We study the fractional Langevin equation with fractional $α$-order and linear friction terms of a system coupled to white and colored thermal baths using both analytical and numerical methods. We find analytical expressions for the position and the mean squared displacement (MSD) of the system using the Prabhakar-Mittag-Leffler function. The MSD exhibits long-term sub-diffusive regimes $t^α$ driven by colored noise and $t^{2α-1}$ driven by white noise. When the linear friction is neglected, periodic ordered phases emerge for small fractional orders $α\lesssim 0.1$. In particular, the zero-linear friction system driven only by colored noise manifests the properties of a time crystal, with a ground state satisfying the fluctuation-dissipation theorem and a periodicity proportional to $2π$. On the other hand, the zero-linear friction system driven only by white noise displays an out-of-equilibrium time glass phase with periodicity proportional to $π$. A mixed phase with contributions from both ground and out-of-equilibrium states is encountered when the system couples to both baths. In that case, the periodicity deviates from $2π$ due to damping effects. We test all the analytical results numerically by implementing a discrete recursive expression, where the random forces of the system are modeled as the derivative of the fractional Brownian motion. A microscopic description for the system is also provided by an extension of the Caldeira-Leggett model.
We present a comprehensive study of the symmetries of the generating functionals of generic Langevin processes with multiplicative colored noise. We treat both Martin-Siggia-Rose-Janssen-deDominicis and supersymmetric formalisms. We summarize the relations between observables that they imply including fluctuation relations, fluctuation-dissipation theorems, and Schwinger-Dyson equations. Newtonian dynamics and their invariances follow in the vanishing friction limit.
We have discussed the dynamics of Langevin model subjected to colored noise, by using the functional-integral method (FIM) combined with equations of motion for mean and variance of the state variable. Two sets of colored noise have been investigated: (a) one additive and one multiplicative colored noise, and (b) one additive and two multiplicative colored noise. The case (b) is examined with the relevance to a recent controversy on the stationary subthreshold voltage distribution of an integrate-and-fire model including stochastic excitatory and inhibitory synapses and a noisy input. We have …
We discuss the use of a Langevin equation with a colored (correlated) noise to perform constant-temperature molecular dynamics simulations. Since the equations of motion are linear in nature, it is easy to predict the response of a Hamiltonian system to such a thermostat and to tune at will the relaxation time of modes of different frequency. This allows one to optimize the time needed to thermalize the system and generate independent configurations. We show how this frequency-dependent response can be exploited to control the temperature of Car-Parrinello-like dynamics, keeping at low tempera…
We propose a variational superposed Gaussian approximation (VSGA) for dynamical solutions of Langevin equations subject to applied signals, determining time-dependent parameters of superposed Gaussian distributions by the variational principle. We apply the proposed VSGA to systems driven by a chaotic signal, where the conventional Fourier method cannot be adopted, and calculate the time evolution of probability density functions (PDFs) and moments. Both white and colored Gaussian noises terms are included to describe fluctuations. Our calculations show that time-dependent PDFs obtained by VSG…
We derive a relation similar to the fluctuation theorem for work done on a system obeying Langevin dynamics with thermal and colored noises. Then, we propose a method of calculating the correlation function of the colored noise by using this fluctuation relation.
The non--static generalized Langevin equation and its corresponding Fokker--Planck equation for the position of a viscous fluid particle were solved in closed form for a time dependent external force. Its solution for a constant external force was obtained analytically. The non--Markovian stochastic differential equation, associated to the dynamics of the position under a colored noise, was then applied to the description of the dynamics and persistence time of particles constrained within absorbing barriers. Comparisons with molecular dynamics were very satisfactory.
