A scaling analysis is presented of the statistics of long DNA confined in nanochannels and nanoslits. It is argued that there are several regimes in between the de Gennes and Odijk limits introduced long ago. The DNA chain folds back on itself giving rise to a global persistence length that may be very large owing to entropic deflection. Moreover, there is an orientational excluded-volume effect between the DNA segments imposed solely by the nanoconfinement. These two effects cause the chain statistics to be intricate leading to nontrivial power laws for the chain extension in the intermediate regimes. It is stressed that DNA confinement within nanochannels differs from that in nanoslits because the respective orientational excluded-volume effects are not the same.

The fractal structure of high-temperature graphs of the three-dimensional Ising and $XY$ models is investigated by simulating these graphs directly on a cubic lattice and analyzing them with the help of percolation observables. The Ising graphs are shown to percolate right at the Curie critical point. The diverging length scale relevant to the graphs in the vicinity of the percolation threshold is shown to be provided by the spin correlation length. The fractal dimension of the high-temperature graphs at criticality is estimated to be $D=1.7349\left(65\right)$ for the Ising and $D=1.7626\left(66\right)$ for the $XY$ models.

We study the hopping transport of a quantum particle through randomly diluted percolation clusters in two dimensions realized both on the square and triangular lattices. We investigate the nature of localization of the particle by calculating the transmission coefficient as a function of energy ($-2<E<2$ in units of the hopping integral in the tight-binding Hamiltonian) and disorder, $q$ (probability that a given site of the lattice is not available to the particle). Our study based on finite-size scaling suggests the existence of delocalized states that depends on energy and the amount of disorder present in the system. For energies away from the band center $\left(E=0\right)$, delocalized states appear only at low disorder $\left(q<15\%\right)$. The transmission near the band center is generally very small for any amount of disorder and therefore makes it difficult to locate the transition to delocalized states if any, but our study does indicate a behavior that is weaker than power-law localization. Apart from this localization-delocalization transition, we also find the existence of two different kinds of localization regimes depending on energy and the amount of disorder. For a given energy, states are exponentially localized for sufficiently high disorder. As the disorder decreases, states first show power-law localization before showing a delocalized behavior.

The spatially uniaxially anisotropic $d=3$ Ising spin glass is solved exactly on a hierarchical lattice. Five different ordered phases, namely, ferromagnetic, columnar, layered, antiferromagnetic, and spin-glass phases, are found in the global phase diagram. The spin-glass phase is more extensive when randomness is introduced within the planes than when it is introduced in lines along one direction. Phase diagram cross sections, with no Nishimori symmetry, with Nishimori symmetry lines, or entirely imbedded into Nishimori symmetry, are studied. The boundary between the ferromagnetic and spin-glass phases can be either reentrant or forward, that is either receding from or penetrating into the spin-glass phase, as temperature is lowered. However, this boundary is always reentrant when the multicritical point terminating it is on the Nishimori symmetry line.

We examine a phase field crystal model for simple liquid-solid systems consisting of a free energy functional related to the Ramakrishnan-Yussouff free energy of classical density functional theory and an equation of motion capable of describing long-time-scale behavior in the deeply supercooled regime. The thermodynamics and dynamics of freezing and glass formation in this model system are studied through large-scale three-dimensional Langevin simulations. At low cooling rates bcc crystals are formed by nucleation and growth from the melt. At large cooling rates no clear glass transition is observed, but a kinetically driven first-order transition from supercooled liquid to a disordered glasslike solid does occur. Despite the peculiarities of the transition, the structure and properties of the resulting disordered solid are shown to strongly resemble those of a typical glass. Consequences of pseudocritical behavior and heterogeneity near the liquid spinodal are also discussed.

