Extended numerical simulations of threshold models have been performed on a human brain network with $N=836733$ connected nodes available from the Open Connectome Project. While in the case of simple threshold models a sharp discontinuous phase transition without any critical dynamics arises, variable threshold models exhibit extended power-law scaling regions. This is attributed to fact that Griffiths effects, stemming from the topological or interaction heterogeneity of the network, can become relevant if the input sensitivity of nodes is equalized. I have studied the effects of link directness, as well as the consequence of inhibitory connections. Nonuniversal power-law avalanche size and time distributions have been found with exponents agreeing with the values obtained in electrode experiments of the human brain. The dynamical critical region occurs in an extended control parameter space without the assumption of self-organized criticality.

Small particles in suspension in a turbulent fluid have trajectories that do not follow the pathlines of the flow exactly. We investigate the statistics of the angle of deviation $\varphi$ between the particle and fluid velocities. We show that, when the effects of particle inertia are small, the probability distribution function (PDF) $P_{\varphi }$ of this deviation angle shows a power-law region in which $P_{\varphi }\sim {\varphi }^{-4}$. We also find that the PDFs of the trajectory curvature $\kappa$ and modulus $\theta$ of the torsion $\vartheta$ have power-law tails that scale, respectively, as $P_{\kappa }\sim {\kappa }^{-5/2}$, as $\kappa \to \infty$, and $P_{\theta }\sim {\theta }^{-3}$, as $\theta \to \infty$: These exponents are in agreement with those previously observed for fluid pathlines. We propose a way to measure the complexity of heavy-particle trajectories by the number $N_{\mathrm{I}}(t,\mathrm{St})$ of points (up until time $t$) at which the torsion changes sign. We present numerical evidence that $n_{\mathrm{I}}\left(\mathrm{St}\right)\equiv {lim}_{t\to \infty }\frac{N_{\mathrm{I}}(t,\mathrm{St})}{t}\sim {\mathrm{St}}^{-\mathrm{\Delta }}$ for large $\mathrm{St}$, with $\mathrm{\Delta }\simeq 0.5$.

Characteristic functions (CFs) provide a very efficient method for evaluating the probability density functions of stochastic thermodynamic quantities and investigating their statistical features in quantum master equations (QMEs). A conventional procedure for obtaining these functions is to resort to a first-principles approach; namely, the evolution equations of the CFs of the combined system and its environment are obtained and then projected into the degrees of freedom of the system. However, the QMEs can be unraveled by a quantum jump trajectory. Thermodynamic quantities such as the heat, work, and entropy production can be well defined along a trajectory. Hence, on the basis of the notion of a trajectory, can we straightforwardly derive these CFs, e.g., their evolution equations? This is essential to establish the self-contained stochastic thermodynamics of a QME. In this paper, we show that it is indeed plausible and also simple. Particularly, these equations are fully consistent with those obtained by the first-principles method. Our results have practical significance; they indicate that the quantum fluctuation relations could be verified by more realistic photocounting experiments.

The linear perturbation renormalization group (LPRG) is used to study the phase transition of the weakly coupled Ising chains with intrachain $\left(J\right)$ and interchain nearest-neighbor $\left(J_1\right)$ and next-nearest-neighbor $\left(J_2\right)$ interactions forming the triangular and rectangular lattices in a field. The phase diagrams with the frustration point at $J_2=-J_1/2$ for a rectangular lattice and $J_2=-J_1$ for a triangular lattice have been found. The LPRG calculations support the idea that the phase transition is always continuous except for the frustration point and is accompanied by a divergence of the specific heat. For the antiferromagnetic chains, the external field does not change substantially the shape of the phase diagram. The critical temperature is suppressed to zero according to the power law when approaching the frustration point with an exponent dependent on the value of the field.

We consider exact and asymptotic solutions of the stationary cubic nonlinear Schrödinger equation on metric graphs. We focus on some basic example graphs. The asymptotic solutions are obtained using the canonical perturbation formalism developed in our earlier paper [S. Gnutzmann and D. Waltner, Phys. Rev. E 93, 032204 (2016)]. For closed example graphs (interval, ring, star graph, tadpole graph), we calculate spectral curves and show how the description of spectra reduces to known characteristic functions of linear quantum graphs in the low-intensity limit. Analogously for open examples, we show how nonlinear scattering of stationary waves arises and how it reduces to known linear scattering amplitudes at low intensities. In the short-wavelength asymptotics we discuss how genuine nonlinear effects may be described using the leading order of canonical perturbation theory: bifurcation of spectral curves (and the corresponding solutions) in closed graphs and multistability in open graphs.

