We develop a technique for calculating three-dimensional classical partition functions using projected entangled-pair states (PEPS). Our method is based on variational PEPS optimization algorithms for two-dimensional quantum spin systems, and allows us to compute free energies directly in the thermodynamic limit. The main focus of this work is classical frustration in three-dimensional many-body systems leading to an extensive ground-state degeneracy. We provide high-accuracy results for the residual entropy of the dimer model on the cubic lattice, water ice $I_h$, and water ice $I_c$. In addition, we show that these systems are in a Coulomb phase by computing the dipolar form of the correlation functions. As a further benchmark of our methods, we calculate the critical temperature and exponents of the Ising model on the cubic lattice.

While the phase transitions in binary propagation on complex networks have been studied a great deal from diverse fields during the past years, the effect of feedback has rarely been systematically addressed in a general manner. Here, we study a minimal network model of binary propagation with a general feedback effect. It is found that when the basic propagation rate is small, moderate positive feedback can induce bistable dynamics, resulting in an abrupt and discontinuous phase transition, which is absent in the case of negative feedback or no feedback. Moreover, the degree heterogeneity of the network further induces a hysteresis loop structure in this phase transition. It is also found that positive feedback accelerates spread before reaching the peak activity, whereas negative feedback has a contrary effect, which may uncover the origin of the accelerating spread observed in sophisticated models [S. V. Scarpino, A. Allard, and L. Hébert-Dufresne, Nat. Phys. 12, 1042 (2016)]. These findings reveal the universal and nontrivial effects of feedback on binary propagation dynamics. It can help to preestimate the phase-transition behavior in specific network propagation models and help to better understand the criticality of other dynamical processes on complex networks arising from different scientific backgrounds.

We study structural changes of adaptive networks in the coevolutionary susceptible-infected-susceptible (SIS) network model along its phase transition. We clarify to what extent these changes can be used as early-warning signs for the transition at the critical infection rate ${\lambda }_c$ at which the network collapses and the system disintegrates. We analyze the interplay between topology and node-state dynamics near criticality. Several network measures exhibit clear maxima or minima close to the critical threshold and could potentially serve as early-warning signs. These measures include the SI link density, triplet densities, clustering, assortativity, and the eigenvalue gap. For the SI link density and triplet densities the maximum is found to originate from the coexistence of two power laws. Other network quantities, such as the degree, the branching ratio, or the harmonic mean distance, show scaling with a singularity at $\lambda =0$ (not at ${\lambda }_c$), which means that they are incapable of detecting the transition.

The geographical distribution of production is getting an increased amount of attention in economics. Distributed trade opens for production where it is cheapest, which in turn is reinforced by economies of scale. Using a simple agent-based model for the geographical interplay between transportation costs, economies of scale, as well as information costs, we address here the transition between local and distributed economies. The model naturally recapitulates that decreased transportation and information costs favor large companies. This suggests history dependence in the sense that new companies typically reemerge in the vicinity of old ones. Further, it suggests that company stability depends on transport costs, and that the transition from a local economy to a global economy is naturally driven by reduced transportation costs and an increased information horizon.

Characteristic diffusion in a Lorentz gas system and its extensions are typically described through a scale-free order parameter. The relative validity of possible order parameters for the diffusion of a thick tracer rod through spherical scatterers in two and three dimensions as a function of the rod length $L$ and scatterer density $\rho$ has been examined by simulation. We find that the often-used two-dimensional order parameter, $\rho L^2$, is accurate. However, its naive generalization to three dimensions, changing the exponent according to dimensionality, does not fit the observed results. A revised order parameter can be generated through a theory based on the behavior of the mean-free path of the tracer. We find that this generalized order parameter captures the multidimensional diffusive behavior.

Slip flow in ducts and porous media is simulated using lattice-Boltzmann method incorporated with interfacial force models. The dependence of the results on the viscosity, LBM scheme (D3Q15 and D3Q19) and the relaxation time model (single- or multirelaxation time) is investigated. The severity of spurious velocities (arisen from classic and advanced interfacial force models) is discussed that leads to entirely nonphysical results for whole flow rates ($0.027<\text{Re}<10.7$). A simple method based on superposition of solutions is proposed to fully rehabilitate the simulations. We validate the method by showing the simulations versus analytical solution of slip flow through circular ducts. The validity of the rehabilitated results for porous media applications are also tested through two approaches: First, we show that the rehabilitation method is independent to the force scheme used, i.e., the rehabilitated results are identical in both pore and macroscales for different force schemes with different distributions of spurious velocities. Second, using an analogy based on the Kozeny-Carman model, we show that the permeability variation in porous media resulted from the flow slippage obtained from rehabilitated simulations is reliable. We argue that to obtain correct results, it is necessary to use the rehabilitation method whenever interfaical force models are used in LB simulations. The results reveal that the permeability of porous media may increase or decrease with positive or negative slippage (repulsive and attractive interfaces), respectively. The permeability enhancement rate increases as the system becomes simpler in its interfaces, i.e., for the same positive slippage of flow, $(\frac{\kappa }{{\kappa }_{\text{NS}}}{)}_{\text{parallel}\text{plates}}>(\frac{\kappa }{{\kappa }_{\text{NS}}}{{)}_{\text{square}\text{ducts}}>(\frac{\kappa }{{\kappa }_{\text{NS}}})}_{\text{porous}\text{media}}$ (where $\kappa$ is the permeability $\mathrm{and}\text{NS}$ denotes no-slip).

