We consider the problem of constructing a discrete differential geometry defined on nonplanar quadrilateral meshes. Physical models on discrete nonflat spaces are of inherent interest, as well as being used in applications such as computation for electromagnetism, fluid mechanics, and image analysis. However, the majority of analysis has focused on triangulated meshes. We consider two approaches: discretizing the tensor calculus, and a discrete mesh version of differential forms. While these two approaches are equivalent in the continuum, we show that this is not true in the discrete case. Nevertheless, we show that it is possible to construct mesh versions of the Levi-Civita connection (and hence the tensorial covariant derivative and the associated covariant exterior derivative), the torsion, and the curvature. We show how discrete analogs of the usual vector integral theorems are constructed in such a way that the appropriate conservation laws hold exactly on the mesh, rather than only as approximations to the continuum limit. We demonstrate the success of our method by constructing a mesh version of classical electromagnetism and discuss how our formalism could be used to deal with other physical models, such as fluids.

The rate of temperature decrease of a cooled quantum bath is studied as its temperature is reduced to absolute zero. The third law of thermodynamics is then quantified dynamically by evaluating the characteristic exponent $\zeta$ of the cooling process $\frac{dT\left(t\right)}{dt}\sim -T^{\zeta }$ when approaching absolute zero, $T\to 0$. A continuous model of a quantum refrigerator is employed consisting of a working medium composed either by two coupled harmonic oscillators or two coupled two-level systems. The refrigerator is a nonlinear device merging three currents from three heat baths: a cold bath to be cooled, a hot bath as an entropy sink, and a driving bath which is the source of cooling power. A heat-driven refrigerator (absorption refrigerator) is compared to a power-driven refrigerator. When optimized, both cases lead to the same exponent $\zeta$, showing a lack of dependence on the form of the working medium and the characteristics of the drivers. The characteristic exponent is therefore determined by the properties of the cold reservoir and its interaction with the system. Two generic heat bath models are considered: a bath composed of harmonic oscillators and a bath composed of ideal Bose/Fermi gas. The restrictions on the interaction Hamiltonian imposed by the third law are discussed. In the Appendices, the theory of periodically driven open systems and its implication for thermodynamics are outlined.

We discuss the relevance of information contained in cross correlations among different degrees of freedom, which is crucial in nonequilibrium systems. In particular we consider a stochastic system where two degrees of freedom $X_1$ and $X_2$—in contact with two different thermostats—are coupled together. The production of entropy and the violation of equilibrium fluctuation-dissipation theorem (FDT) are both related to the cross correlation between $X_1$ and $X_2$. Information about such cross correlation may be lost when single-variable reduced models for $X_1$ are considered. Two different procedures are typically applied: (a) one totally ignores the coupling with $X_2$; and (b) one models the effect of $X_2$ as an average memory effect, obtaining a generalized Langevin equation. In case (a) discrepancies between the system and the model appear both in entropy production and linear response; the latter can be exploited to define effective temperatures, but those are meaningful only when time scales are well separated. In case (b) linear response of the model well reproduces that of the system; however the loss of information is reflected in a loss of entropy production. When only linear forces are present, such a reduction is dramatic and makes the average entropy production vanish, posing problems in interpreting FDT violations.

We show that anomalous diffusion arises in two different models for the motion of randomly forced and weakly damped particles: one is a generalization of the Ornstein-Uhlenbeck process with a random force, which depends on position as well as time, the other is a generalization of the Chandrasekhar-Rosenbluth model of stellar dynamics, encompassing non-Coulombic potentials. We show that both models exhibit anomalous diffusion of position $x$ and momentum $p$ with the same exponents: $\langle x^2\rangle \sim C_x{t}^2$ and $\langle p^2\rangle \sim C_p{t}^{2/5}$. We are able to determine the prefactors $C_x$, $C_p$ analytically.

We investigate the dynamic properties of inhomogeneous nanomaterials, which appear in analytical descriptions typically as a series of $\delta$ functions with corresponding Gibbs weights. We focus on observables relevant for transport theories of Josephson junction arrays and granular systems near the superconductor-insulator transition. Furthermore, our description applies to the theory of tunnel junctions exchanging energy with a “bath,” the latter having a discrete spectrum. Using the matrix $\theta$-function formalism, we find an analytical expression for the transport characteristics capturing the complete temperature-driven transition from the quantum to the classical regime.

Using a continuum model capable of describing the one-component liquid-gas hydrodynamics down to the contact line scale, we carry out numerical simulation and physical analysis for the droplet motion driven by thermal singularity. For liquid droplets in one-component fluids on heated or cooled substrates, the liquid-gas interface is nearly isothermal. Consequently, a thermal singularity occurs at the contact line and the Marangoni effect due to temperature gradient is suppressed. Through evaporation or condensation in the vicinity of the contact line, the thermal singularity makes the contact angle increase with the increasing substrate temperature. This effect on the contact angle can be used to move the droplets on substrates with thermal gradients. Our numerical results for this kind of droplet motion are explained by a simple fluid dynamical model at the droplet length scale. Since the mechanism for droplet motion is based on the change of contact angle, a separation of length scales is exhibited through a comparison between the droplet motion induced by a wettability gradient and that by a thermal gradient. It is shown that the flow field at the droplet length scale is independent of the statics or dynamics at the contact line scale.