We have studied dynamical responses and the Jarzynski equality (JE) of classical open systems described by the generalized Caldeira-Leggett model with the nonlocal system-bath coupling. In the derived non-Markovian Langevin equation, the nonlinear nonlocal dissipative term and state-dependent diffusion term yielding multiplicative colored noise satisfy the fluctuation-dissipation relation. Simulation results for harmonic oscillator systems have shown the following: (a) averaged responses of the system $< x >$ to applied sinusoidal and step forces significantly depend on model parameters of mag…
The Generalized Langevin Equation (GLE) has been recently suggested to simulate the time evolution of classical solid and molecular systems when considering general non-equilibrium processes. In this approach, a part of the whole system (an open system), which interacts and exchanges energy with its dissipative environment, is studied. Because the GLE is derived by projecting out exactly the harmonic environment, the coupling to it is realistic, while the equations of motion are non-Markovian. Although the GLE formalism has already found promising applications, e.g., in nanotribology and as a …
We propose a generalized Langevin formalism to describe transport in combs and similar ramified structures. Our approach consists of a Langevin equation without drift for the motion along the backbone. The motion along the secondary branches may be described either by a Langevin equation or by other types of random processes. The mean square displacement (MSD) along the backbone characterizes the transport through the ramified structure. We derive a general analytical expression for this observable in terms of the probability distribution function of the motion along the secondary branches. We…
Power-law distributions are ubiquitous in nature. Random multiplicative processes are a basic model for the generation of power-law distributions. It is known that, for discrete-time systems, the power-law exponent decreases as the autocorrelation time of the multiplier increases. However, for continuous-time ystems, it has not yet been elucidated as to how the temporal correlation affects the power-law behavior. Herein, we have analytically investigated a multiplicative Langevin equation with colored noise. We show that the power-law exponent depends on the details of the multiplicative noise…
The Langevin system subjected to non-Gaussian noise has been discussed, by using the second-order moment approach with two kinds of models for generating the noise. We have derived the effective differential equation (DE) for a variable $x$, from which the stationary probability distribution $P(x)$ has been calculated with the use of the Fokker-Planck equation. The result of $P(x)$ calculated by the moment method is compared to several expressions obtained by different methods such as the universal colored noise approximation (UCNA) [Jung and Hänggi, Phys. Rev. A {\bf 35}, 4464 (1987)] and the…
A theoretical investigation of magnetic relaxation processes in single domain particles driven by colored noise is presented. Two approaches are considered; the Landau-Lifshitz-Miyazaki-Seki equation, which is a Langevin dynamics model based on the introduction of an Ornstein-Uhlenbeck correlated noise into the Landau-Lifshitz-Gilbert equation and a Generalized Master Equation approach whereby the ordinary Master Equation is modified through the introduction of an explicit memory kernel. It is found that colored noise is likely to become important for high anisotropy materials where the charac…
High-gain microwave amplifiers operating near quantum limit are crucial for development of quantum technology. However, a systematic theoretical modeling and simulations of their performance represent rather challenging tasks due to the occurrence of colored noises and nonlinearities in the underlying circuits. Here, we develop a response theory for such an amplifier whose circuit dynamics is based on nonlinear oscillations of an unshunted Josephson junction. The theory accounts for a subtle interplay between exponentially damped fluctuations around the stable limit cycle and the nonlinear dyn…
We present a method, based on a non-Markovian Langevin equation, to include quantum corrections to the classical dynamics of ions in a quasi-harmonic system. By properly fitting the correlation function of the noise, one can vary the fluctuations in positions and momenta as a function of the vibrational frequency, and fit them so as to reproduce the quantum-mechanical behavior, with minimal a priori knowledge of the details of the system. We discuss the application of the thermostat to diamond and to ice Ih. We find that results in agreement with path-integral molecular dynamics can be obtaine…
When nano-magnets are coupled to random external sources, their magnetization becomes a random variable, whose properties are defined by an induced probability density, that can be reconstructed from its moments, using the Langevin equation, for mapping the noise to the dynamical degrees of freedom. When the spin dynamics is discretized in time, a general fluctuation-dissipation theorem, valid for non-Markovian noise, can be established, even when zero modes are present. We discuss the subtleties that arise, when Gilbert damping is present and the mapping between noise and spin degrees of free…
