We have developed a coarse-grained formulation for modeling the dynamic behavior of cells quantitatively, based on stochasticity and heterogeneity, rather than on biochemical reactions. We treat each reaction as a continuous-time stochastic process, while reducing each biochemical quantity to a binary value at the level of individual cells. The system can be analytically represented by a finite set of ordinary linear differential equations, which provides a continuous time course prediction of each molecular state. Here we introduce our formalism and demonstrate it with several examples.

We study the statistical behavior under random sequential renormalization (RSR) of several network models including Erdös-Rényi (ER) graphs, scale-free networks, and an annealed model related to ER graphs. In RSR the network is locally coarse grained by choosing at each renormalization step a node at random and joining it to all its neighbors. Compared to previous (quasi-)parallel renormalization methods [Song et al., Nature (London) 433, 392 (2005)], RSR allows a more fine-grained analysis of the renormalization group (RG) flow and unravels new features that were not discussed in the previous analyses. In particular, we find that all networks exhibit a second-order transition in their RG flow. This phase transition is associated with the emergence of a giant hub and can be viewed as a new variant of percolation, called agglomerative percolation. We claim that this transition exists also in previous graph renormalization schemes and explains some of the scaling behavior seen there. For critical trees it happens as $N/N_0\to 0$ in the limit of large systems (where $N_0$ is the initial size of the graph and $N$ its size at a given RSR step). In contrast, it happens at finite $N/N_0$ in sparse ER graphs and in the annealed model, while it happens for $N/N_0\to 1$ on scale-free networks. Critical exponents seem to depend on the type of the graph but not on the average degree and obey usual scaling relations for percolation phenomena. For the annealed model they agree with the exponents obtained from a mean-field theory. At late times, the networks exhibit a starlike structure in agreement with the results of Radicchi et al. [Phys. Rev. Lett. 101, 148701 (2008)]. While degree distributions are of main interest when regarding the scheme as network renormalization, mass distributions (which are more relevant when considering “supernodes” as clusters) are much easier to study using the fast Newman-Ziff algorithm for percolation, allowing us to obtain very high statistics.

We report the critical point for site percolation for the “explosive” type for two-dimensional square lattices using Monte Carlo simulations and compare it to the classical well-known percolation. We use similar algorithms as have been recently reported for bond percolation and networks. We calculate the explosive site percolation threshold as $p_c=0.695$ and we find evidence that explosive site percolation surprisingly may belong to a different universality class than bond percolation on lattices, providing that the transitions (a) are continuous and (b) obey the conventional finite size scaling forms. Finally, we study and compare the direct and reverse processes, showing that while the reverse process is different from the direct process for finite size systems, the two cases become equivalent in the thermodynamic limit of large $L$.

We derive a multiplication law for free non-Hermitian random matrices allowing for an easy reconstruction of the two-dimensional eigenvalue distribution of the product ensemble from the characteristics of the individual ensembles. We define the corresponding non-Hermitian $S$ transform being a natural generalization of the Voiculescu $S$ transform. In addition, we extend the classical Hermitian $S$ transform approach to deal with the situation when the random matrix ensemble factors have vanishing mean including the case when both of them are centered. We use planar diagrammatic techniques to derive these results.

Dynamic heterogeneity is now considered to be an intrinsic kinetic feature of a supercooled liquid. Here, we access the nonlocal complex modulus of a glass-forming liquid using molecular dynamics simulations. We find that the shear-stress relaxation exhibits a marked wave number dependence in a supercooled state. This dependence provides direct evidence that slow mechanical relaxation takes place heterogeneously in space. Its characteristic length $\xi$ increases with decreasing temperature and becomes mesoscopic. The response involves particle rearrangements over $\xi$ and takes a time comparable to the structural relaxation time ${\tau }_{\alpha }$. Our finding suggests that the heterogeneous structural relaxation is of fundamental importance in anomalous viscous transport and viscoelasticity in supercooled liquids.

Certain regulatory proteins influence the polymerization dynamics of microtubules by inducing catastrophe with a rate that depends on the microtubule length. Using a discrete formulation, here we show that, for a catastrophe rate proportional to the microtubule length, the steady-state probability distributions of length decay much faster with length than an exponential decay as seen in the absence of these proteins.

By methods of modern spectral analysis, we rigorously find distributions of eigenvalues of linearized operators associated with an ideal hydromagnetic Couette-Taylor flow. The transition to instability in the limit of a vanishing magnetic field has a discontinuous change compared to the Rayleigh stability criterion for hydrodynamical flows, which is known as the Velikhov-Chandrasekhar paradox.