We construct an efficient methodology for calculating wormlike chain statistics in arbitrary $\mathit{D}$ dimensions over all chain rigidities, from fully rigid to completely flexible. The structure of our exact analytical solution for the end-to-end distribution function for a wormlike chain in arbitrary $\mathit{D}$ dimensions in Fourier-Laplace space (i.e., Fourier-transformed end position and Laplace-transformed chain length) adopts the form of an infinite continued fraction, which is advantageous for its compact structure and stability for numerical implementation. We then proceed to present a step-by-step methodology for performing the Fourier-Laplace inversion in order to make full use of our results in general applications. Asymptotic methods for evaluating the Laplace inversion (power-law expansion and Rayleigh-Schrödinger perturbation theory) are employed in order to improve the accuracy of the numerical inversions of the end-to-end distribution function in real space. We adapt our results to the evaluation of the single-chain structure factor, rendering simple, closed-form expressions that facilitate comparison with scattering experiments. Using our techniques, the accuracy of the end-to-end distribution function is enhanced up to the limit of the machine precision. We demonstrate the utility of our methodology with realizations of the chain statistics, giving a general methodology that can be applied to a wide range of biophysical problems.

Using a path integral technique, we show exactly that for a semiflexible biopolymer in constant extension ensemble, no matter how long the polymer and how large the external force, the effects of short-range correlations in the sequence-dependent spontaneous curvatures and torsions can be incorporated into a model with well-defined mean spontaneous curvature and torsion as well as a renormalized persistence length. Moreover, for a long biopolymer with large mean persistence length, the sequence-dependent persistence lengths can be replaced by their mean. However, for a short biopolymer or for a biopolymer with small persistence lengths, inhomogeneity in persistence lengths tends to make physical observables very sensitive to details and therefore less predictable.

We show that a weak transverse spatial modulation in $(2+1)$ nonlinear Schrödinger-type equation can result in nontrivial dynamics of a radially symmetric soliton. We provide examples of chaotic soliton motion in periodic media both for conservative and dissipative cases. We show that complex dynamics can persist even for soliton sizes greater than the modulation period.

We investigate the condensation phase transitions of the conserved-mass aggregation (CA) model on weighted scale-free networks (WSFNs). In WSFNs, the weight $w_{ij}$ is assigned to the link between the nodes $i$ and $j$. We consider the symmetric weight given by $w_{ij}={\left(k_i{k}_j\right)}^{\alpha }$. On WSFNs, we numerically show that a certain critical ${\alpha }_c$ exists below which the CA model undergoes the same type of condensation transitions as those of the CA model on regular lattices. However, for $\alpha \ge {\alpha }_c$, the condensation always occurs for any density $\rho$ and $\omega$. We analytically find ${\alpha }_c=\left(\gamma -3\right)/2$ on the WSFN with the degree exponent $\gamma$. To obtain ${\alpha }_c$, we analytically derive the scaling behavior of the stationary probability distribution $P_k^{\infty }$ of finding a walker at nodes with degree $k$, and the probability $D\left(k\right)$ of finding two walkers simultaneously at the same node with degree $k$. We find $P_k^{\infty }\sim k^{\alpha +1-\gamma }$ and $D\left(k\right)\sim k^{2\left(\alpha +1\right)-\gamma }$, respectively. With $P_k^{\infty }$, we also show analytically and numerically that the average mass $m\left(k\right)$ on a node with degree $k$ scales as $k^{\alpha +1}$ without any jumps at the maximal degree of the network for any $\rho$ as in SFNs with $\alpha =0$.

We systematically develop a technique for reconstructing the phase dynamics equations for coupled oscillators from data. For autonomous oscillators and for two interacting oscillators we demonstrate how phase estimates obtained from general scalar observables can be transformed to genuine phases. This allows us to obtain an invariant description of the phase dynamics in terms of the genuine, observable-independent phases. We discuss the importance of this transformation for characterization of strength and directionality of interaction from bivariate data. Moreover, we demonstrate that natural (autonomous) frequencies of oscillators can be recovered if several observations of coupled systems at different, yet unknown coupling strengths are available. We illustrate our method by several numerical examples and apply it to a human electrocardiogram and to a physical experiment with coupled metronomes.

We use a probabilistic model of microtubule interaction via molecular motors to study microtubule bundle interaction. Our model indicates that initially disordered systems of interacting polar rods exhibit an orientational instability resulting in spontaneous ordering. We study the existence and dynamic interaction of microtubule bundles analytically and numerically. Our results show a long term attraction and coalescing of bundles indicating a clear coarsening in the system; Microtubule bundles concentrate into fewer orientations on a slow logarithmic time scale.