The evaporation of picoliter water and ethanol droplets generated by drop-on-demand inkjet printing was investigated on substrates with apparent contact angles between 10${}^{\circ }$ and 135${}^{\circ }$ and thermal conductivities between 0.25 and 149 $\text{W}{\text{m}}^{-1}{\text{K}}^{-1}$. Drying times were calculated from a diffusion-limited model for droplets with both pinned and moving contact lines as a function of droplet diameter and apparent contact angle. Droplets with a moving contact line take longer to dry on hydrophilic substrates than pinned droplets. The difference in drying times between evaporative modes vanishes at large apparent contact angles. Hence similar drying times are obtained for both modes on hydrophobic substrates. The predicted drying times for glass and silicon substrates were in good quantitative agreement with experimental data, suggesting that thermal effects are negligible for substrates of these base materials. However, on a PTFE substrate which has a lower thermal conductivity more relevant to inkjet printing, evaporative cooling reduces the evaporation rate causing drying times to be underpredicted by isothermal models.

The rock-paper-scissors game is a model example of the ongoing cyclic turnover typical of many ecosystems, ranging from the terrestrial and aquatic to the microbial. Here we explore the evolution of a rock-paper-scissors system where three species compete for space. The species are allowed to mutate and change the speed by which they invade one another. In the case when all species have similar mutation rates, we observe a perpetual arms race where no single species prevails. When only two species mutate, their aggressions increase indefinitely until the ecosystem collapses and only the nonmutating species survives. Finally we show that when only one species mutates, group selection removes individual predators with the fastest growth rates, causing the growth rate of the species to stabilize. We explain this group selection quantitatively.

We demonstrate an improved technique for implementing logic circuits in light-sensitive chemical excitable media. The technique makes use of the constant-speed propagation of waves along defined channels in an excitable medium based on the Belousov-Zhabotinsky reaction, along with the mutual annihilation of colliding waves. What distinguishes this work from previous work in this area is that regions where channels meet at a junction can periodically alternate between permitting the propagation of waves and blocking them. These valvelike areas are used to select waves based on the length of time that it takes waves to propagate from one valve to another. In an experimental implementation, the channels that make up the circuit layout are projected by a digital projector connected to a computer. Excitable channels are projected as dark areas and unexcitable regions as light areas. Valves alternate between dark and light: Every valve has the same period and phase, with a 50$\%$ duty cycle. This scheme can be used to make logic gates based on combinations of or and and-not operations, with few geometrical constraints. Because there are few geometrical constraints, compact circuits can be implemented. Experimental results from an implementation of a four-bit input, two-bit output integer square root circuit are given.

A thread of viscous fluid falling onto a moving surface exhibits a spectacular variety of types of motion as the surface speed and nozzle height are varied. For modest nozzle heights, four clear regimes are observed. For large surface speed, the thread is dragged into a stretched centenary configuration which is confined to a plane. As the surface speed is lowered, the thread exhibits a supercritical bifurcation to a meandering state. At very low surface speeds, the state resembles the usual coiling motion of a viscous thread falling on a stationary surface. In between the meandering and coiling regimes, a window containing a novel multifrequency state, previously called “figures of eight,” is found. Using an improved visualization technique and a fully automated apparatus, we made detailed measurements of the longitudinal and transverse motion of the thread in all these states. We found that the multifrequency state is characterized by a complex pattern of motion whose main frequencies are locked in a 3:2 ratio. This state appears and disappears with finite amplitude at sharp bifurcations without measurable hysteresis.

Strongly thermophilic nanofluids are able to transfer either small or large quantities of heat when subjected to a stable temperature difference. We investigate the bistability diagram of the heat transferred by this class of nanofluids. We show that bistability can be exploited to obtain a controlled switching between a conductive and a convective regime of heat transfer, so as to achieve a controlled modulation of the heat flux.

Linear stability analysis of strongly coupled incompressible dusty plasma in presence of shear flow has been carried out using the generalized hydrodynamical (GH) model. With the proper Galilean invariant GH model, a nonlocal eigenvalue analysis has been done using different velocity profiles. It is shown that the effect of elasticity enhances the growth rate of shear flow driven Kelvin- Helmholtz (KH) instability. The interplay between viscosity and elasticity not only enhances the growth rate but the spatial domain of the instability is also widened. The growth rate in various parameter space and the corresponding eigenfunctions are presented.

In a recent paper [Phys. Rev. E 85, 016308 (2012)], we reported on numerical simulations of swirling two-phase flows in which for cases where the viscosity of the upper fluid is sufficiently smaller than that of the lower fluid, a small region of reversed flow is present on the interface. In this Brief Report, we show that the reported reversed flow is grid converged and comment further on its physical origin.

