Percolation models shed a light on network integrity and functionality and have numerous applications in network theory. This paper studies a targeted percolation ($\alpha$ model) with incomplete knowledge where the highest degree node in a randomly selected set of $n$ nodes is removed at each step, and the model features a tunable probability that the removed node is instead a random one. A “mirror image” process ($\beta$ model) in which the target is the lowest degree node is also investigated. We analytically calculate the giant component size, the critical occupation probability, and the scaling law for the percolation threshold with respect to the knowledge level $n$ under both models. We also derive self-consistency equations to analyze the $k$-core organization including the size of the $k$ core and its corona in the context of attacks under tunable limited knowledge. These percolation models are characterized by some interesting critical phenomena and reveal profound quantitative structure discrepancies between Erdős-Rényi networks and power-law networks.

The internal flow and mixing properties inside deformable droplets, after reaching the steady state within two types of passive droplet traps, are visualized and analyzed as dynamical systems. The first droplet trap (constriction) is formed by three spheres arranged in an equilateral triangle, while the second consists of two parallel spherocylinders (capsules). The systems are assumed to be embedded in a uniform far-field flow at low Reynolds number, and the steady shapes and interfacial velocities on the drops are generated using the boundary-integral method. The internal velocity field is recovered by solving the internal Dirichlet problem, also via a desingularized boundary-integral method. Calculation of 2D streamlines within planes of symmetry reveals the internal equilibria of the flow. The type of each equilibrium is classified in 3D and their interactions probed using passive tracers and their Poincaré maps. For the two-capsule droplet, saddle points located on orthogonal symmetry planes influence the regular flow within the drop. For the three-sphere droplet, large regions of chaos are observed, embedded with simple periodic orbits. Flow is visualized via passive dyes, using material lines and surfaces. In 2D, solely the interface between two passive interior fluids is advected using an adaptive number of linked tracer particles. The reduction in dimension decreases the number of required tracer points, and also resolves arbitrarily thin filaments, in contrast to backward cell-mapping methods. In 3D, the advection of a material surface, bounded by the droplet interface, is enabled using an adaptive mesh scheme. Off-lattice 3D contour advection allows for highly resolved visualizations of the internal flow and quantification of the associated degree of mixing. Analysis of the time-dependent growth of material surfaces and 3D mixing numbers suggests the three-sphere droplet exhibits superior mixing properties compared to the two-capsule droplet.

We point out that the minimal components of the tensorial moments of the distribution that can be independently relaxed in collision without violating rotational symmetry are its irreducible representation (irrep) of SO(3), and a generic multiple-relaxation-time collision model can be constructed by independently relaxing these components. As the simplest example the irreps of the second moment are the traceless deviatoric stress and an isotropic tensor which is conserved in monatomic gases. Applying the decomposition to the thermal lattice Boltzmann model for polyatomic gases [Phys. Rev. E 77, 035701(R) (2008)], the shear and bulk viscosities are decoupled by two independent relaxation times. The hydrodynamic equation of the model is obtained via Chapman-Enskog calculation and verified by numerical simulation.

A class of explicit numerical schemes is developed to solve for the relativistic dynamics and spin of particles in electromagnetic fields, using the Lorentz–Bargmann-Michel-Telegdi equation formulated in the Clifford algebra representation of Baylis. It is demonstrated that these numerical methods, reminiscent of the leapfrog and Verlet methods, share a number of important properties: they are energy conserving, volume conserving, and second-order convergent. These properties are analyzed empirically by benchmarking against known analytical solutions in constant uniform electrodynamic fields. It is demonstrated that the numerical error in a constant magnetic field remains bounded for long-time simulations in contrast to the Boris pusher, whose angular error increases linearly with time. Finally, the intricate spin dynamics of a particle is investigated in a plane-wave field configuration.