We propose a model of a heterogeneous glass-forming liquid and compute the low-temperature behavior of a tagged molecule moving within it. This model exhibits stretched-exponential decay of the wave-number-dependent, self-intermediate scattering function in the limit of long times. At temperatures close to the glass transition, where the heterogeneities are much larger in extent than the molecular spacing, the time dependence of the scattering function crosses over from stretched-exponential decay with an index $b=1/2$ at large wave numbers to normal, diffusive behavior with $b=1$ at small wave numbers. There is a clear separation between early-stage, cage-breaking $\beta$ relaxation and late-stage $\alpha$ relaxation. The spatial representation of the scattering function exhibits an anomalously broad exponential (non-Gaussian) tail for sufficiently large values of the molecular displacement at all finite times.

Nucleation phenomena in lattice gas models of simple and chain molecules confined in slitlike pores are studied using Monte Carlo methods. Finite-size scaling is used to investigate the nature of phase transitions accompanying the formation of layers at the pore walls. It is demonstrated that nucleation leads to the symmetry breaking and the formation of nuclei at one wall only.

We propose a surface treatment allowing one to obtain a sliding planar anchoring of nematic (or cholesteric) liquid crystals. It consists of depositing a thin layer of the polymercaptan hardener of an epoxy resin on an isotropic substrate (bare or ITO-coated glass plates). Microscopic observations of defect annihilations and capacitance measurements show that the molecules align parallel to the surface and slide viscously on it when they change orientation, which implies a zero (or extremely small) azimuthal anchoring energy. In contrast, the zenithal anchoring energy $W_{\theta }$ is found to be larger than $3\times {10}^{-5}\mathrm{J}∕{\mathrm{m}}^2$. We also measured the liquid crystal rotational surface viscosity ${\gamma }_S$ by a thermo-optical method using the large temperature variation of the pitch of a compensated cholesteric mixture. We found that the sliding length ${\gamma }_S∕{\gamma }_1$ (where ${\gamma }_1$ is the bulk rotational viscosity) is very large in comparison with the length of a liquid crystal molecule. This result is explained by a simple model which takes into account the diffusion of the liquid crystal within the polymer layer.

The compound 10OHF has a partially inverted phase sequence, unique among the series of $n\text{OHF}$ homologous compounds and all other known liquid crystals, with the smectic-$C_{FI2}^{\ast }$ $\left(\text{Sm}{C}_{FI2}^{\ast }\right)$ phase occurring at higher temperature than the smectic-$C^{\ast }$ $\left(\text{Sm}{C}^{\ast }\right)$ phase. We present ellipsometric data to identify the phase sequences of 9OHF, 10OHF, 11OHF, and 12OHF. Binary mixtures of 10OHF with C11, a compound with the typical phase sequence among the smectic phases, show that the unusual phase sequence of 10OHF stabilizes upon mixing and that $\text{Sm}{C}_{FI2}^{\ast }$ predominates over $\text{Sm}{C}^{\ast }$ throughout the entire mixing phase diagram. In thin films of some mixtures, surface interactions induce a reentrant $\text{Sm}{C}_{FI2}^{\ast }\text{-Sm}{C}^{\ast }\text{-Sm}{C}_{FI2}^{\ast }$ transition in the rest of the film.

In the present work an operational recipe for the mean square displacement (MSD) determination, highlighting the connection between elastic incoherent neutron scattering (EINS) intensity profiles and the associated self-distribution function, is presented. The determination of the thermal behavior of the total MSD and of its partial contributions is tested on EINS data collected by the backscattering spectrometer IN13 (ILL, Grenoble) on a model system such as PolyEthylene Glycol with a mean molecular weight of 400 Dalton (PEG 400).

