The spherical-model limit in a random field

The spherical-model limitn of then-vector model in a random field, with either a statistically independent distribution or with long-range correlated random fields, is studied to demonstrate the correctness of the replica method in which then and replica limits limits are interchanged, provided the replica and thermodynamic limits are taken in the right order, in the case of long-range correlated random fields. A scaling form for the two-point correlation function relevant to the first-order phase transition below the lower critical dimensionality of the random system is also obtained.     

On the existence of thermodynamics for the generalized random energy model

Derrida's generalized random energy model is considered. Almost sure andL ^p convergence of the free energy at any inverse temperature are proven for an arbitrary numbern of hierarchical levels. The explicit form of the free energy is given in the most general case and the limitn is discussed.     

The free energy of a class of Hopfield models

We show that in the limitp ,N 0,=p/N 0 the limit free energy of the Hopfield model equals in probability the Curie-Weiss free energy. We prove also that the free energy of the Hopfield model is self-averaging for any finite .     

Phase Transformations α → β → γ → δ → ε in Titanium Hydride TiH x with Increase in Hydrogen Concentration

Structural phase transformations of the hcp bcc fct fcc fct ( ) type in titanium hydrides TiH _x experimentally found to occur with increase in the hydrogen concentration x are studied theoretically. Free energy phases are statistically calculated, and their concentration dependence is determined. Phase transformation temperatures are found from intersection points of free energy curves and from points of common tangents to the curves. The energy parameters of the system are estimated. The calculation results are compared with the experimental data.     

The mean spherical model in a random external field and the replica method

We provide a quick elementary solution of the mean spherical model in a random external field. This also allows an immediate proof of the self-averaging property of the free energy. We calculate the free energy by means of the replica method, i.e., for any (not necessarily integer) replica numbern, and show that when a phase transition occurs the limits andn 0 are not interchangeable.     

From monotonic to oscillatory decay of correlations: Analytical approximation for the two-dimensional,one-component plasma

An approximate evaluation of the pair distribution and the structure factor is performed analytically for the two-dimensional, one-component plasma at any value of the coupling constant. The approximate distribution remains positive and satisfies three sum rules, including the compressibility one. When 0 or 2, exact results are found. At=2 the transition from monotonie (<2) to oscillatory (>2) decay of correlations takes place. Comparison with the Monte Carlo simulations shows good agreement for 0<<4.     

A two-dimensional Fokker-Planck equation degenerating on a straight line

By an example of a two-dimensional hydrodynamic system, second-order Langevin equations with two correlated noise sources are investigated. It is shown that the asymptotic expression (t) for the stationary distribution functionP depends on the order in which the limiting transitions;t andN ₂₂0 (N ₂₂ is the power of one of the noises) are made. Using the method of local expansions in trigonometric form, approximate expressions are written for the distribution functionP at small but finiteN ₂₂ tending atN ₂₂0 to the known exact solution.     

Semiclassical quantum mechanics

We consider the 0 limit of the quantum dynamics generated by the HamiltonianH()=–(²/2m)+V. We prove that the evolution of certain Gaussian states is determined asymptotically as 0 by classical mechanics. For suitable potentialsV inn3 dimensions, our estimates are uniform in time and our results hold for scattering theory.Supported in part by the National Science Foundation under Grant PHY 78-08066     

Studies on the microstructure of phase transitions in two kinds of liquid crystals by PAT and DSC

The positron lifetime and DSC measerments for EBBA and DOBAMBC have been made with heating and cooling clycles. The experimental results show that a shorter lifetime (₁) is essentially independent of temperature while the longer lifetime (₂) and the intensity (I ₂) change abruptly double or triple with temperature. Consequently, the EBBA has only nematic phases while the DOBAMBC has two liquid-crystalline phases (smectic C* and smectic A) with transition temperatures as follows: for EBBA, solid nematic (304.5 K), nematic isotropic (356.5 K), isotropic nematic (356.5 K), nematic solid (301 K); and for DOBAMBC, solid smectic C* (346 K), smectic C* smectic A (357.5 K), smectic A isotropic (389 K). These critical temperatures are in accordance with the transition temperatures measured by DSC. In addition, the difference in the solid-nematic transition temperature in the heating and cooling cycles was also observed. A discussion about the correlation of the observed changes in lifetime (₂) with the changes in molecular orientational order (S) and dielectric anisotropy () is presented.     

