The Kuramoto-Sakaguchi model for coupled phase oscillators with phase frustration is often studied in the thermodynamic limit of infinitely many oscillators. Here we extend a model reduction method based on collective coordinates to capture the collective dynamics of finite-size Kuramoto-Sakaguchi models. We find that the inclusion of the effects of rogue oscillators is essential to obtain an accurate description, in contrast to the original Kuramoto model, where we show that their effects can be ignored. We further introduce a more accurate ansatz function to describe the shape of synchronized oscillators. Our results from this extended collective coordinate approach reduce in the thermodynamic limit to the well-known mean-field consistency relations. For finite networks we show that our model reduction describes the collective behavior accurately, reproducing the order parameter, the mean frequency of the synchronized cluster, and the size of the cluster at a given coupling strength, as well as the critical coupling strength for partial and for global synchronization.

Asymmetrically charged, nonspherical colloidal particles in general perform complex rotations and oblique motions under an electric field. The interplay of electrostatic and hydrodynamic forces complicates the prediction of these motions. We demonstrate a method of calculating the body tensors that dictate translational and rotational velocity vectors arising from an external electric field. We treat insulating rigid bodies in the linear-response regime, with indefinitely small electrostatic screening length. The method represents the body as an assembly of point sources of both hydrodynamic drag and surface electric field. We demonstrate agreement with predicted electrophoretic mobility to within a few percent for several shapes with uniform and nonuniform charges. We show that even symmetric shapes can have strong chiral twisting motions. The method applies more generally to active colloidal swimmers.

The present understanding of asymmetric lamellar eutectics focuses on pure diffusive transport, and how the external perturbations cause asymmetric pattern transitions remains unclear. In this work, the effect of external perturbations is discussed in terms of both thermal and convective effects via phase-field modeling. The presence of thermal perturbation distorts eutectic lamellae, while the convective perturbation causes a tilt band. Both can adjust the eutectic trajectory to accommodate newly established thermodynamics by reconstructing the transport equilibrium. Furthermore, how to regulate the eutectic growth (eutectic colony, zigzag, and snakelike patterns) by altering external perturbations is investigated, which provides information on how to control eutectic evolution.

The typicality approach and the Hilbert space averaging method as its technical manifestation are important concepts of quantum statistical mechanics. Extensively used for expectation values we extend them in this paper to transition probabilities. In this context we also find that the transition probability of two random uniformly distributed states is connected to the spectral statistics of the considered operator. Furthermore, within our approach we are capable to consider distributions of matrix elements between states that are not orthogonal. We will demonstrate our quite general result numerically for a kicked spin chain in the integrable resp. chaotic regime.

We investigate the phase diagram and the nature of the phase transitions of three-dimensional monopole-free ${CP}^{N-1}$ models, characterized by a global $U\left(N\right)$ symmetry, a $U\left(1\right)$ gauge symmetry, and the absence of monopoles. We present numerical analyses based on Monte Carlo simulations for $N=2$, 4, 10, 15, and 25. We observe a finite-temperature transition in all cases, related to the condensation of a local gauge-invariant order parameter. For $N=2$ we are unable to draw any definite conclusion on the nature of the transition. The results may be interpreted in terms of either a weak first-order transition or a continuous transition with anomalously large scaling corrections. However, the results allow us to exclude that the transition belongs to the $O\left(3\right)$ vector universality class, as it occurs in the standard three-dimensional ${CP}^1$ model without monopole suppression. For $N=4,10,\text{and}15$, the transition is of first order, and significantly weaker than that observed in the presence of monopoles. For $N=25$ the results are consistent with a conventional continuous transition. We compare our results with the existing literature and with the predictions of different field-theory approaches. They are consistent with the scenario in which the model undergoes continuous transitions for large values of $N$, including $N=\infty$, in agreement with analytic large-$N$ calculations for the $N$-component Abelian-Higgs model.

We examine the structure of the percolating cluster (PC) formed by site percolation on a random clustered network (RCN) model. Using the generating functions, we formulate the clustering coefficient and assortative coefficient of the PC. We analytically and numerically show that the PC in the highly clustered networks is clustered even at the percolation threshold. The assortativity of the PC depends on the details of the RCN. The PC at the percolation threshold is disassortative when the numbers of edges and triangles of each node are assigned by Poisson distributions, but assortative when each node in an RCN has the same small number of edges, most of which form triangles. This result seemingly contradicts the disassortativity of fractal networks, although the renormalization scheme unveils the disassortative nature of a fractal PC.

