986)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Legoll, F., Luskin, M., Moeckel, R.: Non-ergodicity of the Nose–Hoover thermostatted harmonic oscillator. Arch. Ration. Mech. Anal. 184, 449–463 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Patra, P.K., Bhattacharya, B.: Nonergodicity of the Nose–Hoover chain thermostat in computationally achievable time. Phys. Rev. E 90, 043304 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    Hoover, W.G., Hoover, C.G.: Ergodicity of a time-reversibly thermostated harmonic oscillator and the 2014 Ian Snook Prize. CMST 20, 87–92 (2014)CrossRefGoogle Scholar
  14. 14.
    Hoover, W.G., Hoover, C.G., Isbister, D.J.: Chaos, ergodic convergence, and fractal instability for a thermostated canonical harmonic oscillator. Phys. Rev. E 63, 026209-1-5 (2001)Google Scholar
  15. 15.
    Patra, P.K., Sprott, J.C., Hoover, W.G., Hoover, C.G.: Deterministic time-reversible thermostats: chaos, ergodicity, and the zeroth law of thermodynamics. Mol. Phys. 113, 2863–2872 (2015)ADSCrossRefGoogle Scholar
  16. 16.
    Hoover, W.G., Hoover, C.G.: Ergodicity of the Martyna–Klein–Tuckerman thermostat and the 2014 Ian Snook Prize. CMST 21, 5–10 (2015)CrossRefGoogle Scholar
  17. 17.
    Hoover, W.G., Sprott, J.C., Patra, P.K.: Ergodic time-reversible chaos for Gibbs canonical oscillator. Phys. Lett. A 379, 2935–2940 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Samoletov, A.A., Dettman, C.P., Chaplain, C.P.: Thermostats for slow configurational modes. J. Stat. Phys. 128, 1321–1336 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Malakhov, A.N.: Cumulant Analysis of Random Non-Gaussian Processes and their Transformation. Sovetskoe Radio, Moscow (1978). (in Russian)zbMATHGoogle Scholar
  20. 20.
    Primak, S., Kontorovich, V., Lyandres, V.: Stochastic Methods and their Applications to Communications. Stochastic Differential Equations Approach. Wiley, New York (2004)CrossRefzbMATHGoogle Scholar
  21. 21.
    Kontorovich, V.: Applied statistical analysis for strange attractors and related problems. Math. Methods Appl. Sci. 30, 1705–1717 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hockney, R.W.: The potential calculation and some applications. Methods Comput. Phys. 9, 136–211 (1970)Google Scholar
  23. 23.
    Verlet, L.: Computer “experiment” on classical fluids. 1. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 159, 98–103 (1967)ADSCrossRefGoogle Scholar
  24. 24.
    Lichtenberg, A.J., Liberman, M.A.: Regular and Stochastic Motion. Springer, New York (1983)CrossRefGoogle Scholar
  25. 25.
    Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57 (1985)Google Scholar
  26. 26.
    Thompson, J.M.T., Stewar, H.B.: Nonlinear Dynamics and Chaos. Wiley, New York (1986)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Donetsk Institute for Physics and EngineeringDonetskUkraine

Personalised recommendations