Analyticity properties of the surface free energy of the Ising model

For an Ising ferromagnet with nearest-neighbour interactions of strengthK and surface magnetic fieldh, the surface free energy in the presence of a positively (or negatively) magnetized zero-field bulk phase is shown to be analytic inh for Reh<K–/, where =2.96 ... and is the inverse temperature. This puts the lower boundK–/ on the values ofh at which wetting and layering transitions can take place.     

Metastable decay rates,asymptotic expansions,and analytic continuation of thermodynamic functions

The grand potentialP(z)/kT of the cluster model at fugacityz, neglecting interactions between clusters, is defined by a power series _n Q _n z ⁿ , whereQ _n , which depends on the temperatureT, is the partition function of a cluster of sizen. At low temperatures this series has a finite radius of convergencez _s . Some theorems are proved showing that ifQ _n , considered as a function ofn, is the Laplace transform of a function with suitable properties, thenP(z) can be analytically continued into the complexz plane cut along the real axis fromz _s to + and that (a) the imaginary part ofP(z) on the cut is (apart from a relatively unimportant prefactor) equal to the rate of nucleation of the corresponding metastable state, as given by Becker-Döring theory, and (b) the real part ofP(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term.     

Self-avoiding walks and trees in spread-out lattices

LetG _R be the graph obtained by joining all sites ofZ ^d which are separated by a distance of at mostR. Let (G _R ) denote the connective constant for counting the self-avoiding walks in this graph. Let (G _R ) denote the coprresponding constant for counting the trees embedded inG _R . Then asR, (G _R ) is asymptotic to the coordination numberk _R ofG _R , while (G _R ) is asymptotic toek _R. However, ifd is 1 or 2, then (G _R )-k _R diverges to –.Dedicated to Oliver Penrose on this occasion of his 65th birthday.     

Clusters,metastability, and nucleation: Kinetics of first-order phase transitions

We describe and interpret computer simulations of the time evolution of a binary alloy on a cubic lattice, with nearest neighbor interactions favoring like pairs of atoms. Initially the atoms are arranged at random; the time evolution proceeds by random interchanges of nearest neighbor pairs, using probabilities compatible with the equilibrium Gibbs distribution at temperatureT. For temperatures 0.59T_c, 0.81 T_c, and 0.89T _c, with density of A atoms equal to that in the B-rich phase at coexistence, the density C₁ of clusters ofl A atoms approximately satisfies the following empirical formulas: C₁ w(1 –)³ andC ₁, (1 –)⁴Q₁w¹ (2 l 10). Herew is a parameter and we defineQ _l = _K e ^–E(K) , where the sum goes over all translationally nonequivalentl-particle clusters andE(K) is the energy of formation of the clusterK. Forl > 10,Q ₁ is not known exactly; so we use an extrapolation formulaQ _l Aw _s ^–l l ^– exp(–bl ), wherew _s is the value ofw at coexistence. The same formula (withw > w _s) also fits the observed values of C, (for small values ofl) at densities greater than the coexistence density (forT=0.59T_c): When the supersaturation is small, the simulations show apparently metastable states, a theoretical estimate of whose lifetime is compatible with the observations. For higher supersaturation the system is observed to undergo a slow process of segregation into two coexisting phases (andw therefore changes slowly with time). These results may be interpreted as a more quantitative formulation (and confirmation) of ideas used in standard nucleation theory. No evidence for a spinodal transition is found.Supported by AFOSR Grant No. 73-2430D and by ERDA Contract No. EY-76-C-02-3077*000.     

Growth of clusters in a first-order phase transition

The results of computer simulations of phase separation kinetics in a binary alloy quenched from a high temperature are analyzed in detail, using the ideas of Lifshitz and Slyozov. The alloy was modeled by a three-dimensional Ising model with Kawasaki dynamics. The temperature after quenching was 0.59T _c, whereT _c is the critical temperature, and the concentration of minority atoms was=0.075, which is about five times their largest possible single-phase equilibrium concentration at that temperature. The time interval covered by our analysis goes from about 1000 to 6000 attempted interchanges per site. The size distribution of small clusters of minority atoms is fitted approximately byc ₁(1-)³ w(t),c ₁ (1–)⁴ Q _l w(t)^l(2l10); wherec _l is the concentration of clusters of sizel;Q ₂,...,Q ₁₀ are known constants, the cluster partition functions;t is the time; andw(t)=0.015(1+7.17t ^–1/3). The distribution of large clusters (l20) is fitted approximately by the type of distribution proposed by Lifshitz and Slyozov,c _l ,(t)=–(d/dl) [lnt+p (l/t)], where is a function given by those authors and is defined by(x)=C _o e–x-C ₁ e ^–4x/3-C ₂ e ^–5x/3;C ₀,C ₁,C ₂ are constants determined by considering how the total number of particles in large clusters changes with time.Supported by the U.S. Air Force Office of Scientific Research under Grant No. 78-3522 and by the U.S. Department of Energy under Contract No. EY-76-C-02-3077*000.     

