This paper exhibits a Monte Carlo study on site percolation using the Newmann-Ziff algorithm in distorted square and simple cubic lattices where each site is allowed to be directly linked with any other site if the Euclidean separation between the pair is at most a certain distance $d$, called the connection threshold. Distorted lattices are formed from regular lattices by a random but controlled dislocation of the sites with the help of a parameter $\alpha$, called the distortion parameter. The distinctive feature of this study is the relaxation of the restriction of forming bonds with only the nearest neighbors. Owing to this flexibility and the intricate interplay between the two parameters $\alpha$ and $d$, the site percolation threshold may either increase or decrease with distortion. The dependence of the percolation threshold on the average degree of a site has been explored to show that the obtained results are consistent with those on percolation in regular lattices with an extended neighborhood and continuum percolation.

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing semirigid rods from a $L\times L$ square lattice that contains two layers (and $M=L\times L\times 2$ sites). The process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by removing groups of $k$ consecutive monomers according to a generalized random sequential adsorption mechanism. The study is conducted by following the behavior of two critical concentrations with size $k$: (1) jamming coverage ${\theta }_{j,k}$, which represents the concentration of occupied sites at which the jamming state is reached, and (2) inverse percolation threshold ${\theta }_{c,k}$, which corresponds to the maximum concentration of occupied sites for which connectivity disappears. The obtained results indicate that (1) the jamming coverage exhibits an increasing dependence on the size $k$—it rapidly increases for small values of $k$ and asymptotically converges towards a definite value for infinitely large $k$ sizes ${\theta }_{j,k\to \infty }\approx 0.2701$—and (2) the inverse percolation threshold is a decreasing function of $k$ in the range $1\le k\le 17$. For $k\ge 18$, all jammed configurations are percolating states (the lattice remains connected even when the highest allowed concentration of removed sites is reached) and, consequently, there is no nonpercolating phase. This finding contrasts with the results obtained in literature for a complementary problem, where straight rigid $k$-mers are randomly and irreversibly deposited on a square lattice forming two layers. In this case, percolating and nonpercolating phases extend to infinity in the space of the parameter $k$ and the model presents percolation transition for the whole range of $k$. The results obtained in the present study were also compared with those reported for the case of inverse percolation by removal of rigid linear $k$-mers from a square monolayer. The differences observed between monolayer and bilayer problems were discussed in terms of vulnerability and network robustness. Finally, the accurate determination of the critical exponents $\nu$, $\beta$, and $\gamma$ reveals that the percolation phase transition involved in the system has the same universality class as the standard percolation problem.

The efficient detection of chaotic behavior in orbits of a complex dynamical system is an active domain of research. Several indicators have been proposed, and new ones have recently been developed in view of improving the performance of chaos detection by means of numerical simulations. The challenge is to predict chaotic behavior based on the analysis of orbits of limited length. In this paper the performance analysis of past and recent indicators of chaos, in terms of predictive power, is carried out in detail using the dynamical system characterized by a symplectic Hénon-like cubic polynomial map.

The present study investigates the arrangement of hollow pyramidal cone shells and their interactions with degenerate planar anchoring on the inner and outer surfaces of particles within the nematic host. The shell thickness is in order of the nematic coherence length. The numerical behavior of colloids is determined by minimizing the Landau-de Gennes free energy in the presence of the Fournier surface energy and using the finite element method. Colloidal pyramidal cones can orient parallel and perpendicular with the far director orientation. In the parallel alignment, we found the splay director distortion into the pyramid with two boojum defects at the inner and outer tips. The director shows bending distortion without defect patterns when the pyramid is aligned perpendicularly. They induce long-range dipolar interaction and can form nested structures in close contact.

