An important physical property for a stochastic process is how it responds to an external force or spatial confinement. This paper aims to study the relaxation dynamics of a generic process confined in a harmonic potential. We find the dependence of ensemble- and time-averaged mean squared displacements of the confined process on the velocity correlation function $C(t,t+\tau )$ of the original process without any external force. Combining two kinds of scaling forms of $C(t,t+\tau )$ for small $\tau$ and large $\tau$, the stationary value and the relaxation behaviors can be immediately obtained. Our results are valid for a large amount of anomalous diffusion processes, including the ones with single-scaled velocity correlation function (such as fractional Brownian motion and scaled Brownian motion) and the multiscaled ones (like Lévy walk with a broad range of power law exponents of flight time distribution). Note that the latter includes a special case with telegraphic active noise, which could take up athermal energy from the environment.

It is known that introducing time delays into the communication network of mobile-agent swarms produces coherent rotational patterns, from both theory and experiments. Often such spatiotemporal rotations can be bistable with other swarming patterns, such as milling and flocking. Yet, most known bifurcation results related to delay-coupled swarms rely on inaccurate mean-field techniques. As a consequence, the utility of applying macroscopic theory as a guide for predicting and controlling swarms of mobile robots has been limited. To overcome this limitation, we perform an exact stability analysis of two primary swarming patterns in a general model with time-delayed interactions. By correctly identifying the relevant spatiotemporal modes, we are able to accurately predict unstable oscillations beyond the mean-field dynamics and bistability in large swarms—laying the groundwork for comparisons to robotics experiments.

Collective behaviors by self-organization are ubiquitous in nature and human society and extensive efforts have been made to explore the mechanisms behind them. Artificial intelligence (AI) as a rapidly developing field is of great potential for these tasks. By combining reinforcement learning with evolutionary game (RLEG), we numerically discover a rich spectrum of collective behaviors—explosive events, oscillation, and stable states, etc., that are also often observed in the human society. In this work, we aim to provide a theoretical framework to investigate the RLEGs systematically. Specifically, we formalize AI-agents' learning processes in terms of belief switches and behavior modes defined as a series of actions following beliefs. Based on the preliminary results in the time-independent environment, we investigate the stability at the mixed equilibrium points in RLEGs generally, in which agents reside in one of the optimal behavior modes. Moreover, we adopt the maximum entropy principle to infer the composition of agents residing in each mode at a strictly stable point. When the theoretical analysis is applied to the $2\times 2$ game setting, we can explain the uncovered collective behaviors and are able to construct equivalent systems intuitively. Also, the inferred composition of different modes is consistent with simulations. Our work may be helpful to understand the related collective emergence in human society as well as behavioral patterns at the individual level and potentially facilitate human-computer interactions in the future.

The wrinkling transition experimentally identified by Mutz et al. [Phys. Rev. Lett. 67, 923 (1991)] and then thoroughly studied by Chaieb et al. [Phys. Rev. Lett. 96, 078101 (2006)] in partially polymerized lipid membranes is reconsidered. One shows that the features associated with this transition, notably the various scaling behaviors of the height-height correlation functions that have been observed, are qualitatively and quantitatively well described by a recent nonperturbative renormalization group approach to quenched disordered membranes by Coquand et al. [Phys. Rev E 97, 030102(R) (2018)]. As these behaviors are associated with fixed points of renormalization group transformations they are universal and should also be observed in, e.g., defective graphene and graphene-like materials.

Surface-diffusion mediated solid-state dewetting has been observed and studied in a number of different systems during the past two decades. This process can be accompanied by the pinching of the film at a finite distance from the retracting triple line. The repetition of this pinching is often referred to as periodic mass shedding. We show that the disjoining pressure of the film can accelerate mass shedding by orders of magnitude in ultrathin films with nanometric thickness. In the presence of power-law disjoining pressures induced by van der Waals forces, the mass shedding time exhibits an approximate power-law dependence on film thickness $t_{\mathrm{ms}}\sim {\overline{h}}^{\nu }$, with $\nu \approx 6$. Exponentially decaying disjoining forces also give rise to a strong acceleration of mass shedding. However, due to the finite range of the exponential potential, the mass shedding time does not exhibit a simple power-law dependence on the thickness, and is controlled by a cutoff thickness. In addition, two-dimensional simulations indicate that, within the range of thicknesses that we have studied and for isotropic dynamics, the transversal instability of a straight front does not lead to fingering, and mass shedding is the dominant instability of the dewetting front. Finally, we also show that no significant difference is observed in the dewetting dynamics between simulations based on a model with a wetting potential integrated over the film surface area, or over the projected substrate area.

