The thermodynamic properties of disks moving in a channel sufficiently narrow that they can collide only with their nearest neighbors can be solved exactly by determining the eigenvalues and eigenfunctions of an integral equation. Using it, we have determined the correlation length $\xi$ of this system. We have developed an approximate solution which becomes exact in the high-density limit. It describes the system in terms of defects in the regular zigzag arrangement of disks found in the high-density limit. The correlation length is then effectively the spacing between the defects. The time scales for defect creation and annihilation are determined with the help of transition-state theory, as is the diffusion coefficient of the defects, and these results are found to be in good agreement with molecular dynamics simulations. On compressing the system with the Lubachevsky-Stillinger procedure, jammed states are obtained whose packing fractions ${\varphi }_J$ are a function of the compression rate $\gamma$. We find a quantitative explanation of this dependence by making use of the Kibble-Zurek hypothesis. We have also determined the point-to-set length scale ${\xi }_{PS}$ for this system. At a packing fraction $\varphi$ close to its largest value ${\varphi }_{\text{max}}$, ${\xi }_{PS}$ has a simple power law divergence, ${\xi }_{PS}\sim 1/(1-\varphi /{\varphi }_{\text{max}})$, while $\xi$ diverges much faster, $ln\left(\xi \right)\sim 1/(1-\varphi /{\varphi }_{\text{max}})$.

This article explores the applicability of concepts from nonequilibrium thermodynamics to rare event prediction. The Crooks fluctuation theorem is an equality constraint on the probability distribution of a thermodynamical observable. We consider as a prediction target the exceedance of a threshold of such an observable, where the magnitude of the threshold modulates the rareness of the event. A probability forecast is constructed for this event based on a small observational data set. A simple method is proposed that exploits the Crooks fluctuation theorem to estimate this probability. It is shown that this estimator has improved predictive skill compared to the relative frequency of exceedance in the data set. We quantify this improvement in two examples, and study its dependence on the threshold magnitude and sample size in different systems. Further improvements are achieved by combining the Crooks estimator with the frequency estimator.

When conducting bonds are occupied randomly in a two-dimensional square lattice, the conductivity of the system increases continuously as the density of those conducting bonds exceeds the percolation threshold. Such a behavior is well known in percolation theory; however, the conductivity behavior has not been studied yet when the percolation transition is discontinuous. Here we investigate the conductivity behavior through a discontinuous percolation model evolving under a suppressive external bias. Using effective medium theory, we analytically calculate the conductivity behavior as a function of the density of conducting bonds. The conductivity function exhibits a crossover behavior from a drastically to a smoothly increasing function beyond the percolation threshold in the thermodynamic limit. The analytic expression fits well our simulation data.

Generalized measurements of an observable performed on a quantum system during a force protocol are investigated and conditions that guarantee the validity of the Jarzynski equality and the Crooks relation are formulated. In agreement with previous studies by M. Campisi, P. Talkner, and P. Hänggi [Phys. Rev. Lett. 105, 140601 (2010); Phys. Rev. E 83, 041114 (2011)], we find that these fluctuation relations are satisfied for projective measurements; however, for generalized measurements special conditions on the operators determining the measurements need to be met. For the Jarzynski equality to hold, the measurement operators of the forward protocol must be normalized in a particular way. The Crooks relation additionally entails that the backward and forward measurement operators depend on each other. Yet, quite some freedom is left as to how the two sets of operators are interrelated. This ambiguity is removed if one considers selective measurements, which are specified by a joint probability density function of work and measurement results of the considered observable. We find that the respective forward and backward joint probabilities satisfy the Crooks relation only if the measurement operators of the forward and backward protocols are the time-reversed adjoints of each other. In this case, the work probability density function conditioned on the measurement result satisfies a modified Crooks relation. The modification appears as a protocol-dependent factor that can be expressed by the information gained by the measurements during the forward and backward protocols. Finally, detailed fluctuation theorems with an arbitrary number of intervening measurements are obtained.

We present a force-displacement contact model for the compressive loading of elastoplastic spheres. This model builds from the well known Hertz contact law for elastic, quasistatic compression to incorporate a material's strain-rate-dependent plasticity in order to describe collisions between particles. In the quasistatic regime, finite-element analysis is used to derive an empirical function of the material properties. A Johnson-Cook strain rate dependence is then included into the model to study dynamic effects. We validate the model using split Hopkinson bar experiments and show that the model can accurately simulate the force-displacement response of strain-rate-dependent elastoplastic spheres during dynamic compression and unloading.

