Here we consider dynamical problems as in linear response theory but for purely nonlinear systems where acoustic propagation is prohibited by the potential, e.g., the case of an alignment of elastic grains confined between walls. Our simulations suggest that in the absence of acoustic propagation, the system relaxes using only solitary waves and the eventual state does not resemble an equilibrium state. Further, the studies reveal that multiple perturbations could give rise to hot and cold spots in these systems. We first use particle dynamics based simulations to understand how one of the two unequal colliding solitary waves in the chain can gain energy. Specifically, we find that for head-on collisions the smaller wave gains energy, whereas when a more energetic wave overtakes a less energetic wave, the latter gains energy. The balance between the rate at which the solitary waves break down and the rate at which they grow eventually makes it possible for the system to reach a peculiar equilibriumlike phase that is characteristic of these purely nonlinear systems. The study of the features and the robustness of the fluctuations in time has been addressed next. A particular characteristic of this equilibriumlike or quasiequilibrium phase is that very large energy fluctuations are possible—and by very large, we mean that the energy can vary between zero and several times the average energy per grain. We argue that the magnitude of the fluctuations depend on the nature of the nonlinearity in the potential energy function and the feature that any energy must eventually travel as a compact solitary wave in these systems where the solitary wave energies may vary widely. In closing we address whether these fluctuations are peculiar to one dimension or can exist in higher dimensions. The study hence raises the following intriguing possibility. Are there physical or biological systems where these kinds of nonlinear forces exist, and if so, can such large fluctuations actually be seen? Implications of the study are briefly discussed.

We present a method for the evaluation of the interaction potential of an equilibrium classical system starting from the (partial) knowledge of its structure factor. The procedure is divided into two phases, both of which are based on the maximum entropy principle of information theory. First we determine the maximum entropy estimate of the radial distribution function constrained by the information contained in the structure factor. Next we invert the pair function and extract the interaction potential. The method is tested on a Lennard-Jones fluid at high density and the reliability of its results with respect to the missing information in the structure factor data are discussed. Finally, it is applied to the experimental data of liquid sodium at 100 ${}^{\circ }$C.

For one-component fluids, we examine the predrying phase transition between a thin and thick low-density layer in liquid on a wall repelling the fluid. This is the case of a hydrophobic wall for water. A predrying line starts from the coexistence curve and ends at a surface critical point in the phase diagram. We calculate this line numerically using the van der Waals model and analytically using the free-energy expansion up to the quartic order. We also examine the predrying dynamics of a layer created on a hydrophobic spot on a heterogeneous wall. It is from a thin to thick layer during decompression and from a thick to thin layer during compression. Upon the transition, a liquid region above the film is cooled for decompression and heated for compression due to latent heat convection, and a small pressure pulse is emitted from the film into the liquid.

As a generic model for phase equilibria under confinement in a thin-film geometry in the presence of a gradient in the field conjugate to the order parameter, an Ising-lattice gas system is studied by both Monte Carlo simulations and a phenomenological theory. Choosing an $L\times L\times D$ geometry with $L\gg D$ and periodic boundary conditions in the $x,y$ directions, we place competing surface fields on the two $L\times L$ surfaces. In addition, a field gradient $g$ is present in the $z$ direction across the film, in competition with the surface fields. At temperatures $T$ exceeding the critical temperature of the interface localization-delocalization transition, one finds a phase coexistence between oppositely oriented domains, aligned parallel to the surface fields and separated by an interface in the center of the film, for small enough $g$. For a weak gradient, a second-order transition to a monodomain state occurs, but it becomes first order if $g$ exceeds a tricritical threshold. For sufficiently large gradients, another domain state becomes stabilized with domains oriented antiparallel to the surface fields.

