Stationary states and energy cascades in inelastic gases

We find a general class of nontrivial stationary states in inelastic gases where, due to dissipation, energy is transferred from large velocity scales to small velocity scales. These steady states exist for arbitrary collision rules and arbitrary dimension. Their signature is a stationary velocity distribution f(v) with an algebraic high-energy tail, f(v) approximately v(-sigma). The exponent sigma is obtained analytically and it varies continuously with the spatial dimension, the homogeneity index characterizing the collision rate, and the restitution coefficient. We observe these stationary states in numerical simulations in which energy is injected into the system by infrequently boosting particles to high velocities. We propose that these states may be realized experimentally in driven granular systems.     

Natural complexity, computational complexity and depth

Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a typical system state or history starting from simple initial conditions. The properties of depth are discussed and it is compared with other complexity measures. Depth can only be large for systems with embedded computation.     

The computational complexity of pattern formation

The computational complexity of diffusion-limited aggregation and fluid invasion in porous media is studied. The time requirements on an idealized parallel computer for simulating the patterns formed by these models are investigated. It is shown that these growth models are P-complete. These results provide strong evidence that such pattern formation processes are inherently sequential and cannot be simulated efficiently in parallel.     

Long time tails in stationary random media II: Applications

In a previous paper we developed a mode-coupling theory to describe the long time properties of diffusion in stationary, statistically homogeneous, random media. Here the general theory is applied to deterministic and stochastic Lorentz models and several hopping models. The mode-coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse-grained diffusion coefficient and a coarse-grained free volume. For one-dimensional models these amplitudes can be evaluated, and the mode-coupling theory is shown to agree with exact solutions obtained for these models. For higher-dimensional Lorentz models the formal theory yields expressions which are difficult to evaluate. For these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers.Work supported by grant No. CHE 77-16308 from the National Science Foundation and by a Nato Travel Grant.     

Ground states and thermal states of the random field Ising model

The random field Ising model is studied numerically at both zero and positive temperature. Ground states are mapped out in a region of random field and external field strength. Thermal states and thermodynamic properties are obtained for all temperatures using the Wang-Landau algorithm. The specific heat and susceptibility typically display sharp peaks in the critical region for large systems and strong disorder. These sharp peaks result from large domains flipping. For a given realization of disorder, ground states and thermal states near the critical line are found to be strongly correlated--a concrete manifestation of the zero temperature fixed point scenario.     

Mode coupling theory of hydrodynamics and steady state systems

We construct a formal mode coupling theory for hydrodynamic systems which includes contributions from all powers of the hydrodynamic variables. This theory is applied to nonequilibrium steady state systems. A generalization of the local equilibrium distribution is used to describe the nonequilibrium state. This distribution independently constrains all moments of the hydrodynamic variables. The infinite hierarchy of equations for the moments of the hydrodynamic variables is truncated using an inverse system size expansion. Explicit results are obtained for the time correlation functions of fluids with a linear temperature gradient or a linear shear. These results agree with previous studies of these steady states.     

On the behavior of the surface tension for spin systems in a correlated porous medium

We investigate the behavior of various spin-systems that are subject to the highly correlated and extremely diluted quenched disorder as provided by the fractal aerogel model. For these systems, it is (easily) established that, at all temperatures, the free energy is identical to that of the corresponding uniform system. The surface tension, however, behaves quite differently. Foremost, at any fixed temperature corresponding to the low temperature phase in the uniform system, there is a non-trivial curve in the aerogel phase plane dividing high-temperature behavior (zero surface tension) from low-temperature behavior (positive surface tension). The fractal aerogel has two distinctive phases in its own right: gel and sol. In the gel phase, the spin system has zero surface tension at all temperatures. In one region of the sol phase, the surface tension is shown to be equal to its value in the uniform system. Since part of this region borders on the gel phase, a certain portion of the sol/gel phase boundary is also the dividing line between high- and low-temperature behavior. Evidently, in this case, the surface tension is discontinuous at the phase boundary. on the other hand, there are well-defined length scales that diverge as the phase boundary is approached. This demonstrates an absence of scaling in these systems.     

Sampling Chaotic Trajectories Quickly in Parallel

The parallel computational complexity of the quadratic map is studied. A parallel algorithm is described that generates typical pseudotrajectories of length t in a time that scales as log t and increases slowly in the accuracy demanded of the pseudotrajectory. Long pseudotrajectories are created in parallel by putting together many short pseudotrajectories; Monte Carlo procedures are used to eliminate the discontinuities between these short pseudotrajectories and then suitably randomize the resulting long pseudotrajectory. Numerical simulations are presented that show the scaling properties of the parallel algorithm. The existence of the fast parallel algorithm provides a way to formalize the intuitive notion that chaotic systems do not generate complex histories.