We employ the method of virial expansion to compute the retarded density correlation function (generalized diffusion propagator) in the critical random matrix ensemble in the limit of strong multifractality. We find that the long-range nature of the Hamiltonian is a common root of both multifractality and Lévy flights, which show up in the power-law intermediate- and long-distance behaviors, respectively, of the density correlation function. We review certain models of classical random walks on fractals and show the similarity of the density correlation function in them to that for the quantum problem described by the random critical long-range Hamiltonians.

Using a combination of light scattering techniques and broadband dielectric spectroscopy, we have measured the temperature dependence of structural relaxation time and self diffusion in three imidazolium-based room temperature ionic liquids: [bmim][NTf${}_2$], [bmim][PF${}_6$], and [bmim][TFA]. A detailed analysis of the results demonstrates that self diffusion decouples from structural relaxation in these systems as the temperature is decreased toward $T_g$. The degree to which the dynamics are decoupled, however, is shown to be surprisingly weak when compared to other supercooled liquids of similar fragility. In addition to the weak decoupling, we demonstrate that the temperature dependence of the structural relaxation time in all three liquids can be well described by a single Vogel-Fulcher-Tamann function over 13 decades in time from 10${}^{-11}$ s up to 10${}^2$ s. Furthermore, the stretching of the structural relaxation is shown to be temperature independent over the same range of time scales, i.e., time temperature superposition is valid for these ionic liquids from far above the melting point down to the glass transition temperature. We suggest that these phenomena are interconnected and all result from the same underlying mechanism—strong and directional intermolecular interactions.

The ability to predict the collapse of ecological communities is of significant concern in light of global patterns of rapid species extinctions. Here, we use a recently developed dynamic Boolean network-based model of mutualistic plant-pollinator community formation to investigate the stability of simulated ecological communities in the face of sequential species extinctions. We assess communities in terms of the relative change in biodiversity after species loss, and find that communities that experience a significant loss of biodiversity differ from more robust communities according to a number of topological characteristics. Notably, we show that high nestedness, a property commonly believed to promote community stability, may in extreme circumstances promote a critical over-reliance on individual species. Furthermore, the species important to the survival of the rest of the ecosystem occupy different positions in the network than less important species. Our results suggest that network measures may be applied to real ecosystems to yield insight into both their stability and the identity of potentially critical species.

We formalize a theory to help explore the effect of conformational fluctuation on the energy landscape evolution of single-molecule protein. Using this formalization, we investigate the photon emission from single photoactivated fluorescent protein. A bimodal regulation on the energy landscape evolution was discovered, and its origin was attributed to slow conformational fluctuations of the protein matrix.

In cell biology, time-resolved fluctuation analysis of tracer particles has recently gained great importance. One such method is the local mean-square displacement (MSD) analysis, which provides an estimate of two parameters as functions of time: the exponent of growth of the MSD and the diffusion coefficient. Here, we study the joint and marginal distributions of these parameters for Brownian motion with Gaussian velocity fluctuations, including the cases of vanishing correlations (overdamped Brownian motion) and of a finite negative velocity correlation (as observed in intracellular motion). Numerically, we demonstrate that a small number of MSD points is optimal for the estimation of the diffusion measures. Motivated by this observation, we derive an analytic approximation for the joint and marginal probability densities of the exponent and diffusion coefficient for the special case of two MSD points. These analytical results show good agreement with numerical simulations for sufficiently large window sizes. Our results might promote better statistical analysis of intracellular motility.

We report measurements and numerical simulations of the three-dimensional velocity and temperature fields in turbulent Rayleigh-Bénard convection in air. Highly resolved velocity and temperature measurements inside and outside the boundary layers have been directly compared with equivalent data obtained in direct numerical simulations (DNSs). This comparison comprises a set of two Rayleigh numbers at $\mathrm{Ra}=3\times {10}^9$ and $3\times {10}^{10}$ and a fixed aspect ratio; this is the ratio between the diameter and the height of the Rayleigh-Bénard cell of $\Gamma =1$. We find that the measured velocity data are in excellent agreement with the DNS results while the temperature data slightly differ. In particular, the measured mean temperature profile does not show the linear trend as seen in the DNS data, and the measured gradients at the wall are significantly higher than those obtained from the DNS. Both viscous and thermal boundary layer thickness scale with respect to the Rayleigh number as ${\delta }_v\sim {\mathrm{Ra}}^{-0.24}$ and ${\delta }_{\theta }\sim {\mathrm{Ra}}^{-0.24}$, respectively.

