arXiv subjects

N. Leibovich

Publications and source records attributed to N. Leibovich.

5 recordsLinked to original sources

arXiv2018

Infinite Ergodic Theory for Heterogeneous Diffusion Processes

We show the relation between processes which are modeled by a Langevin equation with multiplicative noise and infinite ergodic theory. We concentrate on a spatially dependent diffusion coefficient that behaves as ${D(x)}\sim |x-\tilde{x}|^{2-2/α}$ in the vicinity of a point $\tilde{x}$, where $α$ can be either positive or negative. We find that a nonnormalized state, also called an infinite density, describes statistical properties of the system. For processes under investigation, the time averages of a wide class of observables, are obtained using an ensemble average with respect to the nonnormalized density. A Langevin equation which involves multiplicative noise may take different interpretation; Itô, Stratonovich, or Hänggi-Klimontovich, so the existence of an infinite density, and the density's shape, are both related to the considered interpretation and the structure of $D(x)$.

cond-mat.stat-mech↗

arXiv2017

Conditional $1/f^α$ noise: from single molecules to macroscopic measurements

We demonstrate that the measurement of $1/f^α$ noise at the single molecule or nano-object limit is remarkably distinct from the macroscopic measurement over a large sample. The single particle measurements yield a conditional time-dependent spectrum. However, the number of units fluctuating on the time scale of the experiment is increasing in such a way that the macroscopic measurements appear perfectly stationary. The single particle power spectrum is a conditional spectrum, in the sense that we must make a distinction between idler and non-idler units on the time scale of the experiment. We demonstrate our results based on stochastic and deterministic models, in particular the well known superposition of Lorentzians approach, the blinking quantum dot model, and deterministic dynamics generated by non-linear mapping. Our results show that the $1/f^α$ spectrum is inherently nonstationary even if the macroscopic measurement completely obscures the underlying time dependence of the phenomena.

cond-mat.stat-mech↗

arXiv2016

Aging Wiener-Khinchin Theorem and Critical Exponents of $1/f$ Noise

The power spectrum of a stationary process may be calculated in terms of the autocorrelation function using the Wiener-Khinchin theorem. We here generalize the Wiener-Khinchin theorem for nonstationary processes and introduce a time-dependent power spectrum $\left\langle S_{t_m}(ω)\right\rangle$ where $t_m$ is the measurement time. For processes with an aging correlation function of the form $\left\langle I(t)I(t+τ)\right\rangle=t^Υϕ_{\rm EA}(τ/t)$, where $ϕ_{\rm EA}(x)$ is a nonanalytic function when $x$ is small, we find aging $1/f$ noise. Aging $1/f$ noise is characterized by five critical exponents. We derive the relations between the scaled correlation function and these exponents. We show that our definition of the time-dependent spectrum retains its interpretation as a density of Fourier modes and discuss the relation to the apparent infrared divergence of $1/f$ noise. We illustrate our results for blinking quantum dot models, single-file diffusion and Brownian motion in logarithmic potential.

cond-mat.stat-mech↗

arXiv2015

Aging Wiener-Khinchin Theorem

The Wiener-Khinchin theorem shows how the power spectrum of a stationary random signal $I(t)$ is related to its correlation function $\left\langle I(t)I(t+τ)\right\rangle$. We consider non-stationary processes with the widely observed aging correlation function $\langle I(t) I(t+τ) \rangle \sim t^γϕ_{\rm EN}(τ/t)$ and relate it to the sample spectrum. We formulate two aging Wiener-Khinchin theorems relating the power spectrum to the time and ensemble averaged correlation functions, discussing briefly the advantages of each. When the scaling function $ϕ_{\rm EN}(x)$ exhibits a non-analytical behavior in the vicinity of its small argument we obtain aging $1/f$ type of spectrum. We demonstrate our results with three examples: blinking quantum dots, single file diffusion and Brownian motion in a logarithmic potential, showing that our approach is valid for a wide range of physical mechanisms.

cond-mat.stat-mech↗

arXiv2013

The Lasting Effect of Initial Conditions on Single File Diffusion

We study the dynamics of a tagged particle in an environment of point Brownian particles with hard-core interactions in an infinite one dimensional channel (a single-file model). In particular we examine the influence of initial conditions on the dynamic of the tagged particle. We compare two initial conditions: equal distances between particles and uniform density distribution. The effect is shown by the differences of mean-square-displacement and correlation function for the two ensembles of initial conditions. We discuss the violation of Einstein relation, and its dependence on the initial condition, and the difference between time and ensemble averaging. More specifically, using the Jepsen line, we will discuss how transport coefficients, like diffusivity, depend on the initial state. Our work shows that initial conditions determine the long time limit of the dynamic, and in this sense the system never forgets its initial state in complete contrast with thermal systems (i.e a closed system which attains equilibrium independent of the initial state).

cond-mat.stat-mech↗