The phase behavior of ($A$-comb-$C$)-$b$-($B$-comb-$C$) diblock copolymer melts is investigated using the strong segregation theory approach. Three different regimes are distinguished. In regime 1 both disordered comb blocks are microphase separated from each other, in regime 2 the side chains $C$ are microphase separated from the disordered $A$-$b$-$B$ diblock backbones, and, finally, in regime 3 all species $A$, $B$, and $C$ are microphase separated. In the first regime the behavior is similar to that of a simple diblock copolymer melt with a renormalized Flory-Huggins interaction parameter. In regime 2 the region of stability of the different phases is significantly changed compared to simple diblocks due to the comb architecture. The fully microphase separated case, regime 3, is characterized by hierarchical structure formation. We restrict the analysis to systems where self-assembly results in the formation of alternating $C$ layers and internally microphase separated $AB$ layers. The latter consist of alternating $A$ and $B$ layers or disks of the minority component. In the former case, the $A$ and $B$ layers are generally perpendicular to the $C$ layers. The parallel orientation is only possible for small grafting densities.

Based on simulations of stochastic three-phase and two-phase traffic flow models, we reveal that at a signalized city intersection under small link inflow rates at which a vehicle queue developed during the red phase of the light signal dissolves fully during the green phase, i.e., no traffic gridlock should be expected, nevertheless, spontaneous traffic breakdown with subsequent city gridlock occurs with some probability after a random time delay. In most cases, this traffic breakdown is initiated by a phase transition from free flow to a synchronized flow occurring upstream of the queue at the light signal. The probability of traffic breakdown at the light signal is an increasing function of the link inflow rate and duration of the red phase of the light signal.

We study the Axelrod's cultural adaptation model using the concept of cluster-size entropy $S_c$, which gives information on the variability of the cultural cluster size present in the system. Using networks of different topologies, from regular to random, we find that the critical point of the well-known nonequilibrium monocultural-multicultural (order-disorder) transition of the Axelrod model is given by the maximum of the $S_c\left(q\right)$ distributions. The width of the cluster entropy distributions can be used to qualitatively determine whether the transition is first or second order. By scaling the cluster entropy distributions we were able to obtain a relationship between the critical cultural trait $q_c$ and the number $F$ of cultural features in two-dimensional regular networks. We also analyze the effect of the mass media (external field) on social systems within the Axelrod model in a square network. We find a partially ordered phase whose largest cultural cluster is not aligned with the external field, in contrast with a recent suggestion that this type of phase cannot be formed in regular networks. We draw a $q-B$ phase diagram for the Axelrod model in regular networks.

Waves of excitation through excitable media, such as cardiac tissue, can propagate as plane waves or break up to form reentrant spiral waves. In diseased hearts reentrant waves can be associated with fatal cardiac arrhythmias. In this paper we investigate the conditions that lead to wave break, reentry, and propagation failure in mathematical models of heterogeneous excitable media. Two types of heterogeneities are considered: sinks are regions in space in which the voltage is fixed at its rest value, and breaks are nonconducting regions with no-flux boundary conditions. We find that randomly distributed heterogeneities in the medium have a decremental effect on the velocity, and above a critical density of such heterogeneities the conduction fails. Using numerical and analytical methods we derive the general relationship among the conduction velocity, density of heterogeneities, diffusion coefficient, and the rise time of the excitation in both two and three dimensions. This work helps us understand the factors leading to reduced propagation velocity and the formation of spiral waves in heterogeneous excitable media.

Using extensive Monte Carlo simulations, we study the effective permeability, porosity, and percolation properties of two-dimensional fracture networks in which the fractures are represented by rectangles of finite widths. The parameters of the study are the width of the fractures and their number density. For low and intermediate densities, the average porosity of the network follows a power-law relation with the density. The exponent of the power law itself depends on the fractures’ width through a power law. For an intermediate range of the densities, the effective permeability scales with the fractures’ width as a power law, with an exponent that depends on the density. For high densities the effective permeability also depends on the porosity through a power law, with an exponent that depends on the fractures’ width. In agreement with the results, experimental data also indicate the existence of a power-law relationship between the effective permeability and porosity in consolidated sandstones and sedimentary rocks with a nonuniversal exponent. The percolation threshold or critical number density of the fractures depends on their width and is maximum if they are represented by squares, rather than rectangles.