Using numerical simulations of a model disk system, we demonstrate that a machine learning generated order-parameter-like measure can detect depinning transitions and different dynamic flow phases in systems driven far from equilibrium. We specifically consider monodisperse passive disks with short range interactions undergoing a depinning phase transition when driven over quenched disorder. The machine learning derived order-parameter-like measure identifies the depinning transition as well as different dynamical regimes, such as the transition from a flowing liquid to a phase separated liquid-solid state that is not readily distinguished with traditional measures such as velocity-force curves or Voronoi tessellation. The order-parameter-like measure also shows markedly distinct behavior in the limit of high density where jamming effects occur. Our results should be general to the broad class of particle-based systems that exhibit depinning transitions and nonequilibrium phase transitions.

Transfer entropy is an important index for investigating the causal relationship between random variables. It is known that a phase transition can be detected by transfer entropy. In this work, I propose a method for obtaining the transfer entropy by applying the variational approximation method to a dynamic Ising model. Our method is also effective for dynamics that do not satisfy the detailed balance condition. Our method finds the transfer entropy by solving a transcendental equation. When compared with the transfer entropy obtained by Monte Carlo sampling, it was found that they qualitatively matched near the transition point; moreover, the match was obtained with good accuracy in the region deviated from the transition point.

We consider the statistics of occupation times, the number of visits at the origin, and the survival probability for a wide class of stochastic processes, which can be classified as renewal processes. We show that the distribution of these observables can be characterized by a single exponent, that is connected to a local property of the probability density function of the process, viz., the probability of occupying the origin at time $t$, $P\left(t\right)$. We test our results for two different models of lattice random walks with spatially inhomogeneous transition probabilities, one of which of non-Markovian nature, and find good agreement with theory. We also show that the distributions depend only on the occupation probability of the origin by comparing them for the two systems: When $P\left(t\right)$ shows the same long-time behavior, each observable follows indeed the same distribution.

Multiple-spin coupling is a key tool for investigating magnetic quantum systems. In particular, models with four-spin interactions became of great interest since they have been applied to describe material properties such as superconductivity. In this framework, the dynamics of the spin-1/2 Ising two-leg ladder with four-spin plaquette interaction in a transverse field is investigated. By means of the recurrence relations method, the first four recurrants are determined exactly for the general model, while five exact recurrants are calculated for two particular models. In all cases, higher-order recurrants are computed through extrapolation to obtain time-dependent autocorrelation functions and spectral densities in the long-time regime and infinite-temperature limit. It is found that the relaxation functions decay slower than the expected behavior of the one-dimensional transverse Ising model. The spectral lines, characterized by a Lorentzian behavior in the central body and a Gaussian shape in the tails, display exchange narrowing as the coupling intensities increase.

Spatial networks are a powerful framework for studying a large variety of systems belonging to a broad diversity of contexts: from transportation to biology, from epidemiology to communications, and migrations, to cite a few. Spatial networks can be described in terms of their total cost (i.e., the total amount of resources needed for building or traveling their connections). Here, we address the issue of how to gauge and compare the quality of spatial network designs (i.e., efficiency vs. total cost) by proposing a two-step methodology. First, we assess the network's design by introducing a quality function based on the concept of network's efficiency. Second, we propose an algorithm to estimate computationally the upper bound of our quality function for a given network. Complementarily, we provide a universal expression to obtain an approximated upper bound to any spatial network, regardless of its size. Smaller differences between the upper bound and the empirical value correspond to better designs. Finally, we test the applicability of this analytic tool set on spatial network data-sets of different nature.

This paper deals with electron acceleration by a laser pulse in a plasma with a static uniform magnetic field $B_{*}$. The laser pulse propagates perpendicular to the magnetic field lines with the polarization chosen such that $({\mathit{E}}_{\mathrm{laser}}·{\mathit{B}}_{*})=0$. The focus of the work is on the electrons with an appreciable initial transverse momentum that are unable to gain significant energy from the laser in the absence of the magnetic field due to strong dephasing. It is shown that the magnetic field can initiate an energy increase by rotating such an electron, so that its momentum becomes directed forward. The energy gain continues well beyond this turning point where the dephasing drops to a very small value. In contrast to the case of purely vacuum acceleration, the electron experiences a rapid energy increases with the analytically derived maximum energy gain dependent on the strength of the magnetic field and the phase velocity of the wave. The energy enhancement by the magnetic field can be useful at high laser amplitudes, $a_0\gg 1$, where the acceleration similar to that in the vacuum is unable to produce energetic electrons over just tens of microns. A strong magnetic field helps leverage an increase in $a_0$ without a significant increase in the interaction length.