We investigate a network of coupled phase oscillators whose interactions evolve dynamically depending on the relative phases between the oscillators. We found that this coevolving dynamical system robustly yields three basic states of collective behavior with their self-organized interactions. The first is the two-cluster state, in which the oscillators are organized into two synchronized groups. The second is the coherent state, in which the oscillators are arranged sequentially in time. The third is the chaotic state, in which the relative phases between oscillators and their coupling weights are chaotically shuffled. Furthermore, we demonstrate that self-assembled multiclusters can be designed by controlling the weight dynamics. Note that the phase patterns of the oscillators and the weighted network of interactions between them are simultaneously organized through this coevolving dynamics. We expect that these results will provide new insight into self-assembly mechanisms by which the collective behavior of a rhythmic system emerges as a result of the dynamics of adaptive interactions.

Financial markets provide an ideal frame for the study of crossing or first-passage time events of non-Gaussian correlated dynamics, mainly because large data sets are available. Tick-by-tick data of six futures markets are herein considered, resulting in fat-tailed first-passage time probabilities. The scaling of the return with its standard deviation collapses the probabilities of all markets examined—and also for different time horizons—into single curves, suggesting that first-passage statistics is market independent (at least for high-frequency data). On the other hand, a very closely related quantity, the survival probability, shows, away from the center and tails of the distribution, a hyperbolic $t^{-1/2}$ decay typical of a Markovian dynamics, albeit the existence of memory in markets. Modifications of the Weibull and Student distributions are good candidates for the phenomenological description of first-passage time properties under certain regimes. The scaling strategies shown may be useful for risk control and algorithmic trading.

We analyze the efficiency of available algorithms for the simulation of classical fidelity and show that their computational costs increase exponentially with the number of degrees of freedom. Then we present an algorithm for which the number of trajectories needed for convergence is independent of the system's dimensionality and show that, within a continuous family of algorithms, this algorithm is the only one with this property. Simultaneously we propose a general analytical approach to estimate efficiency of trajectory-based methods and suggest how to use it to accelerate calculations of other classical correlation functions. Converged numerical results are provided for systems with phase space volume $2^{1700}$ times larger than the volume of the initial state.

Slip behavior of three liquids with distinct molecular shapes—linear (hexadecane), branched (pentaerythritol tetra), and a chain of rings (polyphenylether)—is studied using molecular dynamics simulation and reduced-order modeling. Slip at a liquid-solid interface is shown to be affected by the molecular structure of the liquid. A two-dimensional Frenkel-Kontorova model captures the fundamental structural features of the liquid molecules and gives insight into how molecules flex and slip along the surface. We formulate an approximation to the Peierls-Nabarro energy which incorporates both the position of liquid atoms relative to substrate atoms and molecular flexibility. We find that increased molecular flexibility can lead to reduced slip by allowing the liquid to conform epitaxially to the substrate with only a small energetic penalty. Liquid molecules which are less flexible can conform to the substrate only with greater expense of conformational energy, and so exhibit larger slip.

We present an investigation for the generation of a dust-acoustic rogue wave in a dusty plasma composed of negatively charged dust grains, as well as nonextensive electrons and ions. For this purpose, the reductive perturbation technique is used to obtain a nonlinear Schrödinger equation. The critical wave-number threshold $k_c$, which indicates where the modulational instability sets in, has been determined precisely for various regimes. Two different behaviors of $k_c$ against the nonextensive parameter $q$ are found. For small $k_c$, it is found that increasing $q$ would lead to an increase of $k_c$ until $q$ approaches a certain value $q_c$, then further increase of $q$ beyond $q_c$ decreases the value of $k_c$. For large $k_c$, the critical wave-number threshold $k_c$ is always increasing with $q$. Within the modulational instability region, a random perturbation of the amplitude grows and thus creates dust-acoustic rogue waves. In order to show that the characteristics of the rogue waves are influenced by the plasma parameters, the relevant numerical analysis of the appropriate nonlinear solution is presented. The nonlinear structure, as reported here, could be useful for controlling and maximizing highly energetic pulses in dusty plasmas.

This paper investigates the dynamics of the flapping regime of a filament placed in a two-dimensional soap-film flow for different filament lengths and flow speeds. It was found that the onset of flapping is quasiperiodic, with the main flapping amplitude and frequency modulated by low-amplitude, low-frequency oscillation. At higher flow velocities, the oscillation becomes chaotic. The transition to chaos occurs via the quasiperiodic route to chaos. A new bistability phenomenon was discovered in which the system alternates between the stretched-straight and oscillatory states, which is here referred to as “switching oscillation.” Unlike some previously reported forms of bistability, in this case the system alternates between the two states continuously, without any external perturbation.

A scheme is developed for estimating state-dependent drift and diffusion coefficients in a stochastic differential equation from time-series data. The scheme does not require to specify parametric forms for the drift and diffusion coefficients in advance. In order to perform the nonparametric estimation, a maximum likelihood method is combined with a concept based on a kernel density estimation. In order to deal with discrete observation or sparsity of the time-series data, a local linearization method is employed, which enables a fast estimation.

