A nonparametric $k$-nearest-neighbor-based entropy estimator is proposed. It improves on the classical Kozachenko-Leonenko estimator by considering nonuniform probability densities in the region of $k$-nearest neighbors around each sample point. It aims to improve the classical estimators in three situations: first, when the dimensionality of the random variable is large; second, when near-functional relationships leading to high correlation between components of the random variable are present; and third, when the marginal variances of random variable components vary significantly with respect to each other. Heuristics on the error of the proposed and classical estimators are presented. Finally, the proposed estimator is tested for a variety of distributions in successively increasing dimensions and in the presence of a near-functional relationship. Its performance is compared with a classical estimator, and a significant improvement is demonstrated.

A two-dimensional constrained diffusion model is presented and characterized by numerical simulations. The model generalizes the one-dimensional single-file diffusion model by considering a cage diffusion constraint induced by neighboring particles, which is a more stringent condition than volume exclusion. Using numerical simulations we characterize the diffusion process and we particularly show that asymmetric transition probabilities lead to the two-dimensional Kardar-Parisi-Zhang universality class. Therefore, this very simple model effectively generalizes the one-dimensional totally asymmetric simple exclusion process to higher dimensions.

We perform a detailed numerical study of the conductance $G$ through one-dimensional (1D) tight-binding wires with on-site disorder. The random configurations of the on-site energies $\epsilon$ of the tight-binding Hamiltonian are characterized by long-tailed distributions: For large $\epsilon$, $P\left(\epsilon \right)\sim 1/{\epsilon }^{1+\alpha }$ with $\alpha \in (0,2)$. Our model serves as a generalization of the 1D Lloyd model, which corresponds to $\alpha =1$. First, we verify that the ensemble average $〈-lnG〉$ is proportional to the length of the wire $L$ for all values of $\alpha$, providing the localization length $\xi$ from $〈-lnG〉=2L/\xi$. Then, we show that the probability distribution function $P\left(G\right)$ is fully determined by the exponent $\alpha$ and $〈-lnG〉$. In contrast to 1D wires with standard white-noise disorder, our wire model exhibits bimodal distributions of the conductance with peaks at $G=0$ and 1. In addition, we show that $P(lnG)$ is proportional to $G^{\beta }$, for $G\to 0$, with $\beta \le \alpha /2$, in agreement with previous studies.

Suppose that a $d$-dimensional domain is filled with a gas of (in general, interacting) diffusive particles with density $n_0$. A particle is absorbed whenever it reaches the domain boundary. Employing macroscopic fluctuation theory, we evaluate the probability $\mathcal{P}$ that no particles are absorbed during a long time $T$. We argue that the most likely gas density profile, conditional on this event, is stationary throughout most of the time $T$. As a result, $\mathcal{P}$ decays exponentially with $T$ for a whole class of interacting diffusive gases in any dimension. For $d=1$ the stationary gas density profile and $\mathcal{P}$ can be found analytically. In higher dimensions we focus on the simple symmetric exclusion process (SSEP) and show that $-ln\mathcal{P}\simeq D_{0}T{L}^{d-2}s\left(n_0\right)$, where $D_0$ is the gas diffusivity, and $L$ is the linear size of the system. We calculate the rescaled action $s\left(n_0\right)$ for $d=1$, for rectangular domains in $d=2$, and for spherical domains. Near close packing of the SSEP $s\left(n_0\right)$ can be found analytically for domains of any shape and in any dimension.

We have reported in Phys. Rev. Lett. 110, 064103 (2013) that in systems which otherwise do not show oscillatory dynamics, the interplay between pinning to a defect and pulling by drift allows the system to exhibit excitability and oscillations. Here we build on this work and present a detailed bifurcation analysis of the various dynamical instabilities that result from the competition between a pulling force generated by the drift and a pinning of the solitons to spatial defects. We show that oscillatory and excitable dynamics of dissipative solitons find their origin in multiple codimension-2 bifurcation points. Moreover, we demonstrate that the mechanisms leading to these dynamical regimes are generic for any system admitting dissipative solitons.

We investigated the phase-response curve of a coupled system of density oscillators with an analytical approach. The behaviors of two-, three-, and four-coupled systems seen in the experiments were reproduced by the model considering the phase-response curve. Especially in a four-coupled system, the clustering state and its incidence rate as functions of the coupling strength are well reproduced with this approach. Moreover, we confirmed that the shape of the phase-response curve we obtained analytically was close to that observed in the experiment where a perturbation is added to a single-density oscillator. We expect that this approach to obtaining the phase-response curve is general in the sense that it could be applied to coupled systems of other oscillators such as electrical-circuit oscillators, metronomes, and so on.

