Two approaches to describing the thermodynamics of a subsystem that interacts with a thermal bath are considered. Within the first approach, the mean system energy $E_S$ is identified with the expectation value of the system Hamiltonian, which is evaluated with respect to the overall $\left(\text{system}+\text{bath}\right)$ equilibrium distribution. Within the second approach, the system partition function $Z_S$ is considered as the fundamental quantity, which is postulated to be the ratio of the overall $\left(\text{system}+\text{bath}\right)$ and the bath partition functions, and the standard thermodynamic relation $E_S=-d\left(\text{ln}{Z}_S\right)/d\beta$ is used to obtain the mean system energy. Employing both classical and quantum-mechanical treatments, the advantages and shortcomings of the two approaches are analyzed in detail for various different systems. It is shown that already within classical mechanics both approaches predict significantly different results for thermodynamic quantities provided the system-bath interaction is not bilinear or the system of interest consists of more than a single particle. Based on the results, it is concluded that the first approach is superior.

A nearest-neighbor-interaction Ising spin glass, in the presence of an external magnetic field, is studied on different hierarchical lattices that approach the cubic lattice. The magnetic field is considered as uniform or random (following either a bimodal or a Gaussian probability distribution). In all cases, a spin-glass attractor is found, in the plane magnetic field versus temperature, associated with a low-temperature phase. The physical consequences of this attractor are discussed, in view of the present scenario of the spin-glass problem.

In the original formulation of vibration-transit (V-T) theory for monatomic liquid dynamics, the transit contribution to entropy was taken to be a universal constant, calibrated to the constant-volume entropy of melting. This model suffers two deficiencies: (a) it does not account for experimental entropy differences of $\pm 2\%$ among elemental liquids and (b) it implies a value of zero for the transit contribution to internal energy. The purpose of this paper is to correct these deficiencies. To this end, the V-T equation for entropy is fitted to an overall accuracy of $\pm 0.1\%$ to the available experimental high-temperature entropy data for elemental liquids. The theory contains two nuclear motion contributions: (a) the dominant vibrational contribution $S_{vib}\left(T/{\theta }_0\right)$, where $T$ is temperature and ${\theta }_0$ is the vibrational characteristic temperature, and (b) the transit contribution $S_{tr}\left(T/{\theta }_{tr}\right)$, where ${\theta }_{tr}$ is a scaling temperature for each liquid. The appearance of a common functional form of $S_{tr}$ for all the liquids studied is a property of the experimental data, when analyzed via the V-T formula. The resulting $S_{tr}$ implies the correct transit contribution to internal energy. The theoretical entropy of melting is derived in a single formula applying to normal and anomalous melting alike. An ab initio calculation of ${\theta }_0$, based on density-functional theory, is reported for liquid Na and Cu. Comparison of these calculations with the above analysis of experimental entropy data provides verification of V-T theory. In view of the present results, techniques currently being applied in ab initio simulations of liquid properties can be employed to advantage in the further testing and development of V-T theory.

We report a computer-simulation study of the equilibrium phase diagram of a three-dimensional system of particles with a repulsive-shoulder potential. The phase diagram was obtained using free-energy calculations. At low temperatures, we observe a number of distinct crystal phases. We show that at certain values of the potential parameters the system exhibits the waterlike thermodynamic anomalies: a density anomaly and a diffusion anomaly. The anomalies disappear with increasing the repulsive step width: more precisely, their locations move to the region where the crystalline phase is stable.

Spike propagation in chain networks is usually studied in the synfire regime, in which successive groups of neurons are synaptically activated sequentially through the unidirectional excitatory connections. Here we study the dynamics of chain networks with dominant global feedback inhibition that prevents the synfire activity. Neural activity is driven by suprathreshold external inputs. We analytically and numerically demonstrate that spike propagation along the chain is a unique dynamical attractor in a wide parameter regime. The strong inhibition permits a robust winner-take-all propagation in the case of multiple chains competing via the inhibition.

The dynamical reaction-diffusion Selkov system as a model describing the complex traveling wave behavior is presented. The approximate amplitude-phase solution allows us to extract the base properties of the biochemical distributed system, which determines such patterns. It is shown that this relatively simple model could describe qualitatively the main features of the glycolysis waves observed in the experiments.

Here we propose a method, based on detrended fluctuation analysis (DFA), to investigate lagged correlations for nonstationary time series. The aim is to show that the largest correlation can be found at positive lags, reflecting the existence of underlying delays in the evolution of real time sequences. The performance of the lagged DFA method is illustrated by selected real examples.

