Stationary Correlations for the 1D KPZ Equation

We study exact stationary properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation by using the replica approach. The stationary state for the KPZ equation is realized by setting the initial condition the two-sided Brownian motion (BM) with respect to the space variable. Developing techniques for dealing with this initial condition in the replica analysis, we elucidate some exact nature of the height fluctuation for the KPZ equation. In particular, we obtain an explicit representation of the probability distribution of the height in terms of the Fredholm determinants. Furthermore from this expression, we also get the exact expression of the space-time two-point correlation function.     

Universal Nature of Replica Symmetry Breaking in Quantum Systems with Gaussian Disorder

We study quantum spin systems with quenched Gaussian disorder. We prove that the variance of all physical quantities in a certain class vanishes in the infinite volume limit. We study also replica symmetry breaking phenomena, where the variance of an overlap operator in the other class does not vanish in the replica symmetric Gibbs state. On the other hand, it vanishes in a spontaneous replica symmetry breaking Gibbs state defined by applying an infinitesimal replica symmetry breaking field. We prove also that the finite variance of the overlap operator in the replica symmetric Gibbs state implies the existence of a spontaneous replica symmetry breaking.     

Correlations in the sand pile model: From the log-normal distribution to self-organized criticality

We have studied the approach of the Abelian sand pile model towards the stationary, self-organized criticality state. The uncorrelated limit is shown both numerically and by a simple analysis to follow the log-normal distribution. We introduce and evaluate several correlation fuctions to study the correlated region.     

Feed-forward chains of recurrent attractor neural networks with finite dilution near saturation

A stationary state replica analysis for a dual neural network model that interpolates between a fully recurrent symmetric attractor network and a strictly feed-forward layered network, studied by Coolen and Viana, is extended in this work to account for finite dilution of the recurrent Hebbian interactions between binary Ising units within each layer. Gradual dilution is found to suppress part of the phase transitions that arise from the competition between recurrent and feed-forward operation modes of the network. Despite that, a long chain of layers still exhibits a relatively good performance under finite dilution for a balanced ratio between inter-layer and intra-layer interactions.     

Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

 By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick model, and the Derrida p-spin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from below by the value given in the Parisi Ansatz, uniformly in the size of the system. Moreover, the difference between the two expressions is given in the form of a sum rule, extending our previous work on the comparison between the true free energy and its replica symmetric Sherrington-Kirkpatrick approximation. We give also a variational bound for the infinite volume limit of the ground state energy per site. Received: 6 May 2002 / Accepted: 6 September 2002 Published online: 13 January 2003     

N-particle Bogoliubov vacuum state

We consider the Bogoliubov vacuum state in the number-conserving Bogoliubov theory proposed by Castin and Dum [Phys. Rev. A 57, 3008 (1998)]. We show that, in the particle representation, the vacuum can be written in a simple diagonal form. The vacuum state can describe the stationary N-particle ground state of a condensate in a trap, but it can also represent a dynamical state when, for example, a Bose-Einstein condensate initially prepared in the stationary ground state is subject to a time-dependent perturbation. In both cases the diagonal form of the Bogoliubov vacuum can be obtained by basically diagonalizing the reduced single-particle density matrix of the vacuum. We compare N-body states obtained within the Bogoliubov theory with the exact ground states in a 3-site Bose-Hubbard model. In this example, the Bogoliubov theory fails to accurately describe the stationary ground state in the limit when N → ∞ but a small fraction of depleted particles is kept constant.     

Magnetization densities as replica parameters: The dilute ferromagnet

In this paper we compute exactly the ground state energy and entropy of the dilute ferromagnetic Ising model. The two thermodynamic quantities are also computed when a magnetic field with random locations is present. The result is reached in the replica approach frame by a class of replica order parameters introduced by Monasson (1998) [5]. The strategy is first illustrated considering the SK model, for which we will show the complete equivalence with the standard replica approach. Then, we apply to the diluted ferromagnetic Ising model with a random located magnetic field, which is mapped into a Potts model.     

Unitary Transformation in Quantum Teleportation

In the well-known treatment of quantum teleportation, the receiver should convert the state of his EPR particle into the replica of the unknown quantum state by one of four possible unitary transformations. However, the importance of these unitary transformations must be emphasized. We will show in this paper that the receiver cannot transform the state of his particle into an exact replica of the unknown state which the sender wants to transfer if he has not a proper implementation of these unitary transformations. In the procedure of converting state, the inevitable coupling between EPR particle and environment which is needed by the implementation of unitary transformations will reduce the accuracy of the replica.     