We use an extension of the van der Pol oscillator as an example of a system with multiple time scales to study the susceptibility of its trajectory to polynomial perturbations in the dynamics. A striking feature of many nonlinear, multiparameter models is an apparently inherent insensitivity to large-magnitude variations in certain linear combinations of parameters. This phenomenon of “sloppiness” is quantified by calculating the eigenvalues of the Hessian matrix of the least-squares cost function. These typically span many orders of magnitude. The van der Pol system is no exception: Perturbations in its dynamics show that most directions in parameter space weakly affect the limit cycle, whereas only a few directions are stiff. With this study, we show that separating the time scales in the van der Pol system leads to a further separation of eigenvalues. Parameter combinations which perturb the slow manifold are stiffer and those which solely affect the jumps in the dynamics are sloppier.

The enterobacteria lambda phage is a paradigm temperate bacteriophage. Its lysogenic and lytic life cycles echo competition between the DNA binding $\lambda$-repressor (CI) and CRO proteins. Here we scrutinize the structure, stability, and folding pathways of the $\lambda$-repressor protein, which controls the transition from the lysogenic to the lytic state. We first investigate the supersecondary helix-loop helix composition of its backbone. We use a discrete Frenet framing to resolve the backbone spectrum in terms of bond and torsion angles. Instead of four, there appears to be seven individual loops. We model the putative loops using an explicit soliton Ansatz. It is based on the standard soliton profile of the continuum nonlinear Schrödinger equation. The accuracy of the Ansatz far exceeds the B-factor fluctuation distance accuracy of the experimentally determined protein configuration. We then investigate the folding pathways and dynamics of the $\lambda$-repressor protein. We introduce a coarse-grained energy function to model the backbone in terms of the C${}_{\alpha }$ atoms and the side chains in terms of the relative orientation of the C${}_{\beta }$ atoms. We describe the folding dynamics in terms of relaxation dynamics and find that the folded configuration can be reached from a very generic initial configuration. We conclude that folding is dominated by the temporal ordering of soliton formation. In particular, the third soliton should appear before the first and second. Otherwise, the DNA binding turn does not acquire its correct structure. We confirm the stability of the folded configuration by repeated heating and cooling simulations.

We investigate the dynamics of the evacuation process with multiple bottlenecks using the floor field model. To deal with this problem, we first focus on a part of the system and report its microscopic behavior. The system is controlled by parameters of inflow and the competitiveness of the pedestrians, and large inflow leads to a congested situation. Through simulations, the metastable state induced by conflicts of pedestrians is observed. The metastability is related to the phase transition from free flow to congestion. The critical condition of the transition is theoretically derived. In addition, we give simulation results of situations with multiple bottlenecks. They imply that local improvement of pedestrian flow sometimes adversely affects the total evacuation time, and that the total optimization of the system is not straightforward.

When a very flexible wire is dipped into a soapy solution, it collapses onto itself. We consider the regions of high curvature where the wire folds back onto itself, enclosing a capillary film. The shapes of these end loops are measured in experiments using soap films and compared to a known similarity solution. The sizes of these structures provide a simple and reliable way to measure surface tension.

We derive the finite-size dependence of the clustering coefficient of scale-free random graphs generated by the configuration model with degree distribution exponent $2<\gamma <3$. Degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets. We also find that for values of $\gamma \approx 2$, clustering is virtually size independent and, at the same time, becomes a de facto non-self-averaging topological property. This implies that a single-instance network is not representative of the ensemble even for very large network sizes.

Appropriate constructions of Padé approximants are believed to provide reasonable estimates of the asymptotic (large-coupling) amplitude and exponent of an observable, given its weak-coupling expansion to some desired order. In many instances, however, sequences of such approximants are seen to converge very poorly. We outline here a strategy that exploits the idea of fractional calculus to considerably improve the convergence behavior. Pilot calculations on the ground-state perturbative energy series of quartic, sextic, and octic anharmonic oscillators reveal clearly the worth of our endeavor.

We introduce the idea of transformation trajectories to describe the evolution of nematic shells in terms of defect locations and director field when the elastic anisotropy and the shell thickness heterogeneity vary. Experiments are compared to numerical results to clarify the exact role played by these two parameters. We demonstrate that heterogeneity in thickness is a result of a symmetry breaking initiated by buoyancy and enhanced by liquid crystal elasticity, and is irrespective of the elastic anisotropy. In contrast, elastic anisotropy—in particular, disfavored bend distortion—drives an asymmetric defect reorganization. These shell states can be both stable or metastable.

Transport networks are found at the heart of myriad natural systems, yet are poorly understood, except for the case of river networks. The Scheidegger model, in which rivers are convergent random walks, has been studied only in the case of flat topography, ignoring the variety of curved geometries found in nature. Embedding this model on a cone, we find a convergent and a divergent phase, corresponding to few, long basins and many, short basins, respectively, separated by a singularity, indicating a phase transition. Quantifying basin shape using Hacks law $l\sim a^h$ gives distinct values for $h$, providing a method of testing our hypotheses. The generality of our model suggests implications for vascular morphology, in particular, differing number and shapes of arterial and venous trees.