Many real-world complex systems consist of a set of elementary units connected by relationships of different kinds. All such systems are better described in terms of multiplex networks, where the links at each layer represent a different type of interaction between the same set of nodes rather than in terms of (single-layer) networks. In this paper we present a general framework to describe and study multiplex networks, whose links are either unweighted or weighted. In particular, we propose a series of measures to characterize the multiplexicity of the systems in terms of (i) basic node and link properties such as the node degree, and the edge overlap and reinforcement, (ii) local properties such as the clustering coefficient and the transitivity, and (iii) global properties related to the navigability of the multiplex across the different layers. The measures we introduce are validated on a genuinely multiplex data set of Indonesian terrorists, where information among 78 individuals are recorded with respect to mutual trust, common operations, exchanged communications, and business relationships.

In this paper, the electrohydrodynamic stability in an annular liquid layer with a radial electrical conductivity gradient is investigated. A weak shear flow arises from a constant pressure gradient in the axial direction. In the radial direction, an electric field is applied. The three-dimensional linear instability analysis is implemented to study the influence of the inner radius, electrical conductivity gradient, shear flow, and ionic diffusion on the dynamics of the fluid layer. It is found that the critical unstable mode may either be oscillatory or stationary. The system becomes more unstable as the dimensionless inner radius $a$ increases. When the inner radius $a$ is small, the critical unstable mode is stationary, while it is given by three-dimensional oblique waves when $a$ is large. When the conductivity gradient is small, the critical unstable mode is the three-dimensional oblique wave, while when the conductivity gradient is large, it would switch to the stationary mode rather than the oscillatory mode. The system becomes more unstable when the Reynolds number is slightly increased from zero. Additionally, it is found that the electrical Schmidt number has dual effects. The liquid layer becomes either more unstable or stable as the electric Schmidt number increases.

The excitation of atomic gases by short high-intensity optical pulses leads to significant electron ionization. In dilute systems, the generated distribution of ionized electrons is highly anisotropic, reflecting the quantum mechanical properties of the atomic states involved in the many photon transitions. For higher atomic densities, the Coulomb interaction in the electron-ion system leads to the development of an isotropic electron plasma. To study the ionization process in the presence of the many-body interaction, a fully microscopic model is developed that combines a generalized version of the optical Bloch equations describing the optical excitation with a microscopic description of the many-body interactions. The numerical evaluation shows that the Coulomb interaction significantly modifies the distribution anisotropy already during the excitation process. Whereas a reduced anisotropy is still present after the pulse for low ionization degrees and pressures, it is completely absent for elevated gas densities. An ionization degree is predicted that is significantly enhanced by the many-body interactions.

Oscillatory gas flows are commonly generated by micro- and nanoelectromechanical systems. Due to their small size and high operating frequencies, these devices often produce noncontinuum gas flows. Theoretical analysis of such flows requires solution of the unsteady Boltzmann equation, which can present a formidable challenge. In this article, we explore the applicability of the lattice Boltzmann (LB) method to such linear oscillatory noncontinuum flows; this method is derived from the linearized Boltzmann Bhatnagar-Gross-Krook (BGK) equation. We formulate four linearized LB models in the frequency domain, based on Gaussian-Hermite quadratures of different algebraic precision (AP). The performance of each model is assessed by comparison to high-accuracy numerical solutions to the linearized Boltzmann-BGK equation for oscillatory Couette flow. The numerical results demonstrate that high even-order LB models provide superior performance over the greatest noncontinuum range. Our results also highlight intrinsic deficiencies in the current LB framework, which is incapable of capturing noncontinuum behavior at high oscillation frequencies, regardless of quadrature AP and the Knudsen number.

Currently the Boltzmann equation and its model equations are widely used in numerical predictions for dilute gas flows. The nonlinear integro-differential Boltzmann equation is the fundamental equation in the kinetic theory of dilute monatomic gases. By replacing the nonlinear fivefold collision integral term by a nonlinear relaxation term, its model equations such as the famous Bhatnagar-Gross-Krook (BGK) equation are mathematically simple. Since the computational cost of solving model equations is much less than that of solving the full Boltzmann equation, the model equations are widely used in predicting rarefied flows, multiphase flows, chemical flows, and turbulent flows although their predictions are only qualitatively right for highly nonequilibrium flows in transitional regime. In this paper the differences between the Boltzmann equation and its model equations are investigated aiming at giving guidelines for the further development of kinetic models. By comparing the Boltzmann equation and its model equations using test cases with different nonequilibrium types, two factors (the information held by nonequilibrium moments and the different relaxation rates of high- and low-speed molecules) are found useful for adjusting the behaviors of modeled collision terms in kinetic regime. The usefulness of these two factors are confirmed by a generalized model collision term derived from a mathematical relation between the Boltzmann equation and BGK equation that is also derived in this paper. After the analysis of the difference between the Boltzmann equation and the BGK equation, an attempt at approximating the collision term is proposed.