Unitary Transformation in Quantum Teleportation

In the well-known treatment of quantum teleportation, the receiver should convert the state of his EPR particle into the replica of the unknown quantum state by one of four possible unitary transformations. However, the importance of these unitary transformations must be emphasized. We will show in this paper that the receiver cannot transform the state of his particle into an exact replica of the unknown state which the sender wants to transfer if he has not a proper implementation of these unitary transformations. In the procedure of converting state, the inevitable coupling between EPR particle and environment which is needed by the implementation of unitary transformations will reduce the accuracy of the replica.     

Optical absorption and electromagnetically induced transparency in semiconductor quantum well driven by intense terahertz field

An approach for solving the excitonic absorption in a semiconductor quantum well driven by an intense terahertz field is presented.The formalism relies on the stationary single-photon Schro¨dinger equation in the full quantum mechanical framework.The optical absorption dynamics in both weak and strong couplings are discussed and compared.The excitonic absorption spectra show the Autler-Townes doublets for the resonance terahertz field,a replica peak for the non-resonance terahertz field,and the electromagnetically induced transparency phenomenon for modulating the decay rate of the second electron state in the weak coupling.In particular,the electromagnetically induced transparency phenomenon window range is discussed.In the strong coupling region,the multi-order energy level resonance splitting due to the strong optical field is found.There are three(non-resonance terahertz field) or four(resonance terahertz field) peaks in the optical absorption spectra.This work provides a simple and convenient approach to deal with the optical absorption in the exciton system.     

Relativistically covariant Bohm-Bub hidden-variable theory for spin measurement of a single particle

We present a simple first step toward a relativistically covariant generalization of the Bohm-Bub hidden-variable theory. The model is applicable to spin measurement on a single Dirac particle and describes the collapse of the state vector to a spin-up or spin-down state. The essential postulate is that the hidden-variable vector transforms in the same way as the state vector under a Lorentz transformation. This yields a covariant collapse equation, which reduces to the ordinary Bohm-Bub equation for an observer stationary with respect to the particle and shows a time dilated collapse for a moving observer. The model yields the correct quantum transition probabilities for all observers.Deceased.     

Finite Size Corrections to the Parisi Overlap Function in the GREM

We investigate the effects of finite size corrections on the overlap probabilities in the Generalized Random Energy Model in two situations where replica symmetry is broken in the thermodynamic limit. Our calculations do not use replicas, but shed some light on what the replica method should give for finite size corrections. In the gradual freezing situation, which is known to exhibit full replica symmetry breaking, we show that the finite size corrections lead to a modification of the simple relations between the sample averages of the overlaps \( Y_k \) between k configurations predicted by replica theory. This can be interpreted as fluctuations in the replica block size with a negative variance. The mechanism is similar to the one we found recently in the random energy model in Derrida and Mottishaw (J Stat Mech 2015(1): P01021, 2015). We also consider a simultaneous freezing situation, which is known to exhibit one step replica symmetry breaking. We show that finite size corrections lead to full replica symmetry breaking and give a more complete derivation of the results presented in Derrida and Mottishaw (Europhys Lett 115(4): 40005, 2016) for the directed polymer on a tree.     

A Note on the Guerra and Talagrand Theorems for Mean Field Spin Glasses: The Simple Case of Spherical Models

The aim of this paper is to discuss the main ideas of the Talagrand proof of the Parisi Ansatz for the free-energy of Mean Field Spin Glasses with a physicist's approach. We consider the case of the spherical p-spin model, which has the following advantages: (1) the Parisi Ansatz takes the simple “one step replica symmetry breaking form,” (2) the replica free-energy as a function of the order parameters is simple enough to allow for numerical maximization with arbitrary precision. We present the essential ideas of the proof, we stress its connections with the theory of effective potentials for glassy systems, and we reduce the technically more difficult part of the Talagrand's analysis to an explicit evaluation of the solution of a variational problem.     

