We report on numerical simulations of a vibrated granular bed including the effect of the ambient air, generating the famous Faraday heaps known from experiment. A detailed analysis of the forces shows that the heaps are formed and stabilized by the airflow through the bed while the gap between bed and vibrating bottom is growing, confirming the pressure gradient mechanism found experimentally by Thomas and Squires [Phys. Rev. Lett. 81, 574 (1998)], with the addition that the airflow is partly generated by isobars running parallel to the surface of the granular bed. Importantly, the simulations also explain the heaping instability of the initially flat surface and the experimentally observed coarsening of a number of small heaps into a larger one.

A general theoretical background is introduced for characterization of conformational motions in protein molecules, and for building reduced coarse-grained models of proteins, based on the statistical analysis of their phase trajectories. Using the projection operator technique, a system of coupled generalized Langevin equations is derived for essential collective coordinates, which are generated by principal component analysis of molecular dynamic trajectories. The number of essential degrees of freedom is not limited in the theory. An explicit analytic relation is established between the generalized Langevin equation for essential collective coordinates and that for the all-atom phase trajectory projected onto the subspace of essential collective degrees of freedom. The theory introduced is applied to identify correlated dynamic domains in a macromolecule and to construct coarse-grained models representing the conformational motions in a protein through a few interacting domains embedded in a dissipative medium. A rigorous theoretical background is provided for identification of dynamic correlated domains in a macromolecule. Examples of domain identification in protein G are given and employed to interpret NMR experiments. Challenges and potential outcomes of the theory are discussed.

Recent experiments and models for the spreading of liquids laden with nanoparticles have demonstrated particle layering at the three-phase contact line; this is associated with the structural component of the disjoining pressure. Effects driven by structural disjoining pressures occur on scales longer than the diameter of a particle, below which other disjoining pressure components such as van der Waals and electrostatic forces are dominant. Motivated by these experimental observations, we investigate the dynamic spreading of a droplet laden with nanoparticles in the presence of structural disjoining pressure effects. We use lubrication theory to derive evolution equations for the interfacial location and the concentration of particles. These equations account for the presence of the structural component of the disjoining pressure for film thicknesses exceeding the diameter of a nanoparticle; below such thicknesses, van der Waals forces are assumed to be operative. The resulting evolution equations, for the particle motion and free surface position, are solved allowing for the viscosity to vary as a function of nanoparticle concentration. The results of our numerical simulations demonstrate qualitative agreement with experimental observations of a “step” emerging from the contact line. The results are also relevant to a wide range of other phenomena involving layering, or terraced spreading of nanodroplets, or stepwise thinning of micellar thin films.

We present an approach for model reduction of nonlinear dynamical systems based on proper orthogonal decomposition (POD). Our method, derived from the density-matrix renormalization group, provides a significant reduction in computational effort for the calculation of the reduced system, compared to a POD. The efficiency of the algorithm is tested on the one-dimensional Burgers equations and a one-dimensional equation of the Fisher type as nonlinear model systems.

It has recently been observed that a large reverberant cavity admits a classically motivated random $s$ matrix that is not unitary but that can be made so in a minimally invasive manner. A random process with an envelope $⟨s^2\left(t\right)⟩\sim \exp (-t∕T_H)$ representing reflection from a structure having no internal time scales other than Heisenberg time $T_H$ was shown to lead to a unitary $S$ matrix exhibiting mesoscopic behaviors not present in the classically inspired original $⟨s^2\left(t\right)⟩$. These included enhanced backscatter, quantum echo, power law tails, and level repulsion. Here the procedure is extended to two systems having, in addition to Heisenberg times, internal time scales corresponding to conduction and diffusion. The repaired $S$ matrices for coupled rooms and one-dimensional random structures with multiple scattering are found to correspond to Wigner $K$ matrices with signatures of localization.

At nonequilibrium phase transitions into absorbing (trapped) states, it is well known that the directed percolation (DP) critical scaling is shared by two classes of models with a single $\left(S\right)$ absorbing state and with infinitely many (IM) absorbing states. We study the crossover behavior in one dimension, arising from a considerable reduction of the number of absorbing states (typically from the IM-type to the $S$-type DP models) by following two different (excitatory or inhibitory) routes which make the auxiliary field density abruptly jump at the crossover. Along the excitatory route, the system becomes overly activated even for an infinitesimal perturbation and its crossover becomes discontinuous. Along the inhibitory route, we find a continuous crossover with universal crossover exponent $\varphi \simeq 1.78\left(6\right)$, which is argued to be equal to ${\nu }_{\Vert }$, the relaxation time exponent of the DP universality class on a general footing. This conjecture is also confirmed in the case of the directed Ising (parity-conserving) class. Finally, we discuss the effect of diffusion on the IM-type models and suggest an argument why diffusive models with some hybrid-type reactions should belong to the DP class.