We consider the canonical ensemble of $N$-vertex Erdős-Rényi (ER) random topological graphs with quenched vertex degree, and with fugacity $\mu$ for each closed triple of bonds. We claim complete defragmentation of large-$N$ graphs into the collection of $\left[p^{-1}\right]$ almost full subgraphs (cliques) above critical fugacity, ${\mu }_c$, where $p$ is the ER bond formation probability. Evolution of the spectral density, $\rho \left(\lambda \right)$, of the adjacency matrix with increasing $\mu$ leads to the formation of a multizonal support for $\mu >{\mu }_c$. Eigenvalue tunneling from the central zone to the side one means formation of a new clique in the defragmentation process. The adjacency matrix of the network ground state has a block-diagonal form, where the number of vertices in blocks fluctuates around the mean value $Np$. The spectral density of the whole network in this regime has triangular shape. We interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications.

The mechanical behaviors of polycrystalline solids are determined by the interplay between phenomena governed by two different thermodynamic temperatures: the configurational effective temperature that controls the density of dislocations, and the ordinary kinetic-vibrational temperature that controls activated depinning mechanisms and thus deformation rates. This paper contains a review of the effective-temperature theory and its relation to conventional dislocation theories. It includes a simple illustration of how these two thermal effects can combine to produce a predictive theory of spatial heterogeneities such as shear-banding instabilities. Its main message is a plea that conventional dislocation theories be reformulated in a thermodynamically consistent way so that the vast array of observed behaviors can be understood systematically.

We consider a solid spherical particle adsorbed at a flat interface between two immiscible fluids and having arbitrary contact angle at the triple contact line. We derive analytically the flow field corresponding to dilatational surface flow in the case of a large ratio of dynamic shear viscosities of two fluids. Considering a dilute assembly of such particles we calculate numerically the dependence on the contact angle of the effective surface dilatational viscosity particle-laden fluid interface. The effective surface dilatational viscosity is proportional to the size and surface concentration of particles and monotonically increases with the increase in protrusion of particles into the fluid with larger shear viscosity.

We study the probability density function (PDF) of the cover time $t_c$ of a finite interval of size $L$ by $N$ independent one-dimensional Brownian motions, each with diffusion constant $D$. The cover time $t_c$ is the minimum time needed such that each point of the entire interval is visited by at least one of the $N$ walkers. We derive exact results for the full PDF of $t_c$ for arbitrary $N\ge 1$ for both reflecting and periodic boundary conditions. The PDFs depend explicitly on $N$ and on the boundary conditions. In the limit of large $N$, we show that $t_c$ approaches its average value of $\langle t_c\rangle \approx L^2/(16DlnN)$ with fluctuations vanishing as $1/{(lnN)}^2$. We also compute the centered and scaled limiting distributions for large $N$ for both boundary conditions and show that they are given by nontrivial $N$ independent scaling functions.

Domain patterns are simulated by the Landau-Lifshitz-Gilbert (LLG) equation with an easy-axis anisotropy. If the Gilbert damping is removed from the LLG equation, it merely describes the precession of magnetization with a ferromagnetic interaction. However, even without the damping, domains that look similar to those of scalar fields are formed, and they grow with time. It is demonstrated that the damping has no significant effects on domain growth laws and large-scale domain structure. In contrast, small-scale domain structure is affected by the damping. The difference in small-scale structure arises from energy dissipation due to the damping.