Rate equations in the hopping transport

The usual kinetic equations for the site occupation probabilities in an external field are solved exactly in a simple one-dimensional periodic model with two kinds of atoms using a) free boundary conditions and order of limitsN, 0 needed for a proper treatment of the dc conductivity here b) boundary conditions with metallic contacts and order of limitsN, 0 and c) the same boundary conditions but reversed order of limiting processes 0,N typical of e.g. numerical and percolation treatments. (N and are the number of sites and frequency.) It is demonstrated that though the bulk dc conductivity is the same in all three cases, local bulk properties of the material are strongly dependent on the régime used. The role of the order of all three limiting processes 0,N+ andn+ (N–n+) for local shifts of the chemical potential _n in the dc limit is examined (n is the number of the relevant site calculated from a boundary of the chain). It is shown especially that the rate equation treatment (régime a) on the one hand and numerical or percolation treatments (régime c) on the other hand never yield the same bulk values of _r.     

Radiation laws in the vicinity of metallic boundaries

Corrections to Planck's radiation law and to the Stefan Boltzmann law in the vicinity of a dissipative halfspacez<0 are studied. The dissipation is described by a frequency independent conductivity . The halfspacez0 is empty.For a perfectly reflecting wall (=) the proximity corrections of the thermal electric and magnetic energy mutually cancel out. Therefore the space-dependent corrections are only due to the finite conductivity of the wall.The dissipative properties of the system lead to divergencies in the limitz0. In the limitz all corrections vanish. In properly scaledz>0 ranges analytical expressions for the corrections to the radiation laws are calculated.As a by-product the density of states of surface polaritons in the passive medium (z>0) are derived.     

Triple Fluorescence of Substituted Benzanilides in Solution and in Solid States

The competitive triple fluorescence of benzanilide is studied by steady-state fluorescence investigations in dependence on the solvent polarity and the para-substitution of the aniline core as well as by comparison with the fluorescence behavior of 4-methoxy-N-methylbenzanilide. The normal fluorescence of benzanilide S₁(LE) S₀ appears at _max = 345 nm, whereas a superposition of proton transfer (PT) fluorescence S₁(PT) S₀(PT) and intramolecular charge transfer (ICT) flu-orescence S₁(ICT) S₀(FC) is responsible for the long-wavelength fluorescence in the 500-nm region. Different possibilities for the formation of the PT and ICT states are discussed. Investigations of the fluorescence behavior of benzanilides both in solution and as crystals in dependence on the para-substitution of the benzanilide moiety support the PT/ICT model.     

On the phase transition to sheet percolation in random Cantor sets

Thed-dimensional random Cantor set is a generalization of the classical middle-thirds Cantor set. Starting with the unit cube [0, 1] ^d , at every stage of the construction we divide each cube remaining intoM ^d equal subcubes, and select each of these at random with probabilityp. The resulting limit set is a random fractal, which may be crossed by paths or (d–1)-dimensional sheets. We examine the critical probabilityp _s(M, d) marking the existence of these sheet crossings, and show that p_s(M,d)1–p_c(M ^d) asM, where p_c(M ^d) is the critical probability of site percolation on the lattice (M ^d) obtained by adding the diagonal edges to the hypercubic lattice ^d. This result is then used to show that, at least for sufficiently large values ofM, the phases corresponding to the existence of path and sheet crossings are distinct.     

Finite-size scaling study of a first-order temperature-driven symmetry-breaking structural phase transition

We present a study of finite-size effects in a model exhibiting a first-order temperature-driven symmetry-breaking structural phase transition in theL × cylindrical geometry in theL limit. Exact studies demonstrate the applicability of our scaling ansatz even in the one-dimensional limit, making this model ideal for studying finite-size effects. The scaling ansatz, similar to the previously developed ansatz for field-driven transitions, demonstrates that latent heat is crucial in driving these transitions. This ansatz is supported by a 2×2 phenomenological transfer matrix based upon the symmetries of the system; this produces an analytic free energy which has the scaling form. Order parameter probability distributions show that the high- and low-temperature phases coexist only in a small finite-size-affected regime near the bulk transition temperature; this regime vanishes exponentially fast asL diverges.     

