We study the percolation phase transition in hierarchical scale-free nets. Depending on the method of construction, the nets can be fractal or small world (the diameter grows either algebraically or logarithmically with the net size), assortative or disassortative (a measure of the tendency of like-degree nodes to be connected to one another), or possess various degrees of clustering. The percolation phase transition can be analyzed exactly in all these cases, due to the self-similar structure of the hierarchical nets. We find different types of criticality, illustrating the crucial effect of other structural properties aside from the scale-free degree distribution of the nets.

Parsons-Lee and Onsager theories are formulated for the isotropic-nematic transition in a binary mixture of hard rods and hard spheres. Results for the phase coexistence and for the equation of state in both phases for mixtures with different relative sizes and composition are presented. The two theories explain correctly the general behavior observed in experiments and computer simulations for these fluids. In particular, the theory accounts for the destabilization of the nematic phase when spherical or globular macromolecules are added to a system of rodlike colloids, and the entrance of the system into a demixed regime at high volume fractions of the spherical particles. Upon demixing a nematic state rich in rods coexists in equilibrium with an isotropic state much more diluted in the rodlike component. Onsager theory fails on quantitative grounds for aspect ratios of the rodlike molecules smaller than 100, and in the cases where the molar fractions of spheres becomes close to unity. On the contrary, the Parsons-Lee approximation remains accurate down to aspect ratios as small as 5. The spinodal analysis indicates that the isotropic-isotropic and nematic-nematic coexistences become feasible for sufficiently large spheres and long rods, respectively. The latter type of coexistence interferes partially with the isotropic-nematic coexistence regime of interest to the present work. Overall, the study serves to rationalize and control key aspects of the behavior of these binary nematogenic colloidal systems, which can be tuned with an appropriate choice of the relative size and molar fractions of the particles.

Genetic toggle switches (TSs) are one of the best studied small gene regulatory networks (GRNs), due to their simplicity and relevant role. They have been interpreted as decision circuits in cell differentiation, a process long hypothesized to be bistable [1], or as cellular memory units [2]. In these contexts, they must be reliable. Once a “decision” is made, the system must remain stable. One way to gain stability is by duplicating the genes of a TS and coupling the two TSs. Using a recent modeling strategy of GRNs, driven by a delayed stochastic simulation algorithm (delayed SSA) that allows modeling transcription and translation as multidelayed reactions, we analyze the stability of systems of coupled TSs. For this, we introduce the coupling strength $\left(C\right)$, a parameter to characterize the GRN structure, against which we compare the GRN stability $\left(S\right)$. We first show that time delays in transcription, associated to the promoter region release, ensure bistability of a TS, given no cooperative binding or self-activation reactions. Next, we couple two TSs and measure their toggling frequencies as $C$ varies. Three dynamical regimes are observed: (i) for weak coupling, high frequency synchronized oscillations, (ii) for average coupling, low frequency synchronized oscillations, and (iii) for strong coupling the system becomes stable after a transient, in one of two steady states. The system stability, $S$, goes through a first order phase transition as $C$ increases, in the average coupling regime. After, we study the effects of spatial separation in two compartments on the dynamics of two coupled TSs, where spatial separation is modeled as normally distributed random time delayed reactions. The phase transition of $S$, as $C$ increases, occurs for lower values of $C$ than when the two TSs are in the same compartment. Finally, we couple weakly and homogeneously several TSs within a single compartment and observe that as the number of coupled TSs increases, the system goes through the phase transition in $S$, from oscillatory to stable and for $C$ values lower than in the two previous cases.

A theoretical microscopic technique is proposed that may reduce multiphoton interaction in the excitation volume of a two-photon microscope. Since higher-order photobleaching is common in two-photon excitation microscopy, the study of thin samples is limited by increased photobleaching and photodamage. This limitation is elevated by using even coherent state light. The advantage of even coherent state light is that only excitation due to an even number of photons can survive. The very first nonzero even excitation (two-photon) can be isolated from the nearby one- and three-photon excitation. Hence the photobleaching due to one- and three-photon excitation can be eliminated and higher-order processes can be minimized owing to their small molecular cross section.

Nucleosynthesis, on the surface of accreting neutron stars, produces a range of chemical elements. We perform molecular dynamics simulations of crystallization to see how this complex composition forms new neutron star crust. We find chemical separation, with the liquid ocean phase greatly enriched in low atomic number elements compared to the solid crust. This phase separation should change many crust properties such as the thermal conductivity and shear modulus.

