Loopy belief propagation (LBP), which is equivalent to the Bethe approximation in statistical mechanics, is a message-passing-type inference method that is widely used to analyze systems based on Markov random fields (MRFs). In this paper, we propose a message-passing-type method to analytically evaluate the quenched average of LBP in random fields by using the replica cluster variation method. The proposed analytical method is applicable to general pairwise MRFs with random fields whose distributions differ from each other and can give the quenched averages of the Bethe free energies over random fields, which are consistent with numerical results. The order of its computational cost is equivalent to that of standard LBP. In the latter part of this paper, we describe the application of the proposed method to Bayesian image restoration, in which we observed that our theoretical results are in good agreement with the numerical results for natural images.

Colloidal self-replication represents an exciting research frontier in soft matter physics. Currently, all reported self-replication schemes involve coating colloidal particles with stimuli-responsive molecules to allow switchable interactions. In this paper, we introduce a scheme using ferromagnetic dipolar colloids and preprogrammed external magnetic fields to create an autonomous self-replication system. Interparticle dipole-dipole forces and periodically varying weak-strong magnetic fields cooperate to drive colloid monomers from the solute onto templates, bind them into replicas, and dissolve template complexes. We present three general design principles for autonomous linear replicators, derived from a focused study of a minimalist sphere-dimer magnetic system in which single binding sites allow formation of dimeric templates. We show via statistical models and computer simulations that our system exhibits nonlinear growth of templates and produces nearly exponential growth (low error rate) upon adding an optimized competing electrostatic potential. We devise experimental strategies for constructing the required magnetic colloids based on documented laboratory techniques. We also present qualitative ideas about building more complex self-replicating structures utilizing magnetic colloids.

Strongly segregated copolymer mixtures with uneven composition ratio can form hexagonally ordered thin films. A simplified model describing the size and position of micellelike clusters is derived, allowing for investigation of much larger domain sizes than in previous studies. Simulations of this model are performed to study the generation of large scale order and evolution of pattern defects. We find three temporal regimes exhibiting different scaling laws for orientational correlation length and defect number. In the early stage, topological defects are rapidly eliminated by pairwise annihilation. A slower intermediate stage is characterized by the migration of grain boundaries and the elimination of small grains. In the final stage, grain boundaries become pinned and the evolution halts. A scaling law for defect interaction is proposed which is consistent with the crossover between the first and second stages.

We introduce and study a model of time-dependent billiard systems with billiard boundaries undergoing infinitesimal wiggling motions. The so-called quivering billiard is simple to simulate, straightforward to analyze, and is a faithful representation of time-dependent billiards in the limit of small boundary displacements. We assert that when a billiard's wall motion approaches the quivering motion, deterministic particle dynamics become inherently stochastic. Particle ensembles in a quivering billiard are shown to evolve to a universal energy distribution through an energy diffusion process, regardless of the billiard's shape or dimensionality, and as a consequence universally display Fermi acceleration. Our model resolves a known discrepancy between the one-dimensional Fermi-Ulam model and the simplified static wall approximation. We argue that the quivering limit is the true fixed wall limit of the Fermi-Ulam model.

Knowledge of approximate harmonic behavior of crystals is introduced into a new “mapped averaging” framework to yield alternative expressions for the thermodynamic properties of crystalline systems. The expressions separate the known harmonic behavior from residual averages, which thus encapsulate anharmonic contributions to the properties. With harmonic contributions removed, direct measurement of these anharmonic contributions by molecular simulation can be accomplished without contamination by noise produced by the already-known harmonic behavior. We show with application to the Lennard-Jones model that first-derivative properties (pressure, energy) can be obtained to a given precision via this harmonically mapped averaging at least 10 times faster than by using conventional averaging, and second-derivative properties (e.g., heat capacity) are obtained at least 100 times faster; in more favorable cases, the speedup exceeds a millionfold. Free-energy calculations are accelerated by 50 to 1000 times. Data obtained using these formulations are rigorous and not subject to any added approximation, and in fact are less sensitive to inaccuracies relating to finite-size effects, potential truncation, equilibration, and similar considerations. Moreover, the approach does not require any alteration in how sampling is performed during the simulation, so it may be used with standard Monte Carlo or molecular dynamics methods. However, the mapped averages do require evaluation of first and second derivatives of the intermolecular potential, for evaluation of first and second thermodynamic-derivative properties, respectively. Apart from its usefulness to simulation, the formalism developed here may constitute a basis for new theoretical treatments of crystals.