A nonequilibrium entropy for dynamical systems

It is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure. These are chosen by considering the totality of future possibilities for observation on the system. It is shown that this entropy is necessarily a nondecreasing function of the timet. There is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal. Under suitable conditions (which are stated explicitly) this entropy approaches the equilibrium entropy ast+ and the fine-grained entropy ast–. In particular, the conditions can always be satisfied if the system is aK-system, as in the Sinai billiard models. Some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition, and at the same time elucidate the connection between our concept of entropy and the entropy invariant of Kolmogorov and Sinai.Research supported in part by NSF grants PHY78-03816 and PHY78-15920.     

Golden Oldie

A form of initial value problem is considered in which the initial hypersurface is not spacelike but null. This approach has the striking advantage over the more usual Cauchy problem that all constraints (initial data equations) are eliminated from the theory, for a wide class of interacting fields in special relativity and also for general relativity. The theory is most naturally described in terms of the two-component spinor calculus, for which an elementary introduction is given here. A general scheme for interacting fields, which holds both in special and general relativity, is presented which describes all fields in terms of sets of irreducible spinors. The concept of an exact set of such spinors is introduced and it is shown that this concept is the appropriate one for an initial value problem on a null cone without constraints. The initial data can be expressed in the form of a complex number, called a null datum, defined at each point of the null cone, one corresponding to each spinor. There is the curious feature of these null data that apparently it is sufficient here, to have onehalf as much information per point as in the corresponding Cauchy problem. The classical Maxwell-Dirac theory and the Einstein-Maxwell theory are two examples that can be put into the form of exact sets. The Einstein empty-space equations are also of particular note, and in this case the null datum describes essentially the intrinsic geometry of the null cone. The argument given here as applied to a general exact set is incomplete in two important respects. Firstly it depends on the null data being analytic, and secondly the initial hypersurface must be a cone. However, both these restrictions are removed in the case of certain elementary fields called basic free fields, examples of which are the Weyl neutrino field, the free Maxwell field, and the linearized gravitational field. For these cases a simple explicit formula is introduced which expresses the field at any point in terms of the null datum, as an integral taken over the intersection of the initial null hypersurface with the null cone of the point.This article originally appeared in 1963 in Aerospace Research Laboratories 63-56 (P.G. Bergmann). It is an important and oft-cited work, but as it has never been published in a widely distributed journal, it is generally inaccessable to the relativity community. This regrettable situation is hereby rectified-Ed.This work was done while the author was at Princeton, Syracuse, and Cornell Universities, visiting under a NATO Fellowship administered by the Department of Scientific and Industrial Research in London. The work at Syracuse was supported by the Aeronautical Research Laboratory and at Cornell by the National Science Foundation.     

Thermodynamic limit for classical systems with Coulomb interactions in a constant external field

We introduce a new method for studying the thermodynamic limit for systems of particles with Coulomb interactions. The method is based on calculating the potential energy of the Coulomb interactions from the electric or magnetic fields in the system rather than from the energy of the individual particle — particle interactions. We are able to include the effects of a constant external field being imposed at the boundary of the system. The difficulties associated with Coulomb potentials being not even weakly tempered are overcome by imposing the boundary condition that at the boundary of the region containing the particles, the electric or magnetic field has normal component equal to that of the applied field. We prove that the thermodynamic free energy density exists and is independent of the sequence of regions used to define the limit. We introduce sequences of regions all of the same shape and show that for these sequences of regions the thermodynamic free energy density is independent of shape. Finally, we prove that the thermodynamic free energy is a convex function of the density of particles and of the applied field.     

Towards quantum superpositions of a mirror

We propose an experiment for creating quantum superposition states involving of the order of 10(14) atoms via the interaction of a single photon with a tiny mirror. This mirror, mounted on a high-quality mechanical oscillator, is part of a high-finesse optical cavity which forms one arm of a Michelson interferometer. By observing the interference of the photon only, one can study the creation and decoherence of superpositions involving the mirror. A detailed analysis of the requirements shows that the experiment is within reach using a combination of state-of-the-art technologies.