The distance of closest approach of particles with hard cores is a key parameter in statistical theories and computer simulations of liquid crystals and colloidal systems. In this Brief Report, we provide an algorithm to calculate the distance of closest approach of two ellipsoids of arbitrary shape and orientation. This algorithm is based on our previous analytic result for the distance of closest approach of two-dimensional ellipses. The method consists of determining the intersection of the ellipsoids with the plane containing the line joining their centers and rotating the plane. The distance of closest approach of the two ellipses formed by the intersection is a periodic function of the plane orientation, whose maximum corresponds to the distance of closest approach of the two ellipsoids.

For a certain class of two-dimensional lattices, lattice-dual pairs are shown to have the same bond-percolation critical exponents. A computational proof is given for the martini lattice and its dual to illustrate the method. The result is generalized to a class of lattices that allows the equality of bond-percolation critical exponents for lattice-dual pairs to be concluded without performing the computations. The proof uses the substitution method, which involves stochastic ordering of probability measures on partially ordered sets. As a consequence, there is an infinite collection of infinite sets of two-dimensional lattices, such that all lattices in a set have the same critical exponents.

We introduce solvable stochastic dealer models, which can reproduce basic empirical laws of financial markets such as the power law of price change. Starting from the simplest model that is almost equivalent to a Poisson random noise generator, the model becomes fairly realistic by adding only two effects: the self-modulation of transaction intervals and a forecasting tendency, which uses a moving average of the latest market price changes. Based on the present microscopic model of markets, we find a quantitative relation with market potential forces, which have recently been discovered in the study of market price modeling based on random walks.

We present a mean-field theory to describe volume phase transitions of side-chain liquid crystalline (LC) gels, accompanied by isotropic-nematic-smectic-$A$ phase transitions. Three different uniaxial nematic phases (${\text{N}}_1$, ${\text{N}}_2$, and ${\text{N}}_3$) and smectic-$A$ phases (${\text{S}}_1$, ${\text{S}}_2$, and ${\text{S}}_3$) are defined by using orientational order parameter $S_m$ of side-chain liquid crystals (mesogens), $S_b$ of semiflexible backbone chains, and a translational order parameter $\sigma$ for a smectic-$A$ phase. We derive the free energy for smectic-$A$ phases of side-chain LC gels dissolved in an isotropic solvent and examine the swelling curve of the LC gel, the orientational order parameters, and the deformation of the LC gel as a function of temperature. We find that the LC gel discontinuously changes the volume at an isotropic-nematic, an isotropic-smectic-$A$, and a nematic-smectic-$A$ phase transition.

Tononi et al. [Proc. Natl. Acad. Sci. U.S.A. 91, 5033 (1994)] proposed a measure of neural complexity based on mutual information between complementary subsystems of a given neural network, which has attracted much interest in the neuroscience community and beyond. We develop an approximation of the measure for a popular Gaussian model which, applied to a continuous-time process, elucidates the relationship between the complexity of a neural system and its structural connectivity. Moreover, the approximation is accurate for weakly coupled systems and computationally cheap, scaling polynomially with system size in contrast to the full complexity measure, which scales exponentially. We also discuss connectivity normalization and resolve some issues stemming from an ambiguity in the original Gaussian model.

We study pairwise Ising models for describing the statistics of multineuron spike trains, using data from a simulated cortical network. We explore efficient ways of finding the optimal couplings in these models and examine their statistical properties. To do this, we extract the optimal couplings for subsets of size up to 200 neurons, essentially exactly, using Boltzmann learning. We then study the quality of several approximate methods for finding the couplings by comparing their results with those found from Boltzmann learning. Two of these methods—inversion of the Thouless-Anderson-Palmer equations and an approximation proposed by Sessak and Monasson—are remarkably accurate. Using these approximations for larger subsets of neurons, we find that extracting couplings using data from a subset smaller than the full network tends systematically to overestimate their magnitude. This effect is described qualitatively by infinite-range spin-glass theory for the normal phase. We also show that a globally correlated input to the neurons in the network leads to a small increase in the average coupling. However, the pair-to-pair variation in the couplings is much larger than this and reflects intrinsic properties of the network. Finally, we study the quality of these models by comparing their entropies with that of the data. We find that they perform well for small subsets of the neurons in the network, but the fit quality starts to deteriorate as the subset size grows, signaling the need to include higher-order correlations to describe the statistics of large networks.

We propose a kind of reaction-diffusion equations for cell differentiation, which exhibits the Turing instability. If the diffusivity of some variables is set to be infinity, we get coupled competitive reaction-diffusion equations with a global feedback term. The size ratio of each cell type is controlled by a system parameter in the model. Finally, we extend the model to a cascade model of cell differentiation. A hierarchical spatial structure appears as a result of the cell differentiation. The size ratio of each cell type is also controlled by the system parameter.

