Large Fluctuations for Spatial Diffusion of Cold Atoms

Simulation of the continuous time random walk using subordination schemes

The continuous time random walk model has been widely applied in various fields, including physics, biology, chemistry, finance, social phenomena, etc. In this work, we present an algorithm that utilizes a subordinate formula to generate data of the continuous time random walk in the long time limit. The algorithm has been validated using commonly employed observables, such as typical fluctuations of the positional distribution, rare fluctuations, the mean and the variance of the position, and breakthrough curves with time-dependent bias, demonstrating a perfect match.

Anatomy of an extreme event: What can we infer about the history of a heavy-tailed random walk?

Extreme events are by nature rare and difficult to predict, yet are often much more important than frequent, typical events. An interesting counterpoint to the prediction of such events is their retrodiction-given a process in an outlier state, how did the events leading up to this endpoint unfold? In particular, was there only a single, massive event, or was the history a composite of multiple, smaller but still significant events? To investigate this problem we take heavy-tailed stochastic processes (specifically, the symmetric, α-stable Lévy processes) as prototypical random walks. A natural and useful characteristic scale arises from the analysis of processes conditioned to arrive in a particular final state (Lévy bridges). For final displacements longer than this scale, the scenario of a single, long jump is most likely, even though it corresponds to a rare, extreme event. On the other hand, for small final displacements, histories involving extreme events tend to be suppressed. To further illustrate the utility of this analysis, we show how it provides an intuitive framework for understanding three problems related to boundary crossings of heavy-tailed processes. These examples illustrate how intuition fails to carry over from diffusive processes, even very close to the Gaussian limit. One example yields a computationally and conceptually useful representation of Lévy bridges that illustrates how conditioning impacts the extreme-event content of a random walk. The other examples involve the conditioned boundary-crossing problem and the ordinary first-escape problem; we discuss the observability of the latter example in experiments with laser-cooled atoms.

Nonergodicity of d-dimensional generalized Lévy walks and their relation to other space-time coupled models

We investigate the nonergodicity of the generalized Lévy walk introduced by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)PRLTAO0031-900710.1103/PhysRevLett.58.1100] with respect to the squared displacements. We present detailed analytical derivations of our previous findings outlined in a recent letter [Phys. Rev. Lett. 120, 104501 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.104501], give detailed interpretations, and in particular emphasize three surprising results. First, we find that the mean-squared displacements can diverge for a certain range of parameter values. Second, we show that an ensemble of trajectories can spread subdiffusively, whereas individual time-averaged squared displacements show superdiffusion. Third, we recognize that the fluctuations of the time-averaged squared displacements can become so large that the ergodicity breaking parameter diverges, what we call infinitely strong ergodicity breaking. This phenomenon can also occur for paramter values where the lag-time dependence of the mean-squared displacements is linear indicating normal diffusion. In order to numerically determine the full distribution of time-averaged squared displacements, we use importance sampling. For an embedding of our findings into existing results in the literature, we define a more general model which we call variable speed generalized Lévy walk and which includes well-known models from the literature as special cases such as the space-time coupled Lévy flight or the anomalous Drude model. We discuss and interpret our findings regarding the generalized Lévy walk in detail and compare them with the nonergodicity of the other space-time coupled models following from the more general model.

Large deviations of the ballistic Lévy walk model

We study the ballistic Lévy walk stemming from an infinite mean traveling time between collision events. Our study focuses on the density of spreading particles all starting from a common origin, which is limited by a "light" cone -v_{0}t<x<v_{0}t. In particular we study this density close to its maximum in the vicinity of the light cone. The spreading density follows the Lamperti-arcsine law describing typical fluctuations. However, this law blows up in the vicinity of the spreading horizon, which is nonphysical in the sense that any finite-time observation will never diverge. We claim that one can find two laws for the spatial density: The first one is the mentioned Lamperti-arcsine law describing the central part of the distribution, and the second is an infinite density illustrating the dynamics for x≃v_{0}t. We identify the relationship between a large position and the longest traveling time describing the single big jump principle. From the renewal theory we find that the distribution of rare events of the position is related to the derivative of the average of the number of renewals at a short "time" using a rate formalism.

Single-big-jump principle in physical modeling

The big-jump principle is a well-established mathematical result for sums of independent and identically distributed random variables extracted from a fat-tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: Instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big-jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory, and quenched disorder. Doing so we are able to predict rare events using the extended big-jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure, and in a class of Lévy walks with memory. We argue that the generalized big-jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy-tailed distributions, ranging from contamination spreading to active transport in the cell.

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)∼|x-x[over ̃]|^{2-2/α} in the vicinity of a point x[over ̃], 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).

From Non-Normalizable Boltzmann-Gibbs Statistics to Infinite-Ergodic Theory

We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as 1/x, e.g., a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. As time increases, our system approaches a non-normalizable Boltzmann state. We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are normalization, finite averaged energy, and a mean-squared displacement which increases linearly with time. Our work merges infinite-ergodic theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of ergodicity.

One-dimensional superdiffusive heat propagation induced by optical phonon-phonon interactions

One-dimensional anomalous heat propagation is usually characterized by a Lévy walk superdiffusive spreading function with two side peaks located on the fronts due to the finite velocity of acoustic phonons. In the case when the acoustic phonons vanish, e.g., due to the phonon-lattice interactions such that the system's momentum is not conserved, the side peaks will disappear and a normal Gaussian diffusive heat-propagating behavior will be observed. Here we show that there exists another new type of superdiffusive, non-Gaussian heat propagation but without side peaks in a typical nonacoustic, momentum-nonconserving system. It implies that thermal transport in this system disobeys the Fourier law, in clear contrast with the existing theoretical predictions. The underlying mechanism is related to an effect of optical phonon-phonon interactions. These findings may open a new avenue for further exploring thermal transport in low dimensions.

Large deviations in the presence of cooperativity and slow dynamics

We study simple models of intermittency, involving switching between two states, within the dynamical large-deviation formalism. Singularities appear in the formalism when switching is cooperative or when its basic time scale diverges. In the first case the unbiased trajectory distribution undergoes a symmetry breaking, leading to a change in shape of the large-deviation rate function for a particular dynamical observable. In the second case the symmetry of the unbiased trajectory distribution remains unbroken. Comparison of these models suggests that singularities of the dynamical large-deviation formalism can signal the dynamical equivalent of an equilibrium phase transition but do not necessarily do so.

Infinite invariant densities due to intermittency in a nonlinear oscillator

Dynamical intermittency is known to generate anomalous statistical behavior of dynamical systems, a prominent example being the Pomeau-Manneville map. We present a nonlinear oscillator, i.e., a physical model in continuous time, whose properties in terms of weak ergodity breaking and aging have a one-to-one correspondence to the properties of the Pomeau-Manneville map. So for both systems in a wide range of parameters no physical invariant density exists. We show how this regime can be characterized quantitatively using the techniques of infinite invariant densities and the Thaler-Dynkin limit theorem. We see how expectation values exhibit aging in terms of scaling in time.