We investigate the spectral properties of a one-dimensional piecewise linear intermittent map, which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point. We explicitly derive generalized eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map for classes of observables and piecewise constant initial densities, and it is found that the Frobenius-Perron operator has two simple real eigenvalues 1 and ${\lambda }_d∊(-1,0)$ and a continuous spectrum on the real line [0,1]. From these spectral properties, we also found that this system exhibits a power law decay of correlations. This analytical result is found to be in a good agreement with numerical simulations. Moreover, the system can be extended to an area-preserving invertible map defined on the unit square. This extended system is similar to the baker transformation, but does not satisfy hyperbolicity. A relation between this area-preserving map and a billiard system is also discussed.

A low energy non-neutral electron plasma was confined with a magnetic mirror field and an electrostatic potential to investigate the basic confinement properties of a simple magnetic mirror trap. The mirror ratio of the magnetic field was increased up to 5. As expected the confinement time became longer as a function of the mirror ratio. The axially integrated radial density profiles in equilibrium were measured and compared with a theoretical model. The axial electrostatic oscillations of a confined electron plasma were also observed.

Based on the concept of a nonequilibrium steady state, we present a method to experimentally determine energy landscapes acting on colloidal systems. By measuring the stationary probability distribution and the current in the system, we explore potential landscapes with barriers up to several hundred $k_{\mathrm{B}}T$. As an illustration, we use this approach to measure the effective diffusion coefficient of a colloidal particle moving in a tilted potential.

Employing Monte Carlo simulations, we show that the topology of the phase diagram of a single flexible homopolymer chain changes in dependence on the range of an attractive square well interaction between the monomers. For a range of attraction larger than a critical value, the equilibrium phase diagram of the single polymer chain and the corresponding polymer solution phase diagram exhibit vapor (swollen coil, dilute solution), liquid (collapsed globule, dense solution), and solid phases. Otherwise, the liquid-vapor transition vanishes from the equilibrium phase diagram for both the single chain and the polymer solution. This change in topology of the phase diagram resembles the behavior known for colloidal dispersions. The interplay of enthalpy and conformational entropy in the polymer case thus can lead to the same topology of phase diagrams as the interplay of enthalpy and translational entropy in simple liquids.

We study work fluctuation theorems for oscillators in non-Markovian heat baths. By calculating the work distribution function for a harmonic oscillator with motion described by the generalized Langevin equation, the Jarzynski equality (JE), transient fluctuation theorem (TFT), and Crooks’ theorem (CT) are shown to be exact. In addition to this derivation, numerical simulations of anharmonic oscillators indicate that the validity of these nonequilibrium theorems does not depend on the memory of the bath. We find that the JE and the CT are valid under many oscillator potentials and driving forces, whereas the TFT is not applicable when the driving force is asymmetric in time and the potential is asymmetric in position.

We study the distribution function of a three-dimensional wormlike chain with a fixed orientation of one chain end using the exact representation of the distribution function in terms of the Green’s function of the quantum rigid rotator in a homogeneous external field. The transverse one-dimensional distribution function of the free chain end displays a bimodal shape in the intermediate range of chain lengths $(1.3L_p,\dots ,3.5L_p)$. We also present analytical results for short and long chains, which are in complete agreement with the results of previous studies obtained using different methods.

In this paper, we study the oscillatory dynamics of intracellular signaling cascades. We derive a reaction-diffusion model of the mitogen-activated protein kinase cascade, and use it to show how oscillations of the protein kinase concentrations can occur as a function of the depth of the cascade. We find that only cascades with depths of three or more layers undergo oscillatory instabilities. In addition, the oscillatory instability is spatially uniform. Thus, the oscillations synchronize the protein kinase concentrations and result in them being uniformly distributed in the cytosol, despite the presence of protein kinase diffusion. Finally, we show how the oscillations are perturbed when parallel cascades “crosstalk.” We find that the protein kinases in the downstream layers of the cascade are less perturbed than those in the upstream layers. In particular, cascades of three layers are able to maintain the total power of the protein kinase activities at approximately the unperturbed level. Taken together, our results suggest that only cascades of at least three layers can synchronize the oscillations of protein kinases in the cytosol and operate in parallel in the presence of crosstalk without loss of signaling fidelity.

We report on the observed characteristics of the first-order phase transition of F-actin from the isotropic state to the nematic liquid-crystalline state. Solutions of short average filament length F-actin at appropriate concentrations phase separate to form tactoidal droplets. These tactoids are the result of the minimization of their free energy and show a bipolar director field connecting two opposite poles. The tactoids are shown to form through two distinct mechanisms: nucleation and growth and spinodal decomposition. Both mechanisms produce tactoids with final domain sizes that are of the same order of magnitude. Additionally, analysis of the system shows several features of metastability. The solution can exist in a variety of steady states near equilibrium and can be easily perturbed, settling in one prescribed by the path followed in phase space.

