Conditional equilibrium and the equivalence of microcanonical and grandcanonical ensembles in the thermodynamic limit

Equivalence (allowing for convex combinations) of microcanonical, canonical and grandcanonical ensembles for states of classical systems is established under very mild assumptions on the limiting state. We introduce the notion of conditional equilibrium (C.E.), a property of states of infinite systems which characterizes convex combinations of limits of microcanonical ensembles. It is shown that C.E. states are, under quite general conditions, mixtures of Gibbs states.Supported in part by NSF Grant No. MCS 75-21684 A02Supported in part by NSF Grant No. MPS 72-04534Supported in part by NSF Grant No. Phy 77-22302     

Completely integrable gradient flows

In this paper we exhibit the Toda lattice equations in a double bracket form which shows they are gradient flow equations (on their isospectral set) on an adjoint orbit of a compact Lie group. Representations for the flows are given and a convexity result associated with a momentum map is proved. Some general properties of the double bracket equations are demonstrated, including a discussion of their invariant subspaces, and their function as a Lie algebraic sorter.Supported in part by NSF Grant DMS-90-02136, NSF PYI Grant DMS-9157556, and a Seed Grant from Ohio State UniversitySupported in part by AFOSR grant AFOSR-96-0197, by U.S. Army Research Office grant DAAL03-86-K-0171 and by NSF grant CDR-85-00108Supported in part by NSF Grant DMS-8922699     

Curvature of determinant bundles for degenerate families

We calculate the (1, 1) curvature of the Beilinson Schechtman connection for the determinant bundle associated to a family of Riemann surfaces with ordinary singularities. As consequences we obtain generalizations of theorems of Bismut and Bost.Supported in part by NSF Grant No DMS-9201022.Supported in part by NSC Grant No 83-0208-M-002-039, Republic of China.     

Lattice systems with a continuous symmetry

We prove that the expansion in powers of the temperatureT of the correlation functions and the free energy of the plane rotator model on ad-dimensional lattice is asymptotic to all orders inT. The leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series. We also prove the inverse power decay of the pair correlation at low temperatures ford=3.Supported by NSF Grant No. MCS 78-01885Supported by NSF Grant No. PHY 78-15920Supported by NSF Grant No. DMR 73-04355Supported by NSF Grant No. PHY-7825390 A01On leave from: Institut de Physique Théorique, Université de Louvain, BelgiumAlso: Department of Physics     

Nonlinear stability of rarefaction waves for compressible Navier Stokes equations

It is shown that expansion waves for the compressible Navier-Stokes equations are nonlinearly stable. The expansion waves are constructed for the compressible Euler equations based on the inviscid Burgers equation. Our result shows that Navier-Stokes equations and Euler equations are time-asymptotically equivalent on the level of expansion waves. The result is proved using the energy method, making essential use of the expansion of the underlining nonlinear waves and the specific form of the constitutive eqution for a polytropic gas.Supported in part by NSF Grant DMS-87-03971 and Army Grant DAAL03-87-K-0063Supported in part by Army Grant DAAL03-87-K-0063     

The existence of non-minimal solutions of the Yang-Mills-Higgs equations overR 3 with arbitrary positive coupling constant

This paper proves the existence of a non-trivial critical point of theSU(2) Yang-Mills-Higgs functional onR ³ with arbitrary positive coupling constant. The critical point lies in the zero monopole class but has action bounded strictly away from zero.Supported in part by NSF Grant DMS-9200576.Supported in part by NSF Grant DMS-9109491.     

Moduli of super Riemann surfaces

The basic properties of super Riemann surfaces are presented, and their supermoduli spaces are constructed, in a manner suitable for the application of algebro-geometric techniques to string theory.Supported in part by NSF Grant No. DMS-8704401Supported in part by NSF Grants No. DMS-8501783 and No. DMS-86107301(1)     

Bohr-sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula

We show how the moduli space of flatSU(2) connections on a two-manifold can be quantized in the real polarization of [15], using the methods of [6]. The dimension of the quantization, given by the number of integral fibres of the polarization, matches the Verlinde formula, which is known to give the dimension of the quantization of this space in a Kähler polarization.Supported in part by MSRI under NSF Grant 85-05550Supported in part by an NSF Graduate fellowship, and by a grant-in-aid from the J. Seward Johnson Charitable TrustSupported in part by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291. Address as of January 1, 1993: Department of Mathematics, Columbia University, New York, NY 10027, USA     

Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators

We completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi-exactly solvable Schrödinger operator on the line. Methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi-exactly solvable Hamiltonians and explicit normalizability conditions in general coordinate systems.Supported in Part by DGICYT Grant PS 89-0011Supported in Part by an NSERC GrantSupported in Part by NSF Grant DMS 92-04192     

The GHS and other correlation inequalities for a class of even ferromagnets

We prove the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory. These include spin –1/2 Ising models, ⁴ field theories, and other continuous spin models. The proofs are based on the properties of a classG ^– of probability measures which contains all measures of the form const exp(–V(x))dx, whereV is even and continuously differentiable anddV/dx is convex on [0, ). A new proof of the GKS inequalities using similar ideas is also given.Supported in part by National Science Foundation Grant MPS 71-02838 A 04.Supported by National Science Foundation Grant MPS 74-24696.Supported in part by National Science Foundation Grant MPS 74-04870.     

