A linear scale method for calculating electronic properties of large and complex systems is introduced within a local density approximation. The method is based on the Chebyshev polynomial expansion and the time-dependent method, which is tested on the calculation of the electronic structure of a model $n$-type GaAs quantum dot.

Molecular dynamic simulations are performed to investigate the melting process of small three-dimensional clusters (i.e., systems with one and two shells) of classical charged particles trapped in an isotropic parabolic potential. The confined particles interact through a repulsive potential. We find that the ground-state configurations for systems with $N=6$, 12, 13, and 38 particles interacting through a Coulomb potential are magic clusters. Such magic clusters have an octahedral or icosahedral symmetry and are found to have a large stability against intrashell diffusion leading to an intershell melting transition prior to the intrashell and radial melting process. For systems with two shells a local radial melting of subshells is found at low temperatures resulting in a structural transition leading to an increased symmetry of the ordered system. Using Lindemann’s criterion the different melting temperatures are determined and the influence of the screening of the interparticle interaction was investigated. A normal mode analysis is performed and some of the normal modes are found to be determinantal for the melting process.

We studied the nonequilibrium dynamics of the $q$-state Potts model in the square lattice, after a quench to subcritical temperatures. By means of a continuous time Monte Carlo algorithm (nonconserved order parameter dynamics) we analyzed the long term behavior of the energy and relaxation time for a wide range of quench temperatures and system sizes. For $q>4$ we found the existence of different dynamical regimes, according to quench temperature range. At low (but finite) temperatures and very long times the Lifshitz-Allen-Cahn domain growth behavior is interrupted with finite probability when the system gets stuck in highly symmetric nonequilibrium metastable states, which induce activation in the domain growth, in agreement with early predictions of Lifshitz [JETP 42, 1354 (1962)]. Moreover, if the temperature is very low, the system always gets stuck at short times in highly disordered metastable states with finite lifetime, which have been recently identified as glassy states. The finite size scaling properties of the different relaxation times involved, as well as their temperature dependency, are analyzed in detail.

A random walk of $N$ particles on a lattice with $M$ sites is studied under the constraint that each lattice site is coupled to its own mesoscopic heat bath. Such a situation can be conveniently described by using the master equation in a quantized Hamiltonian formulation where the exclusion principle is included by using Pauli operators. If all reservoirs are mutually in contact, giving rise to a temperature gradient, an evolution equation for the particle density with two different currents already results in the mean-field approximation. One is the conventional diffusive current, driven by the density gradient, whereas the other includes a coupling between the local density and the temperature gradient. Due to the competitive currents, the system exhibits a stationary solution, where the local density is determined by the local temperature field and depends on the filling factor $M∕N$. The stability of the solution is related to the eigenvalues of a Schrödinger-like equation. In the case of a fixed temperature gradient the stationary density distribution remains stable. The approach used is totally different from and an alternative to the conventional Onsager ansatz.

A hydrodynamic description for an $s$-component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first part of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth- and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper.

The linear integral equations defining the Navier-Stokes (NS) transport coefficients for polydisperse granular mixtures of smooth inelastic hard disks or spheres are solved by using the leading terms in a Sonine polynomial expansion. Explicit expressions for all the NS transport coefficients are given in terms of the sizes, masses, compositions, density, and restitution coefficients. In addition, the cooling rate is also evaluated to first order in the gradients. The results hold for arbitrary degree of inelasticity and are not limited to specific values of the parameters of the mixture. Finally, a detailed comparison between the derivation of the current theory and previous theories for mixtures is made, with attention paid to the implication of the various treatments employed to date.

We introduce a probability model for gene regulatory networks, based on a system of Chapman-Kolmogorov equations that represent the dynamics of the concentration levels of each agent in the network. This unifying approach includes the representation of excitatory and inhibitory interactions between agents, second-order interactions which allow any two agents to jointly act on other agents, and Boolean dependencies between agents. The probability model represents the concentration or quantity of each agent, and we obtain the equilibrium solution for the joint probability distribution of each of the concentrations. The result is an exact solution in “product form,” where the joint equilibrium probability distribution of the concentration for each gene is the product of the marginal distribution for each of the concentrations. The analysis we present yields the probability distribution of the concentration or quantity of all of the agents in a network that includes both logical dependencies and excitatory-inhibitory relationships between agents.