Shear viscosities of fluid water have been measured to $300°\mathrm{C}$ and $6\mathrm{GPa}$ $\left(60\mathrm{kbar}\right)$. Measurements were made in a diamond-anvil cell with a rolling-ball technique. Enskog’s equation for viscosity, coupled with an ad hoc assumption that increased collision rates are due to an “excluded volume”, yield excellent matches to the data at temperatures of $100°\mathrm{C}$ and over, without any freely variable parameter. The data overlap the pressure-temperature range in which experiments on shocked water have previously been interpreted to indicate extremely high viscosities. It is shown conclusively that viscosities in this region are very close to those at ambient temperature. Further, it is argued that explanations of high apparent viscosities which rely on the putative formation of ice behind the shock front are probably incorrect.

A small-angle neutron–x-ray–light-scattering model for multilamellar vesicles is developed on the basis of a simple geometry. $N$ spherical shells with radii of an arithmetic series are allowed for displacements $\Delta R$ which are limited by $\Delta R<R∕N$ due to sterical reasons, with $R$ being the radius of the vesicle. The model shows many properties over a large $Q$ range which include a Guinier region, a first power law, a correlation peak, and a second power law connected to the surface properties of the bilayer. The first power law is related to the compactness of the vesicle and lies between $Q^{-2}$ for surfaces and $Q^{-4}$ for compact volumes (Porod law). The exact exponent is related to the number of shells $N$. The correlation peak has a maximum sharpness for rather small displacements $\Delta R$, but no second order peak is predicted. Only for rather large displacements the correlation peak widens up and shifts to smaller scattering angles. Then the important bilayer spacing is larger. The predictive power of the model lies in the connection of the compactness with $N$ and in the maximum correlation peak sharpness. This model considers many length scales at a time while existing theories focus on length scales of the bilayer spacing and the bilayer itself.

In eukaryotic genes, the protein coding sequence is split into several fragments, the exons, separated by noncoding DNA stretches, the introns. Prokaryotes do not have introns in their genomes. We report calculations of the stability domains of actin genes for various organisms in the animal, plant, and fungi kingdoms. Actin genes have been chosen because they have been highly conserved during evolution. In these genes, all introns were removed so as to mimic ancient genes at the time of the early eukaryotic development, i.e., before intron insertion. Common stability boundaries are found in evolutionarily distant organisms, which implies that these boundaries date from the early origin of eukaryotes. In general, the boundaries correspond with intron positions in the actins of vertebrates and other animals, but not much for plants and fungi. The sharpest boundary is found in a locus where fungi, algae, and animals have introns in positions separated by one nucleotide only, which identifies a hot spot for insertion. These results suggest that some introns may have been incorporated into the genomes through a thermodynamically driven mechanism, in agreement with previous observations on human genes. They also suggest a different mechanism for intron insertion in plants and animals.

The genetic repressilator circuit consists of three transcription factors, or repressors, which negatively regulate each other in a cyclic manner. This circuit was synthetically constructed on plasmids in Escherichia coli and was found to exhibit oscillations in the concentrations of the three repressors. Since the repressors and their binding sites often appear in low copy numbers, the oscillations are noisy and irregular. Therefore, the repressilator circuit cannot be fully analyzed using deterministic methods such as rate equations. Here we perform stochastic analysis of the repressilator circuit using the master equation and Monte Carlo simulations. It is found that fluctuations modify the range of conditions in which oscillations appear as well as their amplitude and period, compared to the deterministic equations. The deterministic and stochastic approaches coincide only in the limit in which all the relevant components, including free proteins, plasmids, and bound proteins, appear in high copy numbers. We also find that subtle features such as cooperative binding and bound-repressor degradation strongly affect the existence and properties of the oscillations.

The forced rupture of single chemical bonds under external load is addressed. A general framework is put forward to optimally utilize the experimentally observed rupture force data for estimating the parameters of a theoretical model. As an application, we explore to what extent a distinction between several recently proposed models is feasible on the basis of realistic experimental data sets.

This paper describes the economic phenomenon of price spiking in electric power markets and introduces an alternative way to model it. A stochastic FitzHugh-Nagumo dynamics in a special regime is proposed as a basic model for the power market, and an extension of the FitzHugh-Nagumo system is introduced to improve the statistical features of the basic model. Ideas from stochastic and coherence resonance are used to discuss the models.

