Gibbs states of the Hopfield model with extensively many patterns

We consider the Hopfield model withM(N)=N patterns, whereN is the number of neurons. We show that if is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a provlem left open in previous work. The key new ingredient is a self-averaging result on the free energy functional. This result has considerable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given.     

Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model

We prove that if _N is the Sherrington-Kirkpatrick (SK) Hamiltonian and the quantity converges in the variance to a nonrandom limit asN, then the mean free energy of the model converges to the expression obtained by SK. Since this expression is known not to be correct in the low-temperature region, our result implies the non-self-averaging of the order parameter of the SK model. This fact is an important ingredient of the Parisi theory, which is widely believed to be exact. We also prove that the variance of the free energy of the SK model converges to zero asN, i.e., the free energy has the self-averaging property.See the Remarks after the proof of Theorem 1 on the validity of our results for more general distributions ofJ _ij .     

Finite-size effects in random energy models and in the problem of polymers in a random medium

By the use of traveling wave equations we calculate the finite-size corrections to the free energy of random energy models in their low-temperature phases and in the neighborhood of the transition temperature. We find that although the extensive part of the free energy does not show any critical behavior when the temperature approaches its transition value, the finite-size corrections signal the transition by becoming singular. We obtain a scaling form for these finite-size corrections valid in the limitN andTT _c . By considering a generalized random energy model in the limit of a very large number of steps, we obtain results for the finite-size corrections in the problem of a polymer in a random medium.     

Dynamics of the finite SK spin-glass

The dynamical behavior of a Sherrington-Kirkpatrick spin-glass model consisting of a large but finite number of Ising spins with a time evolution given by Glauber dynamics is investigated. Starting from the resummation of a diagrammatic expansion we derive a differential equation for the response function which allows us to handle nonperturbative effects. This enables us to find explicit expressions for the dynamical behavior of response and correlation function on time scales related to those free energy barriers which diverge with system sizeN. For the largest of these barriers we find a behavior proportional toN with =1/3.     

Translation of J. D. van der Waals' “The thermodynamik theory of capillarity under the hypothesis of a continuous variation of density”

Van der Waals justifies the choice of minimization of the (Helmholtz) free energy as the criterion of equilibrium in a liquid-gas system (Sections 1–4). If density is a function of heighth then the local free energy density differs from that of a homogeneous fluid by a term proportional to (d ² /dh ²); the extra term arises from the energy not from the entropy (Section 5). He uses this result to show how varies withh (Section 6), how this variation leads to a stable minimum free energy (Section 7), and to calculate the capillary energy or surface tension (Section 9). Near the critical point varies as ( _k -)^3/2, where _k is the critical temperature (Section 11). The paper closes with short discussions of the thickness of the surface layer (Section 12), of the difficulty of assuming that varies discontinuously with height (Section 14), and of the possible effect of derivatives of higher order than (d ² /dh ²) on the free energy and surface tension (Section 15).Originally published (in Dutch) inVerhandel. Konink. Akad. Weten. Amsterdam (Sect. 1), Vol. 1, No. 8 (1893).     

Some rigorous results on the Sherrington-Kirkpatrick spin glass model

We prove that in the high temperature regime (T/J>1) the deviation of the total free energy of the Sherrington-Kirkpatrick (S-K) spin glass model from the easily computed log Av(Z _N ({J})) converges in distribution, asN , to a (shifted) Gaussian variable. Some weak results about the low temperature regime are also obtained.Dedicated to Walter Thirring on his 60^th birthdayResearch supported in part by the NSF grant PHY-8605164. Present address: Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012, USAResearch supported in part by the NSF grant DMR 86-12369On leave from Institut des Hautes Etudes Scientifiques, Bures-Sur-Yvette, France     

New interpretation of Parisi-Sompolinsky mean-field solution for spin glasses

Using real replicas to describe the many-state phase space of the Sherrington-Kirkpatrick model of a spin glass, we rederive the Sompolinsky free energy for the ground state. We propose an interpretation of the order parameter function q(x), x 0,1, from the Sompolinsky free energy directly within the real replica scheme. We also suggest an iterative way of reaching the ground-state (Sompolinsky) free energy within this scheme and present numerical results for the first iteration going beyond the ergodic (replica symmetric) solution.The authors are grateful to the referee for constructive comments leading to improvements in the construction of the Sompolinsky free energy and the following interpretation.     