Compartmentalized bidisperse granular gases are numerically studied. Molecular-dynamics simulations studying granular clock phenomenon in three dimensions are presented, which complement previously reported two-dimensional simulation results. A flux model for binary mixtures is found to give qualitative descriptions for the oscillations, with no undetermined parameters or functions. Two different states, a degenerate oscillatory state and a state with large particles segregated and small particles homogeneously distributed, are found in our simulations. These features reveal a much more complex phase diagram for the system, which challenges the existing theoretical models.

We propose a strategy for achieving maximum cooperation in evolutionary games on complex networks. Each individual is assigned a weight that is proportional to the power of its degree, where the exponent $\alpha$ is an adjustable parameter that controls the level of diversity among individuals in the network. During the evolution, every individual chooses one of its neighbors as a reference with a probability proportional to the weight of the neighbor, and updates its strategy depending on their payoff difference. It is found that there exists an optimal value of $\alpha$, for which the level of cooperation reaches maximum. This phenomenon indicates that, although high-degree individuals play a prominent role in maintaining the cooperation, too strong influences from the hubs may counterintuitively inhibit the diffusion of cooperation. Other pertinent quantities such as the payoff, the cooperator density as a function of the degree, and the payoff distribution are also investigated computationally and theoretically. Our results suggest that in order to achieve strong cooperation on a complex network, individuals should learn more frequently from neighbors with higher degrees, but only to a certain extent.

In this paper, we propose the ultradiscrete optimal velocity model, a cellular-automaton model for traffic flow, by applying the ultradiscrete method for the optimal velocity model. The optimal velocity model, defined by a differential equation, is one of the most important models; in particular, it successfully reproduces the instability of high-flux traffic. It is often pointed out that there is a close relation between the optimal velocity model and the modified Korteweg–de Vries (mkdV) equation, a soliton equation. Meanwhile, the ultradiscrete method enables one to reduce soliton equations to cellular automata which inherit the solitonic nature, such as an infinite number of conservation laws, and soliton solutions. We find that the theory of soliton equations is available for generic differential equations and the simulation results reveal that the model obtained reproduces both absolutely unstable and convectively unstable flows as well as the optimal velocity model.

We study trapping and propagation of a matter-wave soliton through the interface between uniform medium and a nonlinear optical lattice. Different regimes for transmission of a broad and a narrow solitons are investigated. Reflections and transmissions of solitons are predicted as a function of the lattice phase. The existence of a threshold in the amplitude of the nonlinear optical lattice, separating the transmission and reflection regimes, is verified. The localized nonlinear surface state, corresponding to the soliton trapped by the interface, is found. Variational approach predictions are confirmed by numerical simulations for the original Gross-Pitaevskii equation with nonlinear periodic potentials.

When soluble metal salts are placed in a silicate solution, chemical gardens grow. These gardens are treelike structures formed of long thin hollow tubes. The growth is driven by the increase in internal pressure from osmosis. One particular case is examined here, calcium chloride in a solution of sodium trisilicate. We directly measure the internal pressure of a silicate garden as it grows via a series of relaxation oscillations. From these observations we deduce the stresses in the membrane and discuss how they influence the growth of tubes. Also we estimate the critical stress and the average Young’s modulus for the silicate garden’s membrane.

The ability of two-dimensional (2D) photonic crystals (PC) for high enhancements of nonlinear processes is analyzed in the case of a degenerate band edge when two symmetrical diffracted beams are generated by Bragg diffraction in the 2D-PC. Calculations are performed using the very simple Bragg-coupled wave theory which only involves three coupled waves (the incident wave and the two diffracted waves) for the linear interaction. The validity of the approximation is proved for wavelengths lying at the neighborhood of the band edges of 2D-PC. Very large local-field intensities are predicted around the band-edge wavelengths, in particular for the upper band edge. Nonlinear propagation is studied through the analysis of degenerate four-wave mixing. For counterpropagating pump beams orthogonally sent onto the 2D-PC huge improvement of the phase conjugate reflectivity are predicted at least for small incidence angles of the signal beam. These results represent an improvement by a factor of 20 when compared to the case of a one-dimensional PC of the same thickness made of the same materials. As three intense phase conjugate beams are generated in the four-wave mixing interaction, the 2D-PC could be very interesting for the purpose of dense parallel optical signal processing. Moreover, the simple theoretical analysis developed in the paper can be used for any kind of 2D-PC.