Using hydrodynamic simulations, we study the single polymers flowing through model porous media (close-packed colloidal crystal). In good solvent or high flow rates, the polymer transport is similar to gel electrophoresis, with size-dependent sieving for $L_c/L\lesssim 1$ and size-independent biased reptation for $L_c/L\gtrsim 1$ ($L_c$ is the polymer contour length and $L$ is the diameter of colloids forming the porous media). Importantly, in bad solvent and low flow rates, the polymers show an extra window of size-dependent velocity for $1\lesssim L_c/L\lesssim 2$, where the polymer transport is controlled by a globule-stretch transition at pore throats, and the transport velocity is much slower than reptation.

We consider several limiting cases of the joint probability distribution for a random matrix ensemble with an additional interaction term controlled by an exponent $\gamma$ (called the $\gamma$ ensembles). The effective potential, which is essentially the single-particle confining potential for an equivalent ensemble with $\gamma =1$ (called the Muttalib-Borodin ensemble), is a crucial quantity defined in solution to the Riemann-Hilbert problem associated with the $\gamma$ ensembles. It enables us to numerically compute the eigenvalue density of $\gamma$ ensembles for all $\gamma >0$. We show that one important effect of the two-particle interaction parameter $\gamma$ is to generate or enhance the nonmonotonicity in the effective single-particle potential. For suitable choices of the initial single-particle potentials, reducing $\gamma$ can lead to a large nonmonotonicity in the effective potential, which in turn leads to significant changes in the density of eigenvalues. For a disordered conductor, this corresponds to a systematic decrease in the conductance with increasing disorder. This suggests that appropriate models of $\gamma$ ensembles can be used as a possible framework to study the effects of disorder on the distribution of conductances.

Electric circuits influenced by thermal noise are analogous to confined Brownian particles and can be an alternative and convenient scheme for studying stochastic thermodynamics. Here we experimentally demonstrate an effective technique of generating tunable potentials for Brownian dynamics in an electric circuit, realized by external controlled feedback. We present two illustrative examples of one-dimensional virtual potentials: static harmonic potential and time-varying double-well potential. The thermal noises of both cases undergo equivalent Brownian dynamics as if they were in the authentic potentials as long as the feedback is fast enough to respond to the designed potentials. The results show that the electric circuit provides a simple, effective, and programmable scheme to study the feedback-controlled virtual potential.

The Peierls-Nabarro barrier is a discrete effect that frequently occurs in discrete nonlinear systems. A signature of the barrier is the slowing and eventual stopping of discrete solitary waves. This work examines intense electromagnetic waves propagating through a periodic honeycomb lattice of helically driven waveguides, which serves as a paradigmatic Floquet topological insulator. Here it is shown that discrete topologically protected edge modes do not suffer from the typical slowdown associated with the Peierls-Nabarro barrier. Instead, as a result of their topological nature, the modes always move forward and redistribute their energy: a narrow (discrete) mode transforms into a wide effectively continuous mode. On the other hand, a discrete edge mode that is not topologically protected does eventually slow down and stop propagating. Topological modes that are initially narrow naturally tend to wide envelope states that are described by a generalized nonlinear Schrödinger equation. These results provide insight into the nature of nonlinear topological insulators and their application.

Statistical mechanics is an important tool for understanding polymer electroelasticity because the elasticity of polymers is primarily due to entropy. However, a common approach for the statistical mechanics of polymer chains, the Gaussian chain approximation, misses key physics. By considering the nonlinearities of the problem, we show a strong coupling between the deformation of a polymer chain and its dielectric response, that is, its net dipole. When chains with this coupling are cross linked in an elastomer network and an electric field is applied, the field breaks the symmetry of the elastomer's elastic properties and, combined with electrostatic torque and incompressibility, leads to intrinsic electrostriction. Conversely, deformation can break the symmetry of the dielectric response, leading to volumetric torque and asymmetric actuation. Both phenomena have important implications for designing high-efficiency soft actuators and soft electroactive materials, and the presence of mechanisms for volumetric torque, in particular, can be used to develop higher degree of freedom actuators and to achieve bioinspired locomotion.