Scaling limit of the energy variable for the two-dimensional Ising ferromagnet

The critical point limit law (scaling limit) of the suitably renormalized energy variable is explicitly calculated for the two-dimensional nearest-neighbour Ising cylinder with free edges. It is shown that the renormalization factor has to behave as (2M 2N lnN)^1/2, where 2M denotes the number of rows and 2N the number of columns. By first taking the limitM and thenN, the limit law is proven to be Gaussian.     

Static and dynamic critical phenomena of the two-dimensionalq-state Potts model

Theq-state Potts model on the square lattice is studied by Monte Carlo simulation forq=3, 4, 5, 6. Very good agreement is obtained with exact results of Kiharaet al. and Baxter for energy and free energy at the critical point. Critical exponent estimates forq=3 are0.4,0.1,1.45, in rough agreement with high-temperature series extrapolation and real space renormalization-group methods. The transition forq=5, 6 is found to be a very weakly first-order transition, i.e., pronounced pseudocritical phenomena occur, specific heat, susceptibility, etc. (nearly) diverge at the first-order transition temperature. Dynamics is associated to the model in the same way as for the kinetic Ising model, and the nonlinear slowing down of the order parameter and of the energy is studied. The dynamic exponent is estimated to be (=zv)1.9. Within our accuracy noq dependence is detected. The relaxation is found to be consistent with dynamic scaling predictions, and dynamic scaling functions associated with the nonlinear relaxation are estimated.     

Ising Field Theory in a Magnetic Field: Analytic Properties of the Free Energy

We study the analytic properties of the scaling function associated with the 2D Ising model free energy in the critical domain TT _c , H0. The analysis is based on numerical data obtained through the Truncated Free Fermion Space Approach. We determine the discontinuities across the Yang–Lee and Langer branch cuts. We confirm the standard analyticity assumptions and propose extended analyticity; roughly speaking, the latter states that the Yang–Lee branching point is the nearest singularity under Langer's branch cut. We support the extended analyticity by evaluating numerically the associated extended dispersion relation.     

Critical temperature and energy gap in superconductors with singular density of states

The influence of-functional (symmetric as well as asymmetric) singularities in density of states (DOS) on the critical temperature and zero temperature energy gap is calculated. Surprisingly, we have obtained the same function for the off-symmetry of the peak position in DOS on the corresponding critical temperature as for the temperature dependence of the energy gap in the strong-coupling limit. The enhancement of the critical temperature due to the singularity (compared with the constant DOS near the Fermi surface) is much lower for strong-coupling superconductors than in the weak-coupling limit. Hence, the singularity in DOS cannot be the exclusive reason for large values of critical temperatures in highTc superconductors.This work was supported by the grant GA SAV 188/1991.     

Absence of phase transitions in certain one-dimensional long-range random systems

An Ising chain is considered with a potential of the formJ(i, j)/|i–j|, where theJ(i, j) are independent random variables with mean zero. The chain contains both randomness and frustration, and serves to model a spin glass. A simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if>1. This is to be contrasted with a ferromagnetic interaction which requires>2. The basic idea is to prove that the surfacefree energy between two half-lines is finite, although the surface energy may be unbounded. Ford-dimensional systems, it is shown that the free energy does not depend on the specific boundary conditions if>(1/2)d.     

Gaussian,non-Gaussian critical fluctuations in the Curie-Weiss model

It is known that at the critical temperature the Curie-Weiss mean-field model has non-Gaussian fluctuations and that internal fluctuations can be Gaussian. Here we compute the distribution of theq-mode magnetization fluctuations as a function of the temperature, the wave vectorq, and a fading out external field. We obtain new classes of probability distributions generated by this external field as well as new critical behavior in terms of its rate of fading out. We discuss also the susceptibility as the limitq tending to zero.     

