Dynamical Mean-field Equations Research Articles

An optimization-based equilibrium measure describes fixed points of non-equilibrium dynamics: application to edge of chaos

Abstract Understanding neural dynamics is a central topic in machine learning, non-linear physics and neuroscience. However, the dynamics is non-linear, stochastic and particularly non-gradient, i.e., the driving force can not be written as gradient of a potential. These features make analytic studies very challenging. The common tool is the path integral approach or dynamical mean-field theory, but the drawback is that one has to solve the integro-differential or dynamical mean-field equations, which is computationally expensive and has no closed form solutions in general. From the aspect of associated Fokker-Planck equation, the steady state solution is generally unknown. Here, we treat searching for the steady states as an optimization problem, and construct an approximate potential related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running an approximate stochastic gradient dynamics or Langevin dynamics. Only in the zero temperature limit, the distribution of the original steady states can be achieved. The resultant stationary state of the dynamics follows exactly the canonical Boltzmann measure. Within this framework, the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method, which leads naturally to order parameters for the non-equilibrium steady states. Our theory reproduces the well-known result of edge-of-chaos, and further the order parameters characterizing the continuous transition are derived, and the order parameters are explained as fluctuations and responses of the steady states. Our method thus opens the door to analytically study the steady state landscape of the deterministic or stochastic high dimensional dynamics.

Kinetic phase phenomena of model with the chemical formula [formula omitted] on honeycomb lattice

We investigate the ferromagnetic–ferrimagnetic ternary alloy model with the chemical formula ABpC1−p with mean-field theory based on Glauber-type stochastic dynamics, namely dynamic mean-field theory, on a honeycomb lattice consisting of spins SiA=1/2, SjB=3/2 and SjC=5/2. The model includes a selective site disorder, meaning that SiA can randomly have SjC as its nearest neighbor or SjB with probability (1−p). The mean-field dynamic equations were obtained and numerically solved. To identify the type of dynamic phase transitions (first- or second-order), the thermal behavior of dynamic magnetizations was examined. For specified values of the crystal field parameter, exchange interaction parameter, and concentration ratio, the phase diagrams are calculated in the (d−T) and (T−h) planes. The system's fundamental and mixed-phase areas were identified. The system displayed A,TP,QP,Z special points and dynamic tricritical points. It was found that the content of the ternary alloy continuously affected the critical temperature in our dynamic system. Reliable agreements are observed when our findings are compared to those in the literature.

Improved mean-field dynamical equations are able to detect the two-step relaxation in glassy dynamics at low temperatures

We study the stochastic relaxation dynamics of the Ising p-spin model on a random graph, which is a well-known model with glassy dynamics at low temperatures. We introduce and discuss a new closure scheme for the master equation governing the continuous-time relaxation of the system, which translates into a set of differential equations for the evolution of local probabilities. The solution to these dynamical mean-field equations describes the out-of-equilibrium dynamics at high temperatures very well, notwithstanding the key observation that the off-equilibrium probability measure contains higher-order interaction terms not present in the equilibrium measure. In the low-temperature regime, the solution to the dynamical mean-field equations shows the correct two-step relaxation (a typical feature of glassy dynamics), but with a too-short relaxation timescale. We propose a solution to this problem by identifying the range of energies where entropic barriers play a key role and defining a renormalized microscopic timescale for the dynamical mean-field solution. The final result perfectly matches the complex out-of-equilibrium dynamics computed through extensive Monte Carlo simulations.

Nonequilibrium dynamics in a three-state opinion-formation model with stochastic extreme switches.

We investigate the nonequilibrium dynamics of a three-state kinetic exchange model of opinion formation, where switches between extreme states are possible, depending on the value of a parameter q. The mean field dynamical equationsare derived and analyzed for any q. The fate of the system under the evolutionary rules used in S. Biswas etal. [Physica A 391, 3257 (2012)0378-437110.1016/j.physa.2012.01.046] shows that it is dependent on the value of q and the initial state in general. For q=1, which allows the extreme switches maximally, a quasiconservation in the dynamics is obtained which renders it equivalent to the voter model. For general q values, a "frozen" disordered fixed point is obtained which acts as an attractor for all initially disordered states. For other initial states, the order parameter grows with time t as exp[α(q)t] where α=1-q/3-q for q≠1 and follows a power law behavior for q=1. Numerical simulations using a fully connected agent-based model provide additional results like the system size dependence of the exit probability and consensus times that further accentuate the different behavior of the model for q=1 and q≠1. The results are compared with the nonequilibrium phenomena in other well-known dynamical systems.