The stability of the negative ion of hydrogen (${\mathrm{H}}^{-}$) embedded in nonideal classical plasma has been studied by computing the ground state energy of the ion quite accurately. The interactions among the charged particles in plasma have been modelled by a pseudopotential, derived from a solution of Bogolyubov's hierarchy equations. An extensive basis set is employed in Rayleigh-Ritz variational method to compute the ground state energy of ${\mathrm{H}}^{-}$ for various values of plasma parameters. Effects of nonideality of plasma on the stability of the ion have been investigated in detail for a wide range of nonideality. Particular emphasis is made to compute accurately the critical values of the plasma screening parameters.

Time and cost are two main hurdles to acquiring a large number of digital image I of the microstructure of materials. Thus, use of stochastic methods for producing plausible realizations of materials' morphology based on one or very few images has become an increasingly common practice in their modeling. The accuracy of the realizations is often evaluated using two-point microstructural descriptors or physics-based modeling of certain phenomena in the materials, such as transport processes or fluid flow. In many cases, however, two-point correlation functions do not provide accurate evaluation of the realizations, as they are usually unable to distinguish between high- and low-quality reconstructed models. Calculating flow and transport properties of the realization is an accurate way of checking the quality of the realizations, but it is computationally expensive. In this paper a method based on machine learning is proposed for evaluating stochastic approaches for reconstruction of materials, which is applicable to any of such methods. The method reduces the dimensionality of the realizations using an unsupervised deep-learning algorithm by compressing images and realizations of materials. Two criteria for evaluating the accuracy of a reconstruction algorithm are then introduced. One, referred to as the internal uncertainty space, is based on the recognition that for a reconstruction method to be effective, the differences between the realizations that it produces must be reasonably wide, so that they faithfully represent all the possible spatial variations in the materials' microstructure. The second criterion recognizes that the realizations must be close to the original I and, thus, it quantifies the similarity based on an external uncertainty space. Finally, the ratio of two uncertainty indices associated with the two criteria is considered as the final score of the accuracy of a stochastic algorithm, which provides a quantitative basis for comparing various realizations and the approaches that produce them. The proposed method is tested with images of three types of heterogeneous materials in order to evaluate four stochastic reconstruction algorithms.

We cast the metabolism of interacting cells within a statistical mechanics framework considering both the actual phenotypic capacities of each cell and its interaction with its neighbors. Reaction fluxes will be the components of high-dimensional spin vectors, whose values will be constrained by the stochiometry and the energy requirements of the metabolism. Within this picture, finding the phenotypic states of the population turns out to be equivalent to searching for the equilibrium states of a disordered spin model. We provide a general solution of this problem for arbitrary metabolic networks and interactions. We apply this solution to a simplified model of metabolism and to a complex metabolic network, the central core of Escherichia coli, and demonstrate that the combination of selective pressure and interactions defines a complex phenotypic space. We also present numerical results for cells fixed in a grid. These results reproduce the qualitative picture discussed for the mean-field model. Cells may specialize in producing or consuming metabolites complementing each other, and this is described by an equilibrium phase space with multiple minima, like in a spin-glass model.

We study the evolution of unstable liquid films via numerical solutions of the thin-film equation. The film is placed on a coated substrate with disorder. This is modeled by a random spatial variation of the relative value of the Hamaker constants for the substrate and coating. The free energy consists of (a) the van der Waals term for the substrate/coating interactions with the film and (b) a term due to gravity. This free energy admits a Maxwell double-tangent construction with two coexisting phases, i.e., “thin” and “thick” phases. In the absence of disorder, the film dewets by true morphological phase separation (MPS), i.e., the elimination of domain walls between the coexisting phases. The introduction of disorder may result in the trapping of these domain walls, with a drastic slowdown in growth kinetics. We present detailed numerical results in $D=2$ and $D=3$ to understand this slow coarsening, where $D$ is the dimensionality of the liquid-film system.

