Loss of Ergodicity in the Transition from Annealed to Quenched Disorder in a Finite Kinetic Ising Model

We consider a kinetic Ising model which represents a generic agent-based model for various types of socio-economic systems. We study the case of a finite (and not necessarily large) number of agents N as well as the asymptotic case when the number of agents tends to infinity. The main ingredient are individual decision thresholds which are either fixed over time (corresponding to quenched disorder in the Ising model, leading to nonlinear deterministic dynamics which are generically non-ergodic) or which may change randomly over time (corresponding to annealed disorder, leading to ergodic dynamics). We address the question how increasing the strength of annealed disorder relative to quenched disorder drives the system from non-ergodic behavior to ergodicity. Mathematically rigorous analysis provides an explicit and detailed picture for arbitrary realizations of the quenched initial thresholds, revealing an intriguing “jumpy” transition from non-ergodicity with many absorbing sets to ergodicity. For large N we find a critical strength of annealed randomness, above which the system becomes asymptotically ergodic. Our theoretical results suggests how to drive a system from an undesired socio-economic equilibrium (e.g. high level of corruption) to a desirable one (low level of corruption).     

Phase transitions and interface fluctuations in double wedges and bi-pyramids with competing surface fields

The interplay between surface and interface effects on binary AB mixtures that are confined in unconventional geometries is investigated by Monte Carlo simulations and phenomenological considerations. Both double-wedge and bi-pyramid confinements are considered and competing surface fields are applied at the two opposing halves of the system. Below the bulk critical temperature, domains of opposite order parameter are stabilized at the corresponding corners and an interface runs across the middle of the bi-partite geometry. Upon decreasing the temperature further one encounters a phase transition at which the AB symmetry is broken. The interface is localized in one of the two wedges or pyramids, respectively, and the order parameter is finite. In both cases, the transition becomes discontinuous in the thermodynamic limit but it is not a first-order phase transition. In an antisymmetric double wedge geometry the transition is closely related to the wedge-filling transition. Choosing the ratio of the cross-section L × L of the wedge and its length L _y according to L _y /L ³ = const., simulations and phenomenological consideration show that the new type of phase transition is characterized by critical exponents α = 3/4, β = 0, and γ = 5/4 for the specific heat, order parameter, and susceptibility, respectively. In an antisymmetric bi-pyramid the transition occurs at the cone-filling transition of a single pyramid. The important critical fluctuations are associated with the uniform translation of the interface and they can be described by a Landau-type free energy. Monte Carlo results provide evidence that the coefficients of this Landau-type free energy exhibit a system-size dependence, which gives rise to critical amplitudes that diverge with system size and result in a transition that becomes discontinuous in the thermodynamic limit.     

On the mean-field spin glass transition

In this paper we analyze two main prototypes of disordered mean-field systems, namely the Sherrington-Kirkpatrick (SK) and the Viana-Bray (VB) models, to show that, in the framework of the cavity method, the transition from the annealed regime to a broken replica symmetry phase can be thought of as the failure of the saturability property (detailed explained along the paper) of the overlap fluctuations which act as the order parameters of the theory. We show furthermore how this coincides with the lacking of the commutativity of the infinite volume limit with respect to a, suitably chosen, vanishing perturbing field inducing the transition as prescribed by standard statistical mechanics. This is another step towards a complete theory of disordered systems. As a well known consequence it turns out that the annealed and the replica symmetric regions must coincide, implying that the averaged overlap is zero in this phase. Within our framework the finding of the values of the critical point for the SK and line for the VB becomes available straightforwardly and the method is of a large generality and applicable to several other mean field models     

Langevin dynamics of a π4-model with long range interactions

The Dominicis-Peliti generating functional (GF) method is used for the investigation of a Langevin dynamics of the π⁴-model: the symmetric double-well on-site potential and the infinite range interparticle interaction. We limit ourselves to the range above the temperature of the second order phase transition. The role of the 1/N-fluctuations (where N is the number of particles) is systematically investigated by using the steepest descent method. It is shown that the functional Legendre transformation directly results in the kinetic equation for the complete correlation function. Although this equation resembles the mode coupling equations used to describe the glass transition, it is qualitatively different. The solutions of this non-linear equation are investigated. It is shown that 1/N-fluctuations do not result in a breaking or ergodicity if the mean-field correlator is ergodic. On the other hand, if the mean-field correlator is nonergodic (e.g. if the time is much less than the inverse Kramers rate) then 1/N-fluctuations restore the ergodicity with characteristic relaxation time proportional to N.     

Ergodicity of a Single Particle Confined in a Nanopore

We analyze the dynamics of a gas particle moving through a nanopore of adjustable width with particular emphasis on ergodicity. We give a measure of the portion of phase space that is characterized by quasiperiodic trajectories which break ergodicity. The interactions between particle and wall atoms are mediated by a Lennard-Jones potential, so that an analytical treatment of the dynamics is not feasible, but making the system more physically realistic. In view of recent studies, which proved non-ergodicity for systems with scatterers interacting via smooth potentials, we find that the non-ergodic component of the phase space for energy levels typical of experiments, is surprisingly small, i.e. we conclude that the ergodic hypothesis is a reasonable approximation even for a single particle trapped in a nanopore. Due to the numerical scope of this work, our focus will be the onset of ergodic behavior which is evident on time scales accessible to simulations and experimental observations rather than ergodicity in the infinite time limit.     

Broken ergodicity

The breakdown of ergodic behaviour is discussed as a general phenomenon in condensed matter physics. Broken symmetry is a particular case of this broken ergodicity. In a system that is non-ergodic on physical timescales the phase point is effectively confined in one subregion or component of phase space. Theoretical treatments of such systems should compute thermal averages over one component at a time. The probability distribution of physical properties can then be obtained from occurrence probabilities for different components, and moments of these distributions may be used for predicting the results of typical measurements. A two-level statistical mechanics is therefore proposed. Various aspects of the breakdown of ergodicity are discussed, including: the definition of components; mechanisms for the confinement of components; methods for computing the properties of one component; the choice of occurrence probabilities for different components; and average thermodynamic properties of broken ergodicity systems. The theory is illustrated by application to well-understood systems. It is also applied to the spin glass, which is reviewed in some detail. The broken ergodicity viewpoint is combined with a suggested characterization of the components in a spin glass, involving a bifurcation cascade. Together these provide a qualitative explanation for the irreversibility signature, the long time decays, the apparent failure of linear response theory and Maxwell relations, and blocking.     

Nature of ergodicity breaking in ising spin glasses as revealed by correlation function spectral properties

In this Letter we address the nature of broken ergodicity in the low temperature phase of Ising spin glasses by examining spectral properties of spin correlation functions C(ij) identical with. We argue that more than one extensive [i.e., O(N)] eigenvalue in this matrix signals replica symmetry breaking. Monte Carlo simulations of the infinite-range Ising spin-glass model, above and below the Almeida-Thouless line, support this conclusion. Exchange Monte Carlo simulations for the short-range model in four dimensions find a single extensive eigenvalue and a large subdominant eigenvalue consistent with droplet model expectations.     

Critical behaviour of elastically coupled systems with spatially anisotropic critical fluctuations

The spatial anisotropy of critical fluctuations has strong influence on the temperature dependence of the elastic constants of an elastic medium coupled magneto- or electrostrictive to the order parameter. Under pinned boundary conditions we find for uniaxial dipolar systems of hexagonal and trigonal symmetry a second order phase transition, where only c₁₁ and c₁₂ show critical behaviour. For other symmetries a first order transition is expected. At an uniaxial Lifshitz point only c₃₃ becomes critical and the phase transition remains of second order for any symmetry.     

Radiatively induced spontaneous symmetry breaking and phase transitions in curved spacetime

Local gauge symmetries which are spontaneously broken in flat spacetime are shown to be restored for large spacetime curvatures. The case of symmetry breaking due to radiative quantum corrections in gauge theories with elementary scalar fields is considered explicitly. In spacetimes with a positive Ricci curvature scalar R and a cosmological event horizon, the critical curvature R_C is of O(m_H²) or O(m_W²), depending on whether the theory is formulated with conformal or minimal scalar fields. In Ricci flat spacetimes with a conventional event horizon the symmetry is expected to be restored when the temperature of the Hawking thermal radiation is of O(m_W). This phenomenon is described in detail, using functional integral methods and dimensional renormalization, for massless scalar electro-dynamics in de-Sitter spacetime. For conformal scalars, the symmetry restoring phase transition is first order, the critical curvature being R_C = 0.910 m_H². For minimal scalars, an anomalous, curvature dependent mass counterterm is required. The phase transition in this case is second order, and occurs at R_C = 83.57 m_W². Symmetry restoration at finite temperature in flat spacetime is considered in an appendix. The critical temperature at which a first-order phase transition occurs in the Weinberg-Salam model is found to be T_C = 0.329 m_W.     

Dynamical supersymmetry breaking in a finite field theoretical model

We present a supersymmetric field theory in two or three space-time dimensions with an internal symmetry of the O(N) type. In the large-N limit the model is finite and supersymmetry is spontaneously broken. The fields representing the order parameters of the broken supersymmetry phase acquire dynamics through quantum corrections. In particular the Goldstone fermion is a zero-mass fermionic bound state.     

Mean-field renormalization group for the q-state clock spin-glass model

The mean-field renormalization group is used to study the phase diagrams of a d-dimensional q-state clock spin-glass model. We found, for q = 3 clock, the transition from paramagnet to spin glass is an isotropic spin-glass phase, but for q = 4 clock, the transition from paramagnet to spin glass is an anisotropic spin-glass phase. However, for q ⩾ 5 clock, the result of anisotropic spin-glass phase depends on the temperature and the distribution of random coupling. While the coordinate number approaches infinity, the critical temperature evaluated by the mean-field renormalization group method is equal to that by the replica method.     

