We propose a stochastic approach for the description of the time evolution of the magnetization of nanomagnets, that interpolates between the Landau--Lifshitz--Gilbert and the Landau--Lifshitz--Bloch approximations, by varying the strength of the noise. In addition, we take into account the autocorrelation time of the noise and explore the consequences, when it is finite, on the scale of the response of the magnetization, i.e. when it may be described as colored, rather than white, noise and non-Markovian features become relevant. We close the hierarchy for the moments of the magnetization, by introducing a suitable truncation scheme, whose validity is tested by direct numerical solution of the moment equations and compared to the average deduced from a numerical solution of the corresponding stochastic Langevin equation. In this way we establish a general framework, that allows both coarse-graining simulations and faster calculations beyond the truncation approximation used here.
We study the Lindblad master equation in the space of operators and provide simple criteria for closeness of the hierarchy of equations for correlations. We separately consider the time evolution of closed and open systems and show that open systems satisfying the closeness conditions are not necessary of Gaussian type. In addition, we show that dissipation can induce the closeness of the hierarchy of correlations in interacting quantum systems. As an example we study an interacting optomechanical model, the Fermi-Hubbard model, and the Rabi model, all coupled to a fine-tuned Markovian environ…
The problem of description of the evolution of the microscopic phase density and its generalizations is discussed. With this purpose, the sequence of marginal microscopic phase densities is introduced, and the appropriate BBGKY hierarchy for these microscopic distributions and their average values is formulated. The microscopic derivation of the generalized evolution equation for average value of the microscopic phase density is given and the non-Markovian generalization of the Fokker-Planck collision integral is deduced.
The hierarchical equations of motion technique has found widespread success as a tool to generate the numerically exact dynamics of non-Markovian open quantum systems. However, its application to low temperature environments remains a serious challenge due to the need for a deep hierarchy that arises from the Matsubara expansion of the bath correlation function. Here we present a hybrid stochastic hierarchical equation of motion (sHEOM) approach that alleviates this bottleneck and leads to a numerical cost that is nearly independent of temperature. Additionally, the sHEOM method generally conv…
This paper is concerned with the Fokker-Planck (FP) description of classical stochastic systems with discrete time delay. The non-Markovian character of the corresponding Langevin dynamics naturally leads to a coupled infinite hierarchy of FP equations for the various $n$-time joint distribution functions. Here we present a novel approach to close the hierarchy at the one-time level based on a linearization of the deterministic forces in all members of the hierarchy starting from the second one. This leads to a closed equation for the one-time probability density in the steady state. Consideri…
For stochastic systems with discrete time delay, the Fokker-Planck equation (FPE) of the one-time probability density function (PDF) does not provide a complete, self-contained probabilistic description. It explicitly involves the two-time PDF, and represents, in fact, only the first member of an infinite hierarchy. We here introduce a new approach to obtain a Fokker-Planck description by using a Markovian embedding technique and a subsequent limiting procedure. On this way, we find a closed, complete FPE in an infinite-dimensional space, from which one can derive a hierarchy of FPEs. While th…
The diffusion equation for non-Markovian Gaussian stochastic processes
CNR–Consiglio Nazionale delle Ricerche（意大利国家研究委员会）；BCAM–Basque Center for Applied Mathematics & Ikerbasque–Basque Foundation of Science（巴斯克应用数学中心与伊克尔巴斯基巴斯克科学基金会）；Akhiezer Institute for Theoretical Physics, NSC KIPT（阿基埃泽尔理论物理研究所，NSC KIPT）
We develop a rigorous formalism for the description of the kinetic evolution of many-particle systems with the dissipative interaction. The relationships of the evolution of a hard sphere system with inelastic collisions described within the framework of marginal observables governed by the dual BBGKY hierarchy and the evolution of states described by the Cauchy problem of the Enskog kinetic equation for granular gases are established. Moreover, we consider the Boltzmann--Grad asymptotic behavior of the constructed non-Markovian Enskog kinetic equation for granular gases in a one-dimensional s…