Linearization around unstable traveling waves in excitable systems can be used to approximate strength-extent curves in the problem of initiation of excitation waves for a family of spatially confined perturbations to the rest state. This theory relies on the knowledge of the unstable traveling wave solution as well as the leading left and right eigenfunctions of its linearization. We investigate the asymptotics of these ingredients, and utility of the resulting approximations of the strength-extent curves, in the slow-fast limit in two-component excitable systems of FitzHugh-Nagumo type and test those on four illustrative models. Of these, two are with degenerate dependence of the fast kinetic on the slow variable, a feature which is motivated by a particular model found in the literature. In both cases, the unstable traveling wave solution converges to a stationary “critical nucleus” of the corresponding one-component fast subsystem. We observe that in the full system, the asymptotics of the left and right eigenspaces are distinct. In particular, the slow component of the left eigenfunction corresponding to the translational symmetry does not become negligible in the asymptotic limit. This has a significant detrimental effect on the critical curve predictions. The theory as formulated previously uses an heuristic to address a difficulty related to the translational invariance. We describe two alternatives to that heuristic, which do not use the misbehaving eigenfunction component. These new heuristics show much better predictive properties, including in the asymptotic limit, in all four examples.

The dynamics of a membrane coupled to an active fluid on top of a substrate is considered theoretically. It is assumed that the director field of the active fluid has rotational symmetry in the membrane plane. This situation is likely to be relevant for in vitro reconstructed actomyosin-membrane system. Different from a membrane coupled to a polar active fluid, this model predicts that only when the viscosity of the fluid above the membrane is sufficiently large, a contractile active fluid is able to slow down the relaxation of the membrane for perturbations with wavelength comparable to the thickness of the active fluid. Hence, our model predicts a finite-wavelength instability in the limit of strong contractility, which is different from a membrane coupled to a polar active fluid. However, a membrane coupled to an extensile active fluid is always unstable against long-wavelength perturbations due to active extensile stress enhanced membrane undulation.

The evaporation of a sessile drop in a gaseous environment may be critical to many practical applications. Evaporation dynamics of interacting sessile droplets is strongly influenced by the proximity of adjacent droplets. We study the effects of droplet-droplet vapor-mediated interactions on the evaporation lifetime of two-dimensional arrays of sessile water droplets. We observe that the presence of neighboring droplets acts as a mode of vapor accumulation which slows down the evaporation process. By considering an arbitrarily configured two-dimensional array of droplets, here we provide a simple generalized theoretical limit to their lifetime in an evaporating state. Using a scaling analysis, we put forward that the sessile droplet lifetime in a two-dimensional array is a linear function of the extent of confinement for various surface wettability and droplet geometric parameters (contact angle and contact radius). Notwithstanding the geometrical and physical complexity of the effective confinement generated due to their cooperative interactions, we show that the consequent evaporation characteristics may be remarkably insensitive to the topographical details of the overall droplet organization for a wide range of droplet-substrate combinations. With subsequent deployment of particle-laden droplets, however, our results lead to the discovery of a unique pathway towards tailoring the internal flows within the collective system by harnessing an exclusive topologically driven symmetry-breaking phenomenon, yielding a strategy of patterning particulate matters around the droplet array.

A mixture of Brownian particles at different temperatures has been a useful model for studying the out-of-equilibrium properties of systems made up of microscopic components with differing levels of activity. This model was previously studied analytically for two-particle interactions in the dilute limit, yielding a Boltzmann-like two-particle distribution with an effective temperature. Like the Newtonian two- and three-body problems, we ask here whether the two-particle results can be extended to three-particle interactions to get the three-particle distributions. By considering the special solvable case of pairwise quadratic interactions, we show that, unlike the two-particle distribution, the three-particle distribution cannot in general be Boltzmann-like with an effective temperature. We instead find that the steady-state distribution of any two particles in a triplet depends on the properties of and interactions with the third particle, leading to some unexpected behaviors not present in equilibrium.

We study two versions of the XY model where the spins but also the interaction topology is allowed to change. In the free XY model, the number of links is fixed, but their positions in the network are not. We also study a more relaxed version where even the number of links is allowed to vary, we call it the freer XY model. When the interaction networks are dense enough, both models have phase transitions visible both in spin configurations and the network structure. The low-temperature phase in the free XY model is characterized by tightly connected clusters of spins pointing in the same direction and isolated spins disconnected from the rest. For the freer XY model the low-temperature phase is almost completely connected. In both models, exponents describing the magnetic ordering are mostly consistent with values of the mean-field theory of the standard XY model.