A theoretical description of the radial density profile for charged particles with Yukawa interaction in a harmonic trap is described. At strong Coulomb coupling shell structure is observed in both computer simulations and experiments. Correlations responsible for such shell structure are described here using a recently developed model based in density functional theory. A wide range of particle number, Coulomb coupling, and screening lengths is considered within the fluid phase. A hypernetted chain approximation shows the formation of shell structure, but fails to give quantitative agreement with Monte Carlo simulation results at strong coupling. Significantly better agreement is obtained within the hypernetted chain structure using a renormalized coupling constant, representing bridge function corrections.

We show numerically that the interaction potential or force mediated by a liquid crystalline blue phase (BP) between two parallel plates exhibits oscillatory behavior with variation of the interplate distance, when the parallel plates impose strong normal anchoring. Its periodicity is approximately half of the unit-cell dimension of the bulk BP. The interaction arises from the deformation of the confined BP structure around the midplane of the system. The oscillatory interaction can be regarded as a clear manifestation of the BP ordering, because the cholesteric helical alignment adopted by a chiral liquid crystal cannot yield an oscillatory interaction.

We investigate the electrical response of Milli-Q deionized water by using a fractional diffusion equation of distributed order with the interfaces (i.e., the boundary conditions at the electrodes limiting the sample) governed by integrodifferential equations. We also consider that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies Poisson’s equation. In addition, the good agreement between the experimental data and this approach evidences the presence of anomalous diffusion due to the surface effects in this system.

Hysteretic systems may exhibit a runaway avalanche in which a large fraction of the constituents of the system collectively change state. It would be very valuable to understand the role that interaction strength between constituents plays in the size of such catastrophic runaway avalanches. We use a simple model, the random field Ising model, to study how the size of the runaway avalanche changes as the coupling between spins, $J$, is tuned. In particular, we calculate $P\left(S\right)$, the distribution of size changes $S$ in the runaway avalanche size as $J$ comes close to a critical value $J_c$, and find that the distribution scales as $P\left(S\right)\sim S^{-\tau }\mathcal{D}(S^{\sigma }(J/J_c-1))$, with $\tau$ and $\sigma$ critical exponents and $\mathcal{D}\left(x\right)$ a universal scaling function. In mean field theory we find $\tau =3/2$, $\sigma =1/2$, and $\mathcal{D}\left(x\right)=\exp [-{\left(3x\right)}^{1/\sigma }/2]$. On the basis of these results and previous studies, we also predict that for three dimensions $\tau =1.6$ and $\sigma =0.24$.

We study the thermalization of energetic electrons dilutely dispersed in inert gas atomic moderators, with and without the presence of an external electric field. We investigate the shape relaxation of the electron distribution function relative to the steady-state distribution by means of the Kullback-Leibler entropy. The departure of the distribution function from a local Maxwellian parametrized by the temperature of the electrons is also considered with a functional analogous to the Kullback-Leibler entropy. For neon and argon as moderators, we found no evidence for the formation of a local Maxwellian followed by a slower relaxation to equilibrium. The momentum-transfer cross section for e-Ne collisions is almost constant with energy, whereas the e-Ar momentum-transfer cross section has a deep Ramsauer-Townsend minimum and a strong energy dependence. The role of the Ramsauer minimum in the relaxation processes is investigated. The time-dependent Lorentz-Fokker-Planck equation is solved for the speed distribution of the electrons with a finite difference method. A pseudospectral method is also used to investigate the spectral properties of the Fokker-Planck operator. In spite of the multi-exponential time dependence of the speed distribution function, we show that a single average relaxation time can be defined to characterize the relaxation to equilibrium.