We present a numerical study of the time-dependent motion of a two-dimensional vesicle in a channel under an imposed flow. In a Poiseuille flow the shape of the vesicle depends on the flow strength, the mechanical properties of the membrane, and the width of the channel as reported in the past. This study is focused on the centered snaking (CSn) shape, where the vesicle shows an oscillatory motion like a swimmer flagella even though the flow is stationary. We quantify this behavior by the amplitude and frequency of the oscillations of the vesicle's center of mass. We observe regions in parameter space, where the CSn coexists with the parachute or the unconfined slipper. The influence of an amplitude modulation of the imposed flow on the dynamics and shape of the snaking vesicle is also investigated. For large modulation amplitudes transitions to static shapes are observed. A smaller modulation amplitude induces a modulation in amplitude and frequency of the center of mass of the snaking vesicle. In a certain parameter range we find that the center of mass oscillates with a constant envelope indicating the presence of at least two stable states.

We show via particle-in-cell simulations that small normalized magnetic fields (${\omega }_c/{\omega }_p\ll 1$) can significantly modify the evolution of backward stimulated Raman scattering (SRS) in the kinetic regime due to the enhanced dissipation of nonlinear electron plasma waves propagating perpendicular to magnetic fields. A magnetic field applied perpendicularly to the electron plasma wave (and driving light wave) increases the SRS threshold for kinetic inflation and decreases the amount of reflectivity when SRS is driven significantly above threshold. Analysis indicates that this arises because trapped electrons are accelerated as they surf across the wave, leading to the continual dissipation of the electron plasma waves over a wider range of wave amplitudes. The reduction in SRS reflectivity is most significant for a purely perpendicular field, although reduction also occurs for other angles; a parallel field can slightly increase single-speckle SRS but decreases multispeckle SRS. These simulations demonstrate the significance of magnetic-field contributions to nonlinear electron plasma wave damping with respect to nonlinear parametric decay instabilities; the simulation parameters are directly relevant for SRS in inertial confinement fusion devices and indicate that approximately 30 tesla magnetic fields might significantly reduce SRS backscatter.

A general interfacial treatment for the transport process in multicomponent systems governed by the convection-diffusion equation, together with conjugated interfaces where continuity or discontinuity interfacial conditions with regard to the physical scalar and its flux exist, is developed using the lattice Boltzmann method (LBM) with a multiple-relaxation-time collision operator. Factors accounting for interfacial jumps at conjugated interfaces typically include the difference in solubility, porosity, and transport properties, and the interfacial transport resistance. Difficulties in satisfying the discontinuity conditions, where continuity conditions are inherent in the conventional LBM, are overcome based on a second-order accurate treatment for both Dirichlet and Neumann boundary conditions. As a result, the unknown populations at the interface are obtained with a combination of the existing postcollision populations and the defined interfacial jumps without finite-difference approximations, thereby enabling the interfacial requirements precisely. Since the present treatment considers local intersected link fractions of the interface, it can be applied for curved interfaces naturally. As for straight interfaces located at the midpoints of the nodes, the present treatment can be significantly simplified with only two current nodes required. Validation work on the present treatment are performed with four specific problems, including the steady and unsteady convection diffusion in a fluid-fluid channel, steady convection diffusion in a fluid-solid system, steady and unsteady pure conduction in a composite slab, and steady pure conduction in an annulus. Such an interfacial scheme is finally applied to the conjugated mass transfer in a heterogonous medium, highlighting its practicability in complex systems with complicated interfacial conditions. In addition, detailed error analysis on interior nodes, interfacial values, and interfacial fluxes shows that our model is second-order accurate in space for straight interfaces regardless of various interfacial conditions, indicating no artificial disturbance is introduced to the model. While for curved interface, only superlinear and first-order accuracies are obtained for the interior nodes and the interfacial values or fluxes, respectively. Discussions on the degradation of the accuracy are presented. Note that the present interfacial treatment is mainly focused on two-dimensional problems, extensions of the model to be three-dimensional are straightforward.