Reaction,trapping, and multifractality in one-dimensional systems

In the first part of this paper, we present two variants of the A+AA and A+AP reaction in one dimension that can be investigated analytically. In the first model, pairs of neighboring particles disappear reactively at a rate which is independent of their relative distance. It is shown that the probability density (x) for a nearest neighbor distance equal tox approaches the scaling form(x) c exp(–cx/2)/(cx)^1/2 in the long-time limit, withc being the concentration of particles. The second model is a ballistic analogue of the coagulation reaction A+A A. The model is solved by reducing it to a first-passagetime problem. The anomalous relaxation dynamics can be linked in a direct way to the fractal time properties of random walks. In the second part of this paper, we discuss the complications that arise in systems with disorder. We present a new approach that relates first-passage-time characteristics in a one-dimensional random walk to properties of random maps. In particular, we show that Sinai disorder is a borderline case for the appearance of multifractal properties. Finally, we apply a previously introduced renormalization technique to calculate the survival probability of particles moving on the line in the presence of a background of imperfect traps.     

Multiparticle fractal aggregation

Kinetic fractal aggregation in a particle bath where a fractionf of the sites are initially occupied is studied withd=2 computer simulations. Independent particles diffusing to a fixed cluster produce an aggregate with fractal dimensionD 1.7 up to a correlation length(f). At larger lengthsD2.(f) asf 0. When the particles remain fixed but the cluster undergoes a rigid random walkD appears constant at larger scales but varies withf. D 1.95 at largef andD 1.7 asf 0. In both cases, the aggregate sizeN(t) grows with timet ^(f) . Aggregation on a surface by independently diffusing particles produces shapes reminiscent of electrochemical dendritic growth. The dependence of growth rate and geometry is studied as a function of particle concentration and sticking probability.     

Field Theory of Branching and Annihilating Random Walks

We develop a systematic analytic approach to the problem of branching and annihilating random walks, equivalent to the diffusion-limited reaction processes 2A and A (m + 1) A, where m 1. Starting from the master equation, a field-theoretic representation of the problem is derived, and fluctuation effects are taken into account via diagrammatic and renormalization group methods. For d > 2, the mean-field rate equation, which predicts an active phase as soon as the branching process is switched on, applies qualitatively for both even and odd m, but the behavior in lower dimensions is shown to be quite different for these two cases. For even m, and d near 2, the active phase still appears immediately, but with nontrivial crossover exponents which we compute in an expansion in = 2 – d, and with logarithmic corrections in d = 2. However, there exists a second critical dimension d_c 4/3 below which a nontrivial inactive phase emerges, with asymptotic behavior characteristic of the pure annihilation process. This is confirmed by an exact calculation in d = 1. The subsequent transition to the active phase, which represents a new nontrivial dynamic universality class, is then investigated within a truncated loop expansion, which appears to give a correct qualitative picture. The model with m = 2 is also generalized to N species of particles, which provides yet another universality class and which is exactly solvable in the limit N . For odd m, we show that the fluctuations of the annihilation process are strong enough to create a nontrivial inactive phase for all d 2. In this case, the transition to the active phase is in the directed percolation universality class. Finally, we study the modification when the annihilation reaction is 3A . When m = 0 (mod 3) the system is always in its active phase, but with logarithmic crossover corrections for d = 1, while the other cases should exhibit a directed percolation transition out of a fluctuation-driven inactive phase.     

Two results concerning asymptotic behavior of solutions of the Burgers equation with force

We consider the Burgers equation with an external force. For the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast . If the force is the product of a periodic function ofx and white noise in time, we prove the existence of an invariant distribution concentrated on the space of space-periodic functions which is the limit of a wide class of distributions ast .     

Square lattice Ising antiferromagnet in an external magnetic field

We study the anisotropic square lattice Ising antiferromagnet in terms of three parameters: an external magnetic field B, an effective temperature and an anisotropy parameterK. The model, i.e., partition function and free energy, is solved exactly in the anisotropic limit,K0, for arbitrary temperature and field by using the transfer matrix method. We also calculate the first corrections beyond this limit. The limit is non trivial and the phase transition is completely preserved. It is of the expected Ising type. The transition temperature _c (B, K) is determined exactly for bothK0 andK and the results are used to check the validity of a recently conjectured formula by Müller-Hartmann and the author.     

Fluctuations of extensive functions of quenched random couplings

An extensive quantity is a family of functions _v of random parameters, indexed by the finite regionsV (subsets of ^d) over which _v are additive up to corrections satisfying the boundary estimate stated below. It is shown that unless the randomness is nonessential, in the sense that lim _v/|V| has a unique value in the absolute (i.e., not just probabilistic) sense, the variance of such a quantity grows as the volume ofV. Of particular interest is the free energy of a system with random couplings; for such _v bounds are derived also for the generating functionE(e ^t ). In a separate application, variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment.