Oscar E. Lanford

Time evolution of large classical systems

We begin with some very general and elementary remarks about nonequilibrium statistical mechanics. We then establish our notation for discussing finite systems of classical point particles, construct the microcanonical ensemble, and sketch some of the relations between statistical mechanics and ergodic theory.

Universal properties of maps on an interval

We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied. We investigate rigorously those aspects of these bifurcations which are universal, i.e. independent of the choice of a particular one-parameter family. We point out that this universality extends to many other situations such as certain chaotic regimes. We describe the ergodic properties of the maps for which the parameter value equals the limit of the bifurcation points.

Mean Entropy of States in Quantum‐Statistical Mechanics

The equilibrium states for an infinite system of quantum mechanics may be represented by states over suitably chosen C* algebras. We consider the problem of associating an entropy with these states and finding its properties, such as positivity, subadditivity, etc. For the states of a quantum‐spin system, a mean entropy is defined and it is shown that this entropy is affine and upper semicontinuous.

The classical mechanics of one-dimensional systems of infinitely many particles. I. An existence theorem

We prove a global existence and uniqueness theorem for solutions of the classical equations of motion for a one-dimensional system of infinitely many particles interacting by finite-range two-body forces which satisfy a Lipschitz condition.

The classical mechanics of one-dimensional systems of infinitely many particles. II. Kinetic theory

We apply the existence theorem for solutions of the equations of motion for infinite systems to study the time evolution of measures on the set of locally finite configurations of particles. The set of allowed initial configurations and the time evolution mappings are shown to be measurable. It is shown that infinite volume limit states of thermodynamic ensembles at low activity or for positive potentials are concentrated on the set of allowed initial configurations and are invariant under the time evolution. The total entropy per unit volume is shown to be constant in time for a large class of states, if the potential satisfies a stability condition.We apply the existence theorem for solutions of the equations of motion for infinite systems to study the time evolution of measures on the set of locally finite configurations of particles. The set of allowed initial configurations and the time evolution mappings are shown to be measurable. It is shown that infinite volume limit states of thermodynamic ensembles at low activity or for positive potentials are concentrated on the set of allowed initial configurations and are invariant under the time evolution. The total entropy per unit volume is shown to be constant in time for a large class of states, if the potential satisfies a stability condition.

The hard sphere gas in the Boltzmann-Grad limit

This talk will review what has been rigorously proved about the time-dependent behaviour of a gas of classical hard spheres in the limiting regime where the number n of particles per unit volume becomes infinitely large while the particle diameter ϵ goes to zero in such a way that nϵ2 approaches a finite non-zero limit. (It is in this limiting regime that the Boltzmann equation is expected to become exact.)

Integral Representations of Invariant States on B* Algebras

Let U be a B* algebra with a group G of automorphisms and K be the set of G‐invariant states on U. We discuss conditions under which a G‐invariant state has a unique integral representation in terms of extremal points of K, i.e., extremal invariant states.

Rigorous Derivation of the Phase Shift Formula for the Hilbert Space Scattering Operator of a Single Particle

For a single nonrelativistic particle moving in a spherically symmetric potential, the existence of the Hilbert space wave operators and S operator is proved and phase shift formulas for these operators are deduced. The probability, P(Ω), for scattering into the solid angle Ω is obtained from the time dependent theory. The relation between P(Ω) and the R matrix of the standard plane wave formulation of scattering theory is established. For collimated incoming packets, it is shown that P(Ω) can be expressed as an energy average of the differential cross section.

Equilibrium time correlation functions in the low density limit

We consider a system of hard spheres in thermal equilibrium. Using Lanfords result about the convergence of the solutions of the BBGKY hierarchy to the solutions of the Boltzmann hierarchy, we show that in the low-density limit (Boltzmann-Grad limit): (i) the total time correlation function is governed by the linearized Boltzmann equation (proved to be valid for short times), (ii) the self time correlation function, equivalently the distribution of a tagged particle in an equilibrium fluid, is governed by the Rayleigh-Boltzmann equation (proved to be valid for all times). In the latter case the fluid (not including the tagged particle) is to zeroth order in thermal equilibrium and to first order its distribution is governed by a combination of the Rayleigh-Boltzmann equation and the linearized Boltzmann equation (proved to be valid for short times).

Time evolution of infinite anharmonic systems

We prove the existence of a time evolution for infinite anharmonic crystals for a large class of initial configurations. When there are strong forces tying particles to their equilibrium positions then the class of permissible initial conditions can be specified explicitly; otherwise it can only be shown to have full measure with respect to the appropriate Gibbs state. Uniqueness of the time evolution is also proven under suitable assumptions on the solutions of the equations of motion.