A tomographic or image representation of three-dimensional (3D) porous media is a primary tool for the consistent correlation and prediction of multiple physical properties. For multiscale media a single imaging scale may not be sufficient, and a compromise between field of view and resolution is required. The resulting lower resolution compared to relevant length scales in part of the tomogram impacts on the quality of petrophysical cross-correlations. This situation is overcome in geostatistics by carrying out stochastic simulations with lower resolution constraints and propagating uncertainty from smaller scales through upscaling procedures capturing the behavior of the system at smaller scales. Applying this technique to full tomograms requires a fast high-resolution reconstruction technique. We propose a local-similarity statistic reconstruction (LSSR) method to reconstruct 3D high-resolution porous structure by combining a set of increasing resolution micro-computed-tomography (micro-CT) images with decreasing field of view (FOV) to overcome this limitation. The reconstruction technique is based on two assumptions, universally existent local similarity at a given scale and fixed image degradation when downscaling by the same factor between corresponding cubes of high- and low-resolution images. Utilizing the flexibility of micro-CT images in terms of resolution and FOV, a sample was scanned first with low resolution, and then a subset was taken from this sample and scanned with high resolution. A significant number of small cube pairs are extracted from corresponding parts of the low-resolution and high-resolution images, where both high- and low-resolution images are accessible. These cube pairs contain abundant information about features of local porous structure under different resolutions allowing reconstruction of a higher-resolution micro-CT image. Instead of the “Search-Statistic” strategy popularly used in current reconstruction algorithm a “Decomposition-Reconstruction” strategy is applied to accelerate computations and improve the reconstruction accuracy. Local porosity theory and the Minkowski measures are used to estimate the performance of reconstruction algorithms. Compared to multipoint statistics we improve computational efficiency by employing a sparse representation and principal component analysis for compression, while also improving on reconstruction accuracy.

We present an analytical framework to study the first-passage (FP) and first-return (FR) distributions for the broad family of models described by the one-dimensional Fokker-Planck equation in finite domains, identifying general properties of these distributions for different classes of models. When in the Fokker-Planck equation the diffusion coefficient is positive (nonzero) and the drift term is bounded, as in the case of a Brownian walker, both distributions may exhibit a power-law decay with exponent $-3/2$ for intermediate times. We discuss how the influence of an absorbing state changes this exponent. The absorbing state is characterized by a vanishing diffusion coefficient and/or a diverging drift term. Remarkably, the exponent of the Brownian walker class of models is still found, as long as the departure and arrival regions are far enough from the absorbing state, but the range of times where the power law is observed narrows. Close enough to the absorbing point, though, a new exponent may appear. The particular value of the exponent depends on the behavior of the diffusion and the drift terms of the Fokker-Planck equation. We focus on the case of a diffusion term vanishing linearly at the absorbing point. In this case, the FP and FR distributions are similar to those of the voter model, characterized by a power law with exponent $-2$. As an illustration of the general theory, we compare it with exact analytical solutions and extensive numerical simulations of a two-parameter voterlike family models. We study the behavior of the FP and FR distributions by tuning the importance of the absorbing points throughout changes of the parameters. Finally, the possibility of inferring relevant information about the steady-sate probability distribution of a model from the FP and FR distributions is addressed.

Using the formalism of information theory, we analyze the mechanism of information transduction in a simple one-step signaling cascade $\mathrm{S}\to \mathrm{X}$ representing the gene regulatory network. Approximating the signaling channel to be Gaussian, we describe the dynamics using Langevin equations. Upon discretization, we calculate the associated second moments for linear and nonlinear regulation of the output by the input, which follows the birth-death process. While mutual information between the input and the output characterizes the channel capacity, the Fano factor of the output gives a clear idea of how internal and external fluctuations assemble at the output level. To quantify the contribution of the present state of the input to predict the future output, transfer entropy is computed. We find that a higher amount of transfer entropy is accompanied by a greater magnitude of external fluctuations (quantified by the Fano factor of the output) propagation from the input to the output. We notice that a low input population characterized by the number of signaling molecules S, which fluctuates in a relatively slower fashion compared to its downstream (target) species X, is maximally able to predict (as quantified by transfer entropy) the future state of the output. Our computations also reveal that with increased linear nature of the input-output interaction, all three metrics of mutual information, Fano factor, and transfer entropy achieve relatively larger magnitudes.

Deep learning has become a powerful and popular tool for a variety of machine-learning tasks. However, it is challenging to understand the mechanism of deep learning from a theoretical perspective. In this work, we propose a random active path model to study collective properties of deep neural networks with binary synapses, under the removal perturbation of connections between layers. In the model, the path from input to output is randomly activated, and the corresponding input unit constrains the weights along the path into the form of a $p$-weight interaction glass model. A critical value of the perturbation is observed to separate a spin-glass regime from a paramagnetic regime, with the transition being of the first order. The paramagnetic phase is conjectured to have a poor generalization performance.