The Hawkes process has garnered attention in recent years for its suitability to describe the behavior of online information cascades. Here we present a fully tractable approach to analytically describe the distribution of the number of events in a Hawkes process, which, in contrast to purely empirical studies or simulation-based models, enables the effect of process parameters on cascade dynamics to be analyzed. We show that the presented theory also allows predictions regarding the future distribution of events after a given number of events have been observed during a time window. Our results are derived through a differential-equation approach to attain the governing equations of a general branching process. We confirm our theoretical findings through extensive simulations of such processes. This work provides the basis for more complete analyses of the self-exciting processes that govern the spreading of information through many communication platforms, including the potential to predict cascade dynamics within confidence limits.

We investigate the problem of the predominance and survival of “weak” species in the context of the simplest generalization of the spatial stochastic rock-paper-scissors model to four species by considering models in which one, two, or three species have a reduced predation probability. We show, using lattice based spatial stochastic simulations with random initial conditions, that if only one of the four species has its probability reduced, then the most abundant species is the prey of the “weakest” (assuming that the simulations are large enough for coexistence to prevail). Also, among the remaining cases, we present examples in which “weak” and “strong” species have similar average abundances and others in which either of them dominates—the most abundant species being always a prey of a weak species with which it maintains a unidirectional predator-prey interaction. However, in contrast to the three-species model, we find no systematic difference in the global performance of weak and strong species, and we conjecture that a similar result will hold if the number of species is further increased. We also determine the probability of single species survival and coexistence as a function of the lattice size, discussing its dependence on initial conditions and on the change to the dynamics of the model which results from the extinction of one of the species.

Protein folding is a long-standing problem and has been widely investigated using molecular dynamics simulations with both explicit and implicit solvents. However, to what extent the folding mechanisms observed in two water models agree remains an open question. In this study, ab initio folding simulations of ten proteins with different topologies are performed in two combinations of force fields and water models (ff14SB+TIP3P and ff14SBonlysc+GB-Neck2). Interestingly, the latter combination not only folds more proteins but also provides a better balance of different secondary structures than the former in the same number of integration time steps. More importantly, the folding pathways found in the two types of simulations are conserved and they may only differ in their weights. Our results suggest that simulations with an implicit solvent may also be suitable for the investigation of the mechanism of protein folding.

The run-and-tumble (RT) dynamics followed by bacterial swimmers gives rise first to a ballistic motion due to their persistence and later, through consecutive tumbles, to a diffusive process. Here we investigate how long it takes for a dilute swimmer suspension to reach the diffusive regime as well as what is the amplitude of the deviations from the diffusive dynamics. A linear time dependence of the mean-squared displacement (MSD) is insufficient to characterize diffusion and thus we also focus on the excess kurtosis of the displacement distribution. Four swimming strategies are considered: (i) the conventional RT model with complete reorientation after tumbling; (ii) the case of partial reorientation, characterized by a distribution of tumbling angles; (iii) a run-and-reverse model with rotational diffusion; and (iv) a RT particle where the tumbling rate depends on the stochastic concentration of an internal protein. By analyzing the associated kinetic equations for the probability density function and simulating the models, we find that for models (ii), (iii), and (iv) the relaxation to diffusion can take much longer than the mean time between tumble events, evidencing the existence of large tails in the particle displacements. Moreover, the excess kurtosis can assume large positive values. In model (ii) it is possible for some distributions of tumbling angles that the MSD reaches a linear time dependence but, still, the dynamics remains non-Gaussian for long times. This is also the case in model (iii) for small rotational diffusivity. For all models, the long-time diffusion coefficients are also obtained. The theoretical approach, which relies on eigenvalue and angular Fourier expansions of the van Hove function, is in excellent agreement with the simulations.

Whereas disclination defects are energetically prohibitive in two-dimensional flat crystals, their existence is necessary in crystals with spherical topology, such as viral capsids, colloidosomes, or fullerenes. Such a geometrical frustration gives rise to large elastic stresses, which render the crystal unstable when its size is significantly larger than the typical lattice spacing. Depending on the compliance of the crystal with respect to stretching and bending deformations, these stresses are alleviated either by a local increase of the intrinsic curvature in proximity of the disclinations or by the proliferation of excess dislocations, often organized in the form of one-dimensional chains known as “scars.” The associated strain field of the scars is such as to counterbalance the one resulting from the isolated disclinations. Here we develop a continuum theory of dislocation screening in two-dimensional closed crystals with genus one. Upon modeling the flux of scars emanating from a given disclination as an independent scalar field, we demonstrate that the elastic energy of closed two-dimensional crystals with various degrees of asphericity can be expressed as a simple quadratic function of the screened topological charge of the disclinations, at both zero and finite temperature. This allows us to predict the optimal density of the excess dislocations as well as the minimal stretching energy attained by the crystal.