We consider the transport of overdamped particles in a two-dimensional periodic velocity field. This field possesses extended lines of fixed points where the deterministic motion stops. Additive noise makes the lines penetrable and results in an oscillatory motion along tori. We characterize the stochastic motion by the probability distribution density, the stationary mean velocity, and the mean times of escape from bounded domains. For intermediate noise intensities, the fluctuations enhance the transport of the particles compared to the deterministic case. A fast dichotomic modulation of asymmetry enhances fluxes.

The growth rate of small-scale density inhomogeneities (the entropy production rate) is given by the sum of the Lyapunov exponents in a random flow. We derive an analytic formula for the rate in a flow of weakly interacting waves and show that in most cases it is zero up to the fourth order in the wave amplitude. We then derive an analytic formula for the rate in a flow of waves and currents. Estimates of the rate and the fractal dimension of the density distribution show that the interplay between waves and currents is a realistic mechanism for providing patchiness of the pollutant distribution on the ocean surface.

Nanoscale and microscale confinement of biopolymers naturally occurs in cells and has been recently achieved in artificial structures designed for nanotechnological applications. Here, we present an extensive theoretical investigation of the conformations and shape of a biopolymer with varying stiffness confined to a narrow channel. Combining scaling arguments, analytical calculations, and Monte Carlo simulations, we identify various scaling regimes where master curves quantify the functional dependence of the polymer conformations on the chain stiffness and strength of confinement.

In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersion model gives consistent results for the Casimir free energy at low temperatures. Specifically, for the free energy near $T=0$ we find the leading term proportional to $T^2$ and the next-to-leading term proportional to $T^{5∕2}$. These terms give rise to zero Casimir entropy as $T\to 0$ and are thus in accordance with Nernst’s theorem.

Here, an approach in terms of shot noise is proposed to study and characterize surface diffusion and low vibrational motion when having interacting adsorbates on surfaces. In what we call statistical limit, that is, at long times and high number of collisions, one expects that diffusing particles display an essential Markovian behavior. Accordingly, the action of the pairwise potentials accounting for particle-particle collisions is equivalent to considering a shot noise acting on a single particle. We call this approach the interacting single adsorbate approximation, which gathers three important advantages: (i) the dynamics underlying surface diffusion and low vibrational motion can be easily understood in terms of relatively simple stochastic processes; (ii) from our model, appropriate (and well justified) working formulas are easily obtained, which explain the results arising from more complicated (but commonly used) molecular dynamics simulations within the Langevin formulation; and (iii), at the same time, it is less demanding computationally than the latter type of calculations. In order to illustrate the application of this model, numerical results are presented. Specially, our model reproduces the experimental observation regarding the broadening of the quasielastic peak ruling surface diffusion.

We consider dense rapid shear flow of inelastically colliding hard disks. Navier-Stokes granular hydrodynamics is applied accounting for the recent finding that shear viscosity diverges at a lower density than the rest of the constitutive relations. New interpolation formulas for constitutive relations between dilute and dense cases are proposed and justified in molecular dynamics (MD) simulations. A linear stability analysis of the uniform shear flow is performed and the full phase diagram is presented. It is shown that when the inelasticity of particle collision becomes large enough, the uniform sheared flow gives way to a two-phase flow, where a dense “solidlike” striped cluster is surrounded by two fluid layers. The results of the analysis are verified in event-driven MD simulations, and a good agreement is observed.

We present experimental results on the velocity statistics of a granular fluid with an effective stochastic thermostat, in a quasi-two-dimensional configuration. We find the base state, as measured by the single particle velocity distribution $P\left(c\right)$ in the central high-probability regions, to be well described by $P\left(c\right)=f_{\mathrm{MB}}[1+a_2{S}_2(c^2\left)\right]$: It deviates from a Maxwell-Boltzmann $f_{\mathrm{MB}}$ by a second order Sonine polynomial $S_2\left(c^2\right)$ with a single adjustable parameter $a_2$. We find $a_2$ to be a function of the filling fraction and independent of the driving over a wide range of frequencies and accelerations. Moreover, there is a consistent overpopulation in the distribution’s tails, which scale as $P\propto \exp (-A\times c^{3∕2})$. To our knowledge, this is the first time that Sonine deviations have been measured in an experimental system.