A recently proposed [Ph. Chomaz, F. Gulminelli, and O. Juillet, Ann. Phys. (Paris) 320, 135 (2005)] statistical treatment of finite unbound systems in the presence of collective motions is applied to a classical Lennard-Jones system, numerically simulated through molecular dynamics. In the ideal gas limit, the flow dynamics can be exactly recast into effective time-dependent Lagrange parameters acting on a standard Gibbs ensemble with an extra total energy conservation constraint. Using this same ansatz for the low-density freeze-out configurations of an interacting expanding system, we show that the presence of flow can have a sizable effect on the microstate distribution.

We present a comparison between finite differences schemes and a pseudospectral method applied to the numerical integration of stochastic partial differential equations that model surface growth. We have studied, in $1+1$ dimensions, the Kardar, Parisi, and Zhang model (KPZ) and the Lai, Das Sarma, and Villain model (LDV). The pseudospectral method appears to be the most stable for a given time step for both models. This means that the time up to which we can follow the temporal evolution of a given system is larger for the pseudospectral method. Moreover, for the KPZ model, a pseudospectral scheme gives results closer to the predictions of the continuum model than those obtained through finite difference methods. On the other hand, some numerical instabilities appearing with finite difference methods for the LDV model are absent when a pseudospectral integration is performed. These numerical instabilities give rise to an approximate multiscaling observed in earlier numerical simulations. With the pseudospectral approach no multiscaling is seen in agreement with the continuum model.

We use complete enumeration of self-avoiding chains of up to $N=26$ monomers in two-dimensional lattices to investigate the effect of alternative implementations of backbone hydrogen bonds on the cooperativity of homopolypeptide collapse. Following a recent study on protein folding models, we use the square lattice with $z=3$ local conformations per monomer and lattice extensions containing diagonal steps which result in $z=5$ or $z=7$ and assume that only a subset of $z_h<z$ local conformations is compatible with hydrogen bond formation. As previously observed in heteropolymeric folding, a significant increase in cooperativity, as measured by ${\kappa }_2$ values, results from the coupling between hydrogen bonds and hydrophobic interactions, in such a way that hydrophobic contacts are favorable only when contacting monomers are involved in hydrogen bond formation. For some $z/z_h$ combinations the energy distribution is bimodal at the collapse transition temperature. The situation can be regarded as if all hydrophobic contacts actually decrease the energy by the same amount, $2h$, with the addition of an energetic increase, ${ϵ}_2=h$, as a penalty for each contacting monomer not satisfying the hydrogen bond condition. Cooperativity is little affected and might even decrease, however, when hydrogen bonds produce a decrease in energy by the same amount, ${ϵ}_1=h$, for each bonding monomer. For the more general situation when the hydrogen bond effect is not equal, in modulus, to the hydrophobic interaction, i.e., ${ϵ}_2\ne h$ or ${ϵ}_1\ne h$, we observe a pronounced increase in ${\kappa }_2$ for small ${ϵ}_2$, with a maximum around ${ϵ}_2/h\approx 1.5$, followed by a gradual decrease to a limiting value at large ${ϵ}_2$. The opposite behavior is observed when ${ϵ}_1$ is varied. The observed qualitative difference is shown to arise from opposite effects on the convexity of the total density of states of the system when subdensities corresponding to different numbers of hydrogen bonds are differently favored as opposed to the case when subdensities corresponding to different numbers of contacting monomers not forming hydrogen bonds are differently unfavored. Potential implications for the cooperativity of protein folding and protein unspecific collapse are discussed.

Gene expression and differentiation were shown to be stochastic processes. However, cells in a tissue can coordinate their behavior, including gene expression and differentiation pathways choice. A tissue of coupled cells is modeled as a two-dimensional regular square lattice of identical cells, each a three-dimensional compartment with a gene regulatory network (GRN) and a toggle switch (TS). The dynamics is driven by a delayed stochastic simulation algorithm, nearest neighbor cells are coupled by normally distributed time delayed reactions allowing interchange of proteins, and gene expression is a multiple time delayed reaction. It is defined the coupling strength $\left(C\right)$, to characterize the lattice structure as a function of the rate constants of the reactions coupling nearest neighbor cells. Conditions are investigated for the emergence of synchronization and stable differentiation of cells within a tissue. From the time series of the cells GRNs, the tissue dynamical stability $\left(S\right)$ is computed from the average toggling period of each GRN. The synchronization of cells’ proteins expression levels is measured by their time series entropy $\left(H\right)$. It is shown that the tissue goes through various dynamical regimes as $C$ is increased, by measuring $H$ and $S$. For null $C$, the cells GRNs toggle asynchronously. For weak $C$, cells synchronize by regions of space. Increasing $C$, the tissue becomes homogeneously synchronous. As $C$ is further increased, $S$ goes through a phase transition, from synchronized toggling to stable, where all cells produce one and the same protein. Finally, increasing $C$ even further, a new stable state emerges where both genes of all cells are expressed and bistability is lost. This state, resembling an infinitely long transient, is an emergent behavior not observable in a single TS. The results provide an explanation of how cells with bistable GRNs, inherently stochastic, can synchronize or uniformly differentiate into stable states, by interacting with nearest neighbors.

