Fluctuations of one dimensional Ginzburg-Landau models in nonequilibrium

We study the fluctuation of one dimensional Ginzburg-Landau models in nonequilibrium along the hydrodynamic (diffusion) limit. The hydrodynamic limit has been proved to be a nonlinear diffusion equation by Fritz, Guo-Papanicolaou-Varadhan, etc. We proved that if the potential is uniformly convex then the fluctuation process is governed by an Ornstein-Uhlenbeck process whose drift term is obtained by formally linearizing the hydrodynamic equation.Work partially supported by the National Science Foundation under grant no. DMS 8806731 and DMS 9101196     

Finite-size scaling and surface tension from effective one dimensional systems

We develop a method for precise asymptotic analysis of partition functions near first-order phase transitions. Working in a (+1)-dimensional cylinder of volumeL×...×L×t, we show that leading exponentials int can be determined from a simple matrix calculation providedtv logL. Through a careful surface analysis we relate the off-diagonal matrix elements of this matrix to the surface tension andL, while the diagonal matrix elements of this matrix are related to the metastable free energies of the model. For the off-diagonal matrix elements, which are related to the crossover length from hypercubic (L=t) to cylindrical (t=) scaling, this includes a determination of the pre-exponential power ofL as a function of dimension. The results are applied to supersymmetric field theory and, in a forthcoming paper, to the finite-size scaling of the magnetization and inner energy at field and temperature driven first-order transitions in the crossover region from hypercubic to cylindrical scaling.Research partially supported by the A. P. Sloan Foundation and by the NSF under DMS-8858073Research partially supported by the NSF under DMS-8858073 and DMS-9008827     

An exact scaling result for one dimensional interacting electronic systems

For the one dimensional interacting electron system without umklapp processes, Sólyom has shown that to third order in the invariant charges, the Renormalization Group equations predict that g₁ ? 2g₂ is constant under scaling. We show that for g₁ ? 2g₂ = 0 this result is exact even in the strong coupling regime. The possible invariance of g₁ ? 2g₂ ≠ 0 is discussed.     

Universality and scaling of unstable plastic flow

The statistics of the jumplike plastic deformation of a Cu–Be alloy under the conditions of a low-temperature unstable plastic flow is studied experimentally. At a high strain rate, the parameters of the load jumps are found to be related by power laws, which corresponds to a scale-invariant behavior. A comparison with the data obtained for another mechanism of plastic instability, namely, the Portevin–Le Chatelier effect, points to the existence of universal laws governing the dynamics of a dislocation ensemble in the conditions of plastic instability.     

Universality in three dimensional random-field ground states

We investigate the critical behavior of three-dimensional random-field Ising systems with both Gauss and bimodal distribution of random fields and additional the three-dimensional diluted Ising antiferromagnet in an external field. These models are expected to be in the same universality class. We use exact ground-state calculations with an integer optimization algorithm and by a finite-size scaling analysis we calculate the critical exponents , , and . While the random-field model with Gauss distribution of random fields and the diluted antiferromagnet appear to be in same universality class, the critical exponents of the random-field model with bimodal distribution of random fields seem to be significantly different. Received: 9 July 1998 / Received in final form: 15 July 1998 / Accepted: 20 July 1998     

Uniform andL 2 convergence in one dimensional stochastic Ising models

We study the rate of convergence to equilibrium of one dimensional stochastic Ising models with finite range interactions. We donot assume that the interactions are ferromagnetic or that the flip rates are attractive. The infinitesimal generators of these processes all have gaps between zero and the rest of their spectra. We prove that if one of these processes is observed by means of local observables, then the convergence is seen to be exponentially fast with an exponent that is any number less than the spectral gap. Moreover this exponential convergence is uniform in the initial configuration.The authors were partially supported by N.S.F. Grants DMS-8609944 and DMS-8611487, respectively. Also, the second author acknowledges supports from DAAL 03-86-K-0171     

Anomalous scaling and gapless fermions of d-wave superconductors in a magnetic field

The problem of dx2-y2-wave quasiparticles in a weakly disordered Abrikosov vortex lattice is studied. Starting with a periodic lattice, the topological structure of the magnetic crystal momenta of gapless fermions is found for the particle-hole symmetric case. If in addition the site centered inversion symmetry is present, both the location and the number of the gapless fermions can be determined using an index theorem. In the case of spatially aperiodic vortex array, Simon and Lee scaling is found to be violated due to a quantum anomaly. The electronic density of states is found to scale with the root-mean-square vortex displacement as sqrt[H]f(u2rms/xi2), while thermal conductivity is H independent, but different from the H=0 case.     