The bistable FitzHugh-Nagumo (FHN) neural model driven by two multiplicative noises and one additive noise is investigated. Two different potential mechanisms for enhancing coherence of bistable FHN model are presented, that is, the first multiplicative noise changes the system from the bistable to the oscillatory regime, and the second multiplicative noise can enhance the symmetry of two stable states of the system. The two mechanisms are analytically or numerically explained. At any level of the second multiplicative noise, a maximal coherence have been found at some intermediate noise intensity of the first multiplicative noise. Only when the first multiplicative noise intensity is less than 0.0001 can a maximal coherence be obtained at some intermediate noise intensity of the second multiplicative noise. These coherence resonance phenomena have been understood in terms of the presented mechanisms.

We propose a method to search for signs of causal structure in spatiotemporal data making minimal a priori assumptions about the underlying dynamics. To this end, we generalize the elementary concept of recurrence for a point process in time to recurrent events in space and time. An event is defined to be a recurrence of any previous event if it is closer to it in space than all the intervening events. As such, each sequence of recurrences for a given event is a record breaking process. This definition provides a strictly data driven technique to search for structure. Defining events to be nodes, and linking each event to its recurrences, generates a network of recurrent events. Significant deviations in statistical properties of that network compared to networks arising from (acausal) random processes allows one to infer attributes of the causal dynamics that generate observable correlations in the patterns. We derive analytically a number of properties for the network of recurrent events composed by a random process in space and time. We extend the theory of records to treat not only the variable where records happen, but also time as continuous. In this way, we construct a fully symmetric theory of records leading to a number of results. Those analytic results are compared in detail to the properties of a network synthesized from time series of epicenter locations for earthquakes in Southern California. Significant disparities from the ensemble of acausal networks that can be plausibly attributed to the causal structure of seismicity are as follows. (1) Invariance of network statistics with the time span of the events considered. (2) The appearance of a fundamental length scale for recurrences, independent of the time span of the catalog, which is consistent with observations of the “rupture length.” (3) Hierarchy in the distances and times of subsequent recurrences. As expected, almost all of the statistical properties of a network constructed from a surrogate in which the original magnitudes and locations of earthquake epicenters are randomly “shuffled” are completely consistent with predictions from the acausal null model.

We propose a strategy for the detection of temporal irreversibility in stationary time series based on multiple bidimensional tests. The test is helpful to evaluate the displacement of irreversibility toward high dimensions. The test can be used independently of the theoretical functionals actually utilized to check irreversibility. The method was applied to simulated nonlinear signals generated by the delayed Henon map and a two-loop negative feedback model to show how the presence of a delay could produce the displacement of irreversibility toward higher dimensions. The method was applied also to series of a biological variable (i.e., heart period) that is known to be regulated by multiple feedback loops. Simulations and real data support the need of exploring progressively increasing embedding dimensions when assessing temporal irreversibility.

The pinch-off of a gas bubble from a tiny nozzle immersed vertically in another quiescent viscous fluid due to buoyancy is numerically investigated. The dynamics of bubble growth and pinch-off are described by the full Navier-Stokes equations for both gas and liquid phases. The equations are solved with a finite-volume method based on the SIMPLE scheme, coupled with a front tracking method to locate the interface between the two phases. The effects of liquid viscosity, surface tension, and gas density on the bubble pinch-off dynamics, which are always coupled in experiments, are investigated separately through simulations. The numerical results are compared with experimental observations on the bubble pinch-off for validation purposes. The simulation results show that the radius of the necking region decreases in a power law mode with time as $r\sim {\tau }^{\alpha }$, where $\tau$ is the time to pinch-off and the exponent $\alpha$ varies in the range 0.5–1.0 depending strongly upon the liquid properties such as viscosity and surface tension. In addition, the surface tension can significantly affect the bubble pinch-off exponent $\alpha$ when the surface tension coefficient is smaller than 0.030 N/m with a Bond number higher than 0.72. It is also found that both higher viscosity of the liquid phase and higher surface tension may result in a delayed pinch-off process and a larger bubble. The effect of gas phase density on the pinch-off is also investigated. As reported in the literature, the gas density variation has minimal effect on the necking process because the density ratio of the gas phase to the liquid phase is small.