We explore the dynamical large deviations of a lattice heteropolymer model of a protein by means of path sampling of trajectories. We uncover the existence of nonequilibrium dynamical phase transitions in ensembles of trajectories between active and inactive dynamical phases, whose nature depends on the properties of the interaction potential. We consider three potentials: two heterogeneous interaction potentials and a homogeneous Gō potential. When preserving the full heterogeneity of interactions due to a given amino acid sequence, either in a fully interacting model or in a native contacts interacting model (heterogeneous Gō model), the observed dynamic transitions occur between equilibrium highly native states and highly native but kinetically trapped states. A native activity is defined that allows us to distinguish these dynamic phases. In contrast, for the homogeneous Gō model, where all native interaction energies are uniform and the amino acid sequence plays no role, the dynamical transition is a direct consequence of the static bistability between the unfolded and the native state. In the two heterogeneous interaction models the native-active and native-inactive states, despite their thermodynamic similarity, have widely varying dynamical properties, and the transition between them occurs even in lattice proteins whose sequences are designed to make them optimal folders.

We show that the thermodynamic behavior of relativistic ideal Bose gas, recently studied numerically by Grether et al., can be obtained analytically. Using the analytical results, we obtain the critical behavior of the relativistic Bose gas exactly for all the regimes. We show that these analytical results reduce to those of Grether et al. in different regimes of the Bose gas. Furthermore, we also obtain an analytically closed-form expression for the energy density for the Bose gas that is valid in all regimes.

Interest in thin-film fabrication for industrial applications have driven both theoretical and computational aspects of modeling its growth. One of the earliest attempts toward understanding the morphological structure of a film's surface is through a class of solid-on-solid limited-mobility growth models such as the Family, Wolf-Villain, or Das Sarma–Tamborenea models, which have produced fascinating surface roughening behaviors. These models, however, restrict the motion of an incidence atom to be within the neighborhood of its landing site, which renders them inept for simulating long-distance surface diffusion such as that observed in thin-film growth using a molecular-beam epitaxy technique. Naive extension of these models by repeatedly applying the local diffusion rules for each hop to simulate large diffusion length can be computationally very costly when certain statistical aspects are demanded. We present a graph-theoretic approach to simulating a long-range diffusion-attachment growth model. Using the Markovian assumption and given a local diffusion bias, we derive the transition probabilities for a random walker to traverse from one lattice site to the others after a large, possibly infinite, number of steps. Only computation with linear-time complexity is required for the surface morphology calculation without other probabilistic measures. The formalism is applied, as illustrations, to simulate surface growth on a two-dimensional flat substrate and around a screw dislocation under the modified Wolf-Villain diffusion rule. A rectangular spiral ridge is observed in the latter case with a smooth front feature similar to that obtained from simulations using the well-known multiple registration technique. An algorithm for computing the inverse of a class of substochastic matrices is derived as a corollary.

The dynamics of a neural model for hippocampal place cells storing spatial maps is studied. In the absence of external input, depending on the number of cells and on the values of control parameters (number of environments stored, level of neural noise, average level of activity, connectivity of place cells), a “clump” of spatially localized activity can diffuse or remains pinned due to crosstalk between the environments. In the single-environment case, the macroscopic coefficient of diffusion of the clump and its effective mobility are calculated analytically from first principles and corroborated by numerical simulations. In the multienvironment case the heights and the widths of the pinning barriers are analytically characterized with the replica method; diffusion within one map is then in competition with transitions between different maps. Possible mechanisms enhancing mobility are proposed and tested.

