A simple kinetic model is proposed to investigate the effect of nonlinearity and the role of fluctuations in the metastable behavior of spin systems. Nonlinearity is introduced by a sum-rule condition for spin-correlation functions. There appear a long-lived flat part of the magnetizations and transient enhancements of the long-range fluctuations in the course of relaxations from thermodynamically unstable state. An effective, time-dependent potential picture is also obtained by eliminating the fluctuations. The transient enhancements of long-range fluctuations suppress the central potential barrier. In other words this can be interpreted as a by-pass effect caused by the degrees of freedom of spatial fluctuations.
A simple kinetic model is proposed to investigate the effect of nonlinearity and the role of fluctuations in the metastable behavior of spin systems. Nonlinearity is introduced by a sum-rule condition for spin-correlation functions. There appear a long-lived flat part of the magnetizations and transient enhancements of the long-range fluctuations in the course of relaxations from thermodynamically unstable state. An effective, time-dependent potential picture is also obtained by eliminating the fluctuations. The transient enhancements of long-range fluctuations suppress the central potential barrier. In other words this can be interpreted as a by-pass effect caused by the degrees of freedom of spatial fluctuations.
Metastability, i.e., partial relaxation to long-lived, quasi-stationary states before true asymptotic equilibrium sets in, emerges ubiquitously in classical and quantum dynamical systems as a result of timescales separation. In open quantum systems, an intrinsically nonequilibrium analogue, dynamical metastability, can originate from the spectral geometry of a non-Hermitian operator. In noninteracting models, this mechanism produces boundary-sensitive anomalous relaxation, transient amplification, and topologically mandated long-lived edge modes, all of which are enhanced as system size grows. Here we extend dynamical metastability into the nonlinear, interacting regime and identify magnetic heterostructures as a natural platform for its exploration. We introduce an interacting spin Lindbladian whose linearized magnon dynamics map onto a dynamically metastable Hatano-Nelson chain, and show that dynamical metastability in the noninteracting limit seeds genuinely nonlinear phenomena, including size-dependent spin dipping and anomalous attraction to unstable equilibria. Long-lived edge states associated to topologically mandated Dirac bosons persist under nonlinearities and disorder. We further analyze the magnetization dynamics in magnetic multilayers within the classical Landau-Lifshitz-Gilbert-Slonczewski framework, identifying Dzyaloshinskii-Moriya interaction, nonlocal damping, and spin-transfer torque as control parameters governing bulk-boundary stability mismatch and band topology. While all the distinctive dynamical phenomena previously identified reappear in this experimentally relevant setting, the LLGS framework also supports multistability and limit cycles that are absent in the quantum model. Our results constitute the first systematic study of dynamical metastability in nonlinear dynamics, directly relevant to spin-torque oscillator arrays, magnonic devices, and beyond.
Dynamic behavior of a spin-1 Ising model [1] with arbitrary bilinear (J) and biquadratic (K) pair interactions is studied by using the path probability method [2]. First the equilibrium behavior of the model is given briefly in order to understand the nonequilibrium behavior. Then the path probability method is applied to the spin-1 Ising model with J and K Hamiltonian and the system of nonlinear differential equations, which are also called the dynamic equations or rate equations, is found. The dynamic equations are solved using the Runga-Kutta method and the solutions, namely relaxation of order parameters are examined to see the "flatness" property of metastable states. It is found that the system relaxes to either stable or metastable states depending upon the initial values of the order parameters. Moreover, if there are more than one metastable states, the relaxation occurs according to the initial values of the order parameters or free energy values. Finally, the results are compared with the flow diagrams which are the solutions of the dynamic equations in two-dimensional phase space [3] and it is seen that the same results are found except for the "flatness" property of metastable states.
A rigorous quantum theory is developed on the nonlinear spin relaxation process. With the use of the phase space method, the quantum mechanical equation for a density matrix is mapped onto a c -number space. A quasi-probability function is then expanded in terms of spherical harmonics. It is clarified that the nonlinear relaxation process is characterized by 2 J ( J being the magnitude of a spin) different relaxation times. For J =1, time-evolutions of the first moment, fluctuation and probability function are exactly solved. An enhancement of the fluctuation is observed in the course of time development: This is similar to the phenomenon called “anomalous fluctuation” found in certain nonequilibrium systems.
