Spatial Structure of Stationary Nonequilibrium States in the Thermostatted Periodic Lorentz Gas

We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed |v| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of them are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions.Similarities and differences are noted.Comment: 24 pages with 9 figure

The noise properties of stochastic processes and entropy production

Based on a Fokker-Planck description of external Ornstein-Uhlenbeck noise and cross-correlated noise processes driving a dynamical system we examine the interplay of the properties of noise processes and the dissipative characteristic of the dynamical system in the steady state entropy production and flux. Our analysis is illustrated with appropriate examples.Comment: RevTex, 1 figure, To appear in Phys. Rev.

Lyapunov instability for a periodic Lorentz gas thermostated by deterministic scattering

In recent work a deterministic and time-reversible boundary thermostat called thermostating by deterministic scattering has been introduced for the periodic Lorentz gas [Phys. Rev. Lett. {\bf 84}, 4268 (2000)]. Here we assess the nonlinear properties of this new dynamical system by numerically calculating its Lyapunov exponents. Based on a revised method for computing Lyapunov exponents, which employs periodic orthonormalization with a constraint, we present results for the Lyapunov exponents and related quantities in equilibrium and nonequilibrium. Finally, we check whether we obtain the same relations between quantities characterizing the microscopic chaotic dynamics and quantities characterizing macroscopic transport as obtained for conventional deterministic and time-reversible bulk thermostats.Comment: 18 pages (revtex), 7 figures (postscript

Measurements of integral muon intensity at large zenith angles

High-statistics data on near-horizontal muons collected with Russian-Italian coordinate detector DECOR are analyzed. Precise measurements of muon angular distributions in zenith angle interval from 60 to 90 degrees have been performed. In total, more than 20 million muons are selected. Dependences of the absolute integral muon intensity on zenith angle for several threshold energies ranging from 1.7 GeV to 7.2 GeV are derived. Results for this region of zenith angles and threshold energies have been obtained for the first time. The dependence of integral intensity on zenith angle and threshold energy is well fitted by a simple analytical formula.Comment: 4 pages, 4 figures, 1 tabl

Convergence of invariant densities in the small-noise limit

This paper presents a systematic numerical study of the effects of noise on the invariant probability densities of dynamical systems with varying degrees of hyperbolicity. It is found that the rate of convergence of invariant densities in the small-noise limit is frequently governed by power laws. In addition, a simple heuristic is proposed and found to correctly predict the power law exponent in exponentially mixing systems. In systems which are not exponentially mixing, the heuristic provides only an upper bound on the power law exponent. As this numerical study requires the computation of invariant densities across more than 2 decades of noise amplitudes, it also provides an opportunity to discuss and compare standard numerical methods for computing invariant probability densities.Comment: 27 pages, 19 figures, revised with minor correction

Big Entropy Fluctuations in Nonequilibrium Steady State: A Simple Model with Gauss Heat Bath

Large entropy fluctuations in a nonequilibrium steady state of classical mechanics were studied in extensive numerical experiments on a simple 2-freedom model with the so-called Gauss time-reversible thermostat. The local fluctuations (on a set of fixed trajectory segments) from the average heat entropy absorbed in thermostat were found to be non-Gaussian. Approximately, the fluctuations can be discribed by a two-Gaussian distribution with a crossover independent of the segment length and the number of trajectories ('particles'). The distribution itself does depend on both, approaching the single standard Gaussian distribution as any of those parameters increases. The global time-dependent fluctuations turned out to be qualitatively different in that they have a strict upper bound much less than the average entropy production. Thus, unlike the equilibrium steady state, the recovery of the initial low entropy becomes impossible, after a sufficiently long time, even in the largest fluctuations. However, preliminary numerical experiments and the theoretical estimates in the special case of the critical dynamics with superdiffusion suggest the existence of infinitely many Poincar\'e recurrences to the initial state and beyond. This is a new interesting phenomenon to be farther studied together with some other open questions. Relation of this particular example of nonequilibrium steady state to a long-standing persistent controversy over statistical 'irreversibility', or the notorious 'time arrow', is also discussed. In conclusion, an unsolved problem of the origin of the causality 'principle' is touched upon.Comment: 21 pages, 7 figure

Time evolution and observables in constrained systems

The discussion is limited to first-class parametrized systems, where the definition of time evolution and observables is not trivial, and to finite dimensional systems in order that technicalities do not obscure the conceptual framework. The existence of reasonable true, or physical, degrees of freedom is rigorously defined and called {\em local reducibility}. A proof is given that any locally reducible system admits a complete set of perennials. For locally reducible systems, the most general construction of time evolution in the Schroedinger and Heisenberg form that uses only geometry of the phase space is described. The time shifts are not required to be 1symmetries. A relation between perennials and observables of the Schroedinger or Heisenberg type results: such observables can be identified with certain classes of perennials and the structure of the classes depends on the time evolution. The time evolution between two non-global transversal surfaces is studied. The problem is posed and solved within the framework of the ordinary quantum mechanics. The resulting non-unitarity is different from that known in the field theory (Hawking effect): state norms need not be preserved so that the system can be lost during the evolution of this kind.Comment: 31 pages, Latex fil

Dynamics of a thin shell in the Reissner-Nordstrom metric

We describe the dynamics of a thin spherically symmetric gravitating shell in the Reissner-Nordstrom metric of the electrically charged black hole. The energy-momentum tensor of electrically neutral shell is modelled by the perfect fluid with a polytropic equation of state. The motion of a shell is described fully analytically in the particular case of the dust equation of state. We construct the Carter-Penrose diagrams for the global geometry of the eternal black hole, which illustrate all possible types of solutions for moving shell. It is shown that for some specific range of initial parameters there are possible the stable oscillating motion of the shell transferring it consecutively in infinite series of internal universes. We demonstrate also that this oscillating type of motion is possible for an arbitrary polytropic equation of state on the shell.Comment: 17 pages, 7 figure