Unsupervised neural network learning extracts hidden features from unlabeled training data. This is used as a pretraining step for further supervised learning in deep networks. Hence, understanding unsupervised learning is of fundamental importance. Here, we study the unsupervised learning from a finite number of data, based on the restricted Boltzmann machine where only one hidden neuron is considered. Our study inspires an efficient message-passing algorithm to infer the hidden feature and estimate the entropy of candidate features consistent with the data. Our analysis reveals that the learning requires only a few data if the feature is salient and extensively many if the feature is weak. Moreover, the entropy of candidate features monotonically decreases with data size and becomes negative (i.e., entropy crisis) before the message passing becomes unstable, suggesting a discontinuous phase transition. In terms of convergence time of the message-passing algorithm, the unsupervised learning exhibits an easy-hard-easy phenomenon as the training data size increases. All these properties are reproduced in an approximate Hopfield model, with an exception that the entropy crisis is absent, and only continuous phase transition is observed. This key difference is also confirmed in a handwritten digits dataset. This study deepens our understanding of unsupervised learning from a finite number of data and may provide insights into its role in training deep networks.

In order to study how local disturbances affect the ac grid stability, we start from nonlinear power balance equations and map them to complex linear wave equations. Having obtained stationary solutions with phases ${\phi }_i$ at generator and consumer nodes $i$, we next study the dynamics of deviations. Starting with an initially localized perturbation, it is found to spread in a periodic grid diffusively throughout the grid. We find the parametric dependence of diffusion constant $D$. We apply the same solution strategy to general grid topologies and analyze their stability against local perturbations. The perturbation remains either localized or becomes delocalized, depending on grid topology, power capacity, and distribution of consumers and generator power $P_i$. Delocalization is found to increase the lifetime of perturbations and thereby their influence on grid stability, whereas localization results in an exponentially fast decay of perturbations at all grid sites. These results may therefore lead to new strategies to control the stability of electricity grids.

We study the Braess paradox in the transport network as originally proposed by Braess with totally asymmetric exclusion processes (TASEPs) on the edges. The Braess paradox describes the counterintuitive situation in which adding an edge to a road network leads to a user optimum with higher travel times for all network users. Travel times on the TASEPs are nonlinear in the density, and jammed states can occur due to the microscopic exclusion principle, leading to a more realistic description of trafficlike transport on the network than in previously studied linear macroscopic mathematical models. Furthermore, the stochastic dynamics allows us to explore the effects of fluctuations on network performance. We observe that for low densities, the added edge leads to lower travel times. For slightly higher densities, the Braess paradox occurs in its classical sense. At intermediate densities, strong fluctuations in the travel times dominate the system's behavior due to links that are in a domain-wall state. At high densities, the added link leads to lower travel times. We present a phase diagram that predicts the system's state depending on the global density and crucial path-length ratios.

Bacteria communicate using external chemical signals called autoinducers (AI) in a process known as quorum sensing (QS). QS efficiency is reduced by both limitations of AI diffusion and potential interference from neighboring strains. There is thus a need for predictive theories of how spatial community structure shapes information processing in complex microbial ecosystems. As a step in this direction, we apply a reaction-diffusion model to study autoinducer signaling dynamics in a single-species community as a function of the spatial distribution of colonies in the system. We predict a dynamical transition between a local quorum sensing (LQS) regime, with the AI signaling dynamics primarily controlled by the local population densities of individual colonies, and a global quorum sensing (GQS) regime, with the dynamics being dependent on collective intercolony diffusive interactions. The crossover between LQS to GQS is intimately connected to a trade-off between the signaling network's latency, or speed of activation, and its throughput, or the total spatial range over which all the components of the system communicate.

We probe the onset and effect of contact changes in two-dimensional soft harmonic particle packings which are sheared quasistatically under controlled strain. First, we show that, in the majority of cases, the first contact changes correspond to the creation or breaking of contacts on a single particle, with contact breaking overwhelmingly likely for low pressures and/or small systems, and contact making and breaking equally likely for large pressures and in the thermodynamic limit. The statistics of the corresponding strains are near-Poissonian, in particular for large-enough systems. The mean characteristic strains exhibit scaling with the number of particles $N$ and pressure $P$ and reveal the existence of finite-size effects akin to those seen for linear response quantities [C. P. Goodrich , Phys. Rev. Lett. 109, 095704 (2012); C. P. Goodrich et al., Phys. Rev. E 90, 022138 (2014)]. Second, we show that linear response accurately predicts the strains of the first contact changes, which allows us to accurately study the scaling of the characteristic strains of making and breaking contacts separately. Both of these show finite-size scaling, and we formulate scaling arguments that are consistent with the observed behavior. Third, we probe the effect of the first contact change on the shear modulus $G$ and show in detail how the variation of $G$ remains smooth and bounded in the large-system-size limit: Even though contact changes occur then at vanishingly small strains, their cumulative effect, even at a fixed value of the strain, are limited, so, effectively, linear response remains well defined. Fourth, we explore multiple contact changes under shear and find strong and surprising correlations between alternating making and breaking events. Fifth, we show that by making a link with extremal statistics, our data are consistent with a very slow crossover to self-averaging with system size, so the thermodynamic limit is reached much more slowly than expected based on finite-size scaling of elastic quantities or contact breaking strains.

