Long-time relaxation processes occur in numerous physical systems. They are often regarded as multirelaxation processes, which are a superposition of exponential decays with a certain distribution of relaxation times. The relaxation times spectra often convey information about the underlying physics. Extracting the spectrum of relaxation times from experimental data is, however, difficult. This is partly due to the mathematical properties of the problem and partly due to experimental limitations. In this paper, we perform the inversion of time-series relaxation data into a relaxation spectrum using the singular value decomposition accompanied by the Akaike information criterion estimator. We show that this approach does not need any a priori information on the spectral shape and that it delivers a solution that consistently approximates the best one achievable for given experimental dataset. On the contrary, we show that the solution obtained imposing an optimal fit of experimental data is often far from reconstructing well the distribution of relaxation times.

The thermodynamic as well as optical properties of strongly coupled plasmas depend crucially on the average degree of ionization and the ionic state composition, which, however, cannot be determined by using the normal Saha equation usually used for the ideal plasmas. Hence, an adequate treatment of the ionization balance and the charge state distribution of strongly coupled plasmas is still a challenge for theory due to the interactions between the electrons and ions and among the electrons themselves. Based on a local density temperature-dependent ion-sphere model, the Saha equation approach is extended to the regime of strongly coupled plasmas by taking into account the free-electron-ion interaction, the free-free-electron interaction, the nonuniform free-electron space distribution, and the free-electron quantum partial degeneracy. All the quantities, including the bound orbitals with ionization potential depression, free-electron distribution, and bound and free-electron partition function contributions, are calculated self-consistently in the theoretical formalism. This study shows that the ionization equilibrium is evidently modified by considering the above nonideal characteristics of the free electrons. Our theoretical formalism is validated by the explanation of a recent experimental measurement of the opacity of dense hydrocarbon.

We analyze the power spectral density of a signal composed of nonoverlapping rectangular pulses. First, we derive a general formula for the power spectral density of a signal constructed from the sequence of nonoverlapping pulses. Then we perform a detailed analysis of the rectangular pulse case. We show that pure $1/f$ noise can be observed until extremely low frequencies when the characteristic pulse (or gap) duration is long in comparison to the characteristic gap (or pulse) duration, and gap (or pulse) durations are power-law distributed. The obtained results hold for the ergodic and weakly nonergodic processes.

For systems in equilibrium at a temperature $T$, thermal noise and energy damping are related to $T$ through the fluctuation-dissipation theorem (FDT). We study here an extension of the FDT to an out-of-equilibrium steady state: a microcantilever subject to a constant heat flux. The resulting thermal profile in this spatially extended system interplays with the local energy dissipation field to prescribe the amplitude of mechanical fluctuations. Using three samples with different damping profiles (localized or distributed), we probe this approach and experimentally demonstrate the link between fluctuations and dissipation. The thermal noise can therefore be predicted a priori from the measurement of the dissipation as a function of the maximum temperature of the micro-oscillator.

Previous work showed that the collective activity of large neuronal networks can be tamed to remain near its critical point by a feedback control that maximizes the temporal correlations of the mean-field fluctuations. Since such correlations behave similarly near instabilities across nonlinear dynamical systems, it is expected that the principle should control also low-dimensional dynamical systems exhibiting continuous or discontinuous bifurcations from fixed points to limit cycles. Here we present numerical evidence that the dynamics of a single neuron can be controlled in the vicinity of its bifurcation point. The approach is tested in two models: a two-dimensional generic excitable map and the paradigmatic FitzHugh-Nagumo neuron model. The results show that in both cases, the system can be self-tuned to its bifurcation point by modifying the control parameter according to the first coefficient of the autocorrelation function.

We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find the ensemble-averaged solution of the fractional master equation through subordination involving the fractional Poisson process counting the number of steps at a given time and the underlying discrete random walk with self-reinforcement. We also find the exact solution for the variance which exhibits superdiffusion even as the fractional exponent tends to 1.

Motion of a driven particle in a two-dimensional (2D) periodic potential of square symmetry is studied by means of Brownian dynamics simulations. The average drift velocity and long time diffusion coefficients are obtained as a function of driving force and temperature. For driving forces above the critical depinning force, a reduction of drift velocity is observed as temperature is increased. The drift velocity reaches a minimum for temperatures at which $k_{B}T$ is of the order of the barrier height of the substrate potential and then increases and saturates to the value of drift velocity for the substrate free case. Depending on the driving force, the drop in drift velocity can be as large as $36\%$ of its value at low temperatures. While this phenomenon is observed in 2D for different types of substrate potentials studied and for various drive directions, studies using the exact result show no such dip in drift velocity in one dimension (1D). As in the case of 1D, a peak is observed in the longitudinal diffusion coefficient as the driving force is varied at a fixed temperature. But unlike in 1D, the location of the peak is temperature dependent. Approximate analytical expressions for the average drift velocity and the longitudinal diffusion coefficient are formulated using the exact results in 1D by finding a temperature dependent effective 1D potential to model the motion in the presence of a 2D substrate. This approximate analysis is successful in qualitatively predicting the observations.