Infinite Range Interaction Model of a Structural Glass

In this paper a simple mean-field model for the liquid-glass phase transition is proposed. This is the low density D-dimensional system of N particles interacting via infinite-range oscillating potential. In the framework of the replica approach it is shown that such a system exhibits the phase transition between the high-temperature liquid phase and the low-temperature glass phase. This phase transition is described in terms of the standard one-step replica symmetry breaking scheme.     

Exact critical exponents for quantum spin chains,non-linear σ-models at θ=π and the quantum hall effect

We use non-abelian bosonization to predict critical exponents for quantum chains of arbitrary spin and arbitrary symmetry SU(n). Passing to the large representation limit gives non-linear σ-models on the manifolds U(2n)/U(n)×U(n) at topological angle θ=π. For n=1 this is the familiar “O(3)” model; taking the replica limit n→0 given critical exponents for the localization transition in the quantum Hall effect. Given certain assumptions, these exponents should be exact.     

Asymptotic dynamics,non-critical and critical fluctuations for a geometric long-range interacting model

We study the dynamics of geometric spin system on the torus with long-range interaction. As the number of particles goes to infinity, the process converges to a deterministic, dynamical magnetization field that satisfies an Euler equation (law of large numbers). Its stable steady states are related to the limits of the equilibrium measures (Gibbs states) of the finite particle system. A related equation holds for the magnetization densities, for which the property of propagation of chaos also is established. We prove a dynamical central limit theorem with an infinite-dimensional Ornstein-Uhlenbeck process as a limiting fluctuation process. At the critical temperature of a ferromagnetic phase transition, both a tighter quantity scaling and a time scaling is required to obtain convergence to a one-dimensional critical fluctuation process with constant magnetization fields, which has a non-Gaussian invariant distribution. Similarly, at the phase transition to an antiferromagnetic state with frequencyp ₀, the fluctuation process with critical scaling converges to a two-dimensional critical fluctuation process, which consists of fields with frequencyp ₀ and has a non-Gaussian invariant distribution on these fields. Finally, we compute the critical fluctuation process in the infinite particle limit at a triple point, where a ferromagnetic and an antiferromagnetic phase transition coincide.Work supported by Deutsche Forschungsgemeinschaft     

关于对称破缺的一些体会

考察了对称破缺的数学形式及物理本质,说明了对称破缺、各态历经假设及序参量空间的一些简单联系,并希望通过这种联系能够更深刻地理解相变、亚平衡等概念,增加热力学和统计力学的统一性。     

Phase diagram of random heteropolymers

We propose a new analytic approach to study the phase diagram of random heteropolymers, based on the cavity method. For copolymers we analyze the nature and phenomenology of the glass transition as a function of sequence correlations. Depending on these correlations, we find that two different scenarios for the glass transition can occur. We show that, beside the much studied possibility of an abrupt freezing transition at low temperature, the system can exhibit, upon cooling, a first transition to a soft glass phase with fully broken replica symmetry and a continuously growing degree of freezing as the temperature is lowered.     

Long time limit of equilibrium glassy dynamics and replica calculation

It is shown that the limit t−t^′→∞$t-t^{\prime }\to \infty$ of the equilibrium dynamic self-energy can be computed from the n→1$n\to 1$ limit of the static self-energy of an n  -times replicated system with one step replica symmetry breaking structure. It is also shown that the Dyson equation of the replicated system leads in the n→1$n\to 1$ limit to the bifurcation equation for the glass ergodicity breaking parameter computed from dynamics. The equivalence of the replica formalism to the long time limit of the equilibrium relaxation dynamics is proved to all orders in perturbation for a scalar theory.     

Dynamic critical scattering near the smectic-A—nematic phase transition in CBOOA

High resolution neutron quasielastic scattering has been applied to a study on the critical dynamics near the smectic-A—nematic phase transition in CBOOA. A line narrowing is observed for a certain value of the momentum transfer. We interpret this narrowing as a critical slowing down of the smectic order parameter fluctuations near the transition temperature T_SN. The relaxation times vary between 10^-8 and 10^-7 sec for a temperature interval of 2 K below T_SN.     

An investigation of the phase structure of the two-dimensional O(2) and O(4) symmetric nonlinear sigma-models

We investigate the phase structure of the two-dimensional O(2) and O(4) lattice σ-models by means of a Monte Carlo simulation using Binder's phenomenological renormalization group. For the O(2) model the transition temperature β_c⁻¹ is estimated and the expected line of critical points is found. For the O(4) model no signal of a phase transition is found in the range of couplings considered. This is in contradiction to a recent claim that the O(4) model has a critical point at finite β.