Multipoint correlation functions for continuous-time random walk models of anomalous diffusion

Efficient computation of N-point correlation functions in D dimensions

We present efficient algorithms for computing the N-point correlation functions (NPCFs) of random fields in arbitrary D-dimensional homogeneous and isotropic spaces. Such statistics appear throughout the physical sciences and provide a natural tool to describe stochastic processes. Typically, algorithms for computing the NPCF components have [Formula: see text] complexity (for a dataset containing n particles); their application is thus computationally infeasible unless N is small. By projecting the statistic onto a suitably defined angular basis, we show that the estimators can be written in a separable form, with complexity [Formula: see text] or [Formula: see text] if evaluated using a Fast Fourier Transform on a grid of size [Formula: see text]. Our decomposition is built upon the D-dimensional hyperspherical harmonics; these form a complete basis on the [Formula: see text] sphere and are intrinsically related to angular momentum operators. Concatenation of [Formula: see text] such harmonics gives states of definite combined angular momentum, forming a natural separable basis for the NPCF. As N and D grow, the number of basis components quickly becomes large, providing a practical limitation to this (and all other) approaches: However, the dimensionality is greatly reduced in the presence of symmetries; for example, isotropic correlation functions require only states of zero combined angular momentum. We provide a Julia package implementing our estimators and show how they can be applied to a variety of scenarios within cosmology and fluid dynamics. The efficiency of such estimators will allow higher-order correlators to become a standard tool in the analysis of random fields.

Motional narrowing under Markovian and non-Markovian hopping transitions in inhomogeneous broadened absorption line shape

Inspired by recent experiments showing a minimum of electron spin or paramagnetic resonance (ESR and EPR) line width as a function of inverse temperature, we studied the motional narrowing effect by considering a combined model of carrier transitions and static dispersion of the angular frequency giving rise to an inhomogeneous broadening in the spectrum. The dispersion of the angular frequency results from the distribution of the local field. The transition between the sites under inhomogeneous static local field induces adiabatic relaxation of the spin. We also considered the on-site inherent (nonadiabatic) relaxation of the spin. We obtained the exact solution of the spin correlation function by explicitly considering transitions between two sites for both Markovian and non-Markovian transition processes. The absorption line shape is expressed in terms of the Voigt function, which is a convolution of a Gaussian function and a Lorentzian function. Using the known properties of the Voigt function, we discuss the correlation between the change in the full-width at half-maximum and the change in line shape, both of which are induced by motional narrowing. By assuming thermal activation processes for both the hopping transition and the on-site inherent relaxation, we show that the minimum of the width appears as a function of inverse temperature as observed experimentally in organic materials. Contrary to the general belief, we also show that the narrowing of the Gaussian line shape under a local random field did not necessarily lead to a Lorentzian line shape in particular under the presence of heavy tail property in the waiting time distribution of hopping transitions.

Subordinated stochastic processes with aged operational time

In this paper, subordinated stochastic processes are considered, where the renewal process acting as the operational time. It is assumed that the observation of the process begins at a certain time after the start of the renewal process. A recurrence formula was derived for calculating the multipoint probability density functions of the aged renewal process. Two-point correlation functions for certain subordinated stochastic processes, particularly for the generalized Ornstein-Uhlenbeck process, were calculated. A model of relaxation in a disordered medium with traps and obstacles is proposed.

Langevin formulation of a subdiffusive continuous-time random walk in physical time

Systems living in complex nonequilibrated environments often exhibit subdiffusion characterized by a sublinear power-law scaling of the mean square displacement. One of the most common models to describe such subdiffusive dynamics is the continuous-time random walk (CTRW). Stochastic trajectories of a CTRW can be described in terms of the subordination of a normal diffusive process by an inverse Lévy-stable process. Here, we propose an equivalent Langevin formulation of a force-free CTRW without subordination. By introducing a different type of non-Gaussian noise, we are able to express the CTRW dynamics in terms of a single Langevin equation in physical time with additive noise. We derive the full multipoint statistics of this noise and compare it with the scaled Brownian motion (SBM), an alternative stochastic model describing subdiffusive dynamics. Interestingly, these two noises are identical up to the second order correlation functions, but different in the higher order statistics. We extend our formalism to general waiting time distributions and force fields and compare our results with those of the SBM. In the presence of external forces, our proposed noise generates a different class of stochastic processes, resembling a CTRW but with forces acting at all times.