We developed a statistical model which describes the thermal and volumetric properties of water-like molecules. A molecule is presented as a three-dimensional sphere with four hydrogen-bonding arms. Each water molecule interacts with its neighboring waters through a van der Waals interaction and an orientation-dependent hydrogen-bonding interaction. This model, which is largely analytical, is a variant of a model developed before for a two-dimensional Mercedes-Benz model of water. We explored properties such as molar volume, density, heat capacity, thermal expansion coefficient, and isothermal compressibility as a function of temperature and pressure. We found that the volumetric and thermal properties follow the same trends with temperature as in real water and are in good general agreement with Monte Carlo simulations, including the density anomaly, the minimum in the isothermal compressibility, and the decreased number of hydrogen bonds upon increasing the temperature.

The microscopic nonlinear Langevin equation (NLE) theory of correlated two-particle dynamics in dense fluids of spherical particles is extended to construct a predictive model of multiple correlated hopping and recaging events of a pair of tagged particles as a function of their initial separation. Modest coarse graining over the liquid structural disorder allows contact to be made with various definitions of irreversible particle motion within the context of a multistate Markov model. The correlated space-time hopping process that underlies structural relaxation can also be analyzed in the context of kinetically constrained models. The dependence of microscopically defined mean persistence and exchange times, their distributions, and relaxation-diffusion decoupling on hard-sphere fluid volume fraction is derived from a model in which irreversible jumps serve as the nucleating persistence event. For a subset of questions, the predictions of the two-particle theory are compared with results from the earlier single-particle NLE approach.

The diffusion coefficients of proteins embedded in a lipid membrane are traditionally described by the hydrodynamic Saffman-Delbrück theory, which predicts a weak dependence of the diffusion coefficient on protein radius, $D\sim \ln R$. Recent experiments have observed a stronger dependence, $D\sim 1/R$. This has led to speculation that the primary sources of drag on the protein are not hydrodynamic, but originate in coupling to other fields, such as lipid chain stretching or tilt. We discuss a generic model of a protein coupled to a nonconserved scalar order parameter (e.g., chain stretching), and show that earlier results may not be as universal as previously believed. In particular, we note that the drag depends on the way the protein-order parameter coupling is imposed. In this model, $D\sim 1/R$ can be obtained if the protein is much larger than the order parameter correlation length. However, if we modify the model to include advection of the order parameter, which is a more appropriate assumption for a fluid membrane, we find that the entrainment of the order parameter by the protein's motion significantly changes the scaling of the diffusion coefficient. For parameters appropriate to protein diffusion, the Saffman-Delbrück-like scaling is restored, but with an effective radius for the protein that depends on the order parameter's correlation length. This qualitative difference suggests that hydrodynamic effects cannot be neglected in the computation of drag on a protein interacting with the membrane.

Dynamics of a vesicle under simple shear flow is studied in the limit of small capillary number. A perturbative approach is used to derive the equation of vesicle dynamics. The expansions are shown to converge for significantly deflated vesicles (with excess area from the sphere as high as 2). In particular, we provide an explicit analytical expression for the tank-treading to tumbling bifurcation point. This expression is valid for excess areas up to 2.5. The results are compared with full 3D numerical simulations. The proposed method can be used for analytical or numerical solution of vesicle dynamics under weak flow of general form.

We formulate an effective medium (mean field) theory of a material consisting of randomly distributed nodes connected by straight slender rods, hinged at the nodes. Defining wavelength-dependent effective elastic moduli, we calculate both the static moduli and the dispersion relations of ultrasonic longitudinal and transverse elastic waves. At finite wave vector $k$ the waves are dispersive, with phase and group velocities decreasing with increasing wave vector. These results are directly applicable to networks with empty pore space. They also describe the solid matrix in two-component (Biot) theories of fluid-filled porous media. We suggest the possibility of low density materials with higher ratios of stiffness and strength to density than those of foams, aerogels, or trabecular bone.

We study the transport of excitations on a network of three coupled two-level systems that are subjected to an environment that induces incoherent hopping between the nodes. The two end nodes are coupled to a source while the central node is coupled to a drain. A common feature of these networks is the existence of a dark state that blocks the transport to the drain. Here we propose a means to avoid this state by a suitable initialization of the system, induced by a source that is common to both coupled nodes.

When a complex network is slightly desynchronized, a few of the network nodes will be escaping from the uniform synchronization background frequently with a random fashion, leading to the intermittent network synchronization. Here, based on the eigenvectors of the network coupling matrix, we propose a new method which is able to figure out the unstable nodes in the general case of desynchronized complex networks. Moreover, with this method, we are also able to regulate the seemingly random network dynamics into stable and visible synchronous patterns. The efficiency of this method is verified by a variety of network models, including varying the network structures, the node local dynamics, and the desynchronization types. Our studies show that, even for the complex network systems, synchronous patterns can still be identified and characterized.