Cholesterol is one of the most abundant components in biological membranes. In this paper we apply a detailed state-of-the-art self-consistent field (SCF) theory to predict the influence of cholesterol-look-alikes in the bilayer composed of 1-stearoyl-2-docosahexaenoyl-sn-glycero-3-phosphatidylcholine ($18:0∕22:6\omega 3\mathit{cis}$ PC) lipids with a polyunsaturated 22:6 and a fully saturated 18:0 tail. The cholesterol-like molecule is composed of a hydroxyl group, a rigid chain fragment with length $n$ segments and a branched semiflexible moiety with methylene side groups. We vary both the length of the rigid fragment in the cholesterol-look-alikes and their mole fraction in the tensionless bilayers. We find that these additives significantly increase the order of the saturated tails, but influence the conformational properties of the unsaturated tail much less. With increasing loading the bilayer thickness and the area available per PC head group increase. The hydroxyl group anchors close to the membrane-water interface, but with increasing loading the distribution of this polar group widens. The orientational order of the rigid part is high and we conclude that the cholesterol has significant mobility in the normal direction in the hydrophobic region of the bilayer indicating that one singly hydroxyl group is giving only a weak anchoring to the water-interface. Cholesterol-look-alikes increase the fluctuation of the tail ends and decrease the interdigitation of the tails. Several of our predictions correspond to molecular dynamics (MD) simulation results, but there are also important differences. Most notably the cholesterol-look-alikes can visit the membrane symmetry-plane more easily in SCF than in MD. Possible reasons for this are discussed.

We found that randomly folded thin sheets exhibit unconventional scale invariance, which we termed as an intrinsically anomalous self-similarity, because the self-similarity of the folded configurations and of the set of folded sheets are characterized by different fractal dimensions. Besides, we found that self-avoidance does not affect the scaling properties of folded patterns, because the self-intersections of sheets with finite bending rigidity are restricted by the finite size of crumpling creases, rather than by the condition of self-avoidance. Accordingly, the local fractal dimension of folding structures is found to be universal $(D_l=2.64\pm 0.05)$ and close to expected for a randomly folded phantom sheet with finite bending rigidity. At the same time, self-avoidance is found to play an important role in the scaling properties of the set of randomly folded sheets of different sizes, characterized by the material-dependent global fractal dimension $D<D_l$. So intrinsically anomalous self-similarity is expected to be an essential feature of randomly folded thin matter.

Surface effects on a diffusion process governed by a fractional diffusion equation in a confined region with spatial and time dependent boundary conditions are investigated. First, we consider the one-dimensional case with the boundary conditions $\rho (0,t)={\Phi }_0\left(t\right)$ and $\rho (a,t)={\Phi }_a\left(t\right)$. Subsequently, the two-dimensional case in the cylindrical symmetry with $\rho (a,\theta ,t)={\tilde{\Phi }}_a(\theta ,t)$ and $\rho (b,\theta ,t)={\tilde{\Phi }}_b(\theta ,t)$ is investigated. For these cases, we also obtain exact solutions for an arbitrary initial condition by using the Green’s function approach.

Traveling fronts can become transversally unstable either because of a diffusive instability arising when the key variables diffuse at sufficiently different rates or because of a buoyancy-driven Rayleigh-Taylor mechanism when the density jump across the front is statically unfavorable. The interaction between such diffusive and buoyancy instabilities of fronts is analyzed theoretically for a simple model system. Linear stability analysis and nonlinear simulations show that their interplay changes considerably the stability properties with regard to the pure Rayleigh-Taylor or diffusive instabilities of fronts. In particular, an instability scenario can arise which triggers convection around statically stable fronts as a result of differential diffusion. Moreover, spatiotemporal chaos can be observed when both buoyancy and diffusive effects cooperate to destabilize the front. Experimental conditions to test our predictions are suggested.

We perform extensive simulations of a two-dimensional Lennard-Jones glass subjected to quasistatic shear deformation at $T=0$. We analyze the distribution of nonaffine displacements in terms of contributions of plastic, irreversible events, and elastic, reversible motions. From this, we extract information about correlations between plastic events and about the elastic nonaffine noise. Moreover, we find that nonaffine motion is essentially diffusive, with a clearly size-dependent diffusion constant. These results, supplemented by close inspection of the evolving patterns of the nonaffine tangent displacement field, lead us to propose a phenomenology of plasticity in such amorphous media. It can be schematized in terms of elastic loading and irreversible flips of small, randomly located shear transformation zones, elastically coupled via their quadrupolar fields.

Emergence of hierarchical society is analyzed by use of a simple agent-based model. We extend the mean-field model of Bonabeau et al. [Physica A 217, 373 (1995)] to societies obeying complex diffusion rules where each individual selects a moving direction following their power rankings. We apply this mean-field analysis to the pacifist society model recently investigated by use of Monte Carlo simulation [Physica A 367, 435 (2006)]. We show analytically that the self-organization of hierarchies occurs in two steps as the individual density is increased and there are three phases: one egalitarian and two hierarchical states. We also highlight that the transition from the egalitarian phase to the first hierarchical phase is a continuous change in the order parameter and the second transition causes a discontinuous jump in the order parameter.