We consider the steady-state fingering instability of an elastic membrane separating two fluids of different density under external pressure in a rotating Hele-Shaw cell. Both inextensible and highly extensible membranes are considered, and the role of membrane tension is detailed in each case. Both systems exhibit a centrifugally driven Rayleigh-Taylor–like instability when the density of the inner fluid exceeds that of the outer one, and this instability competes with the restoring forces arising from curvature and tension, thereby setting the finger scale. Numerical continuation is used to compute not only strongly nonlinear primary finger states up to the point of self-contact, but also secondary branches of mixed modes and circumferentially localized folds as a function of the rotation rate and the externally imposed pressure. Both reflection-symmetric and symmetry-broken chiral states are computed. The results are presented in the form of bifurcation diagrams. The ratio of system scale to the natural length scale is found to determine the ordering of the primary bifurcations from the unperturbed circle state as well as the solution profiles and onset of secondary bifurcations.

Using three-dimensional (3D) magnetohydrodynamic simulations, we study how a pit on a metal surface evolves when driven by intense electrical current density $\mathbf{j}$. Redistribution of $\mathbf{j}$ around the pit initiates a feedback loop: $\mathbf{j}$ both reacts to and alters the electrical conductivity $\sigma$, through Joule heating and hydrodynamic expansion, so that $\mathbf{j}$ and $\sigma$ are constantly in flux. Thus, the pit transforms into larger striation and filament structures predicted by the electrothermal instability theory. Both structures are important in applications of current-driven metal: The striation constitutes a density perturbation that can seed the magneto-Rayleigh-Taylor instability, while the filament provides a more rapid path to plasma formation, through 3D j redistribution. Simulations predict distinctive self-emission patterns, thus allowing for experimental observation and comparison.

Unsupervised machine learning applied to the study of phase transitions is an ongoing and interesting research direction. The active contour model, also called the snake model, was initially proposed for target contour extraction in two-dimensional images. In order to obtain a physical phase diagram, the snake model with an artificial neural network is applied in an unsupervised learning way by the authors of [Phys. Rev. Lett. 120, 176401 (2018)]. It guesses the phase boundary as an initial snake and then drives the snake to convergence with forces estimated by the artificial neural network. In this work we extend this unsupervised learning method with one contour to a snake net with multiple contours for the purpose of obtaining several phase boundaries in a phase diagram. For the classical Blume-Capel model, the phase diagram containing three and four phases is obtained. Moreover, a balloon force is introduced, which helps the snake to leave a wrong initial position and thus may allow for greater freedom in the initialization of the snake. Our method is helpful in determining the phase diagram with multiple phases using just snapshots of configurations from cold atoms or other experiments without knowledge of the phases.

The process of frosting is a multiscale problem, which leads to challenges of proposing accurate numerical methods. In this study, a lattice Boltzmann model for predicting frost formation and growth on surfaces of various wettabilities is proposed based on the heterogeneous nucleation and dendrite growth theories. Three lattice Boltzmann equations are used to calculate the velocity, humidity, and temperature distributions. Furthermore, the heterogeneous nucleation theory and dendrite growth theory are used to construct the equations that govern ice production during the frosting process, so that the surface wettability can be considered. After experimental validation, the model was used in the analysis of frosting behaviors on plates and in microchannels with different wettabilities. The effects of the intrinsic contact angles and roughness on the frost layer properties were evaluated. This study will likely facilitate a better understanding of frosting on the mesoscopic level.

We solve the time-dependent Fokker-Planck equation for a two-dimensional active Brownian particle exploring a circular region with an absorbing boundary. Using the passive Brownian particle as basis states and dealing with the activity as a perturbation, we provide a matrix representation of the Fokker-Planck operator and we express the propagator in terms of the perturbed eigenvalues and eigenfunctions. Alternatively, we show that the propagator can be expressed as a combination of the equilibrium eigenstates with weights depending only on time and on the initial conditions, and obeying exact iterative relations. Our solution allows also obtaining the survival probability and the first-passage time distribution. These latter quantities exhibit peculiarities induced by the nonequilibrium character of the dynamics; in particular, they display a strong dependence on the activity of the particle and, to a less extent, also on its rotational diffusivity.

