Relaxation times of kinetically constrained spin models with glassy dynamics

We analyze the density and size dependence of the relaxation time $\tau$ for kinetically constrained spin systems. These have been proposed as models for strong or fragile glasses and for systems undergoing jamming transitions. For the one (FA1f) or two (FA2f) spin facilitated Fredrickson-Andersen model at any density $\rho<1$ and for the Knight model below the critical density at which the glass transition occurs, we show that the persistence and the spin-spin time auto-correlation functions decay exponentially. This excludes the stretched exponential relaxation which was derived by numerical simulations. For FA2f in $d\geq 2$, we also prove a super-Arrhenius scaling of the form $\exp(1/(1-\rho))\leq \tau\leq\exp(1/(1-\rho)^2)$. For FA1f in $d$=$1,2$ we rigorously prove the power law scalings recently derived in \cite{JMS} while in $d\geq 3$ we obtain upper and lower bounds consistent with findings therein. Our results are based on a novel multi-scale approach which allows to analyze $\tau$ in presence of kinetic constraints and to connect time-scales and dynamical heterogeneities. The techniques are flexible enough to allow a variety of constraints and can also be applied to conservative stochastic lattice gases in presence of kinetic constraints.Comment: 4 page

Phase diagram of the ABC model on an interval

The three species asymmetric ABC model was initially defined on a ring by Evans, Kafri, Koduvely, and Mukamel, and the weakly asymmetric version was later studied by Clincy, Derrida, and Evans. Here the latter model is studied on a one-dimensional lattice of N sites with closed (zero flux) boundaries. In this geometry the local particle conserving dynamics satisfies detailed balance with respect to a canonical Gibbs measure with long range asymmetric pair interactions. This generalizes results for the ring case, where detailed balance holds, and in fact the steady state measure is known only for the case of equal densities of the different species: in the latter case the stationary states of the system on a ring and on an interval are the same. We prove that in the N to infinity limit the scaled density profiles are given by (pieces of) the periodic trajectory of a particle moving in a quartic confining potential. We further prove uniqueness of the profiles, i.e., the existence of a single phase, in all regions of the parameter space (of average densities and temperature) except at low temperature with all densities equal; in this case a continuum of phases, differing by translation, coexist. The results for the equal density case apply also to the system on the ring, and there extend results of Clincy et al.Comment: 52 pages, AMS-LaTeX, 8 figures from 10 eps figure files. Revision: minor changes in response to referee reports; paper to appear in J. Stat. Phy

Inhomogeneity-induced second-order phase transitions in Potts model on hierarchical lattices

The thermodynamics of the $q$-state Potts model with arbitrary $q$ on a class of hierarchical lattices is considered. Contrary to the case of the crystal lattices, it has always the second-order phase transitions. The analytical expressions fo the critical indexes are obtained, their dependencies on the structural lattice pararmeters are studied and the scailing relations among them are establised. The structural criterion of the inhomogeneity-induced transformation of the transition order is suggested. The application of the results to a description of critical phenomena in the dilute crystals and substances confined in porous media is discussed.Comment: 9 pages, 2 figure

Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics

Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial non-ergodicity, depending on the constraint parameter and the dimensionality. But in the ergodic cases, the dynamics is shown to be intrinsically cooperative at high densities giving rise to glassy dynamics as observed in simulations. The cooperativity is characterized by two length scales whose behavior controls finite-size effects: these are essential for interpreting simulations. In contrast to hypercubic lattices, on Bethe lattices KA models undergo a dynamical (jamming) phase transition at a critical density: this is characterized by diverging time and length scales and a discontinuous jump in the long-time limit of the density autocorrelation function. By analyzing generalized Bethe lattices (with loops) that interpolate between hypercubic lattices and standard Bethe lattices, the crossover between the dynamical transition that exists on these lattices and its absence in the hypercubic lattice limit is explored. Contact with earlier results are made via analysis of the related Fredrickson-Andersen models, followed by brief discussions of universality, of other approaches to glass transitions, and of some issues relevant for experiments.Comment: 59 page

Jamming percolation and glassy dynamics

We present a detailed physical analysis of the dynamical glass-jamming transition which occurs for the so called Knight models recently introduced and analyzed in a joint work with D.S.Fisher \cite{letterTBF}. Furthermore, we review some of our previous works on Kinetically Constrained Models. The Knights models correspond to a new class of kinetically constrained models which provide the first example of finite dimensional models with an ideal glass-jamming transition. This is due to the underlying percolation transition of particles which are mutually blocked by the constraints. This jamming percolation has unconventional features: it is discontinuous (i.e. the percolating cluster is compact at the transition) and the typical size of the clusters diverges faster than any power law when $\rho\nearrow\rho_c$. These properties give rise for Knight models to an ergodicity breaking transition at $\rho_c$: at and above $\rho_{c}$ a finite fraction of the system is frozen. In turn, this finite jump in the density of frozen sites leads to a two step relaxation for dynamic correlations in the unjammed phase, analogous to that of glass forming liquids. Also, due to the faster than power law divergence of the dynamical correlation length, relaxation times diverge in a way similar to the Vogel-Fulcher law.Comment: Submitted to the special issue of Journal of Statistical Physics on Spin glasses and related topic