Many real-world networks analyzed in modern network theory have a natural spatial element; e.g., the Internet, social networks, neural networks, etc. Yet, aside from a comparatively small number of somewhat specialized and domain-specific studies, the spatial element is mostly ignored and, in particular, its relation to network structure disregarded. In this paper we introduce a model framework to analyze the mediation of network structure by spatial embedding; specifically, we model connectivity as dependent on the distance between network nodes. Our spatially embedded random networks construction is not primarily intended as an accurate model of any specific class of real-world networks, but rather to gain intuition for the effects of spatial embedding on network structure; nevertheless we are able to demonstrate, in a quite general setting, some constraints of spatial embedding on connectivity such as the effects of spatial symmetry, conditions for scale free degree distributions and the existence of small-world spatial networks. We also derive some standard structural statistics for spatially embedded networks and illustrate the application of our model framework with concrete examples.

The thermodynamics and topology of mean-field models with $2+k$ body interaction terms (generalizing $XY$ model) are derived. Focusing on two particular cases ($2+4$ and $2+6$ body interaction terms), a comparison between thermodynamic (phase transition energy, thermodynamically forbidden energy regions) and topological (singularity and curvature of saddle entropy) properties is performed. We find that (i) a topological change is present at the phase transition energy; however, (ii) only one topological change occurs, also for those models exhibiting two phase transitions; (iii) the order of a phase transition is not completely signaled by the curvature of topological quantities.

Alkanes on water show a two-stage wetting transition. Upon raising the temperature, a first-order transition from a molecularly thin to a mesoscopically thick liquid film is followed by a continuous divergence of the film thickness. This second transition is brought about by long-range interactions between adsorbate and substrate and is, therefore, referred to as long-range critical wetting. The divergence of the film thickness is theoretically expected to occur according to the asymptotic power law $l\sim {\left(T_{w,c}-T\right)}^{{\beta }_s}$, with ${\beta }_s=-1$. This value has indeed been found for pentane on pure water; however, for hexane on salt solutions of different concentrations, ${\beta }_s=-0.73$ was found for a $1.5M$ solution of NaCl and ${\beta }_s=-0.57$ for a $2.5M$ salt solution. In addition, for hexane on a $2.5M$ solution of NaCl, an exponent of ${\alpha }_s=0.1$ was found from contact-angle measurements, differing greatly from the theoretically expected value of ${\alpha }_s=-1$. Using Dzyaloshinskii-Lifshitz-Pitaevskii theory, we calculate effective local exponents in order to explain the experimental findings. Taking into account the uncertainty of the exponents derived from experiments as well as the temperature range in which the measurements were carried out, a reasonable agreement between theory and experiment is found, thereby providing a consistent approach to resolving the apparently anomalous behavior of hexane on brine. The experimentally observed exponents ${\beta }_s=-0.57$ and ${\alpha }_s=0.1$ are also compatible with a long-range tricritical wetting transition, which is characterized by ${\beta }_s=-1/2$ and ${\alpha }_s=0$; this alternative explanation of the experimental findings is neither supported nor completely ruled out by our calculations.

We develop a molecular-statistical theory of the smectic-$A$–smectic-$C$ transition which is described as a transition of the order-disorder type. The theory is based on a general expansion of the effective interaction potential and employs a complete set of orientational order parameters. All the order parameters of the smectic-$C$ phase including the tilt angle are calculated numerically as functions of temperature for a number of systems which correspond to different transition scenario. The effective interaction potential and the parameters of the transition are also calculated for specific molecular models based on electrostatic and induction interaction between molecular dipoles. The theory successfully reproduces the main properties of both conventional and so-called “de Vries–type” smectic liquid crystals, clarifies the origin of the anomalously weak layer contraction and describes the tricritical behavior at the smectic-$A$–smectic-$C$ transition. The “de Vries behavior,” i.e., anomalously weak layer contraction is also obtained for a particular molecular model based on interaction between longitudinal molecular dipoles. A simple phenomenological model is presented enabling one to obtain explicit expressions for the layer spacing and the tilt angle which are used to fit the experimental data for a number of materials.