Motivated by a long-standing debate concerning the nature and interrelations of surface-tension variables in fluid membranes, we reformulate the thermodynamics of a membrane vesicle as a generic two-dimensional finite system enclosing a three-dimensional volume. The formulation is shown to require two tension variables, conjugate to the intensive constraints of area per molecule and volume-to-area ratio. We obtain the relation between these two variables in various scenarios, as well as their correspondence to other definitions of tension variables for membranes. Several controversies related to membrane tension are thereby resolved on a model-free thermodynamic level. The thermodynamic formulation may be useful also for treating large-scale properties of vesicles that are insensitive to the membrane's detailed statistical mechanics and interactions.

We discuss the structure and asymptotic long-time properties of coupled equations for the moments of a Brownian particle's momentum $\langle p^n\left(t\right)\rangle$ derived microscopically beyond the lowest approximation in the weak coupling parameter $\lambda$. Generalized fluctuation-dissipation relations are derived and shown to ensure convergence to thermal equilibrium to any order in $\lambda$.

Biphenylene and pyrene were dissolved in the nematic and smectic-$A$ phases of the liquid crystal 4,$4^{\prime }$-di-$n$-heptyl-azoxybenzene and the orientational order parameters of both solutes and solvent measured via proton and deuteron nuclear-magnetic-resonance spectroscopy. This new data set was then merged with the one previously obtained, formed by 4,$4^{\prime }$-di-chloro-benzene and naphthalene as solutes in the same solvent, and the resulting overall data set analyzed with a statistical thermodynamic density-functional theory to provide positional-orientational distribution functions of the various solutes along with the smectic solvent's positional order parameters.

Binary mixtures of chiral liquid crystalline homologs have been studied. One compound designated 9ZBL exhibited reentrancy of a paraelectric smectic-$A^{*}$ phase, Sm$A_{\mathrm{RE}}^{*}$, below the ferroelectric Sm$C^{*}$ phase in the Sm$A^{*}$-Sm$C^{*}$-Sm$A_{\mathrm{RE}}^{*}$ phase sequence. Stabilization of the Sm$A_{\mathrm{RE}}$ phase is established from studying binary mixtures of 9ZBL with its neighboring homologs 8ZBL and 10ZBL. Compound 8ZBL exhibits only Sm$A^{*}$ phase in a wide temperature range and for 10ZBL the Sm$A^{*}$-Sm$C^{*}$ phase sequence is observed on cooling. X-ray studies, dielectric spectroscopy, polarization, and tilt angle measurements have been carried out to characterize studied materials. For binary mixtures 9ZBL-10ZBL the reentrant Sm$A_{\mathrm{RE}}^{*}$ phase is observed for all studied concentrations. For binary mixtures 9ZBL-8ZBL a very small amount of 8ZBL (up to 0.5 mole $\%$) causes disappearance of the Sm$C^{*}$ phase. Nevertheless, a small anomaly in the temperature dependencies of the layer spacing, $d\left(T\right)$, accompanied by a significant decrease in diffracted x-ray intensity occurs within the Sm$A^{*}$ phase for mixtures containing up to 20 mole $\%$ of 8ZBL. This anomaly is evidence of the existence of a boundary between the Sm$A^{*}$ and Sm$A_{\mathrm{RE}}^{*}$ phases, thus proving their different nature.

Randomness is known to affect the dynamical behavior of many systems to a large extent. In this paper we investigate how the nature of randomness affects the dynamics in a zero-temperature quench of the Ising model on two types of random networks. In both networks, which are embedded in a one-dimensional space, the first-neighbor connections exist and the average degree is 4 per node. In random model A the second-neighbor connections are rewired with a probability $p$, while in random model B additional connections between neighbors at a Euclidean distance $l(l>1)$ are introduced with a probability $P\left(l\right)\propto l^{-\alpha }$. We find that for both models, the dynamics leads to freezing such that the system gets locked in a disordered state. The point at which the disorder of the nonequilibrium steady state is maximum is located. The behavior of dynamical quantities such as residual energy, order parameter, and persistence are discussed and compared. Overall, the behavior of physical quantities are similar, although subtle differences are observed due to the difference in the nature of randomness.

Understanding key structural properties of large-scale networks is crucial for analyzing and optimizing their performance and improving their reliability and security. Here, through an analysis of a collection of data networks across the globe as measured and documented by previous researchers, we show that communications networks at the Internet protocol (IP) layer possess global negative curvature. We show that negative curvature is independent of previously studied network properties, and that it has a major impact on core congestion: the load at the core of a finite negatively curved network with $N$ nodes scales as $N^2$, as compared to $N^{1.5}$ for a generic finite flat network.