NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS 137 Anomalous Diffusion and Current Generation in a Driven Hamiltonian System R. Harish and K. P. N. Murthy Abstract—Symmetry considerations have shown the possibility of dc cur- rent generation in the stochastic layer of a system describing the motion of a particle in a one-dimensional potential driven by an ac time periodic force. The current generation is known to be induced by the presence and desym- metrization of ballistic channels inside the stochastic layer. In this paper we numerically obtain the magnitudes of the dc currents as a function of the phase of the ac driving force. We also calculate the mean square displace- ment of trajectories starting out in a small region of the stochastic layer. The diffusion is found to be anomalous. The exponent in is found to be bounded around 1.7 for the entire range of . Keywords— Hamiltonian system, Chaos, Anomalous diffusion, Current generation, L´ evy flights. I. I NTRODUCTION RANSPORT in driven systems has received considerable attention in the recent past [1]-[2]. Nonlinear transport processes in spatially periodic systems are of interest in stochas- tic diffusion in nonlinear systems, transport in randomly driven systems etc. Directed transport or ratchet transport, i.e, noise induced macroscopic current generation in a periodic potential with or without space reflection symmetry, is an interesting ex- ample of chaotic transport in a nonlinear dynamical system. Ratchet diffusion is important in the study of Josephson junc- tions. The origin of statistical laws from deterministic dynamics has been a topic of intense investigation for quite some time. In this connection Hamiltonian systems are studied as model low di- mensional nonlinear dynamical systems. The study of these sys- tems is shown to provide a link between deterministic dynamics and stochastic processes. It is known that driven systems with damping terms included exhibit normal diffusion and Hamilto- nian chaotic systems exhibit anomalous diffusion. Directed transport in a spatially periodic potential can be re- alized when the system is driven away from thermal equilib- rium by deterministic or stochastic perturbation. In computa- tional studies, often, this is done by imposing periodic driving. A further requirement for directed transport is the breaking of the spatial inversion symmetry. Hamiltonian systems which satisfy the requirements for the appearance of directed current also characteristically exhibit mixed phase spaces. The phase space consists of regions of reg- ular behaviour as well as stochastic or chaotic regions. Seperat- ing the phase space into regular and stochastic regions is diffi- cult owing to the complex nature of the boundaries. A trajectory R. Harish (corresponding author) is with Reactor Physics Division, In- dira Gandhi Centre for Atomic Research, Kalpakkam 603 102, e-mail: har- ish@igcar.ernet.in. K. P. N. Murthy is with Materials Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102 when it enters the sticky barriers can remain trapped for a long time near the corresponding regular islands producing ballistic flights. It is known that such correlated motions lead to fail- ure of Brownian type of behaviour. The motion then becomes non-diffusive and is called anomalous. In this paper we study a periodically driven Hamiltonian sys- tem in a space periodic potential. Under certain conditions that lead to symmetry breaking, dc current is generated. We study numerically the current generated and diffusion as a function of the parameter that breaks the symmetry in the system. II. CURRENT GENERATION We consider an example of a particle moving in a spatially periodic potential driven by a time periodic force of the form (1) Here and are canonically conjugate variables. The corre- sponding equation of motion is then given by (2) The system has been studied by several authors [3]- [6]. One of the possible forms for the driving force is [2] -[5] (3) The system with can be written as a system of three first order autonomous coupled equations, (4) The phase space of the system consists of regular regions inter- spersed with regions of stochasticity with complicated bound- aries. The stochastic layer emerges from the destroyed sepera- trix of the undriven system. It is shown in [3] that all possible symmetries that give rise to zero dc current are broken for and . It is easy to see that for , the layer is invariant under the transformation . For these particular values the average velocity defined [2] as (5)