A complete solution of the ground-state problem for an Ising model on the Shastry-Sutherland lattice with an additional interaction along the diagonals of “empty” squares in an applied magnetic field is presented. A rigorous proof is given that this interaction gives rise to a plateau at one-half of the saturation magnetization. Such a fractional plateau has been observed in some rare-earth-metal tetraborides, in particular in strong Ising magnets ErB${}_4$ (where it is the only one) and TmB${}_4$ (where it is the broadest one), but its origin has remained unclear. Our study sheds new light on the solution of this problem.

A free-energy functional for a crystal that contains both the symmetry-conserved and symmetry-broken parts of the direct pair-correlation function has been used to investigate the crystallization of fluids in three dimensions. The symmetry-broken part of the direct pair-correlation function has been calculated using a series in ascending powers of the order parameters and which contains three- and higher-body direct correlation functions of the isotropic phase. It is shown that a very accurate description of freezing transitions for a wide class of potentials is found by considering the first two terms of this series. The results found for freezing parameters including the structure of the frozen phase for fluids interacting via the inverse power potential $u\left(r\right)=\epsilon {\left(\sigma /r\right)}^n$ for $n$ ranging from 4 to $\infty$ are in very good agreement with simulation results. It is found that for $n>6.5$ the fluid freezes into a face-centered cubic (fcc) structure while for $n\le 6$ the body-centered cubic (bcc) structure is preferred. The fluid-bcc-fcc triple point is found to be at $1/n=0.158$, which is in good agreement with simulation result.

It is demonstrated in many thermodynamic textbooks that the equivalence of the different ensembles is achieved in the thermodynamic limit. In this present work we discuss the inequivalence of microcanonical and canonical ensembles in a finite ultracold system at low energies. We calculate the microcanonical momentum distribution function (MDF) in a system of identical fermions (bosons). We find that the microcanonical MDF deviates from the canonical one, which is the Fermi-Dirac (Bose-Einstein) function, in a finite system at low energies where the single-particle density of states and its inverse are finite.

We consider an infinite array of globally coupled overdamped anharmonic oscillators subject to additive Gaussian white noise which is closely related to the mean field ${\Phi }^4$-Ginzburg-Landau model. We prove the existence of a well-behaved critical manifold in the parameter space which separates a symmetric phase from a symmetry broken phase. Given two of the system parameters, there is a unique critical value of the third. The proof exploits that the critical control parameter $a_c$ is bounded by its limit values for weak and strong noise. In these limits, the mechanism of symmetry breaking differs. For weak noise, the distribution is Gaussian and the symmetry is broken as the whole distribution is shifted in either the positive or the negative direction. For strong noise, there is a symmetric double-peak distribution and the symmetry is broken as the weights of the peaks become different. We derive an ordinary differential equation whose solution describes the critical manifold. Using a series ansatz to solve this differential equation, we determine the critical manifold for weak and strong noise and compare it to numerical results. We derive analytic expressions for the order parameter and the susceptibility close to the critical manifold.

For almost a decade the consensus has held that the random walk propagator for the elephant random walk (ERW) model is a Gaussian. Here we present strong numerical evidence that the propagator is, in general, non-Gaussian and, in fact, non-Lévy. Motivated by this surprising finding, we seek a second, non-Gaussian solution to the associated Fokker-Planck equation. We prove mathematically, by calculating the skewness, that the ERW Fokker-Planck equation has a non-Gaussian propagator for the superdiffusive regime. Finally, we discuss some unusual aspects of the propagator in the context of higher order terms needed in the Fokker-Planck equation.

Starting from a robust, nonparametric definition of large returns (“excursions”), we study the statistics of their occurrences, focusing on the recurrence process. The empirical waiting-time distribution between excursions is remarkably invariant to year, stock, and scale (return interval). This invariance is related to self-similarity of the marginal distributions of returns, but the excursion waiting-time distribution is a function of the entire return process and not just its univariate probabilities. Generalized autoregressive conditional heteroskedasticity (GARCH) models, market-time transformations based on volume or trades, and generalized (Lévy) random-walk models all fail to fit the statistical structure of excursions.