We devise an analytical method to deal with a class of nonlinear Schrödinger lattices with random potential and subquadratic power nonlinearity. An iteration algorithm is proposed based on the multinomial theorem, using Diophantine equations and a mapping procedure onto a Cayley graph. Based on this algorithm, we are able to obtain several hard results pertaining to asymptotic spreading of the nonlinear field beyond a perturbation theory approach. In particular, we show that the spreading process is subdiffusive and has complex microscopic organization involving both long-time trapping phenomena on finite clusters and long-distance jumps along the lattice consistent with Lévy flights. The origin of the flights is associated with the occurrence of degenerate states in the system; the latter are found to be a characteristic of the subquadratic model. The limit of quadratic power nonlinearity is also discussed and shown to result in a delocalization border, above which the field can spread to long distances on a stochastic process and below which it is Anderson localized similarly to a linear field.

We first derive the Hamilton-Jacobi theory underlying continuous-time Markov processes, and then we use the construction to develop a variational algorithm for estimating escape (least improbable or first passage) paths for a generic stochastic chemical reaction network that exhibits multiple fixed points. The design of our algorithm is such that it is independent of the underlying dimensionality of the system, the discretization control parameters are updated toward the continuum limit, and there is an easy-to-calculate measure for the correctness of its solution. We consider several applications of the algorithm and verify them against computationally expensive means such as the shooting method and stochastic simulation. While we employ theoretical techniques from mathematical physics, numerical optimization and chemical reaction network theory, we hope that our work finds practical applications with an inter-disciplinary audience including chemists, biologists, optimal control theorists and game theorists.

Ventricular arrhythmias are the leading cause of sudden cardiac death. Understanding the mechanisms of arrhythmia initiation is important for developing effective therapeutics for prevention. Arrhythmias can be induced via premature external stimuli or occur spontaneously via dynamical instabilities. Computer simulations have shown that a large repolarization gradient due to regional prolongation of the action potential duration can result in instabilities leading to premature excitations and arrhythmias, but the bifurcation remains to be elucidated. In this study we carry out numerical simulations and linear stability analyses using a one-dimensional heterogeneous cable consisting of the FitzHugh-Nagumo model. We show that a Hopf bifurcation leads to local oscillations, which, once their amplitudes are large enough, lead to spontaneous propagating excitations. Depending on the degree of heterogeneities, these excitations can range from one to many and to be sustained oscillations, manifesting as premature ventricular contractions (PVCs) and sustained arrhythmias. The dynamics depends on the repolarization gradient and the length of the cable. Complex dynamics is also induced by the repolarization gradient. The mechanistic insights from the simple model may help in the understanding of the genesis of PVCs and arrhythmias in long QT syndrome.

We propose a stochastic process of interacting many agents, which is inspired by rank-based supplanting dynamics commonly observed in a group of Japanese macaques. In order to characterize the breaking of permutation symmetry with respect to agents' rank in the stochastic process, we introduce a rank-dependent quantity, overlap centrality, which quantifies how often a given agent overlaps with the other agents. We give a sufficient condition in a wide class of the models such that overlap centrality shows perfect correlation in terms of the agents' rank in the zero-supplanting limit. We also discuss a singularity of the correlation in the case of interaction induced by a Potts energy.

The synchronization of dust acoustic waves to an external periodic source is studied in the framework of a driven Korteweg–de Vries–Burgers equation that takes into account the appropriate nonlinear and dispersive nature of low-frequency waves in a dusty plasma medium. For a spatiotemporally varying source term, the system is shown to demonstrate harmonic (1:1) and superharmonic (1:2) synchronized states. The existence domains of these states are delineated in the form of Arnold tongue diagrams in the parametric space of the forcing amplitude and forcing frequency and their resemblance to some past experimental results is discussed.

The stochastic dynamics of reinforcement learning is studied using a master equation formalism. We consider two different problems—$Q$ learning for a two-agent game and the multiarmed bandit problem with policy gradient as the learning method. The master equation is constructed by introducing a probability distribution over continuous policy parameters or over both continuous policy parameters and discrete state variables (a more advanced case). We use a version of the moment closure approximation to solve for the stochastic dynamics of the models. Our method gives accurate estimates for the mean and the (co)variance of policy variables. For the case of the two-agent game, we find that the variance terms are finite at steady state and derive a system of algebraic equations for computing them directly.