The three-dimensional random field Ising magnet: interfaces, scaling, and the nature of states

The nature of the zero temperature ordering transition in the 3D Gaussian random field Ising magnet is studied numerically, aided by scaling analyses. In the ferromagnetic phase the scaling of the roughness of the domain walls, $w\sim L^\zeta$, is consistent with the theoretical prediction $\zeta = 2/3$. As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales as for a single second order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions. These dimensions are argued to be related to combinations of the energy scaling exponent, $\theta$, which determines the violation of hyperscaling, the correlation length exponent $\nu$, and the magnetization exponent $\beta$. The value $\beta = 0.017\pm 0.005$ is derived from the magnetization: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.Comment: 32 pp., 2 columns, 32 figure

Interpolating the Stage of Exponential Expansion in the Early Universe: a possible alternative with no reheating

In the standard picture, the inflationary universe is in a supercooled state which ends with a short time, large scale reheating period, after which the universe goes into a radiation dominated stage. An alternative is proposed here in which the radiation energy density smoothly decreases all during an inflation-like stage and with no discontinuity enters the subsequent radiation dominated stage. The scale factor is calculated from standard Friedmann cosmology in the presence of both radiation and vacuum energy density. A large class of solutions confirm the above identified regime of non-reheating inflation-like behavior for observationally consistent expansion factors and not too large a drop in the radiation energy density. One dynamical realization of such inflation without reheating is from warm inflation type scenarios. However the solutions found here are properties of the Einstein equations with generality beyond slow-roll inflation scenarios. The solutions also can be continuously interpolated from the non-reheating type behavior to the standard supercooled limit of exponential expansion, thus giving all intermediate inflation-like behavior between these two extremes. The temperature of the universe and the expansion factor are calculated for various cases. Implications for baryongenesis are discussed. This non-reheating, inflation-like regime also appears to have some natural features for a universe that is between nearly flat and open.Comment: 26 pages, Latex, 2 figures, In press Physical Review

Notes on the Third Law of Thermodynamics.I

We analyze some aspects of the third law of thermodynamics. We first review both the entropic version (N) and the unattainability version (U) and the relation occurring between them. Then, we heuristically interpret (N) as a continuity boundary condition for thermodynamics at the boundary T=0 of the thermodynamic domain. On a rigorous mathematical footing, we discuss the third law both in Carath\'eodory's approach and in Gibbs' one. Carath\'eodory's approach is fundamental in order to understand the nature of the surface T=0. In fact, in this approach, under suitable mathematical conditions, T=0 appears as a leaf of the foliation of the thermodynamic manifold associated with the non-singular integrable Pfaffian form $\delta Q_{rev}$. Being a leaf, it cannot intersect any other leaf $S=$ const. of the foliation. We show that (N) is equivalent to the requirement that T=0 is a leaf. In Gibbs' approach, the peculiar nature of T=0 appears to be less evident because the existence of the entropy is a postulate; nevertheless, it is still possible to conclude that the lowest value of the entropy has to belong to the boundary of the convex set where the function is defined.Comment: 29 pages, 2 figures; RevTex fil

Top Quark and Higgs Boson Masses: Interplay between Infrared and Ultraviolet Physics

We review recent efforts to explore the information on masses of heavy matter particles, notably of the top quark and the Higgs boson, as encoded at the quantum level in the renormalization group (RG) equations. The Standard Model (SM) and the Minimal Supersymmetric Standard Model (MSSM) are considered in parallel. First, the question is addressed to which extent the infrared (IR) physics of the ``top-down'' RG flow is independent of the ultraviolet (UV) physics. The central issues are i) IR attractive fixed point values for the top and the Higgs mass, the most outstanding one being m_t=O(190 GeV)sin(beta) in the MSSM, ii) IR attractive relations between parameters, the most prominent ones being an IR fixed top-Higgs mass relation in the SM, leading to m_H=O(156) GeV for the experimental top mass, and an IR fixed relation between the top mass and tan(beta) in the MSSM, and iii) an analytical assessment of their respective strengths of attraction. The triviality and vacuum stability bounds on the Higgs and top masses in the SM and the upper bound on the lightest Higgs boson mass in the MSSM are reviewed. The mathematical backbone, the rich structure of IR attractive fixed points, lines, surfaces,... in the multiparameter space, is made transparent. Interesting hierarchies emerge, most remarkably: IR attraction in the MSSM is systematically stronger than in the SM. Tau-bottom-(top) Yukawa coupling unification in supersymmetric grand unified theories and its power to focus the ``top-down'' RG flow into the IR top mass fixed point resp. onto the IR fixed line in the m_t-tan(beta) plane is reviewed. The program of reduction of parameters, a search for RG invariant relations between couplings, guided by the requirement of asymptotically free couplings in the UV limit,is summarized; its interrelations with the search forComment: review, 112 pages, 39 figures and 15 figures in a table; one LaTeX file, 50 postscript files; LaTeX uses style files epsfig.sty, rotating.sty, dina4p.sty; to be published in Progress in Particle and Nuclear Physics, Vol. 37, 1996, copyright Elsevier Science Lt