We have obtained analytically the complete set of hydrodynamic modes (HMs) for a two-dimensional (2D) fluid confined within a channel with the Navier slip boundary condition at the hydrodynamic boundary. The HMs are orthogonal to each other and hence each represents an independent degree of freedom. We show that the HMs can be used to recursively generate a time series of random thermal fluctuations of displacement velocity, with identical statistical distributions as those obtained from MD simulations. By projecting the HMs onto molecular dynamics (MD) configurations and evaluating the resulting decay time from the autocorrelation function, we obtain from MD the eigenvalues of the HMs. Multiplying two different HMs and integrating as a function of $z$ from center of the channel towards the fluid-solid interface, the position of the hydrodynamic boundary (HDB) is unambiguously identified as the point at which the integral vanishes. Invariably the HDB is located inside the fluid domain and not on the liquid-solid interface. With the knowledge of the HDB position, the value of the slip length can be obtained directly from HM's dispersion relation. We show that in terms of the complete set of HMs, the fluctuation-dissipation theorem may be expressed in a simple expression involving the average of the inverse of the eigenvalues. Besides offering an alternative perspective on thermal fluctuations and hydrodynamic boundary, the present work opens the possibility of using modulated boundary conditions to manipulate thermal fluctuations in mesoscopic channels, which can lead to interesting statistical mechanical consequences.

In recent years, generative artificial neural networks based on restricted Boltzmann machines (RBMs) have been successfully employed as accurate and flexible variational wave functions for clean quantum many-body systems. In this article, we explore their use in simulations of disordered quantum Ising chains. The standard dense RBM with all-to-all interlayer connectivity is not particularly appropriate for large disordered systems, since in such systems one cannot exploit translational invariance to reduce the amount of parameters to be optimized. To circumvent this problem, we implement sparse RBMs, whereby the visible spins are connected only to a subset of local hidden neurons, thus reducing the amount of parameters. We assess the performance of sparse RBMs as a function of the range of the allowed connections, and we compare it with that of dense RBMs. Benchmark results are provided for two sign-problem-free Hamiltonians, namely pure and random quantum Ising chains. The RBM Ansätzes are trained using the unsupervised learning scheme based on projective quantum Monte Carlo (PQMC) algorithms. We find that the sparse connectivity facilitates the training process and allows sparse RBMs to outperform their dense counterparts. Furthermore, the use of sparse RBMs as guiding functions for PQMC simulations allows us to perform PQMC simulations at a reduced computational cost, avoiding possible biases due to finite random-walker populations. We obtain unbiased predictions for the ground-state energies and the magnetization profiles with fixed boundary conditions, at the ferromagnetic quantum critical point. The magnetization profiles agree with the Fisher–de Gennes scaling relation for conformally invariant systems, including the scaling dimension predicted by the renormalization-group analysis.

Koopman operator theory is shown to be directly related to the renormalization group. This observation allows us, with no assumption of translational invariance, to compute the critical exponents $\eta$ and $\delta$, as well as ratios of critical exponents, of classical spin systems from single observables alone. This broadens the types of problems that the renormalization group framework can be applied to and establish universality classes of. In addition, this connection may allow for a new, data-driven way in which to find the renormalization group fixed point(s), and their relevant and irrelevant directions.

We theoretically study thermally activated “in cage” elementary dynamical processes that precede full structural relaxation in ultradense particle liquids interacting via strong short-range attractive forces. The analysis is based on a microscopic theory formulated at the particle trajectory level built on the dynamic free energy concept and an explicit treatment of how attractive forces control the formation and lifetime of physical bonds. Mean time scales for bond breaking, the early stage of cage escape, and non-Fickian displacement by a fixed amount are analyzed in the repulsive glass, bonded repulsive (attractive) glass, fluid, and dense gel regimes. The theory predicts a strong length-scale-dependent growth of these time scales with attractive force strength at fixed packing fraction, a much weaker slowing down with density at fixed attraction strength, and a strong decoupling of the shorter bond-breaking time with the other two time scales that are controlled mainly by perturbed steric caging. All results are in good accord with simulations, and additional testable predictions are made. The classic statistical mechanical projection approximation of replacing all bare attractive and repulsive forces with a single effective force determined by pair structure incurs major errors for describing processes associated with thermally activated escape from transiently localized states.