We present the variational formulation of a quantitative phase-field model for isothermal low-speed solidification in a binary dilute alloy with diffusion in the solid. In the present formulation, cross-coupling terms between the phase field and composition field, including the so-called antitrapping current, naturally arise in the time evolution equations. One of the essential ingredients in the present formulation is the utilization of tensor diffusivity instead of scalar diffusivity. In an asymptotic analysis, it is shown that the correct mapping between the present variational model and a free-boundary problem for alloy solidification with an arbitrary value of solid diffusivity is successfully achieved in the thin-interface limit due to the cross-coupling terms and tensor diffusivity. Furthermore, we investigate the numerical performance of the variational model and also its nonvariational versions by carrying out two-dimensional simulations of free dendritic growth. The nonvariational model with tensor diffusivity shows excellent convergence of results with respect to the interface thickness.

The rheological properties of granular matter within a two-dimensional flow around a moving disk is investigated experimentally. Using a combination of photoelastic and standard tessellation techniques, the strain and stress tensors are estimated at the grain scale in the time-averaged flow field around a large disk pulled at constant velocity in an assembly of smaller disks. On the one hand, one observes inhomogeneous shear rate and strongly localized shear stress and pressure fields. On the other hand, a significant dilation rate, which has the same magnitude as the shear strain rate, is reported. Significant deviations are observed with local rheology that justify the need of searching for a nonlocal rheology.

We present a mathematical model for Joule heating of an electrolytic solution in a nanopore. The model couples the electrical and thermal dynamics responsible for rapid and extreme superheating of the electrolyte within the nanopore. The model is implemented numerically with a finite element calculation, yielding a time and spatially resolved temperature distribution in the nanopore region. Temperatures near the thermodynamic limit of superheat are predicted to be attained just before the explosive nucleation of a vapor bubble is observed experimentally. Knowledge of this temperature distribution enables the evaluation of related phenomena including bubble nucleation kinetics, relaxation oscillation, and bubble dynamics.

In this paper, based on multicomponent phase-field theory we intend to develop an efficient lattice Boltzmann (LB) model for simulating three-phase incompressible flows. In this model, two LB equations are used to capture the interfaces among three different fluids, and another LB equation is adopted to solve the flow field, where a new distribution function for the forcing term is delicately designed. Different from previous multiphase LB models, the interfacial force is not used in the computation of fluid velocity, which is more reasonable from the perspective of the multiscale analysis. As a result, the computation of fluid velocity can be much simpler. Through the Chapman-Enskog analysis, it is shown that the present model can recover exactly the physical formulations for the three-phase system. Numerical simulations of extensive examples including two circular interfaces, ternary spinodal decomposition, spreading of a liquid lens, and Kelvin-Helmholtz instability are conducted to test the model. It is found that the present model can capture accurate interfaces among three different fluids, which is attributed to its algebraical and dynamical consistency properties with the two-component model. Furthermore, the numerical results of three-phase flows agree well with the theoretical results or some available data, which demonstrates that the present LB model is a reliable and efficient method for simulating three-phase flow problems.

We study the self-averaging properties and ergodicity of the mean square displacement $m\left(t\right)$ of particles diffusing in $d$ dimensional quenched random environments which give rise to subdiffusive average motion. These properties are investigated in terms of the sample to sample fluctuations as measured by the variance of $m\left(t\right)$. We find that $m\left(t\right)$ is not self-averaging for $d<2$ due to the inefficient disorder sampling by random motion in a single realization. For $d\ge 2$ in contrast, the efficient sampling of heterogeneity by the space random walk renders $m\left(t\right)$ self-averaging and thus ergodic. This is remarkable because the average particle motion in $d>2$ obeys a CTRW, which by itself displays weak ergodicity breaking. This paradox is resolved by the observation that the CTRW as an average model does not reflect the disorder sampling by random motion in a single medium realization.

Closed quantum systems obey the Schrödinger equation, whereas nonequilibrium behavior of many systems is routinely described in terms of classical, Markovian stochastic processes. Evidently, there are fundamental differences between those two types of behavior. We discuss the conditions under which the unitary dynamics may be mapped onto pertinent classical stochastic processes. This is first principally addressed based on the notions of “consistency” and “Markovianity.” Numerical data are presented that show that the above conditions are to good approximation fulfilled for Heisenberg-type spin models comprising 12–20 spins. The accuracy to which these conditions are met increases with system size.

We numerically study a Bose-Hubbard ring of finite size with disorder containing a finite number of bosons that are subject to an on-site two-body interaction. Our results show that moderate interactions induce dynamical thermalization in this isolated system. In this regime the individual many-body eigenstates are well described by the standard thermal Bose-Einstein distribution for well-defined values of the temperature and the chemical potential, which depend on the eigenstate under consideration. We show that the dynamical thermalization conjecture works well at both positive and negative temperatures. The relations to quantum chaos, quantum ergodicity, and the Åberg criterion are also discussed.