Infinite-order phase transition in a classical spin system

For an exactly soluble classical spin model with long-range inhomogeneous coupling it is proved that in the absence of external magnetic field the free energy is aC function of the temperature at the critical point.     

Nonlinear excitations in the critical region

The free energy transformation due to fluctuations is investigated in an exactly solvable model. This model accounts for the fluctuation interaction in a reduced manner and leads to a realistic estimation for the free energy. In particular it gives a nice critical exponent=5. It is shown that in spite of the monotonic character of the effective free energy in the critical region the properties of the system should be described on the basis of the ⁶ model. Localized nonlinear excitations are found to be possible with a profile rather like that known as a bump near the point of the first-order phase transition.     

On the hardening in modulated structure

Hardening in modulated structure is evaluated using the periodic approximation. The critical shear stress increment due to the periodic structure is calculated in the constant line energy approximation. The results are applicable to any periodic structure (concentration waves must be neither homophase nor symmetric) exerting on the dislocation local glide forces with an amplitude smaller than ( denotes the line energy of corresponding straight dislocation directed along the concentration variations with the wave vector). In the zero approximation, the critical forceb is then simply the glide force on the straight dislocation averaged along its length in its most hardened position.     

The free energy of the spin-boson model

Forn spins 1/2 coupled linearly to a boson field in a volumeV _n, the existence of the specific free energy is proved in the limitn ,V _n withn/V _n=const. The interaction is essentially of the mean field type, in as much as it is proportional to 1/V _n; the coupling constants are allowed to be spin dependent. A variational expression is obtained for the limiting specific free energy, and a critical temperature is identified above which the system behaves as if there were no coupling at all.     

The classical limit of quantum spin systems

We derive a classical integral representation for the partition function,Z ^Q , of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentumJ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality isZ ^C (J)Z ^Q (J)Z ^C (J+1).On leave from the Department of Mathematics, M.I.T., Cambridge, Mass. 02139, USA. Work partially supported by National Science Foundation Grant GP-31674X and by a Guggenheim Memorial Foundation Fellowship.     

Hubbard-like models in high spatial dimensions

The present paper studies the properties of Hubbard-like models in high spatial dimensionsD. In a first par the limit of infinite dimension and its main features-i.e.i) the mapping onto a generalized atomic model with an additional auxiliary field andii) the validity of the local approximation for the self-energy-are worked out in a systematic (1/D)-expansion. Since the hopping matrix elements have to be properly scaled with the dimensionD, the (1/D)-expansion is also an expansion in the hopping amplitude. Thus for small hopping theD-limit may serve as a proper approximation for finite-dimensional systems. The second part of the paper adopts the hybridisation-perturbation theory of the single impurity Anderson model in order to construct a perturbation theory for the auxiliary field of the generalized atom which can also be interpreted as an expansion in the hopping amplitude. The non-crossing approximation (NCA) is used to study the antiferromagnetic phase transtion of theD-Hubbard model in the case of half filling: the critical temperature, the antiferromagnetic order parameter and the free energy of the lattice system are calculated. The NCA-results are in quite good agreement with recent results from the imaginary-time discretisation method.     

Ising models,julia sets,and similarity of the maximal entropy measures

We study the phase transition of Ising models on diamondlike hierarchical lattices. Following an idea of Lee and Yang, one can make an analytic continuation of free energy of this model to the complex temperature plane. It is known that the Migdal-Kadanoff renormalization group of this model is a rational endomorphism (denoted byf) of and that the singularities of the free energy lie on the Julia setJ(f). The aim of this paper is to prove that the free energy can be represented as the logarithmic potential of the maximal entropy measure onJ(f). Moreover, using this representation, we can show a close relationship between the critical exponent and local similarity of this measure.