Order-disorder transition is an attractive topic in the research field of phase transformation. However, how to describe order-disorder transitions on atomic length scales and diffusional time scales is still challenging. Inspired from high-resolution transmission electron microscopy, we proposed an approach to describe ordered structures by introducing an order parameter into the original phase-field-crystal model to reflect the atomic potential distribution. This new order parameter contains information about kinds of atoms, showing that different kinds of sublattices in ordered structures can be distinguished by the amplitude of the order parameter. Two case studies, growth of ordered precipitations and evolution of antiphase domains, are also presented to demonstrate the capabilities of this approach.

We present an implementation of the cluster variational method (CVM) as a message passing algorithm. The kind of message passing algorithm used for CVM, usually named generalized belief propagation (GBP), is a generalization of the belief propagation algorithm in the same way that CVM is a generalization of the Bethe approximation for estimating the partition function. However, the connection between fixed points of GBP and the extremal points of the CVM free energy is usually not a one-to-one correspondence because of the existence of a gauge transformation involving the GBP messages. Our contribution is twofold. First, we propose a way of defining messages (fields) in a generic CVM approximation, such that messages arrive on a given region from all its ancestors, and not only from its direct parents, as in the standard parent-to-child GBP. We call this approach maximal messages. Second, we focus on the case of binary variables, reinterpreting the messages as fields enforcing the consistency between the moments of the local (marginal) probability distributions. We provide a precise rule to enforce all consistencies, avoiding any redundancy, that would otherwise lead to a gauge transformation on the messages. This moment matching method is gauge free, i.e., it guarantees that the resulting GBP is not gauge invariant. We apply our maximal messages and moment matching GBP to obtain an analytical expression for the critical temperature of the Ising model in general dimensions at the level of plaquette CVM. The values obtained outperform Bethe estimates, and are comparable with loop corrected belief propagation equations. The method allows for a straightforward generalization to disordered systems.

A computational method for the simulation of particulate flows that can efficiently treat the particle-fluid boundary in systems containing many particles was developed based on the smoothed-profile lattice Boltzmann method (SPLBM). In our proposed method, which we call the improved SPLBM (iSPLBM), for an accurate and stable simulation of particulate flows, the hydrodynamic force on a moving solid particle is exactly formulated with consideration of the effect of internal fluid mass. To validate the accuracy and stability of iSPLBM, we conducted numerical simulations of several particulate flow systems and compared our results with those of other simulations and some experiments. In addition, we performed simulations on flotation of many lightweight particles with a wide range of particle size distribution, the results of which demonstrated the effectiveness of iSPLBM. Our proposed model is a promising method to accurately and stably simulate extensive particulate flows.

The quasiempirical Benford law predicts that the distribution of the first significant digit of random numbers obtained from mixed probability distributions is surprisingly meaningful and reveals some universal behavior. We generalize this finding to examine the joint first-digit probability of a pair of two random numbers and show that undetectable correlations by means of the usual covariance-based measure can be identified in the statistics of the corresponding first digits. We illustrate this new measure by analyzing the correlations and anticorrelations of the positions of two interacting particles in their quantum mechanical ground state. This suggests that by using this measure, the presence or absence of correlations can be determined even if only the first digit of noisy experimental data can be measured accurately.

We address the problem of analyzing the radius of convergence of perturbative expansion of nonequilibrium steady states of Lindblad-driven spin chains. A simple formal approach is developed for systematically computing the perturbative expansion of small driven systems. We consider the paradigmatic model of an open $XXZ$ spin-$\frac{1}{2}$ chain with boundary-supported ultralocal Lindblad dissipators and treat two different perturbative cases: (i) expansion in the system-bath coupling parameter and (ii) expansion in the driving (bias) parameter. In the first case (i) we find that the radius of convergence quickly shrinks with increasing the system size, while in the second case (ii) we find that the convergence radius is always larger than 1, and in particular it approaches 1 from above as we change the anisotropy from an easy-plane ($XY$) to an easy-axis (Ising) regime.

We extend a previously proposed resonance phonon approach that is based on the linear response theory. By studying the complex response function in depth, we work out the phonon relaxation time besides the oscillating frequency of the phonons in a few one-dimensional nonlinear lattices. The results in the large wave-number-$k$ regime agree with the expectations of the effective phonon theory. However, in the small-$k$ limit they follow different scaling laws. The phonon mean free path can also be calculated indirectly. It coincides well with that derived from the anharmonic phonon approach. A power-law divergent heat conduction, i.e., the heat conductivity $\kappa$ depends on lattice length $N$ by $\kappa \sim N^{\beta }$ with $\beta >0$, then is supported for the momentum-conserving lattices. Furthermore, this approach can be applied to diatomic lattices. So obtained relaxation time quantitatively agrees with that from the effective phonon theory. As for the mean free path, the resonance phonon approach can detect both the acoustic and the optical branches, whereas the anharmonic phonon approach can only detect a combined branch, i.e., the acoustic branch for small $k$ and the optical branch for large $k$.