We discuss the linear hydrodynamic response of a two-dimensional active chiral compressible fluid with odd viscosity. The viscosity coefficient represents broken time-reversal and parity symmetries in the 2D fluid and characterizes the deviation of the system from a passive fluid. Taking into account the hydrodynamic coupling to the underlying bulk fluid, we obtain the odd viscosity-dependent mobility tensor, which is responsible for the nonreciprocal hydrodynamic response to a point force. Furthermore, we consider a finite-size disk moving laterally in the 2D fluid and demonstrate that the disk experiences a nondissipative lift force in addition to the dissipative drag one.

We calculate the bulk-diffusion coefficient and the conductivity in nonequilibrium conserved-mass aggregation processes on a ring. These processes involve chipping and fragmentation of masses, which diffuse on a lattice and aggregate with their neighboring masses on contact, and, under certain conditions, they exhibit a condensation transition. We find that, even in the absence of microscopic time reversibility, the systems satisfy an Einstein relation, which connects the ratio of the conductivity and the bulk-diffusion coefficient to mass fluctuation. Interestingly, when aggregation dominates over chipping, the conductivity or, equivalently, the mobility of masses, is greatly enhanced. The enhancement in the conductivity, in accordance with the Einstein relation, results in large mass fluctuations and can induce a mobility-driven clustering in the systems. Indeed, in a certain parameter regime, we show that the conductivity, along with the mass fluctuation, diverges beyond a critical density, thus characterizing the previously observed nonequilibrium condensation transition [Phys. Rev. Lett. 81, 3691 (1998)] in terms of an instability in the conductivity. Notably, the bulk-diffusion coefficient remains finite in all cases. We find our analytic results in quite good agreement with simulations.

Jamming and percolation transitions in the standard random sequential adsorption of particles on regular lattices are characterized by a universal set of critical exponents. The universality class is preserved even in the presence of randomly distributed defective sites that are forbidden for particle deposition. However, using large-scale Monte Carlo simulations by depositing dimers on the square lattice and employing finite-size scaling, we provide evidence that the system does not exhibit such well-known universal features when the defects have spatial long-range (power-law) correlations. The critical exponents ${\nu }_j$ and $\nu$ associated with the jamming and percolation transitions, respectively, are found to be nonuniversal for strong spatial correlations and approach systematically their own universal values as the correlation strength is decreased. More crucially, we have found a difference in the values of the percolation correlation length exponent $\nu$ for a small but finite density of defects with strong spatial correlations. Furthermore, for a fixed defect density, it is found that the percolation threshold of the system, at which the largest cluster of absorbed dimers first establishes the global connectivity, gets reduced with increasing the strength of the spatial correlation.

Using renormalization group (RG) analyses and Monte Carlo (MC) simulations, we study the fully packed dimer model on the bilayer square lattice with fugacity equal to $z$ (1) for interlayer (intralayer) dimers, and intralayer interaction $V$ between neighboring parallel dimers on any elementary plaquette in either layer. For a range of not-too-large $z>0$ and repulsive interactions $0<V<V_s$ (with $V_s\approx 2.1$), we demonstrate the existence of a bilayer Coulomb phase with purely dipolar two-point functions, i.e., without the power-law columnar order that characterizes the usual Coulomb phase of square and honeycomb lattice dimer models. The transition line $z_c\left(V\right)$ separating this bilayer Coulomb phase from a large-$z$ disordered phase is argued to be in the inverted Kosterlitz-Thouless universality class. Additionally, we argue for the possibility of a tricritical point at which the bilayer Coulomb phase, the large-$z$ disordered phase and the large-$V$ staggered phase meet in the large-$z$, large-$V$ part of the phase diagram. In contrast, for the attractive case with $V_{cb}<V\le 0$ $(V_{cb}\approx -1.2)$, we argue that any $z>0$ destroys the power-law correlations of the $z=0$ decoupled layers, and leads immediately to a short-range correlated state, albeit with a slow crossover for small $\left|V\right|$. For $V_c<V<V_{cb}$ $(V_c\approx -1.55)$, we predict that any small nonzero $z$ immediately gives rise to long-range bilayer columnar order although the $z=0$ decoupled layers remain power-law correlated in this regime; this implies a nonmonotonic $z$ dependence of the columnar order parameter for fixed $V$ in this regime. Further, our RG arguments predict that this bilayer columnar ordered state is separated from the large-$z$ disordered state by a line of Ashkin-Teller transitions $z_{\mathrm{AT}}\left(V\right)$. Finally, for $V<V_c$, the $z=0$ decoupled layers are already characterized by long-range columnar order, and a small nonzero $z$ leads immediately to a locking of the order parameters of the two layers, giving rise to the same bilayer columnar ordered state for small nonzero $z$.