In aqueous foams, the diffusive gas transfer among neighboring bubbles drives a coarsening process which is accompanied by intermittent rearrangements of the structure. Using time-resolved diffusing-wave spectroscopy, we probe the dynamics of these events as a function of the rigidity of the gas-liquid interfaces, liquid viscosity, bubble size, and confinement pressure. We present in detail two independent techniques for analyzing the light scattering data, from which we extract the rearrangement duration. Our results show that interfacial rheology has a major impact on this duration. In the case of low interfacial rigidity, the rearrangements strongly slow down as the pressure is decreased close to the value zero where the bubble packing unjams. In contrast, if the interfaces are rigid, rearrangement durations are independent of the confinement pressure in the same investigated range. Using scaling arguments, we discuss dissipation mechanisms that may explain the observed dependency of the rearrangement dynamics on foam structure, pressure, and physicochemical solution properties.

In statistical physics any given system can be either at an equilibrium or away from it. Networks are not an exception. Most network models can be classified as either equilibrium or growing. Here we show that under certain conditions there exists an equilibrium formulation for any growing network model, and vice versa. The equivalence between the equilibrium and nonequilibrium formulations is exact not only asymptotically, but even for any finite system size. The required conditions are satisfied in random geometric graphs in general and causal sets in particular, and to a large extent in some real networks.

Droplets splash around when a fluid volume is quickly compressed. This phenomenon has been observed during common activities such as kids clapping with wet hands. The underlying mechanism involves a fluid volume being compressed vertically between two objects. This compression causes the fluid volume to be ejected radially and thereby generate fluid threads and droplets at a high speed. In this study, we designed and performed laboratory experiments to observe the process of thread and drop formation after a fluid is squeezed. A thicker rim at the outer edge forms and moves after the squeezing, and then becomes unstable and breaks into smaller drops. This process differs from previous well-known examples (i.e., transient crown splashes and continuous water bells) in aspects of transient fluid feeding, expanding rim dynamics, or sparsely distributed drops. We compared experimental measurements with theoretical models over three different stages; early squeezing, intermediate sheet-expansion, and later break-up of the liquid thread. In the earlier stage, the fluid is squeezed and its initial velocity is governed by the lubrication force. The outer rim of the liquid sheet forms curved trajectories due to gravity, inertia, drag, and surface tension. At the late stage, drop spacing set by the initial capillary instability does not change in the course of rim expansion, consequently final ejected droplets are very sparse compared to the size of the rim.

We consider a low-Reynolds-number swimmer made from three spheres linked by two slender arms, and explore its motion in a shear flow. This rodlike three-sphere swimmer finally follows the direction dictated by the shear flow. To overcome this shortcoming, we propose a model in which the two outer spheres have permanent magnetic moments along the arms. This magnetic swimmer can be navigated to a desired direction by applying an external static and uniform magnetic field.

We consider fluctuations of the dissipated energy in nonlinear driven diffusive systems subject to bulk dissipation and boundary driving. With this aim, we extend the recently introduced macroscopic fluctuation theory to nonlinear driven dissipative media, starting from the fluctuating hydrodynamic equations describing the system mesoscopic evolution. Interestingly, the action associated with a path in mesoscopic phase space, from which large-deviation functions for macroscopic observables can be derived, has the same simple form as in nondissipative systems. This is a consequence of the quasielasticity of microscopic dynamics, required in order to have a nontrivial competition between diffusion and dissipation at the mesoscale. Euler-Lagrange equations for the optimal density and current fields that sustain an arbitrary dissipation fluctuation are also derived. A perturbative solution thereof shows that the probability distribution of small fluctuations is always Gaussian, as expected from the central limit theorem. On the other hand, strong separation from the Gaussian behavior is observed for large fluctuations, with a distribution which shows no negative branch, thus violating the Gallavotti-Cohen fluctuation theorem, as expected from the irreversibility of the dynamics. The dissipation large-deviation function exhibits simple and general scaling forms for weakly and strongly dissipative systems, with large fluctuations favored in the former case but heavily suppressed in the latter. We apply our results to a general class of diffusive lattice models for which dissipation, nonlinear diffusion, and driving are the key ingredients. The theoretical predictions are compared to extensive numerical simulations of the microscopic models, and excellent agreement is found. Interestingly, the large-deviation function is in some cases nonconvex beyond some dissipation. These results show that a suitable generalization of macroscopic fluctuation theory is capable of describing in detail the fluctuating behavior of nonlinear driven dissipative media.

