Periodic orbit theory of anomalous diffusion.

We introduce a novel technique to find the asymptotic time behaviour of deterministic systems exhibiting anomalous diffusion. The procedure is tested for various classes of simple but physically relevant 1–D maps and possible relevance of our findings for more complicated problems is briefly discussed. In the last few years the phenomenon of deterministic diffusion has been widely investigated: as a matter of fact a major theoretical challenge in the study of dynamical systems is to understand thoroughly the generation of typical stochastic properties in purely deterministic systems. Moreover even the simplest maps for which deterministic diffusion has been observed are supposed to model the behaviour of relevant physical systems (like Josephson junctions in presence of a microwave field, see e.g. [1]). A recent approach, introduced independently in [2] and [3], leads to an expression for the diffusion coefficient in terms of the periodic orbits of the system, in the form of a cycle expansion[4]. Cycle expansions have been applied in a number of different contexts (see for instance [5, 6]): they work remarkably well for low dimensional hyperbolic systems, provided their topology (symbolic dynamics) is under control. The problem of controlling the topology of the system is well illustrated, in the context of diffusion properties via cycle expansions, when one deals with the infinite Lorentz gas with bounded horizon (see [7]). If one relaxes the hypothesis of pure hyperbolicity (absence of marginal stability), much more care has to be taken, and generally the effectiveness of the expansions is regained only if one is able to sum infinite contributions shadowing the marginal fixed point[5]. We emphasize that this problem is of paramount relevance, as it naturally arises when dealing with generic hamiltonian systems, in which elliptic islands of stability and hyperbolic homoclinic webs coexist. Here we address explicitly the problem of marginal stability, which is tightly connected to the appearance of anomalous diffusion (and also represents a crucial feature in establishing a good theory for diffusion in generic two–dimensional area preserving maps[8]). We extend the theory developped in [2, 3] to study deviations from normal diffusion, expressing the asymptotic time behaviour in terms of properties of the periodic orbits of the system. The method is then applied to a class of 1–D maps for which anomalous diffusion has been previously observed. We begin by recalling the essential features of the periodic orbit theory of normal diffusion[2, 3], by considering the simplest context in which it may be applied. We will consider lifts of one–dimensional circle maps xt+1 = f(xt) t ∈ N f(x+ n) = n+ f(x) f(−x) = −f(x) (1) together with corresponding torus maps xt+1 = f(xt)|mod 1 = f̂(xt). Normal diffusion means that asymptotically σ(t) =< (xt − x0) 2 >∼ 2Dt, where the average is over initial conditions (by symmetry x0 may be taken in the unit interval). The set of 1 all periodic orbits of f̂ will be denoted by {p}: each orbit will be characterized by its period np stability Λp (product of the derivatives along the cycle) and integer winding number σp (such that for each cycle point xi(p) we have f (xi(p)) = xi(p) + σp). We focus our attention on the generating function < et0 >= Ωt(β); it has been shown[2, 3] that its asymptotic behaviour is dominated by the leading eigenvalue of an appropriate transfer operator: Ωt(β) ∼ z(β) , where z(β) is the smallest solution of ζ 0 (z(β), β) = ∏