Anomalous Transport in Hamiltonian Systems

Abstract

The importance of Hamiltonian dynamical systems is at the same time practical (e.g., simulations in astronomy and fluid dynamics) and fundamental (e.g., microscopic basis of thermodynamics). The reason for this last point is that systems of many interacting particles considered in statistical physics are in a fundamental sense high-dimensional Hamiltonian systems. An effective description of such systems is performed under the (implicit) assumption of a suited hypothesis of strong chaos [1]. One example is when transport properties are described using kinetic equations like the Fokker-Planck equation. In opposite, low-dimensional Hamiltonian systems are usually not ergodic and correlations do not decay fast, what violates the hypotheses for the derivation of the Fokker-Planck equation and may lead to anomalous transport. The mechanism leading to anomalous transport in low-dimensional Hamiltonian systems is the topic of this contribution. Normal diffusive transport in deterministic systems has its origin in the chaotic nature of generic systems. However, corrections to this are caused by non-hyperbolicities (or weak chaos, see the contribution by Artuso and Cristadoro in Chap. ?? of this book) and by the generic property of stickiness [2, 3]. By this, one denotes a kind of sticking of trajectories to structures in the phase space which causes episodes of regular motion being intertwined into chaotic motion. The characteristics of such intermittent motion depend on the details of the Hamiltonian dynamics. Generic features, however, allow one to establish the analogy between chaotic motion in such systems and a simplified stochastic model, the continuous time random walk (CTRW, outlined in Sec. 1.2). In this model class, anomalous transport results from long (power-law) tails in the probability density functions (PDFs) of flights and traps. From this perspective, the