A systematic method is developed for studying the time-dependent Ising model proposed by Glauber and extended by others. The graphical technique developed for the equilibrium statistical mechanics has been found to be useful in this nonequilibrium problem as well with some modifications. A hierarchy of equations linking various spin cluster functions is obtained. If we choose as the expansion parameter the reciprocal range of the spin interaction which is taken to be of Kac's type, the hierarchy is terminated, and closed sets of nonlinear kinetic equations satisfied by spin cluster functions follow. The dynamical susceptibility is studied using the hierachy, and the polydispersive nature of the spin relaxation is found to be closely connected to the deviation of the spin pair correlation from the Ornstein-Zernike form via multi-spin cluster functions. A sum rule on the distribution of relaxation times is found, which provides a precise expression for the critical slowing-down at the Curie point. The closed set of kinetic equations involving at most three spin cluster functions is used to evaluate explicitly the approximate dynamical susceptibility above the Curie point, and the characteristic time-scale of the polydispersive relaxation is found to increase indefinitely near the Curie point.
For pt.I see ibid., vol.16, no.7, p.1219 (1983). The response of a simple infinite-range model of a spin glass to the imposition of a small external field is studied. The field is turned on at t=0, and the evolution of the spin correlations is followed as processes involving flips of successively larger and larger clusters of spins occur, until equilibrium is achieved at t= infinity . The evolution from one state to the other proceeds through a quasi-continuum of intermediate metastable states labelled by the fraction y of spins that flip simultaneously on that time scale. The time scale for such processes grows very long as the size N of the system increases (possibly like exp((Ny)12/Tg/T)), so in the infinite system, only an infinitesimal fraction of this evolution can be observed. The generalised equation of state describing the evolution of correlations has the form derived by Sompolinsky, but the physical interpretation is different. In particular, the author does not postulate any violation of the fluctuation-dissipation theorem in equilibrium; the terms in the equations which correspond to those which violate it in his theory have the form they do because a non-equilibrium situation is being described. The physical interpretation of the variable y also allows an additional relation between initial- and final-state correlation functions which leads to the Parisi equation for his order parameter q(x), with x=1-y.
An N-sized inertial classical Heisenberg ferromagnet, which consists of a modification of the well-known standard model, where the spins are replaced by classical rotators, is studied in the limit of infinite-range interactions. The usual canonical-ensemble mean-field solution of the inertial classical n-vector ferromagnet (for which n=3 recovers the particular Heisenberg model considered herein) is briefly reviewed, showing the well-known second-order phase transition. This Heisenberg model is studied numerically within the microcanonical ensemble through molecular dynamics. In what concerns the caloric curve, it is shown that, far from criticality, the kinetic temperature obtained at the long-time-limit microcanonical-ensemble simulation recovers well the equilibrium canonical-ensemble estimate, whereas, close to criticality, a discrepancy (presumably due to finite-size effects) is found. The time evolution of the kinetic temperature indicates that a basin of attraction exists for the initial conditions for which the system evolves into a metastable state, whose duration diverges as N--> infinity, before attaining the terminal thermal equilibrium. Such a metastable state is observed for a whole range of energies, which starts right below criticality and extends up to very high energies (in fact, the gap between the kinetic temperatures associated with the metastable and the terminal-equilibrium states is expected to disappear only as one approaches infinite energy). To the best our knowledge, this has never before been observed on similar Hamiltonian models, in a noticeable way, for such a large range of energies. For example, for the XY (n=2) version of the present model, such a behavior was observed only near criticality. It is shown also that the (metastable state) maximum Lyapunov exponent decreases with N like lambda(max) approximately N-kappa, where for the initial conditions employed herein (maximal magnetization), kappa=0.225+/-0.030, both above and below the critical point.