We derive a modification of the Poisson-Boltzmann (PB) equation brought about by the finite and different sizes of mobile ions. The equation we obtain contains explicitly the size asymmetry of the electrolyte, enabling one to apply the PB approach to the cases where this effect may be of importance. We show that the modification we propose predicts the asymmetry of the double-layer capacitance. Our approach thus provides a transparent analytical framework for a previously proposed heuristic modification of the PB equation, which also predicts the same effect. In addition, a detailed exposition of the lattice-based calculation of the entropy, accounting for steric, spatial correlations, enables us to clearly pinpoint some of the drawbacks of such an approach, which are not entirely obvious in previous attempts to include the steric effects in the PB equation.

Using both experiments and finite element simulations, we explore the shape evolution of off-axis twist nematic elastomer ribbons as a function of temperature. The elastomers are prepared by cross-linking the mesogens with planar anchoring of the director at top and bottom surfaces with a 90${}^{\mathrm{o}}$ left-handed twist. Shape evolution depends sensitively on the off-axis director orientation at the sample midplane. Both experiments and theoretical studies show that when the director at midplane is parallel to either the ribbon's long or short axes, ribbons form either helicoids or spirals depending on aspect ratio and temperature. Simulation studies show that if the director at midplane is off-axis, ribbons never form helicoids, instead evolving to distorted spiral shapes. Experimental studies for two samples with off-axis geometry show agreement with this prediction. Samples in all these geometries show a remarkable transition from right- to left-handed chiral shapes on change of temperature. Simulations predict off-axis samples also change their macroscopic chirality at fixed temperature, depending on the angular offset. These results provide insight into the mechanisms driving shape evolution and macroscopic chirality, enabling engineering design of these materials for future applications.

Quantitative understanding of the best possible performance of nanomotors allowed by physical laws pertains to the study of nanomotors from biology as well as nanotechnology. The biological nanomotor F${}_1$ ATPase is the best available model system as it is the only nanomotor known for extreme energy conversion near the limit of energy conservation. Using a unified theoretical framework centered on a concept called directional fidelity, we analyze recent experiments in which the F${}_1$ motor's performance was measured for controlled chemical potentials and expose from the experiments quantitative evidence for the motor's multiple extreme performances in directional fidelity, speed, and catalytic capability close to physical limits. Specifically, the motor nearly exhausts the available energy from the fuel to retain the highest possible directional fidelity for an arbitrary load, encompassing the motor's extreme energy conversion and beyond. The theory-experiment comparison implies a tight chemomechanical coupling up to stalemate as futile steps occur, but unlikely involve fuel consumption. The F${}_1$-motor data also help clarify the relation between directional fidelity and experimentally measured stepping ratio.

Proactive motion in hand tracking and in finger bending, in which the body motion occurs prior to the reference signal, has been previously reported when a periodic target signal was shown to the subjects at relatively high frequencies. These phenomena indicate that the human sensory-motor system tends to choose an anticipatory mode rather than a reactive mode, when the target motion is relatively fast. The present research was undertaken to study what kind of mode appears in the sensory-motor system when two persons were asked to track the hand position of the partner at various mean tracking frequency. The experimental results showed that a transition from a mutual error-correction mode to a synchronization mode occurred in the same region of the transition frequency with the one from a reactive error-correction mode to a proactive anticipatory mode reported previously in the target tracking experiments of the single subjects. Present research indicated that synchronization of body motion occurred only when both of the pair subjects operated in a proactive anticipatory mode. We also presented mathematical models to explain the behavior of the error-correction mode and the synchronization mode.

We study adaptive dynamics in games where players abandon the population at a given rate and are replaced by naive players characterized by a prior distribution over the admitted strategies. We demonstrate how such a process leads macroscopically to a variant of the replicator equation, with an additional term accounting for player turnover. We study how Nash equilibria and the dynamics of the system are modified by this additional term for prototypical examples such as the rock-paper-scissors game and different classes of two-action games played between two distinct populations. We conclude by showing how player turnover can account for nontrivial departures from Nash equilibria observed in data from lowest unique bid auctions.