In a recent paper [Phys. Rev. E 91, 013108 (2015)], Kraeft et al. criticize known exact results on the equation of state of quantum plasmas, which have been obtained independently by several authors. They argue about a difference in the definition of the direct two-body function $Q\left(x\right)$, which appears in virial expansions of thermodynamical quantities, but $Q\left(x\right)$ is not a measurable quantity in itself. Differences in definitions of intermediate quantities are irrelevant, and only differences in physical quantities are meaningful. Beyond Kraeft et al.'s broad statement that there is no agreement at order ${\rho }^{5/2}$ in the virial equation for the pressure, we show that their published results for this quantity are in fact in perfect agreement with previous existing expressions.

The long-standing discrepancy in the equation of state of charge neutral plasmas, the occurrence of an $e^2$ direct term in the second virial coefficient, is dealt with. We state that such a contribution should not appear for a pure Coulomb interaction.

We investigate the small-scale properties of a stochastic cubic-autocatalytic reaction-diffusion (CARD) model using renormalization techniques. We renormalize noise-induced ultraviolet divergences and obtain $\beta$ functions for the decay rate and coupling at one loop. Assuming colored (power-law) noise, our results show that the behavior of both decay rate and coupling with scale depends crucially on the noise exponent. Interpreting the CARD model as a proxy for a (very simple) living system, our results suggest that power-law correlations in environmental fluctuations can both decrease or increase the growth of structures at smaller scales.

We study the effects of next-nearest-neighbor (NNN) interactions on the dynamics of the one-dimensional spin-$1/2$ transverse Ising model in the high-temperature limit. We use exact diagonalization to obtain the time-dependent transverse correlation function and the corresponding spectral density for a tagged spin. Our results for chains of 13 spins with periodic boundary conditions produce results which are valid in the infinite-size limit. In general we find that the NNN coupling produces slower dynamics accompanied by an enhancement of the central mode behavior. Even in the case of a strong transverse field, if the NNN coupling is sufficiently large, then there is a crossover from collective mode to central mode behavior. We also obtain several recurrants for the continued fraction representation of the relaxation function.

As a generation of the classical percolation, clique percolation focuses on the connection of cliques in a graph, where the connection of two $k$ cliques means that they share at least $l<k$ vertices. In this paper we develop a theoretical approach to study clique percolation in Erdős-Rényi graphs, which gives not only the exact solutions of the critical point, but also the corresponding order parameter. Based on this, we prove theoretically that the fraction $\psi$ of cliques in the giant clique cluster always makes a continuous phase transition as the classical percolation. However, the fraction $\varphi$ of vertices in the giant clique cluster for $l>1$ makes a step-function-like discontinuous phase transition in the thermodynamic limit and a continuous phase transition for $l=1$. More interesting, our analysis shows that at the critical point, the order parameter ${\varphi }_c$ for $l>1$ is neither 0 nor 1, but a constant depending on $k$ and $l$. All these theoretical findings are in agreement with the simulation results, which give theoretical support and clarification for previous simulation studies of clique percolation.

We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.

We analyze the critical properties of the three-dimensional Ising model with linear parallel extended defects. Such a form of disorder produces two distinct correlation lengths, a parallel correlation length ${\xi }_{\parallel }$ in the direction along defects and a perpendicular correlation length ${\xi }_{\perp }$ in the direction perpendicular to the lines. Both ${\xi }_{\parallel }$ and ${\xi }_{\perp }$ diverge algebraically in the vicinity of the critical point, but the corresponding critical exponents ${\nu }_{\parallel }$ and ${\nu }_{\perp }$ take different values. This property is specific for anisotropic scaling and the ratio ${\nu }_{\parallel }/{\nu }_{\perp }$ defines the anisotropy exponent $\theta$. Until now, estimates of quantitative characteristics of the critical behavior for such systems have been obtained only within the renormalization group approach. We report a study of the anisotropic scaling in this system via Monte Carlo simulation of the three-dimensional system with Ising spins and nonmagnetic impurities arranged into randomly distributed parallel lines. Several independent estimates for the anisotropy exponent $\theta$ of the system are obtained, as well as an estimate of the susceptibility exponent $\gamma$. Our results corroborate the renormalization group predictions obtained earlier.

Polymer nanocomposites have been an issue of both academic and industrial interest due to promising electrical, mechanical, optical, and magnetic properties. The dynamics of nanoparticles in polymer nanocomposites is a key to understanding those properties of polymer nanocomposites and is important for applications such as self-healing nanocomposites. In this article we investigate the translational and the rotational dynamics of a single nanorod in unentangled polymer melts by employing extensive molecular dynamics simulations. A nanorod and polymers are modeled as semiflexible tangent chains of spherical beads. The stiffness of a nanorod is tuned by changing the bending potential between chemical bonds. When polymers are sufficiently long and the nanorod is stiff, the nanorod translates in an anisotropic fashion along the nanorod axis within time scales of translational relaxation times even in unentangled polymer melts. The rotational diffusion is suppressed more significantly than the translational diffusion as the polymer chain length is increased, thus the translational and rotational diffusion of the nanorod are decoupled. We also estimate the winding numbers of polymers, i.e., how many times a polymer winds the nanorod. The winding number increases with longer polymers but is relatively insensitive to the nanorod stiffness.

