Sheldon Goldstein

Stability and Equilibrium States of Infinite Classical Systems

We prove that any stationary state describing an infinite classical system which is “stable” under local perturbations (and possesses some strong time clustering properties) must satisfy the “classical” KMS condition. (This in turn implies, quite generally, that it is a Gibbs state.) Similar results have been proven previously for quantum systems by Haag et al. and for finite classical systems by Lebowitz et al.

Ergodic Properties of an Infinite One Dimensional Hard Rod System

It is shown that an infinite one dimensional system of hard rods for which the “effective” velocities of the pulses (free velocity plus a drift term due to collisions) are bounded away from some neighborhood of 0 is Bernoulli. This generalizes a result of Sinai who showed that some hard rod systems areK-systems.

Ergodic properties of infinite systems

Macroscopic systems are successfully modeled in statistical mechanics, at least in equilibrium, by infinite systems. We discuss the ergodic theoretic structure of such systems and present results on the ergodic properties of some simple model systems. We argue that these properties, suitably refined by the inclusion of space translations and other structure, are important for an understanding of the nonequilibrium properties of macroscopic systems.

On the equivalence between KMS-states and equilibrium states for classical systems

It is shown that for any KMS-state of a classical system of non-coincident particles, the distribution functions are absolutely continuous with respect to Lebesgue measure; the equivalence between KMS states and Canonical Gibbs States is then established.

Ergodic properties of an infinite system of particles moving independently in a periodic field

We investigate the ergodic properties of a general class of infinite systems of independent particles which undergo nontrivial “collisions” with an external field, e.g. fixed convex barriers (the Lorentz gas). We relate the ergodic properties of these systems to the ergodic properties for a single particle moving in a finite box (with periodic boundary conditions) with the same dynamics. We prove that when the one particle system is mixing or aK-system for a sequence of boxes approaching infinity so is the infinite particle system with an equilibrium measure obtained as a Poisson construction over the one particle phase space.

On the stability of equilibrium states of finite classical systems

The state of a system is characterized, in statistical mechanics, by a measure ω on Γ, the phase space of the system (i.e., by an ensemble). To represent an equilibrium state, the measure must be stationary under the time evolution induced by the systems Hamiltonian H (x), x‐Γ. An example of such a measure is ω (dx) = f (H) dx;dx is the Liouville (Lebesgue) measure and f (H (x)) is the ensemble density. For ’’nonergodic’’ systems there are also other stationary measures with ensemble densities, e.g., for integrable dynamical systems the density can be a function of any of the constants of the motion. We show, however, that the requirement that the equilibrium measure have a certain type of ’’stability’’ singles out, in the typical case, densities which depend only on H.

Space-time ergodic properties of systems of infinitely many independent particles

We investigate the ergodic properties of the equilibrium states of systems of infinitely many particles with respect to the group generated by space translations and time evolution. The particles are assumed to move independently in a periodic external field. We show that insofar as “good thermodynamic behavior” is concerned these properties provide much sharper discrimination than the ergodic properties of the time evolution alone. In particular, we show that though the infinite ideal gas is mixing in the space-time framework, it has vanishing space-time entropy and fails to be a space-timeK-system. On the other hand, if the particles interact with fixed convex scatterers (the Lorentz gas) the system forms a space-timeK-system. Also, the space-time entropy of a system of the type we consider is shown to equal its time entropy per unit volume.

Ergodic properties of simple model system with collisions

We investigate the ergodic properties of the discrete time evolution of a particle in a two‐dimensional torus with velocity in the unit square. The dynamics consists of free motion for a unit time interval followed by a bakers transformation of the velocity.

Pharmacotherapeutic Considerations in Anesthesia

This article focuses on new findings leading to improved understanding of the pathophysiology and mechanisms of potential drug interactions between anesthetic drugs or techniques and cardiovascular medications in patients scheduled for surgery. Only the most frequently used drugs are reviewed. Elective surgery provides the luxury to consider these risks and alter therapy accordingly. Under urgent circumstances, however, the increased risks associated with these agents should be anticipated with the goal to minimize adverse effects while maintaining optimal cardiovascular function in the perioperative period.

Arbitrarily slow approach to limiting behavior

Let f(k,t):R N ×[0,∞)→R be jointly continuous in k and t, with lim t→ ∞f(k,t)=F(k) discontinuous for a dense set of ks. It is proven that there exists a dense set Γ of ks such that, for k∈Γ, |f(k,t)−F(k)| approaches 0 arbitrarily slowly, i.e., more slowly than any expressible function g(t)→0. The result is applied to diffusion and conduction in quasiperiodic media and yields arbitrarily slow approaches to limiting behavior as time or volume becomes infinite. Such a slow approach is in marked contrast to the power laws widely found for random media