We study analytically how noninteracting weakly active particles, for which passive Brownian diffusion cannot be neglected and activity can be treated perturbatively, distribute and behave near boundaries in various geometries. In particular, we develop a perturbative approach for the model of active particles driven by an exponentially correlated random force (active Ornstein-Uhlenbeck particles). This approach involves a relatively simple expansion of the distribution in powers of the Péclet number and in terms of Hermite polynomials. We use this approach to cleanly formulate boundary conditions, which allows us to study weakly active particles in several geometries: confinement by a single wall or between two walls in 1D, confinement in a circular or wedge-shaped region in 2D, motion near a corrugated boundary, and, finally, absorption onto a sphere. We consider how quantities such as the density, pressure, and flow of the active particles change as we gradually increase the activity away from a purely passive system. These results for the limit of weak activity help us gain insight into how active particles behave in the presence of various types of boundaries.

We report on direct measurements of the basal force components for granular material flowing down a smooth incline. We investigate granular flows for a large range of inclination angles from $\theta =13.4^{\circ }$ to 83.6° and various gate openings of the chute. We find that the effective basal friction coefficient ${\mu }_B$, obtained from the ratio of the longitudinal force to the normal one, exhibits a systematic increase with increasing slope angle and a significant weakening with increasing particle holdup $H$ (the depth-integrated particle volume fraction). At low angles, the basal friction is slightly less than or equal to tanθ. The deviation from tanθ can be interpreted as a contribution from the sidewall to the overall friction. At larger angles, the basal friction ${\mu }_B$ saturates at an asymptotic value that is dependent on the gate opening of the chute. Importantly, our data confirm the outcomes of recent discrete numerical simulations. First, for steady and fully developed flows as well as for moderately accelerated ones, the variation of the basal friction can be captured through a unique dimensionless number, the Froude number Fr, defined as $\mathrm{Fr}=\overline{U}/{\left(gH\cos \theta \right)}^{1/2}$, where $\overline{U}$ is the mean flow velocity. Second, the mean velocity scales with the particle holdup $H$ with a power exponent close to $\frac{1}{4}$, contrasting with the Bagnold scaling $\left(\overline{U}\sim H^{3/2}\right)$.

We consider the evolution of arbitrarily large perturbations of a prescribed pure hydrodynamical flow of an electrically conducting fluid. We study whether the flow perturbations as well as the generated magnetic fields decay or grow with time and constitute a dynamo process. For that purpose we derive a generalized Reynolds-Orr equation for the sum of the kinetic energy of the hydrodynamic perturbation and the magnetic energy. The flow is confined in a finite volume so the normal component of the velocity at the boundary is zero. The tangential component is left arbitrary in contrast with previous works. For the magnetic field we mostly employ the classical boundary conditions where the field extends in the whole space. We establish critical values of hydrodynamic and magnetic Reynolds numbers below which arbitrarily large initial perturbations of the hydrodynamic flow decay. This involves generalization of the Rayleigh-Faber-Krahn inequality for the smallest eigenvalue of an elliptic operator. For high Reynolds number turbulence we provide an estimate of critical magnetic Reynolds number below which arbitrarily large fluctuations of the magnetic field decay.