Centrifuge force is used to study granular materials in low gravity conditions. We consider a monolayer of noncohesive spherical grains placed on a plate. Reduced gravity conditions can be simulated in the plane by tilting or by rotating the plate. We compare both approaches experimentally. The volume fraction is found to increase with the apparent gravity and saturates. A model based on the exponential distribution of the Voronoi cell areas has been built and is in excellent agreement with the experimental data by extrapolating the fits of the data. Moreover, numerical simulations exhibit that more arches can be maintained at low apparent gravities than at high.

RNA polymerase II (RNAPII) is the responsible motor protein for transcription. Here we report the formulation and results of a cellular automaton model of the RNAPII dynamics of gene transcription that takes account the effect of the velocity change according to the gene position, such as occurs in introns and exons. We describe RNAPII dynamics in terms of the properties in the time domain, such as elapsed time, residence time, and time intervals. We found that the RNAPII molecules move as a free-flow state, though regions of reduced velocity do exist such as exons, as far as the time interval between nearest RNAPII molecules is larger than the time required for an RNAPII passing the exclusion length in the velocity reduction region. On the other hand, if the reduction is strong enough to reach a certain threshold, at the maximally reductive velocity region, a transition occurs from the RNAPII free-flow state to the states with congested and repetitive flows. We analytically obtained the conditions for these flow states and the transition threshold. From simulations of high-density RNAPII in the SAMD4A gene with the strong blockade, we confirmed the transition from free flow to the repetitive and congested flows, suggesting that the transition may serve as a regulatory mechanism of gene expression. By fitting the experimentally observed RNAPII density profile of the SAMD4A gene during the course of transcription of the normal and altered gene (in knock-down cells) with or without roadblock, we found that the RNAPII density flow is a free state. However, even in this free state, there is a long-range correlation between RNAPII molecules, ranging from 1 to 20 min, with the corresponding distance from 3 to 80 kbp, during transcription in normal cells. This long-range correlation probably relates to the higher-order DNA loop structure.

The motility of a microorganism which tries to avoid a poisoned environment by chemotaxis is studied within a simple model which couples its velocity to the concentration field of the poison. The latter is time independent but inhomogeneous in space. The presence of the poison is assumed to irreversibly reduce the propulsion speed. The model is solved analytically for different couplings of the total poison dose experienced by the microbe to the propulsion mechanism. In a stationary poison field resulting from a constant emission of a fixed point source, we find a power law for the distance traveled by the microbe as a function of time with a nonuniversal exponent which depends on the coupling in the model. With an inverted sign in the couplings, the acceleration of microbe motion induced by a food field can also be described.

The aspect ratio ($\Gamma$) dependence of the heat transfer (Nusselt number Nu in dimensionless form) in turbulent (two-dimensional) Rayleigh-Bénard convection is numerically studied in the regime $0.4⩽\Gamma ⩽1.25$ for Rayleigh numbers ${10}^7⩽\text{Ra}⩽{\text{Ra}}^9$ and Prandtl numbers $\text{Pr}=0.7$ (gas) and $4.3$ (water). $\text{Nu}\left(\Gamma \right)$ shows a very rich structure with sudden jumps and sharp transitions. We connect these structures to the way the flow organizes itself in the sample and explain why the aspect ratio dependence of Nu is more pronounced for small Pr. Even for fixed $\Gamma$ different turbulent states (with different resulting Nu) can exist, between which the flow can or cannot switch. In the latter case the heat transfer thus depends on the initial conditions.

The mesoscopic structure of complex networks has proven a powerful level of description to understand the linchpins of the system represented by the network. Nevertheless, the mapping of a series of relationships between elements, in terms of a graph, is sometimes not straightforward. Given that all the information we would extract using complex network tools depend on this initial graph, it is mandatory to preprocess the data to build it on in the most accurate manner. Here we propose a procedure to build a network, attending only to statistically significant relations between constituents. We use a paradigmatic example of word associations to show the development of our approach. Analyzing the modular structure of the obtained network we are able to disentangle categorical relations, disambiguating words with success that is comparable to the best algorithms designed to the same end.