Attraction and condensation of driven tracers in a narrow channel.

Emergent bath-mediated attraction and condensation arise when multiple particles are simultaneously driven through an equilibrated bath under geometric constraints. While such scenarios are observed in a variety of nonequilibrium phenomena with an abundance of experimental and numerical evidence, little quantitative understanding of how these interactions arise is currently available. Here we approach the problem by studying the behavior of two driven "tracer" particles, propagating through a bath in a 1D lattice with excluded-volume interactions. We apply the mean-field approximation to analytically explore the mechanism responsible for the tracers' emergent interactions and compute the resulting effective attractive potential. This mechanism is then numerically shown to extend to a realistic model of hard driven Brownian disks confined to a narrow 2D channel.

Covariant Lyapunov vectors for rigid disk systems.

We carry out extensive computer simulations to study the Lyapunov instability of a two-dimensional hard-disk system in a rectangular box with periodic boundary conditions. The system is large enough to allow the formation of Lyapunov modes parallel to the x-axis of the box. The Oseledec splitting into covariant subspaces of the tangent space is considered by computing the full set of covariant perturbation vectors co-moving with the flow in tangent space. These vectors are shown to be transversal, but generally not orthogonal to each other. Only the angle between covariant vectors associated with immediate adjacent Lyapunov exponents in the Lyapunov spectrum may become small, but the probability of this angle to vanish approaches zero. The stable and unstable manifolds are transverse to each other and the system is hyperbolic.

Time-reversal symmetry and covariant Lyapunov vectors for simple particle models in and out of thermal equilibrium.

Recently, a new algorithm for the computation of covariant Lyapunov vectors and of corresponding local Lyapunov exponents has become available. Here we study the properties of these still unfamiliar quantities for a simple model representing a harmonic oscillator coupled to a thermal gradient with a two-stage thermostat, which leaves the system ergodic and fully time reversible. We explicitly demonstrate how time-reversal invariance affects the perturbation vectors in tangent space and the associated local Lyapunov exponents. We also find that the local covariant exponents vary discontinuously along directions transverse to the phase flow.

Lyapunov instability of rough hard-disk fluids.

The dynamical instability of rough hard-disk fluids in two dimensions is characterized through the Lyapunov spectrum and the Kolmogorov-Sinai entropy h{KS} for a wide range of densities and moments of inertia I . For small I the spectrum separates into translation-dominated and rotation-dominated parts. With increasing I the rotation-dominated part is gradually filled in at the expense of translation until such a separation becomes meaningless. At any density, the rate of phase-space mixing, given by h{KS} , becomes less and less effective the more the rotation affects the dynamics. However, the degree of dynamical chaos, measured by the maximum Lyapunov exponent, is only enhanced by the rotational degrees of freedom for high-density gases but is diminished for lower densities. Surprisingly, no traces of Lyapunov modes were found in the spectrum for larger moments of inertia. The spatial localization of the perturbation vector associated with the maximum exponent however persists for any I .

Emergence of order in quantum extensions of the classical quasispecies evolution.

We study evolution equations which model selection and mutation within the framework of quantum mechanics. The main question is to what extent order is achieved for an ensemble of typical systems. As an indicator for mixing or purification, a quadratic entropy is used which assumes values between zero for pure states and (d-1)/d for fully mixed states. Here, d is the dimension. Whereas the classical counterpart, the quasispecies dynamics, has previously been found to be predominantly mixing, the quantum quasispecies (QS) evolution surprisingly is found to be strictly purifying for all dimensions. This is also typically true for an alternative formulation (AQS) of this quantum mechanical flow. We compare this also to analogous results for the Lindblad evolution. Although the latter may be viewed as a simple linear superposition of the purifying QS and AQS evolutions, it is found to be predominantly mixing. The reason for this behavior may be explained by the fact that the two subprocesses by themselves converge to different pure states, such that the combined process is mixing. These results also apply to high-dimensional systems.

Emergence of order in selection-mutation dynamics.