A stochastic approach for the description of the time evolution of the magnetization of nanomagnets is proposed, that interpolates between the Landau-Lifshitz-Gilbert and the Landau-Lifshitz-Bloch approximations, by varying the strength of the noise. Its finite autocorrelation time, i.e. when it may be described as colored, rather than white, is, also, taken into account and the consequences, on the scale of the response of the magnetization are investigated. It is shown that the hierarchy for the moments of the magnetization can be closed, by introducing a suitable truncation scheme, whose va…
We present a systematic way to analyze and model systems having many characteristic time-scales. The method we propose is employed for a test-case of a meandering jet model manifesting chaotic tracer dispersion with long time-correlations. We first choose a suitable state space partition and analyze the symbolic dynamics associated to the fluid particle position. In a second step we construct a stochastic process in terms of a multi-time Markovian model. This corresponds to a hierarchy of random travelers on a graph where each traveler moves at his own time scale. The results are compared on t…
The Fokker--Planck equation describes the evolution of a probability distribution towards equilibrium--the flow parameter is the equilibration time. Assuming the distribution remains normalizable for all times, it is equivalent to an open hierarchy of equations for the moments. Ways of closing this hierarchy have been proposed; ways of explicitly solving the hierarchy equations have received much less attention. In this paper we show that much insight can be gained by mapping the Fokker--Planck equation to a Schrödinger equation, where Planck's constant is identified with the diffusion coeffic…
The Fractional Langevin Equation (FLE) describes a non-Markovian Generalized Brownian Motion with long time persistence (superdiffusion), or anti-persistence (subdiffusion) of both velocity-velocity correlations, and position increments. It presents a case of the Generalized Langevin Equation (GLE) with a singular power law memory kernel. We propose and numerically realize a numerically efficient and reliable Markovian embedding of this superdiffusive GLE, which accurately approximates the FLE over many, about r=N lg b-2, time decades, where N denotes the number of exponentials used to approxi…
The time evolution of $\ell$-spin reduced density operators is studied for a class of Heisenberg-type quantum spin models with long-range interactions. In the framework of the quantum Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, we introduce an unconventional representation, different from the usual cluster expansion, which casts the hierarchy into the form of a second-order recursion. This structure suggests a scaling of the expansion coefficients and the corresponding time scales in powers of $N^{1/2}$ with the system size $N$, implying a separation of time scales in the large syst…
We define and analyze measures of correlations for bipartite states based on trace distance. For Bell diagonal states of two qubits, in addition to the known expression for quantum correlations using this metric, we provide analytic expressions for the classical and total correlations. The ensuing hierarchy of correlations based on trace distance is compared to the ones based on relative entropy and Hilbert-Schmidt norm. Although some common features can be found, the trace distance measure is shown to differentiate from the others in that the closest uncorrelated state to a given bipartite qu…
Time reversal symmetry is a fundamental property of many quantum mechanical systems. The relation between statistical physics and time reversal is subtle and not all statistical theories conserve this particular symmetry, most notably hydrodynamic equations and kinetic equations such as the Boltzmann equation. Here we consider quantum kinetic generalizations of the Boltzmann equation by using the method of reduced density operators leading to the quantum generalization of the BBGKY-(Bogolyubov, Born, Green, Kirkwood, Yvon) hierachy. We demonstrate that all commonly used approximations, includi…
Inspired by holographic Wilsonian renormalization, we consider coarse graining a quantum system divided between short distance and long distance degrees of freedom, coupled via the Hamiltonian. Observations using purely long distance observables are described by the reduced density matrix that arises from tracing out the short-distance degrees of freedom. The dynamics of this density matrix is non-Hamiltonian and nonlocal in time, on the order of some short time scale. We describe this dynamics in a model system with a simple hierarchy of energy gaps $ΔE_{UV} > ΔE_{IR}$, in which the coupling …
Capturing reduced-order quantum many-body dynamics out of equilibrium via neural ordinary differential equations
Stuttgart Center for Simulation Science Cluster of Excellence EXC 2075（斯图加特模拟科学中心卓越中心EXC 2075）；University of Stuttgart（斯图加特大学）；Institute for Theoretical Physics（理论物理研究所）；Institute of Information Systems Engineering（信息系统工程研究所）；Institute of Solid State Physics（固态物理研究所）