Strong field line shapes and photon statistics from a single molecule under anomalous noise

We revisit the line-shape theory of a single molecule with anomalous stochastic spectral diffusion. Waiting time profiles for bath induced spectral jumps in the ground and excited states become different when a molecule, probed by continuous-wave laser field, reaches the steady state. This effect is studied for the stationary dichotomic continuous-time-random-walk spectral diffusion of a single two-level chromophore with power-law distributions of waiting times. Correlated waiting time distributions, line shapes, two-point fluorescence correlation function, and Mandel Q parameter are calculated for arbitrary magnitude of laser field. We extended previous weak field results and examined the breakdown of the central limit theorem in photon statistics, indicated by asymptotic power-law growth of Mandel Q parameter. Frequency profile of the Mandel Q parameter identifies the peaks of spectrum, which are related to anomalous spectral diffusion dynamics.

Joint probability distributions and multipoint correlations of the continuous-time random walk

We present an efficient method to determine the Fourier-Laplace transform of the joint n-point probability distribution of a continuous-time random walk for arbitrary finite n. Additionally, we devise a recursive procedure with which it is possible to calculate the Laplace transforms of the multipoint correlation functions without having to determine the joint probability distributions first. The methods are used on several examples with both independent and dependent distributions for the waiting time and the spatial step size.

Kubo-Anderson oscillator and NMR of solid state

The analytical solution for the Kubo-Anderson oscillator with a fluctuating frequency omega for arbitrary distribution function p(omega) has been obtained. The obtained theoretical expression has been applied to consideration of some dynamical problems of solid state NMR, namely (1) dynamical transformation of NMR line shape and spin-echo signal and (2) the temperature transformation of the second moment of NMR line for the case, when the potential barrier for the mobility of magnetic nuclei is a stochastic function of time.

Anomalous lineshapes and aging effects in two-dimensional correlation spectroscopy

Multitime correlation functions provide useful probes for the ensembles of trajectories underlying the stochastic dynamics of complex systems. These can be obtained by measuring their optical response to sequences of ultrashort optical pulse. Using the continuous time random walk model for spectral diffusion, we analyze the signatures of anomalous relaxation in two-dimensional four wave mixing signals. Different models which share the same two point joint probability distribution show markedly different lineshapes and may be distinguished. Aging random walks corresponding to waiting time distributions with diverging first moment show dependence of 2D lineshapes on initial observation time, which persist for long times.

Strong correlations between fluctuations and response in aging transport

The correlations between the response of a random walker to an external driving field F switched on at time t_{a}, with the particle's fluctuations in the aging period (0,t_{a}) are investigated. Using the continuous time random walk and the quenched trap model, it is shown that these correlations remain finite even in the asymptotic limit t_{a}-->infinity. Linear response theory gives a relation between the correlations, the fractional diffusion coefficient, and the field F , thus generalizing the Einstein relation. In systems which exhibit aging, fluctuations in the aging period can be used to statistically predict the nonidentical response of particles to an external field.

Anomalous continuous-time random-walk spectral diffusion in coherent third-order optical response

Recursive relations are derived for nonlinear optical response functions of a two-level chromophore with stochastic frequency fluctuations described by a continuous-time random walk. Stationary ensembles are constructed and signatures of anomalous relaxation in the photon echo signal are discussed for a two state jump modulation with a power law waiting time density function psi(t) approximately t(-alpha-1). Stretched exponential decay of the photon echo signal is predicted for 0<alpha<1 and power law asymptotics for 1<alpha<2.