We characterize the time evolution of a d -dimensional probability distribution by the value of its final entropy. If it is near the maximally possible value we call the evolution mixing, if it is near zero we say it is purifying. The evolution is determined by the simplest nonlinear equation and contains a d x d matrix as input. Since we are not interested in a particular evolution but in the general features of evolutions of this type, we take the matrix elements as uniformly distributed random numbers between zero and some specified upper bound. Computer simulations show how the final entropies are distributed over this field of random numbers. The result is that the distribution crowds at the maximum entropy, if the upper bound is unity. If we restrict the dynamical matrices to certain regions in matrix space, to diagonal or triangular matrices, for instance, then the entropy distribution is maximal near zero, and the dynamics typically becomes purifying.

Thermodynamic instability of a confined gas.

The best possible cooling agent is a system with negative specific heat. If in thermal contact with a second system, any acquisition of energy due to a random fluctuation lowers its temperature, and the energy transfer in this direction is further enhanced. It continues until all the energy is extracted from the second system and their temperatures are at par. We exhibit these microcanonical features with a simple mechanical model of interacting classical gas particles in a specially confined domain and subjected to gravitation. As predicted, most of the gas particles are cooled and collect in the lowest part of the container, where the energy is carried away by a few remaining particles.

Stellar stability by thermodynamic instability.

For gravity-dominated systems the three features shrinking <=> energy decrease <=> temperature increase are dynamically linked together. So are their inverses: expansion <=> energy increase <=> temperature decrease. We exhibit these features by one classical particle in a suitable environment, and by many particles with purely attractive interactions. We then show how the ensuing negative heat capacity tames an explosive energy input.

Fluctuations, convergence times, correlation functions, and power laws from many-body Lyapunov spectra for soft and hard disks and spheres.

The dynamical instability of many-body systems is best characterized through the time-dependent local Lyapunov spectrum [lambda(j)], its associated comoving eigenvectors [delta(j)], and the "global" time-averaged spectrum []. We study the fluctuations of the local spectra as well as the convergence rates and correlation functions associated with the delta vectors as functions of j and system size N. All the number dependences can be described by simple power laws. The various powers depend on the thermodynamic state and force law as well as system dimensionality.

Chaos and irreversibility in simple model systems.

The multifractal link between chaotic time-reversible mechanics and thermodynamic irreversibility is illustrated for three simple chaotic model systems: the Baker Map, the Galton Board, and many-body color conductivity. By scaling time, or the momenta, or the driving forces, it can be shown that the dissipative nature of the three thermostated model systems has analogs in conservative Hamiltonian and Lagrangian mechanics. Links between the microscopic nonequilibrium Lyapunov spectra and macroscopic thermodynamic dissipation are also pointed out. (c) 1998 American Institute of Physics.

Thermal heat reservoirs via Gauss' principle of least constraint; Dissipation, chaos, and phase-space dimensionality loss in one-dimensional chains.

We use Gauss' principle of least constraint to impose different kinetic temperatures on the two halves of a periodic one-dimensional chain. The thermodynamic result is heat flow, as predicted by the Second Law of Thermodynamics. The statistical-mechanical result can be either a phase-space limit cycle or a strange attractor, depending on the chain length and the size of the temperature difference. We document the sensitivity of the Lyapunov spectrum and the underlying phase-space topology by varying the chain length and the size of the kinetic-temperature difference.

Fractal dimension of steady nonequilibrium flows.

The Kaplan-Yorke information dimension of phase-space attractors for two kinds of steady nonequilibrium many-body flows is evaluated. In both cases a set of Newtonian particles is considered which interacts with boundary particles. Time-averaged boundary temperatures are imposed by Nose-Hoover thermostat forces. For both kinds of nonequilibrium systems, it is demonstrated numerically that external isothermal boundaries can drive the otherwise purely Newtonian flow onto a multifractal attractor with a phase-space information dimension significantly less than that of the corresponding equilibrium flow. Thus the Gibbs' entropy of such nonequilibrium flows can diverge.

Heat transfer between two degrees of freedom.

A particularly simple chaotic nonequilibrium open system with two Cartesian degrees of freedom, characterized by two distinct temperatures T(x) and T(y), is introduced. The two temperatures are maintained by Nose-Hoover canonical-ensemble thermostats. Both the equilibrium (no net heat transfer) and nonequilibrium (dissipative) Lyapunov spectra are characterized for this simple system.