When intense $p$-polarized laser light is incident on a plasma with an electron density many times the critical density, the flux of fast electrons created by Brunel absorption excites plasma oscillations. These oscillations may in turn affect the spectrum of high harmonics by modulating the spectrum at the plasma frequency, ${\omega }_p$, and by coupling to the radiation field through the steep density gradient at the plasma-vacuum interface, so generating plasma line emission (PLE) at ${\omega }_p$ and harmonics of ${\omega }_p$. Both aspects depend sensitively on a range of plasma and laser pulse parameters, including the initial electron density, the density profile at the plasma-vacuum interface, and the intensity, pulse shape, and pulse length of the incident laser light. These various dependences have been characterised for moderately relativistic laser-plasma interactions by means of a series of particle-in-cell (PIC) simulations.

We study equilibrium states of quantum spin systems with nonadditive long-range interactions by adopting an appropriate scaling of the interaction strength, i.e., the so-called Kac prescription. In classical spin systems, it is known that the equilibrium free energy is obtained by minimizing the free-energy functional over the coarse-grained magnetization. Here we show that it is also true for quantum spin systems. From this observation, it is found that when the canonical ensemble and the microcanonical ensemble are not equivalent in some parameter region, it is not necessarily justified to replace the actual long-range interaction by the infinite-range interaction (Curie-Weiss-type interaction). On the other hand, in the parameter region where the two ensembles are equivalent, this replacement is always justified. We examine the Heisenberg $XXZ$ model as an illustrative example and discuss the relation to experiments.

We generalize the quantum heat engine (QHE) model which was first proposed by Bender et al. [J. Phys. A 33, 4427 (2000)] to the case in which an ideal Fermi gas with an arbitrary number $N$ of particles in a box trap is used as the working substance. Besides two quantum adiabatic processes, the engine model contains two isoenergetic processes, during which the particles are coupled to energy baths at a high constant energy $E_h$ and a low constant energy $E_c$, respectively. Directly employing the finite-time thermodynamics, we find that the power output is enhanced by increasing particle number $N$ (or decreasing minimum trap size $L_A$) for given $L_A$ (or $N$), without reduction in the efficiency. By use of global optimization, the efficiency at possible maximum power output (EPMP) is found to be universal and independent of any parameter contained in the engine model. For an engine model with any particle-number $N$, the efficiency at maximum power output (EMP) can be determined under the condition that it should be closest to the EPMP. Moreover, we extend the heat engine to a more general multilevel engine model with an arbitrary 1D power-law potential. Comparison between our engine model and the Carnot cycle shows that, under the same conditions, the efficiency $\eta =1-\frac{E_c}{E_h}$ of the engine cycle is bounded from above the Carnot value ${\eta }_c=1-\frac{T_c}{T_h}$.

We present theoretical studies of syntectic and monotectic solidification scenarios. Steady-state solidification along the liquid-liquid interface in a syntectic system is considered by means of a boundary-integral technique developed previously. We study the case of small asymmetry of the pattern and extract from the results the scaling relations in terms of the undercooling and the asymmetry parameter. We also investigate monotectic solidification using the phase-field method. We present two kinds of two-phase fingers, with the solid phase being either the exterior phase or the interior phase, and the pattern corresponding to the growth along the solid-liquid interface. We finally analyze the asymptotic shape of these new morphologies far behind their tip.

We introduce the $\varepsilon$-susceptible-infected-susceptible (SIS) spreading model, which is taken as a benchmark for the comparison between the $N$-intertwined approximation and the Pastor-Satorras and Vespignani heterogeneous mean-field (HMF) approximation of the SIS model. The $N$-intertwined approximation, the HMF approximation, and the $\varepsilon$-SIS spreading model are compared for different graph types. We focus on the epidemic threshold and the steady-state fraction of infected nodes in networks with different degree distributions. Overall, the $N$-intertwined approximation is superior to the HMF approximation. The $N$-intertwined approximation is exactly the same as the HMF approximation in regular graphs. However, for some special graph types, such as the square lattice graph and the path graph, the two mean-field approximations are both very different from the $\varepsilon$-SIS spreading model.

We show that generic, slow dynamics can occur in the contact process on complex networks with a tree-like structure and a superimposed weight pattern, in the absence of additional (nontopological) sources of quenched disorder. The slow dynamics is induced by rare-region effects occurring on correlated subspaces of vertices connected by large weight edges and manifests in the form of a smeared phase transition. We conjecture that more sophisticated network motifs could be able to induce Griffiths phases, as a consequence of purely topological disorder.