Chaotic Diffusion on Periodic Orbits: The Perturbed Arnol'd Cat Map

Chaotic diffusion on periodic orbits (POs) is studied for the perturbed Arnol'd cat map on a cylinder, in a range of perturbation parameters corresponding to an extended structural-stability regime of the system on the torus. The diffusion coefficient is calculated using the following PO formulas: (a) The curvature expansion of the Ruelle zeta function. (b) The average of the PO winding-number squared, $w^{2}$, weighted by a stability factor. (c) The uniform (nonweighted) average of $w^{2}$. The results from formulas (a) and (b) agree very well with those obtained by standard methods, for all the perturbation parameters considered. Formula (c) gives reasonably accurate results for sufficiently small parameters corresponding also to cases of a considerably nonuniform hyperbolicity. This is due to {\em uniformity sum rules} satisfied by the PO Lyapunov eigenvalues at {\em fixed} $w$. These sum rules follow from general arguments and are supported by much numerical evidence.Comment: 6 Tables, 2 Figures (postscript); To appear in Physical Review

On the Fluctuation Relation for Nose-Hoover Boundary Thermostated Systems

We discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures. The system is a modified Lorentz gas in which the fixed scatterers exchange energy with the gas of particles, and the thermostats are modelled by two Nos\'e-Hoover thermostats applied at the boundaries of the system. The transient fluctuation relation, which holds only for a precise choice of the initial ensemble, is verified at all times, as expected. Times longer than the mesoscopic scale, needed for local equilibrium to be settled, are required if a different initial ensemble is considered. This shows how the transient fluctuation relation asymptotically leads to the steady state relation when, as explicitly checked in our systems, the condition found in [D.J. Searles, {\em et al.}, J. Stat. Phys. 128, 1337 (2007)], for the validity of the steady state fluctuation relation, is verified. For the steady state fluctuations of the phase space contraction rate \zL and of the dissipation function \zW, a similar relaxation regime at shorter averaging times is found. The quantity \zW satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale; the quantity \zL appears to begin a monotonic convergence after such times. This is consistent with the fact that \zW and \zL differ by a total time derivative, and that the tails of the probability distribution function of \zL are Gaussian.Comment: Major revision. Fig.10 was added. Version to appear in Journal of Statistical Physic

The Steady State Fluctuation Relation for the Dissipation Function

We give a proof of transient fluctuation relations for the entropy production (dissipation function) in nonequilibrium systems, which is valid for most time reversible dynamics. We then consider the conditions under which a transient fluctuation relation yields a steady state fluctuation relation for driven nonequilibrium systems whose transients relax, producing a unique nonequilibrium steady state. Although the necessary and sufficient conditions for the production of a unique nonequilibrium steady state are unknown, if such a steady state exists, the generation of the steady state fluctuation relation from the transient relation is shown to be very general. It is essentially a consequence of time reversibility and of a form of decay of correlations in the dissipation, which is needed also for, e.g., the existence of transport coefficients. Because of this generality the resulting steady state fluctuation relation has the same degree of robustness as do equilibrium thermodynamic equalities. The steady state fluctuation relation for the dissipation stands in contrast with the one for the phase space compression factor, whose convergence is problematic, for systems close to equilibrium. We examine some model dynamics that have been considered previously, and show how they are described in the context of this work.Comment: 30 pages, 1 figur

Simple deterministic dynamical systems with fractal diffusion coefficients

We analyze a simple model of deterministic diffusion. The model consists of a one-dimensional periodic array of scatterers in which point particles move from cell to cell as defined by a piecewise linear map. The microscopic chaotic scattering process of the map can be changed by a control parameter. This induces a parameter dependence for the macroscopic diffusion coefficient. We calculate the diffusion coefficent and the largest eigenmodes of the system by using Markov partitions and by solving the eigenvalue problems of respective topological transition matrices. For different boundary conditions we find that the largest eigenmodes of the map match to the ones of the simple phenomenological diffusion equation. Our main result is that the difffusion coefficient exhibits a fractal structure by varying the system parameter. To understand the origin of this fractal structure, we give qualitative and quantitative arguments. These arguments relate the sequence of oscillations in the strength of the parameter-dependent diffusion coefficient to the microscopic coupling of the single scatterers which changes by varying the control parameter.Comment: 28 pages (revtex), 12 figures (postscript), submitted to Phys. Rev.

Allopatric distribution of juvenile red-legged banana prawns (Penaeus indicus H. Milne Edwards, 1837) and juvenile white banana prawns (Penaeus merguiensis De Man, 1888), and inferred extensive migration, in the Joseph Bonaparte Gulf, northwest Australia

During October to December 1997, we trawled estuarine habitats in the Joseph Bonaparte Gulf (JBG) to determine the distribution of juvenile red-legged banana prawns, Penaeus indicus (H. Milne Edwards, 1837) and white banana prawns, Penaeus merguiensis (de Man, 1888). We made 229 beam-trawls at 185 sites, mostly over a 100-m path (3-min duration). A Global Positioning System (GPS) receiver was used to verify our location. During October to December 1998, we intensively resampled three of the rivers that were sampled in 1997 to confirm the gulf-wide distribution of P. indicus and P. merguiensis and to investigate the microhabitat use of P. indicus. We chose previously sampled and new sites in Forsyth Creek (eastern JBG), the Lyne River (Cambridge Gulf), and the Berkeley River (western JBG). We made 249 trawls at 21 sites, mostly over 100 m. Juvenile banana prawns were abundant in eastern JBG, Cambridge Gulf and western JBG. They were not abundant in southern JBG, although fewer trawls were made there, due to its inaccessibility. In eastern JBG and Cambridge Gulf, over 96% and 73% (respectively) of juvenile banana prawns were P. indicus and they were more abundant there than in the western JBG. Conversely, in the western JBG over 93% of the juvenile banana prawns were P. merguiensis and they were more abundant than in the eastern JBG and Cambridge Gulf. The Lyne River in the northwestern Cambridge Gulf seems to be the transition zone; both P. indicus and P. merguiensis are equally abundant. P. indicus are most abundant on the mangrove-lined muddy banks of waterways within mangrove forests, similar habitats to P. merguiensis. Within these habitats, they were most abundant in gutters and small creeks, rather than rivers and large creeks. Few P. indicus or P. merguiensis were caught in 100 m 2 trawls undertaken midriver (on the channel bottom and on emergent banks), although these habitats may be only 100 m from the mangrove-lined habitats. In all creek and river habitats, both species are most catchable at low tide (irrespective of daylight or darkness) when they move out of the mangrove forests and accumulate in the remnant water bodies. The offshore fishery for P. indicus is in northwestern JBG in waters 50-80 m deep, about 300 and 200 km, respectively, from where juveniles are abundant in their extensive inshore habitats in east JBG and in Cambridge Gulf, demonstrating a geographical separation of the juvenile and adult phases. Postlarval P. indicus, spawned offshore, must use tides and currents to travel south and east to reach nursery habitats. Emigrant subadults must migrate north and west, across relatively shallow inshore sand substrates (30-40 m deep) to reach their offshore habitats