We investigate the diffusio-phoretic motion of a deformable vesicle. A vesicle is built from the linked catalytic and noncatalytic vertices that consumes fuel in the environment and utilize the resulting self-generated concentration gradient to exhibit propulsive motion. Under nonequilibrium conditions it is found that the self-propulsion velocity of the vesicle depends on its shape, which in turn is controlled by the bending rigidity of the membrane and solvent density around it. The self-propulsion velocity of the vesicle for different shapes has been calculated and the factors which affect the velocity are identified.

The irregular firing of a cortical neuron is thought to result from a highly fluctuating drive that is generated by the balance of excitatory and inhibitory synaptic inputs. A previous study reported anomalous responses of the Hodgkin-Huxley neuron to the fluctuated inputs where an irregularity of spike trains is inversely proportional to an input irregularity. In the current study, we investigated the origin of these anomalous responses with the Hindmarsh-Rose neuron model, map-based models, and a simple mixture of interspike interval distributions. First, we specified the parameter regions for the bifurcations in the Hindmarsh-Rose model, and we confirmed that the model reproduced the anomalous responses in the dynamics of the saddle-node and subcritical Hopf bifurcations. For both bifurcations, the Hindmarsh-Rose model shows bistability in the resting state and the repetitive firing state, which indicated that the bistability was the origin of the anomalous input-output relationship. Similarly, the map-based model that contained bistability reproduced the anomalous responses, while the model without bistability did not. These results were supported by additional findings that the anomalous responses were reproduced by mimicking the bistable firing with a mixture of two different interspike interval distributions. Decorrelation of spike trains is important for neural information processing. For such spike train decorrelation, irregular firing is key. Our results indicated that irregular firing can emerge from fluctuating drives, even weak ones, under conditions involving bistability. The anomalous responses, therefore, contribute to efficient processing in the brain.

Detecting communities in a network, based only on the adjacency matrix, is a problem of interest to several scientific disciplines. Recently, Zhang and Moore have introduced an algorithm [Proc. Natl. Acad. Sci. USA 111, 18144 (2014)], called mod-bp, that avoids overfitting the data by optimizing a weighted average of modularity (a popular goodness-of-fit measure in community detection) and entropy (i.e., number of configurations with a given modularity). The adjustment of the relative weight, the “temperature” of the model, is crucial for getting a correct result from mod-bp. In this work we study the many phase transitions that mod-bp may undergo by changing the two parameters of the algorithm: the temperature $T$ and the maximum number of groups $q$. We introduce a new set of order parameters that allow us to determine the actual number of groups $\widehat{q}$, and we observe on both synthetic and real networks the existence of phases with any $\widehat{q}\in \{1,q\}$, which were unknown before. We discuss how to interpret the results of mod-bp and how to make the optimal choice for the problem of detecting significant communities.

The role of committed minorities in shaping public opinion has been recently addressed with the help of multiagent models. However, previous studies focused on homogeneous populations where zealots stand out only for their stubbornness. Here we consider the more general case in which individuals are characterized by different propensities to communicate. In particular, we correlate commitment with a higher tendency to push an opinion, acknowledging the fact that individuals with unwavering dedication to a cause are also more active in their attempts to promote their message. We show that these activists are not only more efficient in spreading their message but that their efforts require an order of magnitude fewer individuals than a randomly selected committed minority to bring the population over to a new consensus. Finally, we address the role of communities, showing that partisan divisions in the society can make it harder for committed individuals to flip the status-quo social consensus.

Extreme events occur in many spatially extended dynamical systems, often devastatingly affecting human life, which makes their reliable prediction and efficient prevention highly desirable. We study the prediction and prevention of extreme events in a spatially extended system, a system of coupled FitzHugh-Nagumo units, in which extreme events occur in a spatially and temporally irregular way. Mimicking typical constraints faced in field studies, we assume not to know the governing equations of motion and to be able to observe only a subset of all phase-space variables for a limited period of time. Based on reconstructing the local dynamics from data and despite being challenged by the rareness of events, we are able to predict extreme events remarkably well. With small, rare, and spatiotemporally localized perturbations which are guided by our predictions, we are able to completely suppress extreme events in this system.