We consider a costly bilingualism model in which one can take two strategies in parallel. We investigate how a single zealot triggers the cascading behavior and how the compatibility of the two strategies affects when interacting patterns change. First, the role of the interaction range in the cascading is studied by increasing the range from local to global. We find that people sometimes do not favor taking the superior strategy even though its payoff is higher than that of the inferior one. This is found to be caused by the local interactions rather than the global ones. Applying this model to social networks, we find that the location of the zealot is also important for larger cascading in heterogeneous networks.

We discuss the flow past a flat heterogeneous solid surface decorated by slipping stripes. The spatially varying slip length, $b\left(y\right)$, is assumed to be small compared to the scale of the heterogeneities, $L$, but finite. For such weakly slipping surfaces, earlier analyses have predicted that the effective slip length is simply given by the surface-averaged slip length, which implies that the effective slip-length tensor becomes isotropic. Here we show that a different scenario is expected if the local slip length has steplike jumps at the edges of slipping heterogeneities. In this case, the next-to-leading term in an expansion of the effective slip-length tensor in powers of $\text{max}\left[b\right(y)/L]$ becomes comparable to the leading-order term, but anisotropic, even at very small $b\left(y\right)/L$. This leads to an anisotropy of the effective slip and to its significant reduction compared to the surface-averaged value. The asymptotic formulas are tested by numerical solutions and are in agreement with results of dissipative particle dynamics simulations.

This study is motivated by recent experimental results dealing with the transition to turbulence in a pipe flow of shear-thinning fluids, where a streaky flow with an azimuthal wave number $n=1$ is observed in the transitional regime. Here, a linear stability analysis of pipe flow of shear-thinning fluids modulated azimuthally by finite amplitude streaks is performed. The shear-thinning behavior of the fluid is described by the Carreau model. The streaky base flows considered are obtained from two-dimensional direct numerical simulation using finite amplitude longitudinal rolls as the initial condition and by extracting the velocity field at time $t_{\max }$, where the amplitude of the streaks reaches its maximum, denoted by ${\displaystyle A_{\max }}$. It is found that the amplitude $A_{\max }$ increases with increasing Reynolds number as well as with increasing amplitude ${\mathcal{E}}_0$ of the initial longitudinal rolls. For sufficiently large streaks amplitude, streamwise velocity profiles develop inflection points, leading to instabilities. Depending on the threshold amplitude $A_c$, two different modes may trigger the instability of the streaks. If $A_c$ exceeds approximately $41.5\%$ of the centerline velocity, the instability mode is located near the axis of the pipe, i.e., it is a “center mode.” For weaker amplitude $A_c$, the instability mode is located near the pipe wall, in the region of highest wall normal shear, i.e., it is a “wall mode.” The threshold amplitude $A_c$ decreases with increasing shear-thinning effects. The energy equation analysis indicates that (i) wall modes are driven mainly by the work of the Reynolds stress against the wall normal shear and (ii) for center modes, the contribution of the normal wall shear remains dominant; however, it is noted that the contribution of the Reynolds stress against the azimuthal shear increases with increasing shear-thinning effects.

We derive analytical expressions for the three-dimensional (3D) acoustophoretic motion of spherical microparticles in rectangular microchannels. The motion is generated by the acoustic radiation force and the acoustic streaming-induced drag force. In contrast to the classical theory of Rayleigh streaming in shallow, infinite, parallel-plate channels, our theory does include the effect of the microchannel side walls. The resulting predictions agree well with numerics and experimental measurements of the acoustophoretic motion of polystyrene spheres with nominal diameters of 0.537 and 5.33 $\mu \text{m}$. The 3D particle motion was recorded using astigmatism particle tracking velocimetry under controlled thermal and acoustic conditions in a long, straight, rectangular microchannel actuated in one of its transverse standing ultrasound-wave resonance modes with one or two half-wavelengths. The acoustic energy density is calibrated in situ based on measurements of the radiation dominated motion of large 5-$\mu \text{m}$-diameter particles, allowing for quantitative comparison between theoretical predictions and measurements of the streaming-induced motion of small 0.5-$\mu \text{m}$-diameter particles.