We develop a stochastic kinetic model of a preformed attachment of a microtubule (MT) with a cell cortex, in which the MT is tethered to the cell by a group of active motor proteins. Such an attachment is a particularly unique case of ligand-receptor bonds: The MT ligand changes its length (and thus binding sites) with time by polymerization-depolymerization kinetics, while multiple motor receptors tend to walk actively along the MT length. These processes, combined with force-mediated unbinding of the motors, result in an elaborate behavior of the MT connection to the cell cortex. A fundamental challenge in this context is to understand how such a preformed attachment maintains its integrity long enough in spite of the ongoing turnover of the MT subunits from its depolymerizing plus end and withstands potentially disruptive effects arising from enhanced rates of detachment of the tethering motors because of external tensions. We present results for the strength and lifetime of the system through the well-established force-clamp and force-ramp protocols when external tension is applied to the MT. The simulation results reveal that the MT-cell attachment behaves as a catch bond or slip bond depending on system parameters. We provide analytical approximations of the lifetime and discuss implications of our results on in vitro experiments.

The distributions of the times to the most recent common ancestor$t_{\mathrm{mrca}}$ is numerically studied for an ecological population model, the extended Moran model. This model has a fixed population size $N$. The number of descendants is drawn from a beta distribution $\mathrm{Beta}(\alpha ,2-\alpha )$ for various choices of $\alpha$. This includes also the classical Moran model ($\alpha \to 0$) as well as the uniform distribution ($\alpha =1$). Using a statistical mechanics-based large-deviation approach, the distributions can be studied over an extended range of the support, down to probabilities like ${10}^{-70}$, which allowed us to study the change of the tails of the distribution when varying the value of $\alpha \in [0,2]$. We find exponential distributions $p\left(t_{\mathrm{mrca}}\right)\sim {\delta }^{t_{\mathrm{mrca}}}$ in all cases, with systematically varying values for the base $\delta$. Only for the cases $\alpha =0$ and $\alpha =1$, analytical results are known, i.e., $\delta =exp(-2/N^2)$ and $\delta =2/3$, respectively. We recover these values, confirming the validity of our approach. Finally, we also study the correlations between $t_{\mathrm{mrca}}$ and the number of descendants.

Motivated by experimental observations, we look for the diffusion of Brownian particles in a medium where they can be either trapped in randomly disposed deep traps or where they diffuse by the regular Fick's law outside of the traps. This process can be represented by two coupled equations—one valid inside the traps and another one outside —yielding the probability distribution of the distance run as a function of time. This probability depends on a unique dimensionless parameter $\lambda$ which is proportional to the product of the (small) density of the traps times the long time of staying in the traps. The mean-square displacement is proportional to the lag time, and for finite or large $\lambda$ the probability density is no longer Gaussian but exponential on intermediate time scales.

A simple two-dimensional (2D) model of the Feynman-Smoluchowski ratchet is studied. Motion of the wheel, driven by stochastic hits of the surrounding molecules, is described as diffusion along the longitudinal coordinate $x$; the stochastic motion of the pawl is represented in the transverse coordinate $y$. Different temperatures of the reservoirs connected to the particular degrees of freedom, together with asymmetry of the energetic landscape of the system, give rise to the ratchet effect. We apply mapping of the corresponding 2D Fokker-Planck equation onto the longitudinal coordinate $x$. The mapped 1D equation is of the generalized Fick-Jacobs type with an effective potential containing a part increasing or decreasing with $x$, connected with vorticity of the scaled driving force and the stationary probability current. It is responsible for the final rectified motion in the longitudinal direction.

From Kawasaki-exchange Monte Carlo simulations of the $q$-state Potts model, we present results for the kinetics of phase separation in multicomponent mixtures, for $q\le 10$, in space dimension $d=2$. A particular focus has been on the quantification of finite-size scaling functions for various values of $q$ and quench depths. For a range of final quench temperatures, our analyses, via finite-size scaling and other state-of-the-art methods, show that the growth follows the Lifshitz-Slyozov behavior, expected for a diffusive mechanism, irrespective of the number of components. We show that the growth for different $q$ values and quench temperatures, in finite systems, can be described by a universal scaling function with a nonuniversal metric factor, originating from the differences in the amplitudes. We also demonstrate the morphological and kinetic equivalence between a $q$-component equal proportion mixture and an off-critical binary mixture, in the framework of the Ising model, with relative concentration of the minority component in the latter being $x_c=1/q$.