I present a technical report indicating that the two methods used for calculating characteristic functions for the work distribution in weakly driven quantum master equations are equivalent. One involves applying the notion of quantum jump trajectory [Phys. Rev. E 89, 042122 (2014)], while the other is based on two energy measurements on the combined system and reservoir [Silaev et al., Phys. Rev. E 90, 022103 (2014)]. These represent backward and forward methods, respectively, which adopt a very similar approach to that of the Kolmogorov backward and forward equations used in classical stochastic theory. The microscopic basis for the former method is also clarified. In addition, a previously unnoticed equality related to the heat is also revealed.

We introduce a probability-current analysis of excitation energy transfer between states of an open quantum system. Expressing the energy transfer through currents of excitation probability between the states in a site representation enables us to gain key insights into the energy transfer dynamics. In particular, the analysis yields direct identification of the pathways of energy transport in large networks of sites and quantifies their relative weights, as well as the respective contributions of unitary dynamics, coherence, dephasing, and relaxation and dissipation processes to the energy transfer. It thus provides much more information than studying only excitation probabilities of the states as a function of time. Our analysis is general and can be readily applied to a broad range of dynamical descriptions of open quantum system dynamics with coupling to non-Markovian or Markovian environments.

We consider the optimization of Markovian dynamics to pursue the fastest convergence to the stationary state. The brachistochrone method is applied to the continuous-time master equation for finite-size systems. The principle of least action leads to a brachistochrone equation for the transition-rate matrix. Three-state systems are explicitly analyzed, and we find that the solution violates the detailed balance condition. The properties of the solution are studied in detail to observe the optimality of the solution. We also discuss the counterdiabatic driving for the Markovian dynamics. The transition-rate matrix is then divided into two parts, and the state is given by an eigenstate of the first part. The second part violates the detailed balance condition and plays the role of a counterdiabatic term.

Time-periodic driving provides a promising route toward engineering nontrivial states in quantum many-body systems. However, while it has been shown that the dynamics of integrable, noninteracting systems can synchronize with the driving into a nontrivial periodic motion, generic nonintegrable systems are expected to heat up until they display a trivial infinite-temperature behavior. In this paper we show that a quasiperiodic time evolution over many periods can also emerge in weakly interacting systems, with a clear separation of the timescales for synchronization and the eventual approach of the infinite-temperature state. This behavior is the analog of prethermalization in quenched systems. The synchronized state can be described using a macroscopic number of approximate constants of motion. We corroborate these findings with numerical simulations for the driven Hubbard model.

Classical correlation functions of ground states typically decay exponentially and polynomially, respectively, for gapped and gapless short-range quantum spin systems. In such systems, entanglement decays exponentially even at the quantum critical points. However, quantum discord, an information-theoretic quantum correlation measure, survives long lattice distances. We investigate the effects of quenched disorder on quantum correlation lengths of quenched averaged entanglement and quantum discord, in the anisotropic $XY$ and $XYZ$ spin glass and random field chains. We find that there is virtually neither reduction nor enhancement in entanglement length while quantum discord length increases significantly with the introduction of the quenched disorder.

We analyze the translational and rotational motion of an ellipsoidal Brownian particle from the viewpoint of stochastic thermodynamics. The particle's Brownian motion is driven by external forces and torques and takes place in an heterogeneous thermal environment where friction coefficients and (local) temperature depend on space and time. Our analysis of the particle's stochastic thermodynamics is based on the entropy production associated with single particle trajectories. It is motivated by the recent discovery that the overdamped limit of vanishing inertia effects (as compared to viscous fricion) produces a so-called “anomalous” contribution to the entropy production, which has no counterpart in the overdamped approximation, when inertia effects are simply discarded. Here we show that rotational Brownian motion in the overdamped limit generates an additional contribution to the “anomalous” entropy. We calculate its specific form by performing a systematic singular perturbation analysis for the generating function of the entropy production. As a side result, we also obtain the (well-known) equations of motion in the overdamped limit. We furthermore investigate the effects of particle shape and give explicit expressions of the “anomalous entropy” for prolate and oblate spheroids and for near-spherical Brownian particles.

We show the link between U(1) lattice gauge theories and the off-equilibrium thermodynamics of a large class of nonlinear oscillators networks. The coupling between the oscillators plays the role of a gauge field, or connection, on the network. The thermodynamical forces that drive energy flows are expressed in terms of the curvature of the connection, analogous to a geometric phase. The model, which holds both close and far from equilibrium, predicts the existence of persistent energy and particle currents circulating in closed loops through the network. The predictions are confirmed by numerical simulations. Possible extension of the theory and experimental applications to nanoscale devices are briefly discussed.