Evaporation of streams of liquid droplets in environments at vacuum pressures below the vapor pressure has not been widely studied. Here, experiments and simulations are reported that quantify the change in droplet diameter when a steady stream of ≈100 μm diameter drops is injected into a chamber initially evacuated to $<{10}^{–8}\mathrm{bar}$. In experiments, droplets fall through the center of a 0.8 m long liquid nitrogen cooled shroud, simulating infinity radiation and vapor mass flux boundary conditions. Experimentally measured changes in drop diameters vary from ≈0 to 6 μm when the initial vapor pressure is increased from ${10}^{–6}$ to ${10}^{–3}$ bar by heating the liquid. Measured diameter changes are predicted by a model based on the Hertz-Knudsen equation. One uncertainty in the calculation is the “sticking coefficient” β. Assuming a constant β for all conditions studied here, predicted diameter changes best match measurements with $\beta \approx 0.3$. This value falls within the range of β reported in the literature for organic liquids. Finally, at the higher vapor pressure conditions considered here, the drop stream disperses transverse to the main flow direction. This spread is attributed to forces imparted by an absolute pressure gradient produced by the evaporating stream.

This Letter presents a numerical study across parameter space to calculate the aspect ratio (ratio of length to diameter) of a fair “three-sided coin”: a cylinder that when tossed, has equal probabilities of landing heads, tails, or sideways. The results are cast in the context of previous analytical studies, and the various mechanisms that govern the dynamics of coin tossing are compared and contrasted. After more than $7\times {10}^8$ tosses of coins of various aspect ratios, this study finds the critical aspect ratio to be slightly less than (but not exactly equal to) $\sqrt{3}/2\approx 0.866$.

Considering viscous friction that varies spatially and temporally, the general expressions for entropy production, free energy, and entropy extraction rates are derived to a Brownian particle that walks in overdamped and underdamped media. Via the well known stochastic approaches to underdamped and overdamped media, the thermodynamic expressions are first derived at a trajectory level then generalized to an ensemble level. To study the nonequilibrium thermodynamic features of a Brownian particle that hops in a medium where its viscosity varies on time, a Brownian particle that walks on a periodic isothermal medium (in the presence or absence of load) is considered. The exact analytical results depict that in the absence of load $f=0$, the entropy production rate ${\dot{e}}_p$ approaches the entropy extraction rate ${\dot{h}}_d=0$. This is reasonable since any system which is in contact with a uniform temperature should obey the detail balance condition in a long time limit. In the presence of load and when the viscous friction decreases either spatially or temporally, the entropy $S\left(t\right)$ monotonously increases with time and saturates to a constant value as $t$ further steps up. The entropy production rate ${\dot{e}}_p$ decreases in time and at steady state (in the presence of load) ${\dot{e}}_p={\dot{h}}_d>0$. On the contrary, when the viscous friction increases either spatially or temporally, the rate of entropy production as well as the rate of entropy extraction monotonously steps up showing that such systems are inherently irreversible. Furthermore, considering a spatially varying viscosity, the nonequilibrium thermodynamic features of a Brownian particle that hops in a ratchet potential with load is explored. In this case, the direction of the particle velocity is dictated by the magnitude of the external load of $f$. Far from the stall load, ${\dot{e}}_p={\dot{h}}_d>0$ and at stall force ${\dot{e}}_p={\dot{h}}_d=0$ revealing the system is reversible at this particular choice of parameter. In the absence of load, ${\dot{e}}_p={\dot{h}}_d>0$ as long as a distinct temperature difference is retained between the hot and cold baths. Moreover, considering a multiplicative noise, we explore the thermodynamic features of the model system.

We show that viscoelastic effects play a crucial role in the damping of vibrational modes in harmonic amorphous solids. The relaxation of a given plane elastic wave is described by a memory function of a semi-infinite one-dimensional mass-spring chain. The initial vibrational energy spreads from the first site of the chain to infinity. In the beginning of the chain, there is a barrier, which significantly reduces the decay of vibrational energy below the Ioffe-Regel frequency. To obtain the parameters of the chain, we present a numerically stable method, based on the Chebyshev expansion of the local vibrational density of states.