We study the Langevin equation with both a white noise and a colored noise. We construct the Lagrangian as well as the Hamiltonian for the generalized Langevin equation which leads naturally to a path integral description from first principles. This derivation clarifies the meaning of the additional fields introduced by Martin, Siggia and Rose in their functional formalism. We show that the transition amplitude, in this case, is the generating functional for correlation functions. We work out explicitly the correlation functions for the Markovian process of the Brownian motion of a free partic…
The stochastic resonance (SR) in bistable systems has been extensively discussed with the use of {\it phenomenological} Langevin models. By using the {\it microscopic}, generalized Caldeira-Leggett (CL) model, we study in this paper, SR of an open bistable system coupled to a bath with a nonlinear system-bath interaction. The adopted CL model yields the non-Markovian Langevin equation with nonlinear dissipation and state-dependent diffusion which preserve the fluctuation-dissipation relation (FDR). From numerical calculations, we find the following: (1) the spectral power amplification (SPA) e…
Simplified neuronal models capture the essence of the electrical activity of a generic neuron, besides being more interesting from the computational point of view when compared to higher dimensional models such as the Hodgkin-Huxley one. In this work, we propose a generalized resonate-and-fire model described by a generalized Langevin equation that takes into account memory effects and colored noise. We perform a comprehensive numerical analysis to study the dynamics and the point process statistics of the proposed model, highlighting interesting new features like: i) non-monotonic behavior (e…
Many applications in computational science require computing the elements of a function of a large matrix. A commonly used approach is based on the the evaluation of the eigenvalue decomposition, a task that, in general, involves a computing time that scales with the cube of the size of the matrix. We present here a method that can be used to evaluate the elements of a function of a positive-definite matrix with a scaling that is linear for sparse matrices and quadratic in the general case. This methodology is based on the properties of the dynamics of a multidimensional harmonic potential cou…
Low temperature surface diffusion is driven by the thermally activated hopping of adatoms between adsorption sites. Helium spin-echo techniques, capable of measuring the sub-picosecond motion of individual adatoms, have enabled the bench-marking of many important adsorbate-substrate properties. The well-known Markovian Langevin equation has emerged as the standard tool for the interpretation of such experimental data, replacing adatom-substrate interactions with stochastic white noise and linear friction. However, the consequences of ignoring the colored noise spectrum and non-linearities inhe…
We relate the anomalous noise found experimentally in spin ice to the subdiffusion of magnetic monopoles. Because monopoles are emergent particles, they do not move in a structureless vacuum. Rather, the underlying spin ensemble filters the thermal white noise, leading to non-trivial coevolution. Thus, monopoles can be considered as random walkers under the effect of stochastic forces only as long as those are not trivially white, but instead subsume the evolution of the spin vacuum. Via this conceptualization, we conjecture relations between the color of the noise and other observables, such …
Anomalous transport of non-Markovian, thermal Brownian particle dynamics in spatially-periodic symmetric systems that is driven by time-periodic symmetric driving and constant bias is investigated numerically. The Brownian dynamics is modeled by a Generalized Langevin equation with exponentially correlated Gaussian thermal noise, obeying the fluctuation-dissipation theorem. We study the role of non-zero correlation time of thermal fluctuations for the occurrence of absolute negative (linear) mobility (ANM) near zero bias, negative-valued, nonlinear mobility (NNM) and negative differential mobi…
Many physical systems characterized by nonlinear multiscale interactions can be effectively modeled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behavior are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like Gaussian white noise, introducing non-Markovian behavior to the dynamics. We address these challenges with a framework for identifying interpret…
We consider the non-Markovian Langevin evolution of a dissipative dynamical system in quantum mechanics in the path integral formalism. After discussing the role of the frequency cutoff for the interaction of the system with the heat bath and the kernel and noise correlator that follow from the most common choices, we derive an analytic expansion for the exact non-Markovian dissipation kernel and the corresponding colored noise in the general case that is consistent with the fluctuation-dissipation theorem and incorporates systematically non-local corrections. We illustrate the modifications t…