Critical metallic phase in the overdoped random t-J model

We investigate a model of electrons with random and all-to-all hopping and spin exchange interactions, with a constraint of no double occupancy. The model is studied in a Sachdev-Ye-Kitaev-like large-M limit with SU(M) spin symmetry. The saddle-point equations of this model are similar to approximate dynamic mean-field equations of realistic, nonrandom, t-J models. We use numerical studies on both real and imaginary frequency axes, along with asymptotic analyses, to establish the existence of a critical non-Fermi-liquid metallic ground state at large doping, with the spin correlation exponent varying with doping. This critical solution possesses a time-reparameterization symmetry, akin to Sachdev-Ye-Kitaev (SYK) models, which contributes a linear-in-temperature resistivity over the full range of doping where the solution is present. It is therefore an attractive mean-field description of the overdoped region of cuprates, where experiments have observed a linear-T resistivity in a broad region. The critical metal also displays a strong particle-hole asymmetry, which is relevant to Seebeck coefficient measurements. We show that the critical metal has an instability to a low-doping spin-glass phase and compute a critical doping value. We also describe the properties of this metallic spin-glass phase.

Analysis of random sequential message passing algorithms for approximate inference

We analyze the dynamics of a random sequential message passing algorithm for approximate inference with large Gaussian latent variable models in a student–teacher scenario. To model nontrivial dependencies between the latent variables, we assume random covariance matrices drawn from rotation invariant ensembles. Moreover, we consider a model mismatching setting, where the teacher model and the one used by the student may be different. By means of dynamical functional approach, we obtain exact dynamical mean-field equations characterizing the dynamics of the inference algorithm. We also derive a range of model parameters for which the sequential algorithm does not converge. The boundary of this parameter range coincides with the de Almeida Thouless (AT) stability condition of the replica-symmetric ansatz for the static probabilistic model.

Tree tensor-network real-time multiorbital impurity solver: Spin-orbit coupling and correlation functions in Sr2RuO4

We present a tree tensor-network impurity solver suited for general multiorbital systems. The network is constructed to efficiently capture the entanglement structure and symmetry of an impurity problem. The solver works directly on the real-time/frequency axis and generates spectral functions with energy-independent resolution of the order of one percent of the correlated bandwidth. Combined with an optimized representation of the impurity bath, it efficiently solves self-consistent dynamical mean-field equations and calculates various dynamical correlation functions for systems with off-diagonal Green's functions. For the archetypal correlated Hund's metal Sr$_2$RuO$_4$, we show that both the low-energy quasiparticle spectra related to the van Hove singularity and the high-energy atomic multiplet excitations can be faithfully resolved. In particular, we show that while the spin-orbit coupling has only minor effects on the orbital-diagonal one-particle spectral functions, it has a more profound impact on the low-energy spin and orbital response functions.

Nonequilibrium magnetic properties of the Sr2FeMoO6 type double perovskite structure

The Glauber-type stochastic dynamic interpreted by the mean-field theory has been applied to investigate the dynamic magnetic properties of the Sr2FeMoO6 type double perovskite structure under the time varying magnetic field. First, we used the Glauber dynamics to obtain the dynamic mean-field equations. The time varying average Fe and Mo magnetizations are examined to find the phase region of the system. The dynamic Fe and Mo magnetizations, hysteresis loop areas and correlations are calculated depending on the temperature in order to determine the nature of the first and second order phase transitions, as well as to get the dynamic transition points for different ratio of the system parameters. The dynamic phase diagrams (DPDs) are constructed in the plane of reduced temperature and external magnetic field amplitude (T, h). The DPDs contain paramagnetic (p), ferromagnetic (f), ferrimagnetic-1 (i1), ferrimagnetic-2 (i2) phases and six mixed regions, (f + p), (f + i1), (f + i2), (i1 + p), (i2 + p) and (i1 + i2). The DPDs also exhibit dynamic tricritical points and reentrant phenomena, which strongly depend on interaction parameters.