The interaction of convection and reaction front propagation is known to exhibit a nonlinear phenomenons in the case of an adverse flow, i.e., when the gas velocity ($\mathbf{V}={[u,v]}^{\prime }$) and the front velocity (${\mathbf{V}}_{\mathbf{f}})$ are of opposite directions. This was demonstrated both experimentally and analytically under isothermal conditions with ${\mathbf{V}}_{\mathbf{f}}$ constant independent of $\mathbf{V}$. Here we analyze thermal reaction front propagation in a porous medium in which an exothermic reaction of Arrhenius kinetics occurs. Numerical simulations of a packed-bed reactor model accounting for variable temperature, concentration, and hydrodynamic fields following the Euler-Darcy equations revealed emergence of spinning transversal patterns. Such solution cannot emerge in a two-variable ($C,T$) model, assuming “frozen” hydrodynamics, within a feasible domain of parameters. Two models are employed for analysis: a qualitative model based on a learning one-tube and two-tube consideration and an extended Landau-Darries instability analysis to account for the momentum losses and for a variable front propagation velocity. Both models revealed the important role of the $V_f\left(u\right)$ dependence which can be presented as $V_f=u-V_{ch}$, where the chemical component of the front velocity ($V_{ch}$) depends on the main governing parameters such as the adiabatic temperature rise ($\Delta T_a$) and the inlet velocity ($u_{in}$). The instability takes place if the parameter $\beta =\partial V_{ch}/\partial u<1$. The effect of $\Delta T_a$, $u_{in}$ on the instability domain obtained in simulations can be translated to the effect of the parameter $\beta$.

We calculate the mean electromotive force in plane Couette flows of a nonrotating conducting fluid under the influence of a large-scale magnetic field for driven turbulence. A vertical stratification of the turbulence intensity results in an $\alpha$ effect owing to the presence of horizontal shear. Here we discuss the possibility of an experimental determination of the components of the $\alpha$ tensor using both quasilinear theory and nonlinear numerical simulations. For magnetic Prandtl numbers of the order of unity, we find that in the high-conductivity limit the $\alpha$ effect in the direction of the flow clearly exceeds the component in spanwise direction. In this limit, $\alpha$ runs linearly with the magnetic Reynolds number $\mathrm{Rm},$ while in the low-conductivity limit it runs with the product $\mathrm{Rm}·\mathrm{Re}$, where $\mathrm{Re}$ is the kinetic Reynolds number, so that for a given $\mathrm{Rm}$ the $\alpha$ effect grows with decreasing magnetic Prandtl number. For the small magnetic Prandtl numbers of liquid metals, a common value for the horizontal elements of the $\alpha$ tensor appears, which makes it unimportant whether the $\alpha$ effect is measured in the spanwise or the streamwise directions. The resulting effect should lead to an observable voltage of about 0.5 mV in both directions for magnetic fields of 1 kG and velocity fluctuations of about 1 m/s in a channel of 50-cm height (independent of its width).

A wide range of networked systems exhibit highly connected nodes (hubs) as prominent structural elements. The functional roles of hubs in the collective nonlinear dynamics of many such networks, however, are not well understood. Here, we propose that hubs in neural circuits may activate local signal transmission along sequences of specific subnetworks. Intriguingly, in contrast to previous suggestions of the functional roles of hubs, here, not the hubs themselves, but nonhub subnetworks transfer the signals. The core mechanism relies on hubs and nonhubs providing activating feedback to each other. It may, thus, induce the propagation of specific pulse and rate signals in neuronal and other communication networks.

The $q$-state Potts model with long-range correlated disorder is studied by means of large-scale Monte Carlo simulations for $q=2$, 4, 8, and 16. Evidence is given of the existence of a Griffiths phase, where the thermodynamic quantities display an algebraic finite-size scaling, in a finite range of temperatures. The critical exponents are shown to depend on both the temperature and the exponent of the algebraic decay of disorder correlations, but not on the number of states of the Potts model. The mechanism leading to the violation of hyperscaling relations is observed in the entire Griffiths phase.

We study the joint probability density of the eigenvalues of a product of rectangular real, complex, or quaternion random matrices in a unified way. The random matrices are distributed according to arbitrary probability densities, whose only restriction is the invariance under left and right multiplication by orthogonal, unitary, or unitary symplectic matrices, respectively. We show that a product of rectangular matrices is statistically equivalent to a product of square matrices. Hereby we prove a weak commutation relation of the random matrices at finite matrix sizes, which previously has been discussed for infinite matrix size. Moreover, we derive the joint probability densities of the eigenvalues. To illustrate our results, we apply them to a product of random matrices drawn from Ginibre ensembles and Jacobi ensembles as well as a mixed version thereof. For these weights, we show that the product of complex random matrices yields a determinantal point process, while the real and quaternion matrix ensembles correspond to Pfaffian point processes. Our results are visualized by numerical simulations. Furthermore, we present an application to a transport on a closed, disordered chain coupled to a particle bath.