PHYSICAL REVIEW E VOLUME 52, NUMBER 3 SEPTEMBER 1995 Fluctuations and decoherence in classical chaos: A model study of a Kubo oscillator generated by a chaotic system Shanta Chaudhuri, Gautam Gangopadhyay, * and Deb Shankar Ray Indian Association for the Cultiuation of Science, Jadaupur, Calcutta 700032, India (Received 28 July 1994; revised manuscript received 2 May 1995) We consider a Kubo oscillator whose stochasticity in frequency is generated by a 1.5-degree-of- freedom chaotic system. Based on the theory of multiplicative noise we show how fluctuation and decoherence and their relationship, which is analogous to the fluctuation-dissipation relation in many- body physics, can be realized in classical chaos. We numerically verify the basic theoretical proposi- tions. PACS number(s): 05.45. +b, 05.40. +j I. INTRODUCTION A key theme in nonlinear physics today is chaos in dynamical systems [1]. Recent experimental and theoret- ical research on few-degree-of-freedom systems has led to the discovery of new generic features in classical motion; regular and chaotic Rows in phase space are, of course, the key notions. The chaotic motion is not associated with any random parameter or forces but is due to the unstable character of the trajectories in phase space. The chaos in dynamical systems, although determinis- tic, is stochastic in nature in the statistical sense. It is therefore expected that statistical mechanical formalism [2 — 9] might be useful for description of classical chaos. For specific discrete systems such a formalism was con- sidered by Kai and Tomita [2] and Oono and Takahashi [3]. Kadanoff and co-workers [4] have introduced a powerful method for characterizing multifractals based on certain partition functions. Widom et al. [5] dis- cussed the example of Julia sets. The method of statisti- cal mechanics was also followed by Kohmoto [6] to intro- duce entropy function and free energy function for mul- tifractals which are a prerequisite for the existence of thermodynamics. These functions are related to Kolmogorov-Sinai entropy and Lyapunov exponents in the case of dynamical systems. For continuous low- dimensional chaotic systems, such as the Henon-Heiles Hamiltonian, the method of equilibrium statistical mechanics has recently been employed [7] to define tem- perature and entropy analogous to thermodynamics. Apart from implementation of the methods of equilib- rium statistical mechanics, kinetic description [9 — 14, 16, 17] has also been used over the years. Ever since the early numerical study of Chirikov mapping re- vealed that [10] the motion of a phase variable can be characterized by a simple random walk diffusion equa- tion, attempts have been made to describe chaotic motion in terms of the Langevin or equivalently Fokker-Planck 'Present address: S. N. Bose National Centre for Basic Sci- ences, DB-17, Sector-I, Salt Lake City, Calcutta 700064, India. equation. It has now been realized that deterministic maps can result in long time diffusional processes and methods have been developed to predict successfully the corresponding difFusion coefficients [1]. While these stud- ies are based on maps, identification of a noise term in the Lorenz equations after recasting it to an approximate Langevin form has been achieved by Nicolis and Nicolis [12] by successfully separating the distinct time scales in- volved in the dynamics using a center manifold method. Based on a strategy of separation of time scales Bianucci, West, and Grigolini [13] have considered a closed Hamil- tonian system and shown that the system of interest fol- lowing a slower dynamics obeys a Fokker-Planck equa- tion having a canonical distribution which defines a tern- perature like quantity. Very recently we have proposed a general fluctuation-diffusion relation [14] (a Kubo rela- tion in chaotic dynamics) for Hamiltonian systems which relates the largest Lyapunov exponent to the Fourier transform of the curvature-curvature (curvature of the potential) correlation function and shown that the theory of multiplicative noise can be a good natural description for classical chaos on several occasions. In spite of a great deal of effort to derive the stochastic processes (for example, Brownian motion) from a purely deterministic dynamical model the problem has largely remained unsolved. From a theoretical standpoint it is worthwhile to distinguish two different situations. In the first case one is considered with the derivation of Browni- an motion from a well-known system heat-bath mode1 [15] comprised of 1+N linear oscillators (1 system oscil- lator and N bath oscillators. N~ co ) In the second case [16,17] one is concerned with the derivation of a similar kind of stochastic process from a system — chaotic-bath model where the ¹ scillator bath is replaced by a low- dimensional chaotic system. In the later context we specifically mention the work of Bianucci and co-workers [16,17] who, based on a simple two-dimensional (2D) map, have given a derivation of Fokker-Planck equation for the system of interest using only dynamical argu- ments where both the friction and diffusion terms are de- rived from the dynamical properties of the chaotic bath. In this paper we have addressed a related issue. We con- sider a Kubo oscillator, i.e. , a harmonic oscillator with 1063-651X/95/52(3)/2262(6)/$06. 00 52 2262 1995 The American Physical Society