Specific Features of Particle Scattering and Excitation of Quasilocal States by a Stationary Flux in a Two-Level System

For a simple model it is demonstrated that the Fano or conventional resonance occurs in systems with a two-level defect depending on the defect parameters. Conditions of the Fano and conventional resonance are analyzed. The problem of a system response to a stationary flux specified on the entire axis by a traveling wave has been solved. It has been found that this flux excites a quasilocal state in the continuous spectrum. Bands of amplitudes of the functions of system response to the stationary flux completely determine the quasilocal state spectrum.     

Aggregation kinetics for a one-dimensional zero-degree Kelvin model of spinodal decomposition

We study analytically the approach to equilibrium in a simple zero-temperature model for phase separation in a binary alloy, in which nearest neighbor interchange can occur only if the portion of AB bonds is thereby decreased. The approach to equilibrium is found analytically. Because of the existence of infinitely many possible stationary states, the asymptotic distribution of AB pairs depends on the details of the initial state and must be obtained by a recursion method.Chargé de recherches FNRS.     

Transients and basins of attraction in neutral network models

Approximate dynamic mean field equations for a generalized Hopfield model are derived, which allow to calculate transient properties of this model. These equations are exact for short times and yield the replica symmetric solution as a stationary solution. They allow reliable computation of retrieval trajectories and basins of attraction of retrieval states, as demonstrated by comparison with simulations. The equations are derived for networks with arbitrary mean activity and results are given for the standard model and for low activities.     

Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators

We consider a generalization of the Kuramoto model in which the oscillators are coupled to the mean field with random signs. Oscillators with positive coupling are "conformists"; they are attracted to the mean field and tend to synchronize with it. Oscillators with negative coupling are "contrarians"; they are repelled by the mean field and prefer a phase diametrically opposed to it. The model is simple and exactly solvable, yet some of its behavior is surprising. Along with the stationary states one might have expected (a desynchronized state, and a partially-synchronized state, with conformists and contrarians locked in antiphase), it also displays a traveling wave, in which the mean field oscillates at a frequency different from the population's mean natural frequency.     

Self organization of interacting polya urns

We introduce a simple model which shows non-trivial self organized critical properties. The model describes a system of interacting units, modelled by Polya urns, subject to perturbations and which occasionally break down. Three equivalent formulations - stochastic, quenched and deterministic - are shown to reproduce the same dynamics. Among the novel features of the model are a non-homogeneous stationary state, the presence of a non-stationary critical phase and non-trivial exponents even in mean field. We discuss simple interpretations in term of biological evolution and earthquake dynamics and we report on extensive numerical simulations in dimensions d=1,2 as well as in the random neighbors limit. Received: 18 February 1998 / Revised: 20 March 1998 / Accepted: 29 March 1998     

Entropy of Non-equilibrium Stationary Measures of Boundary Driven TASEP

We examine the entropy of non-equilibrium stationary states of boundary driven totally asymmetric simple exclusion processes. As a consequence, we obtain that the Gibbs–Shannon entropy of the non equilibrium stationary state converges to the Gibbs–Shannon entropy of the local equilibrium state. Moreover, we prove that its fluctuations are Gaussian, except when the mean displacement of particles produced by the bulk dynamics agrees with the particle flux induced by the density reservoirs in the maximal phase regime.     

Estimating Distributions of Parameters in Nonlinear State Space Models with Replica Exchange Particle Marginal Metropolis–Hastings Method

Extracting latent nonlinear dynamics from observed time-series data is important for understanding a dynamic system against the background of the observed data. A state space model is a probabilistic graphical model for time-series data, which describes the probabilistic dependence between latent variables at subsequent times and between latent variables and observations. Since, in many situations, the values of the parameters in the state space model are unknown, estimating the parameters from observations is an important task. The particle marginal Metropolis–Hastings (PMMH) method is a method for estimating the marginal posterior distribution of parameters obtained by marginalization over the distribution of latent variables in the state space model. Although, in principle, we can estimate the marginal posterior distribution of parameters by iterating this method infinitely, the estimated result depends on the initial values for a finite number of times in practice. In this paper, we propose a replica exchange particle marginal Metropolis–Hastings (REPMMH) method as a method to improve this problem by combining the PMMH method with the replica exchange method. By using the proposed method, we simultaneously realize a global search at a high temperature and a local fine search at a low temperature. We evaluate the proposed method using simulated data obtained from the Izhikevich neuron model and Lévy-driven stochastic volatility model, and we show that the proposed REPMMH method improves the problem of the initial value dependence in the PMMH method, and realizes efficient sampling of parameters in the state space models compared with existing methods.