Time-dependent Random Field Research Articles

Crossover from the macroscopic fluctuation theory to the Kardar-Parisi-Zhang equationcontrols the large deviations beyond Einstein's diffusion.

We study the crossover from the macroscopic fluctuation theory (MFT), which describes one-dimensional stochastic diffusive systems at late times, to the weak noise theory (WNT), which describes the Kardar-Parisi-Zhang (KPZ) equationat early times. We focus on the example of the diffusion in a time-dependent random field, observed in an atypical direction which induces an asymmetry. The crossover is described by a nonlinear system which interpolates between the derivative and the standard nonlinear Schrodinger equationsin imaginary time. We solve this system using the inverse scattering method for mixed-time boundary conditions introduced by us to solve the WNT. We obtain the rate function which describes the large deviations of the sample-to-sample fluctuations of the cumulative distribution of the tracer position. It exhibits a crossover as the asymmetry is varied, recovering both MFT and KPZ limits. We sketch how it is consistent with extracting the asymptotics of a Fredholm determinant formula, recently derived for sticky Brownian motions. The crossover mechanism studied here should generalize to a larger class of models described by the MFT. Our results apply to study extremal diffusion beyond Einstein's theory.

Деякi застосування узагальнених дробових похiдних

The paper presents a concise summary of main properties of generalized fractional derivatives, so-called convolution type derivatives with respect to Bernstein functions. Applications are considered to modeling time dependent random fields on the sphere as solutions to partial differential equations with the generalized fractional derivative in time and random initial condition.

Fractional spherical random fields

In this paper we study the solutions of different forms of fractional equations on the unit sphere S12⊂R3 possessing the structure of time-dependent random fields. We study the correlation structures of the random fields emerging in the analysis of the solutions of two kinds of fractional equations displaying (Theorem 1) a long-range behaviour and (Theorem 2) a short-range behaviour.

Fractional Cauchy problems on compact manifolds

ABSTRACTWe investigate anomalous diffusion on compact Riemannian manifolds, modeled by time-changed Brownian motions. These stochastic processes are governed by equations involving the Laplace–Beltrami operator and a time-fractional derivative of order β ∈ (0, 1). We also consider time dependent random fields that can be viewed as random fields on randomly varying manifolds.

Time dependent random fields on spherical non-homogeneous surfaces

We introduce a class of isotropic time dependent random fields on the non-homogeneous sphere which is represented by a time-changed spherical Brownian motion of order ν∈(0,1]. We can capture some anisotropies in Cosmology with this model. This process is a time-changed rotational diffusion (TRD) or the stochastic solution to the equation involving the spherical Laplace operator and a time-fractional derivative of order ν. TRD is a diffusion on the non-homogeneous sphere and therefore, the spherical coordinates given by TRD represent the coordinates of a non-homogeneous sphere by means of which an isotropic random field is indexed. The time dependent random fields we present in this work are therefore realized through composition and can be viewed as isotropic random field on randomly varying sphere.

Coordinates Changed Random Fields on the Sphere

We introduce a class of time dependent random fields on compact Riemannian monifolds. These are represented by time-changed Brownian motions. These processes are time-changed diffusion, or the stochastic solution to the equation involving the Laplace-Beltrami operator and a time-fractional derivative of order $\beta\in (0,1)$. The time dependent random fields we present in this work can therefore be realized through composition and can be viewed as random fields on randomly varying manifolds.

Limit theorems for measure-valued processes of the level-exceedance type

Let, for each t ∈ T , ψ (t , ۔) be a random measure on the Borel σ -algebra in ℝd such that Eψ (t , ℝd )k and let (t , ۔) be its characteristic function. We call the function (t 1 ,…, t l ; z 1 ,…, z l ) = of arguments l ∈ ℕ, t 1 , t 2 … ∈ T , z 1 , z 2 ∈ ℝd the covaristic of the measure-valued random function (MVRF) ψ (۔, ۔). A general limit theorem for MVRF's in terms of covaristics is proved and applied to functions of the kind ψn (t, B ) = µ {x : ξn (t, x ) ∈ B }, where μ is a nonrandom finite measure and, for each n, ξn is a time-dependent random field.

Interacting diffusions in a random medium: comparison and longtime behavior

We consider a collection of linearly interacting diffusions (indexed by a countable space) in a random medium. The diffusion coefficients are the product of a space–time dependent random field (the random medium) and a function depending on the local state. The main focus of the present work is to establish a comparison technique for systems in the same medium but with different state dependence in the diffusion terms. The technique is applied to generalize statements on the longtime behavior, previously known only for special choices of the diffusion function. One of these special choices, which we employ as a reference model, is that of interacting Fisher–Wright diffusions in a catalytic medium where duality was used to obtain detailed results. The other choice is that of interacting Feller's branching diffusions in a catalytic medium which is itself an (autonomous) branching process and where infinite divisibility was used as the main tool.

Generalization of one problem of stochastic geometry and related measure-valued processes

We prove a functional limit theorem for the measure of a domain in which the values of a time-dependent random field do not exceed a given level. We illustrate this theorem by a geometric model.

Spin correlations and reentrant spin-glass behavior in amorphous Fe-Mn alloys. II. Dynamics

Inelastic-neutron-scattering measurements have been performed on amorphous ${({\mathrm{Fe}}_{1\ensuremath{-}x}{\mathrm{Mn}}_{x})}_{75}{\mathrm{P}}_{16}{\mathrm{B}}_{6}{\mathrm{Al}}_{3}$ alloys for several concentrations $x$ bracketing the spin-glass---ferromagnetic multicritical point found from magnetization measurements. For $x=0.35$, the alloy is nonferromagnetic, and the inelastic spectra are dominated at all temperatures by a resolution-limited quasielastic peak. We find no evidence for propagating modes of any kind. In a 10-kG applied field, the central peak is substantially reduced, and dispersionless sidebands, centered at the Larmor precession frequency, appear. Three samples ($x=0.30,0.25, \mathrm{and} 0.20$) on the ferromagnetic side of the phase diagram have also been studied. Spectra were collected for one of these ($x=0.25$) above its Curie point. These data are consistent with ordinary spin diffusion. For all three samples, resolvable spin-wave peaks exist at temperatures below the Curie points but above the spin-glass transitions established in bulk measurements. As the temperature is reduced, the spin-wave stiffness first increases, as in normal ferromagnets, and then decreases. At the lowest temperatures, resolvable spin-wave peaks are absent, and the spectra are dominated by a resolution-limited (30 \ensuremath{\mu}eV full width at half maximum) quasielastic peak. For the most iron-rich material ($x=0.20$), true inelastic scattering coexists with the central peak, even at 5 K. This inelastic scattering broadens and decreases in intensity with increasing momentum transfer $Q$, in a manner consistent with simple spin-wave theory. The central-peak intensity decays according to a power law ${Q}^{\ensuremath{-}\ensuremath{\alpha}}$, with $\ensuremath{\alpha}\ensuremath{\gtrsim}2$. At temperatures near the spin-glass---to---ferromagnetic crossover, direct inspection of the raw data for $x=0.20$ unambiguously demonstrates the coexistence of the central peak with well-resolved spin-wave peaks. Our experimental results are discussed in terms of existing theories on the dynamics of spin-glasses and ferromagnets with frozen-in disorder. Also included are speculations on time-dependent random fields, and their relevance to the crossover from ferromagnetic to spin-glass---like behavior.