Heat rectifiers would facilitate energy management operations such as cooling or energy harvesting, but devices of practical interest are still missing. Understanding heat rectification at a fundamental level is key to helping us find or design such devices. The match or mismatch of the phonon band spectrum of device segments for forward or reverse temperature bias of the thermal baths at device boundaries was proposed as the mechanism behind rectification. However, no explicit theoretical relation derived from first principles had been found so far between heat fluxes and spectral matching. We study heat rectification in a minimalistic chain of two coupled ions. The fluxes and rectification can be calculated analytically. We propose a definition of the matching that sets an upper bound for the heat flux. In a regime where the device rectifies optimally, matching and flux ratios for forward and reverse configurations are found to be proportional. The results can be extended to a system of $N$ particles in arbitrary traps with nearest-neighbor linear interactions.

A quantum thermal diode is designed based on three pairwise coupled qubits, two connected to a common reservoir and the other to an independent reservoir. It is found that the internal couplings between qubits can enhance heat currents. If the two identical qubits uniformly couple with the common reservoir, the crossing dissipation will occur, leading to the initial-state-dependent steady state, which can be decomposed into the mixture of two particular steady states: the heat-conducting state generating maximum heat current and the heat-resisting state not transporting heat. However, the rectification factor doesn't depend on the initial state. In particular, we find that neither quantum entanglement nor quantum discord is present in the steady state, but the pure classical correlation shows a remarkably consistent behavior as the heat rectification factor, which reveals the vital role of classical correlation in the system.

Through extensive molecular simulations we determine a phase diagram of attractive, fully flexible polymer chains in two and three dimensions. A rich collection of distinct crystal morphologies appear, which can be finely tuned through the range of attraction. In three dimensions these include the face-centered cubic, hexagonal close packed, simple hexagonal, and body-centered cubic crystals and the Frank-Kasper phase. In two dimensions the dominant structures are the triangular and square crystals. A simple geometric model is proposed, based on the concept of cumulative neighbors of ideal crystals, which can accurately predict most of the observed structures and the corresponding transitions. The attraction range can thus be considered as an adjustable parameter for the design of colloidal polymer crystals with tailored morphologies.

The system of driven dense colloid mixtures is studied in one-, two-, and three-dimensional geometries. We calculate the diffusion coefficients and mobilities for each particle type, including cross-terms, in a hydrodynamic limit, using a mean-field-type approximation. The set of nonlinear diffusion equations are then solved. In one dimension, analytical results are possible. We show that in mixtures, the “Brazil nut” phenomenon, or depletion of larger particles by force of smaller ones, appears quite generically. We calculate the ratchet current and quantify the capability of sorting particles according to their size. We also indicate that the “Brazil nut” effect lies behind the possibility of perfect separation, where large and big particles travel in strictly opposite direction.

We present analytical and numerical investigations of energy propagation in systems of massive particles that interact via harmonic (linear) forces. The particle motion is described by a scalar displacement, and the particles are arranged in a simple crystal lattice. For the systems under consideration we prove the conservation of the total energy flux analytically. Then, using a sample case of a square lattice, we confirm the analytical results numerically. We create disturbances of a special kind which can move with a predefined velocity with a minor change in their shape. We show that a clot of energy, associated with each disturbance, moves similarly to a free body of matter in classical mechanics. We also numerically study a simultaneous propagation of a number of energy clots as an analogy to the motion of point masses in the conventional mechanics of particles. The obtained results demonstrate that an energy flow in lattices can be described in terms of numerous separated energy bodies, making a step towards a linkage between lattice dynamics and the kinetic theory of heat transfer in solids.

Cell mechanosensing is implicated in the control of a broad range of cell behaviors, with cytoskeletal contractility a key component. Experimentally, it is observed that the contractility of the cell responds to increasing substrate stiffness, showing increased contractile force and changing the distribution of cytoskeletal elements. Here, we show using a theoretical model of active cell contractility that upregulation of contractility need not be energetically expensive, especially when combined with changes in adhesion and contractile distribution. Indeed, we show that a feedback mechanism based on the maintenance of strain energy would require an upregulation in contractile pressure on all but the softest substrates. We consider both the commonly reported substrate strain energy and active work done. We demonstrate substrate strain energy would preferentially select for the experimentally observed clustering of cell adhesions on stiffer substrates which effectively soften the substrate and enable an upregulation of total contractile pressure, while the localization of contractility has the greatest impact on the internal work.