We study a generalization of the random walk (RW) based on a deformed translation of the unitary step, inherited by the $q$ algebra, a mathematical structure underlying nonextensive statistics. The RW with deformed step implies an associated deformed random walk (DRW) provided with a deformed Pascal triangle along with an inhomogeneous diffusion. The paths of the RW in deformed space are divergent, while those corresponding to the DRW converge to a fixed point. Standard random walk is recovered for $q\to 1$ and a suppression of randomness is manifested for the DRW with $-1<{\gamma }_q<1$ and ${\gamma }_q=1-q$. The passage to the continuum of the master equation associated to the DRW led to a van Kampen inhomogeneous diffusion equation when the mobility and the temperature are proportional to $1+{\gamma }_{q}x$, and provided with an exponential hyperdiffusion that exhibits a localization of the particle at $x=-1/{\gamma }_q$ consistent with the fixed point of the DRW. Complementarily, a comparison with the Plastino-Plastino Fokker-Planck equation is discussed. The two-dimensional case is also studied, by obtaining a 2D deformed random walk and its associated deformed 2D Fokker-Planck equation, which give place to a convergence of the 2D paths for $-1<{\gamma }_{q_1},{\gamma }_{q_2}<1$ and a diffusion with inhomogeneities controlled by two deformation parameters ${\gamma }_{q_1},{\gamma }_{q_2}$ in the directions $x$ and $y$. In both the one-dimensional and the two-dimensional cases, the transformation ${\gamma }_q\to -{\gamma }_q$ implies a change of sign of the corresponding limits of the random walk paths, as a property of the deformation employed.

Experiments have shown that charge distributions of granular materials are non-Gaussian, with broad tails that indicate many particles with high charge. This observation has consequences for the behavior of granular materials in many settings, and may bear relevance to the underlying charge transfer mechanism. However, there is the unaddressed possibility that broad tails arise due to experimental uncertainties, as determining the shapes of tails is nontrivial. Here we show that measurement uncertainties can indeed account for most of the tail broadening previously observed. The clue that reveals this is that distributions are sensitive to the electric field at which they are measured; ones measured at low (high) fields have larger (smaller) tails. Accounting for sources of uncertainty, we reproduce this broadening in silico. Finally, we use our results to back out the true charge distribution without broadening, which we find is still non-Guassian, though with substantially different behavior at the tails and indicating significantly fewer highly charged particles. These results have implications in many natural settings where electrostatic interactions, especially among highly charged particles, strongly affect granular behavior.

Recently, a number of physical models have emerged described by a random process with increments given by a quadratic form of a fast Gaussian process. We find that the rate function which describes sample-path large deviations for such a process can be computed from the large domain size asymptotic of a certain Fredholm determinant. The latter can be evaluated analytically using a theorem of Widom which generalizes the celebrated Szegő-Kac formula to the multidimensional case. This provides a large class of random dynamical systems with timescale separation for which an explicit sample-path large-deviation functional can be found. Inspired by problems in hydrodynamics and atmosphere dynamics, we construct a simple example with a single slow degree of freedom driven by the square of a fast multivariate Gaussian process and analyze its large-deviation functional using our general results. Even though the noiseless limit of this example has a single fixed point, the corresponding large-deviation effective potential has multiple fixed points. In other words, it is the addition of noise that leads to metastability. We use the explicit answers for the rate function to construct instanton trajectories connecting the metastable states.

A well-known feature of a propagating localized excitation in a discrete lattice is the generation of a backwave in the extended normal mode spectrum. To quantify the parameter-dependent amplitude of such a backwave, the properties of a running intrinsic localized mode (ILM) in electric, cyclic, dissipative, nonlinear 1D transmission lines, containing balanced nonlinear capacitive and inductive terms, are studied via simulations. Both balanced and unbalanced damping and driving conditions are treated. The introduction of a unit cell duplex driver, with a voltage source driving the nonlinear capacitor and a synchronized current source, the nonlinear inductor, provides an opportunity to design a cyclic, dissipative self-dual nonlinear transmission line. When the self-dual conditions are satisfied, the dynamical voltage and current equations of motion within a cell become the same, the strength of the fundamental, resonant coupling between the ILM and the lattice modes collapses, and the associated fundamental backwave is no longer observed.

Self-assembly of colloidal particles into predefined structures is a promising way to design inexpensive manmade materials with advanced macroscopic properties. Doping of nematic liquid crystals (LCs) with nanoparticles has a series of advantages in addressing these grand scientific and engineering challenges. It also provides a very rich soft matter platform for the discovery of unique condensed matter phases. The LC host naturally allows the realization of diverse anisotropic interparticle interactions, enriched by the spontaneous alignment of anisotropic particles due to the boundary conditions of the LC director. Here we demonstrate theoretically and experimentally that the ability of LC media to host topological defect lines can be used as a tool to probe the behavior of individual nanoparticles as well as effective interactions between them. LC defect lines irreversibly trap nanoparticles enabling controlled particle movement along the defect line with the use of a laser tweezer. Minimization of Landau–de Gennes free energy reveals a sensitivity of the ensuing effective nanoparticle interaction to the shape of the particle, surface anchoring strength, and temperature, which determine not only the strength of the interaction but also its repulsive or attractive character. Theoretical results are supported qualitatively by experimental observations. This work may pave the way toward designing controlled linear assemblies as well as one-dimensional crystals of nanoparticles such as gold nanorods or quantum dots with tunable interparticle spacing.