The steepest entropy ascent (SEA) dynamical principle provides a general framework for modeling the dynamics of nonequilibrium (NE) phenomena at any level of description, including the atomistic one. It has recently been shown to provide a precise implementation and meaning to the maximum entropy production principle and to encompass many well-established theories of nonequilibrium thermodynamics into a single unifying geometrical framework. Its original formulation in the framework of quantum thermodynamics (QT) assumes the simplest and most natural Fisher-Rao metric to geometrize from a dynamical standpoint the manifold of density operators, which represent the thermodynamic NE states of the system. This simplest SEAQT formulation is used here to develop a general mathematical framework for modeling the NE time evolution of the quantum state of a chemically reactive mixture at an atomistic level. The method is illustrated for a simple two-reaction kinetic scheme of the overall reaction $\mathrm{F}+{\mathrm{H}}_2\iff \mathrm{HF}+\mathrm{F}$ in an isolated tank of fixed volume. However, the general formalism is developed for a reactive system subject to multiple reaction mechanisms. To explicitly implement the SEAQT nonlinear law of evolution for the density operator, both the energy and the particle number eigenvalue problems are set up and solved analytically under the dilute gas approximation. The system-level energy and particle number eigenvalues and eigenstates are used in the SEAQT equation of motion to determine the time evolution of the density operator, thus effectively describing the overall kinetics of the reacting system as it relaxes toward stable chemical equilibrium. The predicted time evolution in the near-equilibrium limit is compared to the reaction rates given by a standard detailed kinetic model so as to extract the single time constant needed by the present SEA model.

Many organisms capitalize on their ability to predict the environment to maximize available free energy and reinvest this energy to create new complex structures. This functionality relies on the manipulation of patterns—temporally ordered sequences of data. Here, we propose a framework to describe pattern manipulators—devices that convert thermodynamic work to patterns or vice versa—and use them to build a “pattern engine” that facilitates a thermodynamic cycle of pattern creation and consumption. We show that the least heat dissipation is achieved by the provably simplest devices, the ones that exhibit desired operational behavior while maintaining the least internal memory. We derive the ultimate limits of this heat dissipation and show that it is generally nonzero and connected with the pattern's intrinsic crypticity—a complexity theoretic quantity that captures the puzzling difference between the amount of information the pattern's past behavior reveals about its future and the amount one needs to communicate about this past to optimally predict the future.

We present an investigation on the breakup of free and pinned spiral waves under an applied electrical current in the Belousov-Zhabotinsky reaction. Spiral fronts propagating towards the negative electrode are decelerated. A breakup of the spiral waves occurs when some segments of the fronts are stopped by a sufficiently strong electrical current. In the absence of obstacles (i.e., free spiral waves), the critical value of the electrical current for the wave breakup increases with the excitability of the medium. For spiral waves pinned to circular obstacles, the critical electrical current increases with the obstacle diameter. Analysis of spiral dynamics shows that the enhancement of the robustness against the breakup of both free and pinned spiral waves is originated by the increment of wave speed when either the excitability is strengthened or the obstacle size is enlarged. The experimental findings are reproduced by numerical simulations using the Oregonator model. In addition, the simulations reveal that the robustness against the forced breakup increases with the activator level in both cases of free and pinned spiral waves.

We present a method for attractor selection in multistable dynamical systems. It involves a feedback term that is active only when the dynamics of the system is in a particular fraction of state space of the attractor. We implement this method first on a simplest symmetric chaotic flow and then on a bistable neuronal system. We find that adding this space-dependent feedback term to the dynamical equations of these systems will drive the dynamics to the desired attractor by annihilating the other. We further demonstrate that the attractor selection due to this feedback term can be used in construction of logic gates, which is one of the practical applications of the proposed method.

Complex systems are often characterized by distinct types of interactions between the same entities. These can be described as a multilayer network where each layer represents one type of interaction. These layers may be interdependent in complicated ways, revealing different kinds of structure in the network. In this work we present a generative model, and an efficient expectation-maximization algorithm, which allows us to perform inference tasks such as community detection and link prediction in this setting. Our model assumes overlapping communities that are common between the layers, while allowing these communities to affect each layer in a different way, including arbitrary mixtures of assortative, disassortative, or directed structure. It also gives us a mathematically principled way to define the interdependence between layers, by measuring how much information about one layer helps us predict links in another layer. In particular, this allows us to bundle layers together to compress redundant information and identify small groups of layers which suffice to predict the remaining layers accurately. We illustrate these findings by analyzing synthetic data and two real multilayer networks, one representing social support relationships among villagers in South India and the other representing shared genetic substring material between genes of the malaria parasite.