Given a quantum many-body system and the expectation-value dynamics of some operator, we study how this reference dynamics is altered due to a perturbation of the system's Hamiltonian. Based on projection operator techniques, we unveil that if the perturbation exhibits a random-matrix structure in the eigenbasis of the unperturbed Hamiltonian, then this perturbation effectively leads to an exponential damping of the original dynamics. Employing a combination of dynamical quantum typicality and numerical linked cluster expansions, we demonstrate that our theoretical findings for random matrices can, in some cases, be relevant for the dynamics of realistic quantum many-body models as well. Specifically, we study the decay of current autocorrelation functions in spin-$1/2$ ladder systems, where the rungs of the ladder are treated as a perturbation to the otherwise uncoupled legs. We find a convincing agreement between the exact dynamics and the lowest-order prediction over a wide range of interchain couplings.

We have performed studies of the three-dimensional random-field $XY$ model on 32 samples of $L\times L\times L$ simple cubic lattices with periodic boundary conditions, with a random field strength of $h_r$ = 1.5, for $L=$ 128, using a parallelized Monte Carlo algorithm. We present results for the sample-averaged magnetic structure factor $S\left(\vec{\mathbf{k}}\right)$ over a range of temperature, using both random hot start and ferromagnetic cold start initial states, and $\vec{\mathbf{k}}$ along the [1,0,0] and [1,1,1] directions. At $T=$ 1.875, $S\left(\vec{\mathbf{k}}\right)$ shows a broad peak near $|\vec{\mathbf{k}}|=0$, with a correlation length which is limited by thermal fluctuations, rather than the lattice size. As $T$ is lowered, this peak grows and sharpens. By $T=$ 1.5, it is clear that the correlation length is larger than $L=$ 128. The lowest temperature for which $S\left(\vec{\mathbf{k}}\right)$ was calculated is $T=$ 1.421875, where the hot start and cold start initial conditions usually do not find the same local minimum in the phase space. Our results are consistent with the idea that there is a finite value of $T$ below which $S\left(\vec{\mathbf{k}}\right)$ diverges slowly as $|\vec{\mathbf{k}}|$ goes to zero. This divergence would imply that the relaxation time of the spins is also diverging. That is the signature of an ergodicity-breaking phase transition.

We investigate a low-dimensional slow-fast model to understand the dynamical origin of El Niño southern oscillation. A close inspection of the system dynamics using several bifurcation plots reveals that a sudden large expansion of the attractor occurs at a critical system parameter via a type of interior crisis. This interior crisis evolves through merging of a cascade of period-doubling and period-adding bifurcations that leads to the origin of occasional amplitude-modulated extremely large events. More categorically, a situation similar to homoclinic chaos arises near the critical point; however, atypical global instability evolves as a channellike structure in phase space of the system that modulates variability of amplitude and return time of the occasional large events and makes a difference from the homoclinic chaos. The slow-fast timescale of the low-dimensional model plays an important role on the onset of occasional extremely large events. Such extreme events are characterized by their heights when they exceed a threshold level measured by a mean-excess function. The probability density of events' height displays multimodal distribution with an upper-bounded tail. We identify the dependence structure of interevent intervals to understand the predictability of return time of such extreme events using autoregressive integrated moving average model and box-plot analysis.

We analyze the strategy equilibrium of dilemma games considering a payoff matrix affected by small and random perturbations on the off-diagonal. Notably, a recent work [Proc. R. Soc. A 476, 20200116 (2020)] reported that while cooperation is sustained by perturbations acting on the main diagonal, a less clear scenario emerges when perturbations act on the off-diagonal. Thus, the second case represents the core of this investigation, aimed at completing the description of the effects that payoff perturbations have on the dynamics of evolutionary games. Our results, achieved by analyzing the proposed model under a variety of configurations as different update rules, suggest that off-diagonal perturbations actually constitute a nontrivial form of noise. In particular, the most interesting effects are detected near the phase transition, as perturbations tend to move the strategy distribution towards nonordered states of equilibrium, supporting cooperation when defection is pervading the population, and supporting defection in the opposite case. To conclude, we identified a form of noise that, under controlled conditions, could be used to enhance cooperation and greatly delay its extinction.