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

Abstract

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.

A The Expressions that are Referenced in the Text

The opening of the cumulant brackets used in the text.

$$\begin{aligned} \langle x^2\rangle= & {} \kappa _{2,0,0(,0)}+\kappa _{1,0,0(,0)}^2,\end{aligned}$$

(36)

$$\begin{aligned} \langle p^2\rangle= & {} \kappa _{0,2,0(,0)}+\kappa _{0,1,0(,0)}^2, \end{aligned}$$

(37)

$$\begin{aligned} \langle x,\zeta p\rangle= & {} \kappa _{1,1,1,0}+\kappa _{1,1,0,0}\kappa _{0,0,1,0} + \kappa _{1,0,1,0}\kappa _{0,1,0,0},\end{aligned}$$

(38)

$$\begin{aligned} \langle p,\xi x\rangle&= \kappa _{1,1,0,1}+\kappa _{0,1,0,1}\kappa _{1,0,0,0} + \kappa _{1,1,0,0}\kappa _{0,0,0,1}, \end{aligned}$$

(39)

$$\begin{aligned} \langle p^4\rangle&= \kappa _{0,4,0,0}+3 \kappa _{0,2,0,0}^2 + 4\kappa _{0,1,0,0}\kappa _{0,3,0,0}\nonumber \\&\quad +\, 6\,\kappa _{0,1,0,0}^2\kappa _{0,2,0,0} + \kappa _{0,1,0,0}^4, \end{aligned}$$

(40)

$$\begin{aligned} \langle x,\xi p^3\rangle&= \kappa _{1,3,0,1} + 3\kappa _{1,2,0,1}\kappa _{0,1,0,0} + \kappa _{1,3,0,0}\kappa _{0,0,0,1} \nonumber \\&\quad +\,3\kappa _{1,1,0,1}\kappa _{0,1,0,0}^2 + 3\kappa _{1,2,0,0}\kappa _{0,1,0,0}\kappa _{0,0,0,1} \nonumber \\&\quad +\,\kappa _{1,0,0,1}\kappa _{0,1,0,0}^3 + 3\kappa _{1,1,0,0}\kappa _{0,1,0,0}^2\kappa _{0,0,0,1} \nonumber \\&\quad +\, 3\kappa _{1,0,0,1}\kappa _{0,2,0,0}\kappa _{0,1,0,0} + 3\kappa _{1,1,0,0}\kappa _{0,1,0,1}\kappa _{0,1,0,0}, \end{aligned}$$

(41)

The example of the cumulants expressed in terms of the moments

$$\begin{aligned} \kappa _{0,2,0}&= \langle p,p\rangle = \langle p^2\rangle -\langle p\rangle ^2, \end{aligned}$$

(42)

$$\begin{aligned} \kappa _{1,1,1}&= \langle x,p,\zeta \rangle = \langle xp\zeta \rangle -\langle x\rangle \langle p\zeta \rangle - \langle p\rangle \langle x\zeta \rangle \nonumber \\&\quad -\,\langle \zeta \rangle \langle xp\rangle + 2\langle x\rangle \langle p\rangle \langle \zeta \rangle . \end{aligned}$$

(43)