The time-dependent barrier passage of an anomalous damping system is studied via the generalized Langevin equation (GLE) with non-Ohmic memory damping friction tensor and corresponding thermal colored noise tensor describing a particle passing over the saddle point of a two-dimensional quadratic potential energy surface. The time-dependent passing probability and transmission coefficient are analytically obtained by using of the reactive flux method. The long memory aspect of friction is revealed to originate a non-monotonic $δ$(power exponent of the friction) dependence of the passing probabi…
Macroscopic parameters as well as precise information on the random force characterizing the Langevin type description of the nuclear fusion process around the Coulomb barrier are extracted from the microscopic dynamics of individual nucleons by exploiting the numerical simulation of the improved quantum molecular dynamics. It turns out that the dissipation dynamics of the relative motion between two fusing nuclei is caused by a non-Gaussian distribution of the random force. We find that the friction coefficient as well as the time correlation function of the random force takes particularly la…
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…
Positive feedback and cooperativity in the regulation of gene expression are generally considered to be necessary for obtaining bistable expression states. Recently, a novel mechanism of bistability termed emergent bistability has been proposed which involves only positive feedback and no cooperativity in the regulation. An additional positive feedback loop is effectively generated due to the inhibition of cellular growth by the synthesized proteins. The mechanism, demonstrated for a synthetic circuit, may be prevalent in natural systems also as some recent experimental results appear to sugge…
Emergent hydrodynamics of chiral active fluids: vortices, bubbles and odd diffusion
School of Sciences and Technology, University of Camerino（卡梅里诺大学科学与技术学院）；Sapienza University of Rome（罗马第一大学）；Paris-Saclay University, CNRS, Laboratoire de Physique des Solides（巴黎萨克雷大学，法国国家科学研究中心，固体物理实验室）
Reduced models for the (defocusing) nonlinear Schrödinger equation are developed. In particular, we develop reduced models that only involve the low-frequency modes given noisy observations of these modes. The ansatz of the reduced parametric models are obtained by employing a rational approximation and a colored noise approximation, respectively, on the memory terms and the random noise of a generalized Langevin equation that is derived from the standard Mori-Zwanzig formalism. The parameters in the resulting reduced models are inferred from noisy observations with a recently developed ensemb…
In this paper we have studied a model for self-induced aggregation in Brownian particle incorporating the non-Markovian and non-Gaussian character of the associated random noise process. In this model the time evolution of each individual is guided by an over-damped Langevin equation of motion with a non-local drift resulting from the local unbalance distributions of the other individuals. Our simulation result shows that colored nose can induce the cluster formation even at large noise strength. Another observation is that critical noise strength grows very rapidly with increase of noise corr…
This paper is concerned with correlation functions of stochastic systems with memory, a prominent example being a molecule or colloid moving through a complex (e.g., viscoelastic) fluid environment. Analytical investigations of such systems based on non-Markovian stochastic equations are notoriously difficult. A common approximation is that of a single-exponential memory, corresponding to the introduction of one auxiliary variable coupled to the Markovian dynamics of the main variable. As a generalization, we here investigate a class of "toy" models with altogether three degrees of freedom, gi…
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…
The driven lattice gas (DLG) evolving at low temperature helps understanding the kinetics of pattern formation in unstable mixtures under anisotropic conditions. We here develop a simple theoretical description of kinetics in Monte Carlo simulations of the DLG. A Langevin continuum analog is also studied which is shown to exhibit the same behavior. We demonstrate that pattern growth is mainly a consequence of single-particle processes and that, after a short transient time, in which a surface evaporation/condensation mechanism is important, hole diffusion in the bulk becomes dominant. Conseque…
We present a hybrid quantum-classical simulation of charge-polaron transport in conjugated polymers. The charge, which couples to the angular rotations of the monomers, is modeled via the time-dependent Schrödinger equation, while the monomers are treated classically via the Ehrenfest equations of motion. In addition, the system is thermalized by assuming that the monomers are subject to Brownian fluctuations modeled by the Langevin equation. Charge coupling to the monomer rotations localizes the particle into a Landau polaron, while the thermal fluctuations of the monomers causes polaron dyna…
