Squared singular values of a product of $s$ square random Ginibre matrices are asymptotically characterized by probability distributions $P_s(x)$, such that their moments are equal to the Fuss–Catalan numbers of order $s$. We find a representation of the Fuss-Catalan distributions $P_s(x)$ in terms of a combination of $s$ hypergeometric functions of the type ${}_s{F}_{s-1}$. The explicit formula derived here is exact for an arbitrary positive integer $s$, and for $s=1$ it reduces to the Marchenko-Pastur distribution. Using similar techniques, involving the Mellin transform and the Meijer $G$ function, we find exact expressions for the Raney probability distributions, the moments of which are given by a two-parameter generalization of the Fuss-Catalan numbers. These distributions can also be considered as a two-parameter generalization of the Wigner semicircle law.

Generalized Langevin equations (GLE) with multiplicative white Poisson noise pose the usual prescription dilemma leading to different evolution equations (master equations) for the probability distribution. Contrary to the case of multiplicative Gaussian white noise, the Stratonovich prescription does not correspond to the well-known midpoint (or any other intermediate) prescription. By introducing an inertial term in the GLE, we show that the Itô and Stratonovich prescriptions naturally arise depending on two time scales, one induced by the inertial term and the other determined by the jump event. We also show that, when the multiplicative noise is linear in the random variable, one prescription can be made equivalent to the other by a suitable transformation in the jump probability distribution. We apply these results to a recently proposed stochastic model describing the dynamics of primary soil salinization, in which the salt mass balance within the soil root zone requires the analysis of different prescriptions arising from the resulting stochastic differential equation forced by multiplicative white Poisson noise, the features of which are tailored to the characters of the daily precipitation. A method is finally suggested to infer the most appropriate prescription from the data.

Following recent work by Marathe and Parrondo [Phys. Rev. Lett. 104, 245704 (2010)], we construct a classical Hamiltonian system whose energy is reduced during the adiabatic cycling of external parameters when initial conditions are sampled microcanonically. Combining our system with a device that measures its energy, we propose a cyclic procedure during which energy is extracted from a heat bath and converted to work, in apparent violation of the second law of thermodynamics. This paradox is resolved by deriving an explicit relationship between the average work delivered during one cycle of operation and the average information gained when measuring the system’s energy.

We present a linearized shear-transformation-zone (STZ) theory of glassy dynamics in which the internal STZ transition rates are characterized by a broad distribution of activation barriers. For slowly aging or fully aged systems, the main features of the barrier-height distribution are determined by the effective temperature and other near-equilibrium properties of the configurational degrees of freedom. Our theory accounts for the wide range of relaxation rates observed in both metallic glasses and soft glassy materials such as colloidal suspensions. We find that the frequency-dependent loss modulus is not just a superposition of Maxwell modes. Rather, it exhibits an $\alpha$ peak that rises near the viscous relaxation rate and, for nearly jammed, glassy systems, extends to much higher frequencies in accord with experimental observations. We also use this theory to compute strain recovery following a period of large, persistent deformation and then abrupt unloading. We find that strain recovery is determined in part by the initial barrier-height distribution, but that true structural aging also occurs during this process and determines the system’s response to subsequent perturbations. In particular, we find by comparison with experimental data that the initial deformation produces a highly disordered state with a large population of low activation barriers, and that this state relaxes quickly toward one in which the distribution is dominated by the high barriers predicted by the near-equilibrium analysis. The nonequilibrium dynamics of the barrier-height distribution is the most important of the issues raised and left unresolved in this paper.

We develop a transfer matrix formalism to visualize the framing of discrete piecewise linear curves in three-dimensional space. Our approach is based on the concept of an intrinsically discrete curve. This enables us to more effectively describe curves that in the limit where the length of line segments vanishes approach fractal structures in lieu of continuous curves. We verify that in the case of differentiable curves the continuum limit of our discrete equation reproduces the generalized Frenet equation. In particular, we draw attention to the conceptual similarity between inflection points where the curvature vanishes and topologically stable solitons. As an application we consider folded proteins, their Hausdorff dimension is known to be fractal. We explain how to employ the orientation of ${\mathrm{C}}_{\beta }$ carbons of amino acids along a protein backbone to introduce a preferred framing along the backbone. By analyzing the experimentally resolved fold geometries in the Protein Data Bank we observe that this ${\mathrm{C}}_{\beta }$ framing relates intimately to the discrete Frenet framing. We also explain how inflection points (a.k.a. soliton centers) can be located in the loops and clarify their distinctive rôle in determining the loop structure of folded proteins.