Renormalization enables a systematic scale-by-scale analysis of multiscale systems. In this paper, we employ renormalization group (RG) to the shell model of turbulence and show that the RG equation is satisfied by $|u_n{|}^2=K_{\mathrm{Ko}}{\epsilon }^{2/3}{k}_n^{-2/3}$, where Ko is the Kolmogorov constant and ${\nu }_n={\nu }_{*}\sqrt{K_{\mathrm{Ko}}}{\epsilon }^{1/3}{k}_n^{-4/3}$, where $k_n\mathrm{and}{u}_n$ are the wave number and velocity of shell $n$; ${\nu }_{*}\mathrm{and}{K}_{\mathrm{Ko}}$ are RG and Kolmogorov's constants; and $\epsilon$ is the energy dissipation rate. We find that ${\nu }_{*}\approx 0.5$ and $K_{\mathrm{Ko}}\approx 1.7$, consistent with earlier RG works on the Navier-Stokes equation. We verify the theoretical predictions using numerical simulations.

To which degree the average entanglement entropy of midspectrum eigenstates of quantum-chaotic interacting Hamiltonians agrees with that of random pure states is a question that has attracted considerable attention in the recent years. While there is substantial evidence that the leading (volume-law) terms are identical, which and how subleading terms differ between them is less clear. Here we carry out state-of-the-art full exact diagonalization calculations of clean spin-1/2 XYZ and XXZ chains with integrability breaking terms to address this question in the absence and presence of $U\left(1\right)$ symmetry, respectively. We first introduce the notion of maximally chaotic regime, for the chain sizes amenable to full exact diagonalization calculations, as the regime in Hamiltonian parameters in which the level spacing ratio, the distribution of eigenstate coefficients, and the entanglement entropy are closest to the random matrix theory predictions. In this regime, we carry out a finite-size scaling analysis of the subleading terms of the average entanglement entropy of midspectrum eigenstates when different fractions $\nu$ of the spectrum are included in the average. We find indications that, for $\nu \to 0$, the magnitude of the negative $O\left(1\right)$ correction is only slightly greater than the one predicted for random pure states. For finite $\nu$, following a phenomenological approach, we derive a simple expression that describes the numerically observed $\nu$ dependence of the $O\left(1\right)$ deviation from the prediction for random pure states.

Encounter-based models of diffusion provide a probabilistic framework for analyzing the effects of a partially absorbing reactive surface, in which the probability of absorption depends upon the amount of surface-particle contact time. In this paper we develop a class of encounter-based models that deal with absorption at sticky boundaries. Sticky boundaries occur in a diverse range of applications, including cell biology, colloidal physics, finance, and human crowd dynamics. They also naturally arise in active matter, where confined active particles tend to spontaneously accumulate at boundaries even in the absence of any particle-particle interactions. We begin by constructing a one-dimensional encounter-based model of sticky Brownian motion (BM), which is based on the zero-range limit of nonsticky BM with a short-range attractive potential well near the origin. In this limit, the boundary-contact time is given by the amount of time (occupation time) that the particle spends at the origin. We calculate the joint probability density or propagator for the particle position and the occupation time, and then identify an absorption event as the first time that the occupation time crosses a randomly generated threshold. We illustrate the theory by considering diffusion in a finite interval with a partially absorbing sticky boundary at one end. We show how various quantities, such as the mean first passage time (MFPT) for single-particle absorption and the relaxation to steady state at the multiparticle level, depend on moments of the random threshold distribution. Finally, we determine how sticky BM can be obtained by taking a particular diffusion limit of a sticky run-and-tumble particle (RTP).