Gradient descent dynamics in the mixed p-spin spherical model: finite-size simulations and comparison with mean-field integration

We perform numerical simulations of a long-range spherical spin glass with two and three body interaction terms. We study the gradient descent dynamics and the inherent structures found after a quench from initial conditions well thermalized at temperature T in. In very large systems, the dynamics perfectly agrees with the integration of the mean-field dynamical equations. In particular, we confirm the existence of an onset initial temperature, within the liquid phase, below which the energy of the inherent structures undoubtedly depends on T in. This behavior is in contrast with that of pure models, where there is a ‘threshold energy’ that attracts all the initial configurations in the liquid. Our results strengthen the analogy between mean-field spin glass models and supercooled liquids.

Dynamical instantons and activated processes in mean-field glass models

We focus on the energy landscape of a simple mean-field model of glasses and analyze activated barrier-crossing by combining the Kac-Rice method for high-dimensional Gaussian landscapes with dynamical field theory. In particular, we consider Langevin dynamics at low temperature in the energy landscape of the pure spherical p-spin model. We select as initial condition for the dynamics one of the many unstable index-1 saddles in the vicinity of a reference local minimum. We show that the associated dynamical mean-field equations admit two solutions: one corresponds to falling back to the original reference minimum, and the other to reaching a new minimum past the barrier. By varying the saddle we scan and characterize the properties of such minima reachable by activated barrier-crossing. Finally, using time-reversal transformations, we construct the two-point function dynamical instanton of the corresponding activated process.

Dynamic magnetic properties in 2-dimensional kinetic spin-7/2 Ising system

The dynamic magnetic properties, such as dynamic phase transitions (DPTs) and dynamic phase diagrams (DPDs), are reported in 2-Dimensional (2D) kinetic spin-7/2 Ising system under an oscillating external magnetic field. With this aim in mind, the Glauber type stochastic dynamics are utilized to obtain the mean-field dynamic equation determining kinetic system's motion on a square lattice. The obtained mean-field dynamic equation is solved numerically. First, to observe the kinetic system's dynamic phases, the average sublattice magnetization time variations are investigated. After that, to determine the type (first- or second-order) of the DPTs and obtain the DPTs, the dynamic sublattice magnetizations' thermal behavior is examined. Finally, the DPDs are calculated in two different planes. The DPDs contains paramagnetic (P), ferromagnetic-1/2 (F1/2), ferromagnetic-7/2 (F7/2) fundamental phases and six mixed regions (or phases) as well as private points and reentrant phenomena. It has been appeared that the DPTs and DPDs are strongly dependent on the interaction parameters.

Laser-induced ultrafast insulator-metal transition in BaBiO3

We investigate ultra-fast coherent quantum dynamics of undoped $\text{BaBiO}_{3}$ driven by a strong laser pulse. Our calculations demonstrate that in a wide range of radiation frequencies and intensities the system undergoes a transient change from the insulating to the metallic state, where the charge density wave and the corresponding energy spectrum gap vanish. The transition takes place on the ultra-fast time scale of tens femtoseconds, comparable to the period of the corresponding lattice vibrations. The dynamics are determined by a complex interplay of the particle-hole excitation over the gap and of the tunnelling through it, giving rise to the highly non-trivial time evolution which comprises high harmonics and reveals periodic reappearance of the gap. The time evolution is obtained by solving the dynamical mean-field theory equations with the realistic parameters for the system and radiation. Results are summarized in the phase diagram, helpful for a possible experimental setup to achieve a dynamical control over the conduction state of this and other materials with the similarly strong electron-phonon interaction.

Rethinking Mean-Field Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed p -Spin Model

The spherical p-spin model is not only a fundamental model in statistical mechanics of disordered system, but has recently gained popularity since many hard problems in machine learning can be mapped on it. Thus the study of the out of equilibrium dynamics in this model is interesting both for the glass physics and for its implications on algorithms solving NP-hard problems. We revisit the long-time limit of the out of equilibrium dynamics of mean-field spherical mixed p-spin models. We consider quenches (gradient descent dynamics) starting from initial conditions thermalized at some temperature in the ergodic phase. We perform numerical integration of the dynamical mean-field equations of the model and we find an unexpected dynamical phase transition. Below an onset temperature, higher than the dynamical transition temperature, the asymptotic energy goes below the "threshold energy" of the dominant marginal minima of the energy function and memory of the initial condition is kept. This behavior, not present in the pure spherical p-spin model, resembles closely the one observed in simulations of glass-forming liquids. We then investigate the nature of the asymptotic dynamics, finding an aging solution that relaxes towards deep marginal minima, evolving on a restricted marginal manifold. Careful analysis, however, rules out simple aging solutions. We compute the constrained complexity in the aim of connecting the asymptotic solution to the energy landscape.

