Local Thermodynamic Equilibrium for some Stochastic Models of Hamiltonian Origin

We consider a class of 1-D stochastic models that are realizations of Hamiltonian models of heat conduction and prove that in the infinite volume limit local thermodynamic equilibrium is attained with linear energy profile.     

Ergodicity and Energy Distributions for Some Boundary Driven Integrable Hamiltonian Chains

We consider systems of moving particles in 1-dimensional space interacting through energy storage sites. The ends of the systems are coupled to heat baths, and resulting steady states are studied. When the two heat baths are equal, an explicit formula for the (unique) equilibrium distribution is given. The bulk of the paper concerns nonequilibrium steady states, i.e., when the chain is coupled to two unequal heat baths. Rigorous results including ergodicity are proved. Numerical studies are carried out for two types of bath distributions. For chains driven by exponential baths, our main finding is that the system does not approach local thermodynamic equilibrium as system size tends to infinity. For bath distributions that are sharply peaked Gaussians, in spite of the near-integrable dynamics, transport properties are found to be more normal than expected.     

Strange Attractors in Periodically-Kicked Limit Cycles and Hopf Bifurcations

We prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with SRB measures when certain simple dynamical systems are kicked at periodic time intervals. The settings considered include limit cycles and stationary points undergoing Hopf bifurcations.     

Nonequilibrium Steady States for Certain Hamiltonian Models

We report the results of a numerical study of nonequilibrium steady states for a class of Hamiltonian models. In these models of coupled matter-energy transport, particles exchange energy through collisions with pinned-down rotating disks. In Commun. Math. Phys. 262, 237–267, 2006, Eckmann and Young studied 1D chains and showed that certain simple formulas give excellent approximations of energy and particle density profiles. Keeping the basic mode of interaction in Commun. Math. Phys. 262, 237–267, 2006, we extend their prediction scheme to a number of new settings: 2D systems on different lattices, driven by a variety of boundary (heat bath) conditions including the use of thermostats. Particle-conserving models of the same type are shown to behave similarly. The second half of this paper examines memory and finite-size effects, which appear to impact only minimally the profiles of the models tested in Commun. Math. Phys. 262, 237–267, 2006. We demonstrate that these effects can be significant or insignificant depending on the local geometry. Dynamical mechanisms are proposed, and in the case of directional bias in particle trajectories due to memory, correction schemes are derived and shown to give accurate predictions.     

Extended Systems with Deterministic Local Dynamics and Random Jumps

We consider particle systems on lattices with internal dynamics at each site and random jumps between sites. Models with simple chaotic local dynamics, namely expanding circle maps, are considered. Results on mean drift rates, central limit theorems and dependences on jump parameters are proved. A version of most of the results in this paper is contained in this author’s Ph.D. thesis [K]. This research is partially supported by a grant from the NSF.     

Strange Attractors with One Direction of Instability

We give simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors. Only the d= 2 case is treated in this paper, although our approach is by no means limited to two phase-dimensions. We develop a dynamical picture for the attractors in this class, proving they have many of the statistical properties associated with chaos: positive Lyapunov exponents, existence of SRB measures, and exponential decay of correlations. Other results include the geometry of fractal critical sets, nonuniform hyperbolic behavior, symbolic coding of orbits, and formulas for topological entropy. Received: 25 April 2000 / Accepted: 17 October 2000