Chaos synchronization in coupled chaotic oscillator systems with diffusive and gradient couplings forced by only one local feedback injection signal (boundary pinning control) is studied. By using eigenvalue analysis, we obtain controllable regions directly in control parameter space for different types of coupling links (including diagonal coupling and nondiagonal couplings). The effects of both diffusive and gradient couplings on chaos synchronization become clear. Some relevant factors on control efficiency such as coupled system size, transient process, and feedback signal intensity are also studied.

Temperature and velocity oscillations have been found in a rectangular Rayleigh-Bénard convection cell, in which one large-scale convection roll exists. At Rayleigh number $\mathrm{Ra}=8.9\times {10}^{11}$ and Prandtl number $\mathrm{Pr}=4$, temperature oscillations can be observed in most parts of the system and the oscillation period remains almost constant, $t_T=74\pm 2\mathrm{s}$. Velocity oscillation can only be found in its horizontal component $v_y$ (perpendicular to the large-scale circulation plane) near the cell sidewall, its oscillation period is also constant, $t_v=65\pm 2\mathrm{s}$, at these positions. Temperature and velocity oscillations have different Ra dependences, which are, respectively, indicated by the Péclect number ${\mathrm{Pe}}_T=0.55{\mathrm{Ra}}^{0.47}$ and ${\mathrm{Pe}}_v=0.28{\mathrm{Ra}}^{0.50}$. In comparison to the case of a cylindrical cell, we find that velocity oscillations are affected by the system geometry.

The Soret-driven instability in binary mixture heated from above is analyzed by using the energy method and its modification. The horizontal fluid layer placed between two plates is in initially quiescent state but the Soret diffusion can induce buoyancy-driven convection in the case of the negative Soret coefficient. For the case of highly unstable density stratification the buoyancy-driven motion sets in during the transient diffusion stage. Here the stability limits which are related to the onset time of instabilities are presented as a function of the Rayleigh number Ra, the Lewis number Le, and the separation ratio $\psi$. The present stability analysis predicts that the onset time of convective instability decreases with increasing buoyancy parameter $\mathrm{Ra}{(\mathrm{Le}∕\psi )}^{-1}$. The relaxed energy method shows that the first visible motion can be detected from a certain time five times larger than the predicted onset time and the critical wave number is not zero but a finite value.

The Poisson-Boltzmann problem in spherical symmetry has been considered using the distribution of a self-consistent potential around a charged grain in a thermal collisional plasma as an example. The qualitative patterns of all possible solutions have been presented and a study of their asymptotics has been carried out. It has been demonstrated that for large potentials it is possible to neglect the curvature of the grain surface and to use the solution of the plane problem. It has also been demonstrated that the electrical interaction of the grains is possible only at distances smaller than eight screening lengths.

Motivated by recent studies of colloidal particles in optical-tweezer arrays, we study a two-dimensional model of a colloidal suspension in a periodic potential. The particles tend to stay near potential minima, approximating a lattice gas. The interparticle interaction, a sum of Yukawa terms, features short-range repulsion and attraction at somewhat larger separations, such that two particles cannot occupy the same potential well, but occupation of adjacent cells is energetically favored. Monte Carlo simulation reveals that the equilibrium system exhibits condensation, as in the Ising model or lattice gas with conserved magnetization; the transition appears to be continuous at one-half occupancy. We study the effect of biased hopping, favoring motion along one lattice direction, as might be generated by a steady flow relative to the potential array. This system is found to exhibit features of the driven lattice gas: the interface is oriented along the drive, and appears to be smooth. A weak drive facilitates ordering of the particles into high- and low-density regions, while stronger bias tends to destroy order, and leads to very large energy fluctuations. We also study ordering in a moving periodic potential. Our results suggest possible realizations of equilibrium and driven lattice gases in a colloidal suspension subject to an optical tweezer array.

Reversible random sequential adsorption of binary mixtures of extended objects on a two-dimensional triangular lattice is studied numerically by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding lattice steps. We concentrate here on the influence of the symmetry properties of the shapes on the kinetics of the adsorption-desorption processes in two-component mixtures. We provide a detailed discussion of the significance of collective events for governing the time coverage behavior of component shapes with different rotational symmetries. We also investigate the role that the mixture composition plays in the deposition process. For the mixtures of equal sized objects, we propose a simple formula for predicting the value of the steady-state coverage fraction of a mixture from the values of the steady-state coverage fractions of pure component shapes.

In order to describe quantum heat engines, here we systematically study isothermal and isochoric processes for quantum thermodynamic cycles. Based on these results the quantum versions of both the Carnot heat engine and the Otto heat engine are defined without ambiguities. We also study the properties of quantum Carnot and Otto heat engines in comparison with their classical counterparts. Relations and mappings between these two quantum heat engines are also investigated by considering their respective quantum thermodynamic processes. In addition, we discuss the role of Maxwell’s demon in quantum thermodynamic cycles. We find that there is no violation of the second law, even in the existence of such a demon, when the demon is included correctly as part of the working substance of the heat engine.