Cluster dynamics in the open-boundary heterogeneous ASEPs coupled with interacting energies

Cluster dynamics possess the promising studies in statistical physics and complexity science. While proposing physical models, applying analytical methods and exploring macroscopic properties and microscopic interactions are critical to investigate physical mechanisms of cluster dynamics. Among these physical models, totally asymmetric simple exclusion process is an important non-equilibrium statistical physics model, as it is essential enough to lattice models. Besides, it is also abbreviated as TASEP. Additionally, asymmetric simple exclusion processes, ASEPs, are derivatives, whose dynamics are based on TASEP with other nonlinear physical processes. Actually, heterogeneity of ASEPs is available to depict the real physical and engineering phenomena in a more reasonable manner. Different with previous researches, connected ASEPs are proposed here, which are constituted by two isolated ASEP coupled with a conjunction. As for each individual ASEP, its dynamics are dominated by interacting energies. Different with previous work, two completely different energies are introduced in system, which are used to describe the competition between subsystems. By analyzing complete transition processes among different particle configurations, mean-field dynamical equations are established. Phase diagrams under different energies are obtained, which yield to findings of coexistence phase of low densities, coexistence phase of high densities and coexistence phase of low and high densities. These local densities are qualitatively different. Hereafter, evolution regularities of phase diagrams obtained from local densities and currents are obtained to deeply understand physical mechanisms of complexity science. Expanded results and discussions about potential applications and fewer basic results are also presented. Expanded model with different Langmuir kinetics and internal interacting energies is established. Governing partially differential equations of the whole system are derived. Four steady stacking states of kinesin-II and OSM-3 on bovine brain tubulins affected by two different kinds of buffers are found. This work will help and promote the understanding of non-equilibrium statistical evolutions and nonlinear dynamic behaviors of such multibody system.

Nonequilibrium magnetic properties of the mixed spin (1/2, 1) Ising nanowire with core-shell structure

Abstract The nonequilibrium magnetic properties (phase transition temperatures, phase diagrams, hysteresis loop areas and correlations) are investigated in the kinetic mixed spin (1/2, 1) Ising nanowire system under the time varying magnetic field. The Glauber-type stochastic dynamics are employed to construct the set of mean field dynamic equations. The time variation of the core/shell magnetizations and the thermal behavior of the dynamic core/shell magnetizations are examined, extensively. The dynamic core/shell magnetizations, hysteresis loop areas and correlations are studied as a function of temperature in order to characterize the nature (continuous or discontinuous) of the phase transitions as well as to find the dynamic phase transition temperatures. The dynamic phase diagrams are presented in the magnetic field amplitude and temperature plane. The dynamic phase diagrams exhibit paramagnetic (p), ferrimagnetic (i), nonmagnetic (nm) phases, three mixed regions, (i + nm), (i + p) and (nm + p). The dynamic phase diagrams contain a dynamic tricritical point and reentrant phenomena, which strongly depend on interaction parameters.

Real-space dynamical mean-field theory of Friedel oscillations in strongly correlated electron systems

We study Friedel oscillations and screening effects of the impurity potential in the Hubbard model. Electronic correlations are accounted for by solving the real-space dynamical mean-field theory equations using the continuous time quantum Monte-Carlo simulations at finite temperatures and using a homogeneous self-energy approximation with the numerical renormalization group at zero temperature. We find that in the Fermi liquid phase both the amplitudes of Friedel oscillations and the screening charge decrease with increasing the interaction and follow the behavior of the Fermi liquid renormalization factor. Inside the Mott insulator regime the Friedel oscillations are absent but the residual screening charge remains finite.