Charged spin-1 gluons,parton model and the Archimedes effect

In a gauge theory of (SU(2)×U(1))_flavour×SU(3)_colour with un-confined integer-charged quarks and massive integer-charged gluons both quarks and gluons contribute to electro- and neutrino-production. The gluon parton contribution to the lepto-production of colour is considered. Supported in part by NSF Grant No. GP43662X, and in part by UGC Research Assistantship, U.K.; Supported in part by a Senior Visiting Fellowship awarded by the Science Research Council, U.K., and by NSF Grant No. GP43662X;     

String structures on loop bundles

Differential geometry and topology of principal loop bundles (bundles of loop groups over loop spaces) are investigated. String structures, defined as bundle extensions corresponding to the central extension of the structure group, do not always exist. Various methods for deriving the obstruction to the existence of string structures are discussed.Supported in part by the U.S. Department of Energy under Grant #DE-FG02-84ER40158 with Harvard University.Supported in part by the NSF Grant #87-08447 and by the U.S. Department of Energy under Contract #DE-AC02-76ER-03069.On leave from the University of Wroclaw, Wroclaw, Poland     

New compactifications of ten and eleven dimensional supergravity on manifolds which are not direct products

Solutions of ten and eleven dimensional supergravity are presented for which the space-time metric is a product of an anti-de Sitter metric and a strictly positive scalar function of the internal coordinates. The critical point of the potential of seven dimensional maximal supergravity with SO(4) symmetry is identified with such a solution.Supported in part by NSF Grant PHY 8109110 A-03Supported in part by the U.S. Department of Energy under Contract No. DE-AC-03-81ER40050, and by the Weingart Foundation     

Non-abelian bosonization in two dimensions

A non-abelian generalization of the usual formulas for bosonization of fermions in 1+1 dimensions is presented. Any fermi theory in 1+1 dimensions is equivalent to a local bose theory which manifestly possesses all the symmetries of the fermi theory.Supported in part by NSF Grant PHY-80-19754     

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.Supported by N.S.F. Grant MPS 71-03375 A03. Part of this work carried out at the Courant Institute where it was supported by N.S.F. Grant GP-37069X.Supported in part by AFOSR Grant #73-2430 and N.S.F. Grant MP S75-20638.Supported by N.S.F. Grant # GP33136X-2. Part of this work was carried out at the Institute for Advanced Study.     

The geometry of real sine-Gordon wavetrains

The characterization ofreal, N phase, quasiperiodic solutions of the sine-Gordon equation has been an open problem. In this paper we achieve this result, employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature. A significant by-product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds, and an understanding of how the real isospectral manifolds are embedded. By placing the problem in this general context, these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions.Supported in part by NSF Grant No. MCS-8202288Supported in part by NSF Grant No. MCS-8002969     

The phase transition in the one-dimensional Ising Model with 1/r 2 interaction energy

We prove the existence of a spontaneous magnetization at low temperature for the one-dimensional Ising Model with 1/r ² interaction energy.Supported in part by NSF Grant DMR 81-00417     

A conventional proof of Kerr's Theorem

A proof of Kerr's Theorem for generating geodesic and shearfree null congruences in Minkowski space is given in the Newman-Penrose null tetrad formalism.Supported in part by NSF Grant GP-43844-X.     

A mechanical model of Brownian motion

We consider a dynamical system consisting of one large massive particle and an infinite number of light point particles. We prove that the motion of the massive particle is, in a suitable limit, described by the Ornstein-Uhlenbeck process. This extends to three dimensions previous results by Holley in one dimension.On leave of the Institut für Theoretische Physik I der Universität Münster. Supported by a Nato fellowshipSupported by NSF Grant, No. PHY 78-03816Supported by NSF Grant, Phy 78-15920     

Time-displaced correlation functions in an infinite one-dimensional mixture of hard rods with different diameters

Time-displaced conditional distribution functions are calculated for an infinite, one-dimensional mixture of equal-mass hard rods of different diameters. The kinetic equation that describes the time dependence of the one-particle total distribution function is found to be non-Markovian, in contrast with the situation in systems of identical rods. The correlation function does not contain any isolated damped oscillation, except for systems of equal-diameter rods with discrete velocities. Thus, we generalize the one-component results of Lebowitz, Perçus, and Sykes, removing some nontypical features of that system.Supported by NSF grant No. MCS 75-21684 A01 (M. A.), NSF grant No. MPS 75-20638 (J. L.), and USAFOSR grant No. 73-2430 B (J. M.)John Guggenheim Fellow on sabbatical leave from Belfer Graduate School of Science, Yeshiva University, New York.