When controlling a complex networked system it is not feasible to control the full network because many networks, including biological, technological, and social systems, are massive in size and complexity. But neither is it necessary to control the full network. In complex networks, the giant connected components provide the essential information about the entire system. How to control these giant connected components of a network remains an open question. We derive the mathematical expression of the degree distributions for four types of giant connected components and develop an analytic tool for studying the controllability of these giant connected components. We find that for both Erdős-Rényi (ER) networks and scale-free (SF) networks with $p$ fraction of remaining nodes, the minimum driver node density to control the giant component first increases and then decreases as $p$ increases from zero to one, showing a peak at a critical point $p=p_{\mathrm{m}}$. We find that, for ER networks, the peak value of the driver node density remains the same regardless of its average degree $\langle k\rangle$ and that it is determined by $p_{\mathrm{m}}\langle k\rangle$. In addition, we find that for SF networks the minimum driver node densities needed to control the giant components of networks decrease as the degree distribution exponents increase. Comparing the controllability of the giant components of ER networks and SF networks, we find that when the fraction of remaining nodes $p$ is low, the giant in-connected, out-connected, and strong-connected components in ER networks have lower controllability than those in SF networks.

The embryonic metabolisms of the ornithischian dinosaurs Protoceratops andrewsi and Hypacrosaurus stebingeri have been determined and are in the range observed in extant reptiles. The average value of the measured embryonic metabolic rates for P. andrewsi and H. stebingeri are then used to calculate the incubation times for 21 dinosaurs from both Sauischia and Ornithischia using a mass growth model based on conservation of energy. The calculated incubation times vary from about 70 days for Archaeopteryx lithographica to about 180 days for Alamosaurus sanjuanensis. Such long incubation times seem unlikely, particularly for the sauropods and large theropods. Incubation times are also predicted with the assumption that the saurischian dinosaurs had embryonic metabolisms in the range observed in extant birds.

Active particle tracking microrheometers have the potential to perform accurate broadband measurements of viscoelasticity within microscopic systems. Generally, their largest possible precision is limited by Brownian motion and low frequency changes to the system. The signal to noise ratio is usually improved by increasing the size of the driven motion compared to the Brownian as well as averaging over repeated measurements. New theory is presented here whereby error in measurements of the complex shear modulus can be significantly reduced by analyzing the motion of a spherical particle driven by nonlinear forces. In some scenarios error can be further reduced by applying a variable transformation which linearizes the equation of motion. This enables normalization that eliminates error introduced by low frequency drift in the particle's equilibrium position. Our measurements indicate that this can further resolve an additional decade of viscoelasticity at high frequencies. Using this method will easily increase the signal strength enough to significantly reduce the measurement time for the same error. Thus the method is more conducive to measuring viscoelasticity in slowly changing microscopic systems, such as a living cell.

Using molecular dynamics simulations, we study the crystallization of supercooled liquids in two dimensions in which particles interact with other particles via the Lennard-Jones-Gauss potential. We first prepare supercooled liquids at various temperatures by rapid quenching from the melt. The simulations are performed with a crystalline seed inserted at the center of the initial system. We investigate the time evolution of the inserted nucleus and its surroundings and determine the critical nucleus size $n_c$ defined as the smallest nucleus which survives. The results show that $n_c$ scales as $\sim {(T_m-T)}^{-2}$ with the melting temperature $T_m$, as expected in the classical nucleation theory. We also obtain the crystallization time at various temperatures as a function of nucleus size and show that the presence of a crystalline seed significantly affects the crystallization time when the temperature is higher than the characteristic temperature $T^{*}$ at which the crystallization time becomes the shortest. This indicates that the crystallization is controlled by thermodynamics in this temperature range. When the temperature is lower than $T^{*}$, the effect of the inserted nucleus on crystallization is less significant, which indicates that crystallization is controlled by emergence and merging of small crystalline nuclei.

We study models of grain nucleation and coarsening during the deposition of a thin film using numerical simulations and scaling approaches. The incorporation of new particles in the film is determined by lattice growth models in three different universality classes, with no effect of the grain structure. The first model of grain coarsening is similar to that proposed by Saito and Omura [Phys. Rev. E 84, 021601 (2011)], in which nucleation occurs only at the substrate, and the grain boundary evolution at the film surface is determined by a probabilistic competition of neighboring grains. The surface grain density has a power-law decay, with an exponent related to the dynamical exponent of the underlying growth kinetics, and the average radius of gyration scales with the film thickness with the same exponent. This model is extended by allowing nucleation of new grains during the deposition, with constant but small rates. The surface grain density crosses over from the initial power law decay to a saturation; at the crossover, the time, grain mass, and surface grain density are estimated as a function of the nucleation rate. The distributions of grain mass, height, and radius of gyration show remarkable power law decays, similar to other systems with coarsening and particle injection, with exponents also related to the dynamical exponent. The scaling of the radius of gyration with the height $h$ relative to the base of the grain show clearly different exponents in growth dominated by surface tension and growth dominated by surface diffusion; thus it may be interesting for investigating the effects of kinetic roughening on grain morphology. In growth dominated by surface diffusion, the increase of grain size with temperature is observed.

