Monte Carlo calculations on critical behavior in two-state, steady-state Ising systems

The thermodynamic treatment of double-pseudobinary solutions of the type (A xBy) r o(M uNv) s o presented in a preceding publication was extended to include the conditions defining the critical points for the asymmetric case (r o≠s o) and approximations for the spinodal and binodal surface near the critical solution point. Closed solutions for the coordinates of the critical points were obtained only for systems with ideal mixing behavior, and the isothermal binodal and spinodal near the critical solution point in such systems are adequately approximated by circles and ellipses, respectively. An axes ratio of $$\sqrt 3$$ is nearly independent of the relative sublattice abundance and the major effect of changes in the ratios o/r o is a rotation of the binodal by an angle tg $$\alpha = \sqrt {\frac{{s_0 }}{{r_0 }}} .$$ . The principal features of nonideal regular systems for temperatures close to the critical solution point are described by expressions derived from small term expansions of the conditional equations, but generalizations are not possible to the same extent as for the case with ideal boundary systems. The results are discussed and the application of the equations demonstrated on model examples.

We study the effect of polydispersity on the structure of polymer brushes by analytical theory, a numerical self-consistent field approach, and Monte Carlo simulations. The polydispersity is represented by the Schulz-Zimm chain-length distribution. We specifically focus on three different polydispersities representing sharp, moderate and extremely wide chain length distributions and derive explicit analytical expressions for the chain end distributions in these brushes. The results are in very good agreement with numerical data obtained with self-consistent field calculations and Monte Carlo simulations. With increasing polydispersity, the brush density profile changes from convex to concave, and for given average chain length $N_n$ and grafting density $\sigma$, the brush height $H$ is found to scale as $(H/H_{\mathrm{mono}}-1)\propto(N_w/N_n-1)^{1/2}$ over a wide range of polydispersity indices $N_w/N_n$ (here $H_{\mathrm{mono}}$ is the height of the corresponding monodisperse brush. Chain end fluctuations are found to be strongly suppressed already at very small polydispersity. Based on this observation, we introduce the concept of the brush as a near-critical system with two parameters (scaling variables), $(N_{n}\sigma^{2/3})^{-1}$ and $(N_w/N_n-1)^{1/2}$, controlling the distance from the critical point. This approach provides a good description of the simulation data. Finally we study the hydrodynamic penetration length $l_\mathrm{p}$ for brush-coated surfaces in flow. We find that it generally increases with polydispersity. The scaling behavior crosses over from $l_\mathrm{p}\sim N_\mathrm{n}^{1/2}\sigma^{-1/6}$ for monodisperse and weakly polydisperse brushes to $l_\mathrm{p}\sim N_{n}^{2/3}$ for strongly polydisperse brushes.

Purified myelin lipids display a critical mixing point at low surface pressure

The q model, a random walk model rich in behavior and applications, is investigated. We introduce and motivate the q model via its application proposed by Coppersmith et al. to the flow of stress through granular matter at rest. For a special value of its parameters the q model has a critical point that we analyze. To characterize the critical point we imagine that a uniform load has been applied to the top of the granular medium and we study the evolution with depth of fluctuations in the distribution of load. Close to the critical point explicit calculation reveals that the evolution of load exhibits scaling behavior analogous to thermodynamic critical phenomena. The critical behavior is remarkably tractable: the harvest of analytic results includes scaling functions that describe the evolution of the variance of the load distribution close to the critical point and of the entire load distribution right at the critical point, values of the associated critical exponents, and determination of the upper critical dimension. These results are of intrinsic interest as a tractable example of a random critical point. Of the many applications of the q model, the critical behavior is particularly relevant to network models of river basins, as we briefly discuss. Finally we discuss circumstances under which quantum network models that describe the surface electronic states of a quantum Hall multilayer can be mapped onto the classical q model. For mesoscopic multilayers of finite circumference the mapping fails; instead a mapping to a ferromagnetic supersymmetric spin chain has proved fruitful. We discuss aspects of the superspin mapping and give an elementary derivation of it making use of operator rather than functional methods.

Effect of pore morphology on vapor–liquid phase transition and crossover behavior of critical properties from 3D to 2D

We study the non-equilibrium steady state (NESS) of a driven dissipative one-dimensional system near a critical point, and explore how the quantum correlations compare to the known critical behavior in the ground state. The model we study corresponds to a cavity array driven parametrically at a two photon resonance, equivalent in a rotating frame to a transverse field anisotropic XY model [C. E. Bardyn and A. Imamo\u{g}lu, Phys. Rev. Lett {\bf 109} 253606 (2012)]. Depending on the sign of transverse field, the steady state of the open system can be either related to the ground state or to the maximum energy state. In both cases, many properties of the entanglement are similar to the ground state, although no critical behavior occurs. As one varies from the Ising limit to the isotropic XY limit, entanglement range grows. The isotropic limit of the NESS is however singular, with simultaneously diverging range and vanishing magnitude of entanglement. This singular limiting behavior is quite distinct from the ground state behavior, it can however be understood analytically within spin-wave theory.

The basic principles of an efficient new methodology for the calculation of the nonsinusoidal periodic steady state in power systems with nonlinear and time-varying components are described. All linear parts, including the network and part of the loads, are represented in the frequency domain, while nonlinear and time-varying components, mainly loads, are represented in the time domain. This hybrid process is iterative, with periodic, nonsinusoidal, bus voltages as inputs for both frequency domain solutions and time domain simulations: a current mismatch is calculated at each bus and used to update the voltages until convergence is reached. Thus the process, but not the solution, is decoupled for the individual harmonics. Its efficiency is enhanced by the use of Newton type algorithms for fast convergence to the periodic steady state in the time domain simulations. Potential applications of this methodology are in the computation of harmonic power flow and in the steady state initialization needed in the calculation of electromagnetic transients.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stability and Pattern of Bifurcating Steady States in Systems with Double Involution

The analogy between equilibrium phase transitions and chemical instabilities is studied in detail in the vicinity of the critical point of a nonequilibrium reacting diffusing system characterized by multiple homogeneous steady states. The critical point of such a system is defined and its mathematical properties are discussed. These properties are shown to be essential in the subsequent reduction of the equations of motion near the critical point. The order parameter characterizing the transition is defined, and its equation of motion near the critical point is obtained in the form of a time-dependent Ginzburg-Landau equation. Fluctuations are taken into account phenomenologically using a Langevin-equation approach. Fluctuations originating from diffusion processes are shown not to be important near the critical point. The size of the critical region for chemical instabilities is estimated using an equivalent of the Ginzburg criterion of equilibrium critical phenomena. The reduction method fails within this region. It is concluded that critical points of chemical instabilities can in principle exhibit "nonclassical" critic\`al behavior. Systems involving the photothermochemical instability and computer simulations of model systems seem currently to be the best candidates for studying the critical properties of chemical instabilities, and for experimental or numerical tests of the predictions of the present theory.

The Gibbs approach to study of the critical state is expanded. It is noted that the well known critical state condition D=0 was obtained by Gibbs as a necessary condition for a non-zero solution of a system of homogeneous linear equations and defines a line of spinoidal solutions. To determine the critical point on this line the explicit form of solutions of a system of homogeneous equations is considered and their relationship to the slope of the phase equilibrium line is established. For the condition D=0 four possible cases of critical system thermodynamic behavior are considered. The relationship between each of these cases and the slope of the phase equilibrium line at the critical point is established. General thermodynamic stability conditions for the critical point are considered as a function of the type of critical behavior at the point. It is shown that one of the consequences of critical point stability is a merger of the binodal line with the spinodal at the critical point.

Computer simulation techniques such as Monte Carlo (MC) and Molecular Dynamics (MD) methods yield numerically exact information (apart from statistical errors) on model systems of classical statistical mechanics. However, a systematic limitation is the restriction to a finite (and often rather small) particle number N (or box linear dimension L, respectively). This limitation is particularly restrictive near critical points (due to the divergence of the correlation length of the order parameter) and for the study of phase equilibria (possibly involving interfaces, droplets, etc.). Starting out with simple lattice gas (Ising) models, finite size scaling analyses have been developed to overcome this limitation. These techniques work for both simple Lennard-Jones fluids and their mixtures, including generalizations to approximate models for quadrupolar fluids such as carbon dioxide, benzene etc. and various mixtures, whose phase behaviour can be predicted. A combination of MC and MD allows the study of dynamic critical phenomena, and specialised techniques (umbrella sampling plus thermodynamic integration) yield the surface free energy of droplets as function of droplet size. Thus, computer simulation has become a versatile and widely applicable tool for the study of fluids.