We study how polymers affect the heat flux in turbulent Rayleigh-Bénard convection at moderate Rayleigh numbers using direct numerical simulations with polymers of different relaxation times. We find that heat flux is enhanced by polymers and the amount of heat enhancement first increases and then decreases with the Weissenberg number, which is the ratio of the polymer relaxation time to the typical time scale of the flow. We show that this nonmonotonic behavior of the heat flux enhancement is the combined effect of the decrease in the viscous energy dissipation rate due to the viscosity of the Newtonian fluid and the increase in the energy dissipation rate due to polymers when Weissenberg number is increased. We explain why the viscous energy dissipation rate decreases with the Weissenberg number. Then by carrying out a generalized boundary layer analysis supplemented by a space-dependent effective viscosity from the numerical simulations, we provide a theoretical understanding of the change of the heat flux when the viscous energy dissipation rate is held constant. Our analysis thus provides a physical way to understand the numerical results.

Motivated by proposed thermometry measurement on an open quantum system, we present a simple model of an externally driven qubit interacting with a finite-sized fermion environment acting as a calorimeter. The derived dynamics is governed by a stochastic Schrödinger equation coupled to the temperature change of the calorimeter. We prove a fluctuation relation and deduce from it a notion of entropy production. Finally, we discuss the first and second law associated with the dynamics.

The Debye-Hückel approximation to the free energy of a simple fluid is written as a functional of the pair correlation function. This functional can be seen as the Debye-Hückel equivalent to the functional derived in the hypernetted chain framework by Morita and Hiroike, as well as by Lado. It allows one to obtain the Debye-Hückel integral equation through a minimization with respect to the pair correlation function, leads to the correct form of the internal energy, and fulfills the virial theorem.

We investigate the form of equilibrium spatiotemporal correlation functions of conserved quantities in the Toda lattice and in other integrable models. From numerical simulations we find that the correlations satisfy ballistic scaling with a remarkable collapse of data from different times. We examine special limiting choices of parameter values, for which the Toda lattice tends to either the harmonic chain or the equal mass hard-particle gas. In both these limiting cases, one can obtain the correlations exactly and we find excellent agreement with the direct Toda simulation results. We also discuss a transformation to “normal mode” variables, as commonly done in hydrodynamic theory of nonintegrable systems, and find that this is useful, to some extent, even for the integrable system. The striking differences between the Toda chain and a truncated version, expected to be nonintegrable, are pointed out.

We derive the Kuramoto-Sakaguchi model from the basic circuit equations governing two coupled Wien-bridge oscillators. A Wien-bridge oscillator is a particular realization of a tunable autonomous oscillator that makes use of frequency filtering (via an $RC$ bandpass filter) and positive feedback (via an operational amplifier). In the past few years, such oscillators have started to be utilized in synchronization studies. We first show that the Wien-bridge circuit equations can be cast in the form of a coupled pair of van der Pol equations. Subsequently, by applying the method of multiple time scales, we derive the differential equations that govern the slow evolution of the oscillator phases and amplitudes. These equations are directly reminiscent of the Kuramoto-Sakaguchi-type models for the study of synchronization. We analyze the resulting system in terms of the existence and stability of various coupled oscillator solutions and explain on that basis how their synchronization emerges. The phase-amplitude equations are also compared numerically to the original circuit equations and good agreement is found. Finally, we report on experimental measurements of two coupled Wien-bridge oscillators and relate the results to the theoretical predictions.