We used molecular simulation to study the static behavior of polymer droplets in vacuum and on solid surfaces, namely the size of the droplet and the contact angle, respectively. The effects of the polymer chain length and the total number of particles were calculated by the many-body dissipative particle dynamics method. For the spherical droplet containing the same number of particles, we show that its radius depends on the polymer chain length. The radius of the droplet is also proportional to one-third power of the total number of particles for all given chain lengths. For the hemispherical droplet, the contact angle increases with the number of particles in the droplet, and this effect is relatively strong, especially for longer polymer chains. The effect of wettability of the solid surface was also investigated by using polymerphobic (low-affinity) and polymerphilic (high-affinity) surfaces. As the chain length increases, the contact angle on the low-affinity surface decreases, while that on the hydrophilic surface increases. The simulation reveals that there is a critical affinity for the monomer on the solid surface; above and below which the wettability increases and decreases as the molecular length increases, respectively.

This is the first part of a two-part study on a partially miscible liquid-liquid flow (liquid carbon dioxide and deionized water) which is highly pressurized and confined in a microfluidic T-junction. Our main focuses are to understand the flow regimes as a result of varying flow conditions and investigate the characteristics of drop flow distinct from coflow, with a capillary number, $\mathrm{C}{\mathrm{a}}_c$, that is calculated based on the continuous liquid, ranging from ${10}^{-3}$ to ${10}^{-2}$ (${10}^{-4}$ for coflow). Here in part I, we present our experimental observation of drop formation cycle by tracking drop length, spacing, frequency, and after-generation speed using high-speed video and image analysis. The drop flow is chronologically composed of a stagnating and filling stage, an elongating and squeezing stage, and a truncating stage. The common “necking” time during the elongating and squeezing stage (with $\mathrm{C}{\mathrm{a}}_c\sim {10}^{-3}$) for the truncation of the dispersed liquid stream is extended, and the truncation point is subsequently shifted downstream from the T-junction corner. This temporal postponement effect modifies the scaling function reported in the literature for droplet formation with two immiscible fluids. Our experimental measurements also demonstrate the drop speed immediately following their generations can be approximated by the mean velocity from averaging the total flow rate over the channel cross section. Further justifications of the quantitative analysis by considering the mass transfer at the interface of the two partially miscible fluids are provided in part II.

This is the second part of a two-part study on a partially miscible liquid-liquid flow (carbon dioxide and deionized water) that is highly pressurized and confined in a microfluidic T-junction. In the first part of this study, we reported experimental observations of the development of flow regimes under various flow conditions and the quantitative characteristics of the drop flow including the drop length, after-generation drop speed, and periodic spacing development between an emerging drop and the newly produced one. Here in part II we provide theoretical justifications to our quantitative studies on the drop flow by considering (1) $\mathrm{C}{\mathrm{O}}_2$ hydration at the interface with water, (2) the diffusion-controlled dissolution of $\mathrm{C}{\mathrm{O}}_2$ molecules in water, and (3) the diffusion distance of the dissolved $\mathrm{C}{\mathrm{O}}_2$ molecules. Our analyses show that (1) the $\mathrm{C}{\mathrm{O}}_2$ hydration at the interface is overall negligible, (2) a saturation scenario of the dissolved $\mathrm{C}{\mathrm{O}}_2$ molecules in the vicinity of the interface will not be reached within the contact time between the two fluids, and (3) molecular diffusion does play a role in transferring the dissolved molecules, but the diffusion distance is very limited compared with the channel geometry. In addition, mathematical models for the drop length and the drop spacing are developed based on the observations in part I, and their predictions are compared to our experimental results.

Many biological materials consist of sparse networks of disordered fibers, embedded in a soft elastic matrix. The interplay between rigid and soft elements in such composite networks leads to mechanical properties that can go far beyond the sum of those of the constituents. Here we present lattice-based simulations to unravel the microscopic origins of this mechanical synergy. We show that the competition between fiber stretching and bending and elastic deformations of the matrix gives rise to distinct mechanical regimes, with phase transitions between them that are characterized by critical behavior and diverging strain fluctuations and with different mechanisms leading to mechanical enhancement.