Broken Ergodicity in Classically Chaotic Spin Systems

A one dimensional classically chaotic spin chain with asymmetric coupling and two different inter-spin interactions, nearest neighbors and all-to-all, has been considered. Depending on the interaction range, dynamical properties such as ergodicity and chaoticity are very different. Indeed, even in the presence of chaoticity, the model displays a lack of ergodicity only in presence of all to all interaction and below an energy threshold, that persists in the thermodynamical limit. The energy threshold can be found analytically and results can be generalized for a generic XY model with asymmetric coupling.     

Large Deviations and Ergodicity for Spin Particle Systems

In this paper we investigate the large deviation principle (LDP) for spin particle systems with possibly vanishing flip rates. The situation turns out to be much more complicated if the flip rates are allowed to be zero than the one considered by Dai, where the systems are assumed to have strictly positive flip rates. The upper and lower large-deviation bounds are studied, respectively. The two governing rate functions are compared and a variational principle is given. We then apply the results to obtain some new large-deviation estimates for the occupation times of attractive systems. In particular, we prove a strong form of exponential convergence for ergodic systems.     

Gravitational Collapse and Ergodicity in Confined Gravitational Systems

The ergodic properties of many-body systems with repulsive-core interactions are the basis of classical statistical mechanics and are well established. This is not the case for systems of purely-attractive or gravitational particles. Here we consider two examples, (i) a family of one-dimensional systems with attractive power-law interactions, , and (ii) a system of N gravitating particles confined to a finite compact domain. For (i) we deduce from the numerically-computed Lyapunov spectra that chaos, measured by the maximum Lyapunov exponent or by the Kolmogorov–Sinai entropy, increases linearly for positive and negative deviations of ν from the case of a non-chaotic harmonic chain (ν = 2). For there is numerical evidence for two additional hitherto unknown phase-space constraints. For the theoretical interpretation of model (ii) we assume ergodicity and show that for a small-enough system the reduction of the allowed phase space due to any other conserved quantity, in addition to the total energy, renders the system asymptotically stable. Without this additional dynamical constraint the particle collapse would continue forever. These predictions are supported by computer simulations. PACS numbers: 05.45.Pq, Numerical simulation of chaotic systems, 05.20.−y, Classical statistical mechanics, 36.40.Qv, Stability and fragmentation of clusters, 95.10.Fh, Chaotic dynamics.     

Spectral Gaps of Quantum Hall Systems with Interactions

A two-dimensional quantum Hall system without disorder for a wide class of interactions including any two-body interaction with finite range is studied by using the Lieb–Schultz–Mattis method [Ann. Phys. (N.Y.) 16:407 (1961)]. The model is defined on an infinitely long strip with a fixed, large width, and the Hilbert space is restricted to the lowest (n _max+1) Landau levels with a large integer n _max. We prove that, for a noninteger filling of the Landau levels, either (i) there is a symmetry breaking at zero temperature or (ii) there is only one infinite-volume ground state with a gapless excitation. We also prove the following two theorems: (a) If a pure infinite-volume ground state has a nonzero excitation gap for a noninteger filling , then a translational symmetry breaking occurs at zero temperature. (b) Suppose that there is no non-translationally invariant infinite-volume ground state. Then, if a pure infinite-volume ground state has a nonzero excitation gap, the filling factor must be equal to a rational number. Here the ground state is allowed to have a periodic structure which is a consequence of the translational symmetry breaking. We also discuss the relation between our results and the quantized Hall conductance, and phenomenologically explain why odd denominators of filling fractions giving the quantized Hall conductance are favored exclusively.     

On the Truncation of Systems with Non-Summable Interactions

In this note we consider long-range q-states Potts models on Z ^d , d≥ 2. For various families of non-summable ferromagnetic pair potentials φ(x)≥ 0, we show that there exists, for all inverse temperature β > 0, an integer N such that the truncated model, in which all interactions between spins at distance larger than N are suppressed, has at least q distinct infinite-volume Gibbs states. This holds, in particular, for all potentials whose asymptotic behaviour is of the type φ(x)∼ ‖x‖^−α, 0≤α≤ d. These results are obtained using simple percolation arguments. Work supported by Swiss National Foundation for Science, Conselho Nacional de Desenvolvimento Cientìfico e Tecnològico, and Programa de Auxìlio para Recèm Doutores PRPq-UFMG.     

Generalized Langevin Equations for Systems with Local Interactions

Journal of Statistical Physics - We present a new method to approximate the Mori–Zwanzig (MZ) memory integral in generalized Langevin equations describing the evolution of smooth observables...     

Ergodicity and Energy Distributions for Some Boundary Driven Integrable Hamiltonian Chains

We consider systems of moving particles in 1-dimensional space interacting through energy storage sites. The ends of the systems are coupled to heat baths, and resulting steady states are studied. When the two heat baths are equal, an explicit formula for the (unique) equilibrium distribution is given. The bulk of the paper concerns nonequilibrium steady states, i.e., when the chain is coupled to two unequal heat baths. Rigorous results including ergodicity are proved. Numerical studies are carried out for two types of bath distributions. For chains driven by exponential baths, our main finding is that the system does not approach local thermodynamic equilibrium as system size tends to infinity. For bath distributions that are sharply peaked Gaussians, in spite of the near-integrable dynamics, transport properties are found to be more normal than expected.     

Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps

To identify and to explain coupling-induced phase transitions in coupled map lattices (CML) has been a lingering enigma for about two decades. In numerical simulations, this phenomenon has always been observed preceded by a lowering of the Lyapunov dimension, suggesting that the transition might require changes of linear stability. Yet, recent proofs of co-existence of several phases in specially designed models work in the expanding regime where all Lyapunov exponents remain positive. In this paper, we consider a family of CML composed by piecewise expanding individual map, global interaction and finite number $N$ of sites, in the weak coupling regime where the CML is uniformly expanding. We show, mathematically for $N=3$ and numerically for $N\ge 3$ , that a transition in the asymptotic dynamics occurs as the coupling strength increases. The transition breaks the (Milnor) attractor into several chaotic pieces of positive Lebesgue measure, with distinct empiric averages. It goes along with various symmetry breaking, quantified by means of magnetization-type characteristics. Despite that it only addresses finite-dimensional systems, to some extend, this result reconciles the previous ones as it shows that loss of ergodicity/symmetry breaking can occur in basic CML, independently of any decay in the Lyapunov dimension.     

Entropy of Open Lattice Systems

We investigate the behavior of the Gibbs-Shannon entropy of the stationary nonequilibrium measure describing a one-dimensional lattice gas, of L sites, with symmetric exclusion dynamics and in contact with particle reservoirs at different densities. In the hydrodynamic scaling limit, L → ∞, the leading order (O(L)) behavior of this entropy has been shown by Bahadoran to be that of a product measure corresponding to strict local equilibrium; we compute the first correction, which is O(1). The computation uses a formal expansion of the entropy in terms of truncated correlation functions; for this system the k ^th such correlation is shown to be O(L ^−k+1). This entropy correction depends only on the scaled truncated pair correlation, which describes the covariance of the density field. It coincides, in the large L limit, with the corresponding correction obtained from a Gaussian measure with the same covariance.     

From Open Quantum Systems to Open Quantum Maps

For a class of quantized open chaotic systems satisfying a natural dynamical assumption we show that the study of the resolvent, and hence of scattering and resonances, can be reduced to the study of a family of open quantum maps, that is of finite dimensional operators obtained by quantizing the Poincaré map associated with the flow near the set of trapped trajectories.     

Large Deviation Techniques Applied to Systems with Long-Range Interactions

We discuss a method to solve models with long-range interactions in the microcanonical and canonical ensemble. The method closely follows the one introduced by R.S. Ellis, Physica D 133:106 (1999), which uses large deviation techniques. We show how it can be adapted to obtain the solution of a large class of simple models, which can show ensemble inequivalence. The model Hamiltonian can have both discrete (Ising, Potts) and continuous (HMF, Free Electron Laser) state variables. This latter extension gives access to the comparison with dynamics and to the study of non-equilibrium effects. We treat both infinite range and slowly decreasing interactions and, in particular, we present the solution of the α-Ising model in one-dimension with 0 ⩽ α < 1.     

The Two-Particle Correlation Function for Systems with Long-Range Interactions

Pemantle and Steif provided a sharp threshold for the existence of a robust phase transition (RPT) for the continuous rotator model and the Potts model in terms of the branching number and the second eigenvalue of the transfer matrix whose kernel describes the nearest neighbor interaction along the edges of the tree. Here a RPT is said to occur if an arbitrarily weak coupling with symmetry-breaking boundary conditions suffices to induce symmetry breaking in the bulk. They further showed that for the Potts model RPT occurs at a different threshold than PT (phase transition in the sense of multiple Gibbs measures), and conjectured that RPT and PT should occur at the same threshold in the continuous rotator model. We consider the class of four- and five-state rotation-invariant spin models with reflection symmetry on general trees which contains the Potts model and the clock model with scalarproduct-interaction as limiting cases. The clock model can be viewed as a particular discretization which is obtained from the classical rotator model with state space \(S^1\). We analyze the transition between \(\hbox {PT}=\hbox {RPT}\) and \(\hbox {PT}\ne \hbox {RPT}\), in terms of the eigenvalues of the transfer matrix of the model at the critical threshold value for the existence of RPT. The transition between the two regimes depends sensitively on the third largest eigenvalue.     

Cosmological Models with Matter Creation in Open Thermodynamic Systems

Regarding the universe as an open thermodynamicsystem, the creation of matter/radiation particles outof gravitational energy is investigated. A new class ofFRW models with creation of matter is obtained and their properties are examined. A suitablechoice of the particle number density function n(t) =(A/t)^3/2 leads toinflationary solutions during the particle creationphase; subsequently the universe enters the Friedmann era. It is found that fora physically acceptable solution > 1. Acomparative study is made for = 4/3, 2, 8/3, and10/3 in order to find a viable model of theuniverse.     

Dynamics of Multi-Level Open Quantum Systems with Initial System-Environment Correlations

Physics of Particles and Nuclei Letters - Exact and approximate master equations were derived by the projection operator method for the reduced statistical operator of a multi-level open quantum...     

Variational Functions in Open Systems

This paper looks for an entropy-like quantity having a monotonic time development. In the case of spontaneous emission, the final state usually consists of a single ground state assigning zero to the ordinary expressions for entropy. Thus entropy ceases to be a monotonic measure of the direction of time. The point is illustrated by a simple test case consisting of three levels coupled by spontaneous emission. It is shown how this case allows the definition of a monotonic function. Using the theory of non-Hermitian operators, the paper shows how such a function may be constructed in the general case, and it explores the main consequences of the expressions suggested. The generalization of the entropy concept is found to relate to time-reversal properties of the dynamics. The paper concludes by discussing open questions and possible further explorations.     

Initial Decoherence of Open Quantum Systems

We present a new short-time approximation scheme for evaluation of decoherence. At low temperatures, the approximation is argued to apply at intermediate times as well. It then provides a tractable approach complementary to Markovian-type approximations, and appropriate for evaluation of deviations from pure states in quantum computing models.     

Developing Entropy of Open Finite-Level Systems

A method to compute the time dependence of theentropy in the reduced dynamics is suggested. As a testit is applied to the Jaynes-Cummings model.