Universality in quantum chaos and the one-parameter scaling theory

The one-parameter scaling theory is adapted to the context of quantum chaos. We define a generalized dimensionless conductance, g, semiclassically and then study Anderson localization corrections by renormalization group techniques. This analysis permits a characterization of the universality classes associated to a metal (g-->infinity), an insulator (g-->0), and the metal-insulator transition (g-->g(c)) in quantum chaos provided that the classical phase space is not mixed. According to our results the universality class related to the metallic limit includes all the systems in which the Bohigas-Giannoni-Schmit conjecture holds but automatically excludes those in which dynamical localization effects are important. The universality class related to the metal-insulator transition is characterized by classical superdiffusion or a fractal spectrum in low dimensions (d < or = 2). Several examples are discussed in detail.     

Universality classes of Ising models

We discuss a transformation of Ising spins which maps a d-dimentional Ising problem into a series of different problems in the same universality class.     

Threshold singularities in the dynamic response of gapless integrable models

We develop a method of an asymptotically exact treatment of threshold singularities in dynamic response functions of gapless integrable models. The method utilizes the integrability to recast the original problem in terms of the low-energy properties of a certain deformed Hamiltonian. The deformed Hamiltonian is local; hence, it can be analyzed using the conventional field theory methods. We apply the technique to spinless fermions on a lattice with nearest-neighbors repulsion, and evaluate the exponent characterizing the threshold singularity in the dynamic structure factor.     

Universality,revisions of and corrections to scaling in fluids

PVT and internal energy data in the critical region of steam are accurately described by a thermodynamic potential based on renormalization-group calculations for Ising-like spin systems. The potential includes both “corrections-to-scaling” and “mixing-of-variables” terms.     

Microcanonical Monte Carlo study of one dimensional self-gravitating lattice gas models

In this study we present a microcanonical Monte Carlo investigation of one dimensional (1 ? d) self-gravitating toy models. We study the effect of hard-core potentials and compare to the results obtained with softening parameters and also the effect of the topology on these systems. In order to study the effect of the topology in the system we introduce a model with the symmetry of motion in a line instead of a circle, which we denominate as 1 /r model. The hard-core particle potential introduces the effect of the size of particles and, consequently, the effect of the density of the system that is redefined in terms of the packing fraction of the system. The latter plays a role similar to the softening parameter ? in the softened particles’ case. In the case of low packing fractions both models with hard-core particles show a behavior that keeps the intrinsic properties of the three dimensional gravitational systems such as negative heat capacity. For higher values of the packing fraction the ring model behaves as the potential for the standard cosine Hamiltonian Mean Field model while for the 1 /r model it is similar to the one-dimensional systems. In the present paper we intend to show that a further simplification level is possible by introducing the lattice-gas counterpart of such models, where a drastic simplification of the microscopic state is obtained by considering a local average of the exact N-body dynamics.     

Universality test for critical amplitudes in two dimensional percolation

From series expansions estimates of Sykes et al. it is concluded that the ratio $\text{B}{}^{\mathrm{\delta -1}}\text{ГE}{}^{\mathrm{-\delta }}$ for the critical amplitudes corresponding to the critical exponents β, γ and δ, respectively, behaves like a universal quantity, within reasonable bounds, for the site and bond percolation problem.     

Universality and scaling of period-doubling bifurcations in a dissipative distributed medium

A homogeneous medium, consisting of nonlinear elements, demonstrating transition to chaos via period-doubling bifurcations, is considered. The coupling between the elements is supposed to be of a dissipative type, i.e. it tends to equalize their instantaneous states. Using the renormalization group approach, the following scaling law for weakly inhomogeneous states near the critical point is obtained: at each period doubling the spatial scale increases by β=√2. On the basis of this law the scaling hypotheses for the transition to chaos in the semi-infinite and finite systems are proposed. The scaling properties are verified by the numerical calculations with a simple model.