We study thermodynamics and critical behaviors of higher-dimensional Lovelock black holes with non-maximally symmetric horizons in the canonical ensemble of extended phase space. The effects from non-constancy of the horizon of the black hole via appearing two chargelike parameters in thermodynamic quantities of third-order Lovelock black holes are investigated. We find that Ricci flat black holes with nonconstant curvature horizon show critical behavior. This is an interesting feature that is not seen for any kind of black hole in Einstein or Lovelock gravity in the literature. We examine how various interesting thermodynamic phenomena such as standard first-order small-large black hole phase transition, a reentrant phase transition, or zeroth order phase transition happens for Ricci flat, spherical, or hyperbolic black holes with nonconstant curvature horizon depending on the values of Lovelock coefficient and chargelike parameters. While for a spherical black hole of third order Lovelock gravity with constant curvature horizon phase transition is observed only for $7\leq d \leq11$, for our solution criticality and phase transition exist in every dimension. With a proper choice of the free parameters, a large-small-large black hole phase transition occurs. This process is accompanied by a finite jump of the Gibbs free energy referred to as a zeroth-order phase transition. For the case $\kappa=-1$ a novel behavior is found for which three critical points could exist.

Pathria's approach has been used to deal with the properties of the finite cubic system of $^{4}\mathrm{He}$ under constant pressure. The analytic expressions for the total number of particles N and the total pressure p near the critical point are obtained for mixture, antiperiodic, Neumann, and periodic boundary conditions. Influences of various boundary conditions upon low-temperature and critical characters of finite systems are discussed. For five boundary conditions, the relationship between the superfluid transition temperature T' and the size of the finite system ${\mathit{L}}_{0}$ has been obtained. The results of this paper can be used in the case of superconductivity. Besides, from this work and others, we have obtained the formula for the phase-transition temperature T' in a finite system, t=\ensuremath{\sigma}${\mathit{L}}_{0}^{\mathrm{\ensuremath{-}}\mathit{b}}$; here b is described as the finiteness constant, and \ensuremath{\sigma} is determined by the boundary conditions, the properties of the system, and the interaction between the system and the walls of the container.

This work used Monte Carlo simulation to investigate the effect of lattice strain on critical temperature of the ferromagnetic Ising ultra-thin-film. Using Bethe-Slater interpretation, exchange interaction was assumed Lennard-Jones (LJ) potential function, and becomes functions of interatomic distance. The cluster flip algorithm was used to update the spin configuration, where energy and magnetization were measure to extract critical temperature via the fourth order cumulant of the magnetization. Results show that initial interatomic distance and strain have strong effect on magnetic critical point behavior. The critical point can be enhanced or suppressed depending on how exchange interaction is modified by the values of initial interatomic distance and strain. A scaling function in predicting critical temperatures is also given in this work. Results agree well with previous applicable works, which confirms the reliability of the reported results.

Light-tissue interactions are commonly studied for near infra-red (NIR) spectroscopy for imaging and diagnosis. While imaging is bound to the surface due to the high tissue scattering, many diagnosis methods, such as the photoplethysmograph (PPG) and pulse oximeter, focus on sensing instead of imaging. There are two approaches for the investigation of light-tissue interactions: one numerical and the other analytical. The most common numerical method is the Monte Carlo (MC) simulation, which is a statistical study of photon migration from the optical properties of the different tissue regions. However, the yield of such statistical approaches is limited. Analytical methods are commonly based on the diffusion theory, yet they are inaccurate near the light source. There are several solution methods, where extrapolated boundary conditions lead to a more accurate solution. Previously, we proposed measuring the full scattering profile (FSP), which is the angular intensity distribution, of cylindrical tissues. MC simulations revealed that the FSP has a fixed intensity point, named the iso-pathlength (IPL) point, which does not depend on the reduced scattering coefficient. The location of this point depends solely on the geometry. In this study, we derived the FSP from the steady state diffusion theory based on the extrapolated zeroboundary condition. A comparison of different reduced scattering coefficients revealed the IPL point, where the intensity remains constant. Furthermore, a MC simulation was performed under the same geometry and optical properties. We show that the position of the IPL according to the diffusion theory is in agreement with the MC simulation.