Out-of-equilibrium dynamical equations of infinite-dimensional particle systems. II. The anisotropic case under shear strain

As an extension of the isotropic setting presented in the companion paper Agoritsas et al (2019 J. Phys. A: Math. Theor. 52 144002), we consider the Langevin dynamics of a many-body system of pairwise interacting particles in d dimensions, submitted to an external shear strain. We show that the anisotropy introduced by the shear strain can be simply addressed by moving into the co-shearing frame, leading to simple dynamical mean field equations in the limit . The dynamics is then controlled by a single one-dimensional effective stochastic process which depends on three distinct strain-dependent kernels—self-consistently determined by the process itself—encoding the effective restoring force, friction and noise terms due to the particle interactions. From there one can compute dynamical observables such as particle mean-square displacements and shear stress fluctuations, and eventually aim at providing an exact benchmark for liquid and glass rheology. As an application of our results, we derive dynamically the ‘state-following’ equations that describe the static response of a glass to a finite shear strain until it yields.

Path integral approach to random neural networks

In this work we study of the dynamics of large size random neural networks. Different methods have been developed to analyse their behavior, most of them rely on heuristic methods based on Gaussian assumptions regarding the fluctuations in the limit of infinite sizes. These approaches, however, do not justify the underlying assumptions systematically. Furthermore, they are incapable of deriving in general the stability of the derived mean field equations, and they are not amenable to analysis of finite size corrections. Here we present a systematic method based on Path Integrals which overcomes these limitations. We apply the method to a large non-linear rate based neural network with random asymmetric connectivity matrix. We derive the Dynamic Mean Field (DMF) equations for the system, and derive the Lyapunov exponent of the system. Although the main results are well known, here for the first time, we calculate the spectrum of fluctuations around the mean field equations from which we derive the general stability conditions for the DMF states. The methods presented here, can be applied to neural networks with more complex dynamics and architectures. In addition, the theory can be used to compute systematic finite size corrections to the mean field equations.

Coherent chaos in a recurrent neural network with structured connectivity.

We present a simple model for coherent, spatially correlated chaos in a recurrent neural network. Networks of randomly connected neurons exhibit chaotic fluctuations and have been studied as a model for capturing the temporal variability of cortical activity. The dynamics generated by such networks, however, are spatially uncorrelated and do not generate coherent fluctuations, which are commonly observed across spatial scales of the neocortex. In our model we introduce a structured component of connectivity, in addition to random connections, which effectively embeds a feedforward structure via unidirectional coupling between a pair of orthogonal modes. Local fluctuations driven by the random connectivity are summed by an output mode and drive coherent activity along an input mode. The orthogonality between input and output mode preserves chaotic fluctuations by preventing feedback loops. In the regime of weak structured connectivity we apply a perturbative approach to solve the dynamic mean-field equations, showing that in this regime coherent fluctuations are driven passively by the chaos of local residual fluctuations. When we introduce a row balance constraint on the random connectivity, stronger structured connectivity puts the network in a distinct dynamical regime of self-tuned coherent chaos. In this regime the coherent component of the dynamics self-adjusts intermittently to yield periods of slow, highly coherent chaos. The dynamics display longer time-scales and switching-like activity. We show how in this regime the dynamics depend qualitatively on the particular realization of the connectivity matrix: a complex leading eigenvalue can yield coherent oscillatory chaos while a real leading eigenvalue can yield chaos with broken symmetry. The level of coherence grows with increasing strength of structured connectivity until the dynamics are almost entirely constrained to a single spatial mode. We examine the effects of network-size scaling and show that these results are not finite-size effects. Finally, we show that in the regime of weak structured connectivity, coherent chaos emerges also for a generalized structured connectivity with multiple input-output modes.

Out-of-equilibrium dynamical mean-field equations for the perceptron model

Perceptrons are the building blocks of many theoretical approaches to a wide range of complex systems, ranging from neural networks and deep learning machines, to constraint satisfaction problems, glasses and ecosystems. Despite their applicability and importance, a detailed study of their Langevin dynamics has never been performed yet. Here we derive the mean-field dynamical equations that describe the continuous random perceptron in the thermodynamic limit, in a very general setting with arbitrary noise and friction kernels, not necessarily related by equilibrium relations. We derive the equations in two ways: via a dynamical cavity method, and via a path-integral approach in its supersymmetric formulation. The end point of both approaches is the reduction of the dynamics of the system to an effective stochastic process for a representative dynamical variable. Because the perceptron is formally very close to a system of interacting particles in a high dimensional space, the methods we develop here can be transferred to the study of liquid and glasses in high dimensions. Potentially interesting applications are thus the study of the glass transition in active matter, the study of the dynamics around the jamming transition, and the calculation of rheological properties in driven systems.