We investigated the dynamics of quadrupedal locomotion by constructing a simple quadruped model that consists of a body mechanical model and an oscillator network model. The quadruped model has front and rear bodies connected by a waist joint with a torsional spring and damper system and four limbs controlled by command signals from the oscillator network model. The simulation results reveal that the quadruped model produces various gait patterns through dynamic interactions among the body mechanical system, the oscillator network system, and the environment. They also show that it undergoes a gait transition induced by changes in the waist joint stiffness and the walking speed. In addition, the gait pattern transition exhibits a hysteresis similar to that observed in human and animal locomotion. We examined the hysteresis mechanism from a dynamic viewpoint.

The protein folding problem is one of the fundamental and important questions in molecular biology. However, the all-atom molecular dynamics studies of protein folding and unfolding are still computationally expensive and severely limited by the time scale of simulation. In this paper, a simple and fast protein unfolding method is proposed based on the conformational stability analyses and structure modeling. In this method, two structure-based conditions are considered to identify the unstable regions of proteins during the unfolding processes. The protein unfolding trajectories are mimicked through iterative structure modeling according to conformational stability analyses. Two proteins, chymotrypsin inhibitor 2 (CI2) and $\alpha$ -spectrin SH3 domain (SH3) were simulated by this method. Their unfolding pathways are consistent with the previous molecular dynamics simulations. Furthermore, the transition states of the two proteins were identified in unfolding processes and the theoretical $\Phi$ values of these transition states showed significant correlations with the experimental data (the correlation coefficients are >0.8). The results indicate that this method is effective in studying protein unfolding. Moreover, we analyzed and discussed the influence of parameters on the unfolding simulation. This simple coarse-grained model may provide a general and fast approach for the mechanism studies of protein folding.

Lignin, a major polymeric component of plant cell walls, forms aggregates in vivo and poses a barrier to cellulosic ethanol production. Here, neutron scattering experiments and molecular dynamics simulations reveal that lignin aggregates are characterized by a surface fractal dimension that is invariant under change of scale from $~$1–1000 Å. The simulations also reveal extensive water penetration of the aggregates and heterogeneous chain dynamics corresponding to a rigid core with a fluid surface.

We analyze the conformations and mechanical properties of closed diblock fibers. In our model, the length fraction of each component and the total fiber length are controlled by tunable chemical potentials. Our formalism can describe fibers in which one block is a bare polymer while the other is an adsorbed protein-filament complex; these blocks maintain different bending rigidities and spontaneous curvatures. We analytically calculate the shape of two-component polymers for all values of the material parameters and chemical potentials. Our results yield a complete analytical description of all possible two-component polymer conformations, a phase portrait detailing the parameter spaces in which these shapes occur, and the identification of spontaneous transitions between shapes driven by environmental changes.

We investigate continuous time random walks with truncated $\alpha$-stable trapping times. We prove distributional ergodicity for a class of observables; namely, the time-averaged observables follow the probability density function called the Mittag-Leffler distribution. This distributional ergodic behavior persists for a long time, and thus the convergence to the ordinary ergodicity is considerably slower than in the case in which the trapping-time distribution is given by common distributions. We also find a crossover from the distributional ergodic behavior to the ordinary ergodic behavior.

In some social and biological networks, the majority of nodes belong to multiple communities. It has recently been shown that a number of the algorithms specifically designed to detect overlapping communities do not perform well in such highly overlapping settings. Here, we consider one class of these algorithms, those which optimize a local fitness measure, typically by using a greedy heuristic to expand a seed into a community. We perform synthetic benchmarks which indicate that an appropriate seeding strategy becomes more important as the extent of community overlap increases. We find that distinct cliques provide the best seeds. We find further support for this seeding strategy with benchmarks on a Facebook network and the yeast interactome.

We present results of direct numerical simulations of passive scalar advection and diffusion in turbulent rotating flows. Scaling laws and the development of anisotropy are studied in spectral space, and in real space using an axisymmetric decomposition of velocity and passive scalar structure functions. The passive scalar is more anisotropic than the velocity field, and its power spectrum follows a spectral law consistent with $~k_{\perp }^{-3/2}$. This scaling is explained with phenomenological arguments that consider the effect of rotation. Intermittency is characterized using scaling exponents and probability density functions of velocity and passive scalar increments. In the presence of rotation, intermittency in the velocity field decreases more noticeably than in the passive scalar. The scaling exponents show good agreement with Kraichnan’s prediction for passive scalar intermittency in two dimensions, after correcting for the observed scaling of the second-order exponent.

We present measurements performed in a spherical shell filled with liquid sodium, where a 74-mm-radius inner sphere is rotated while a 210-mm-radius outer sphere is at rest. The inner sphere holds a dipolar magnetic field and acts as a magnetic propeller when rotated. In this experimental setup called “Derviche Tourneur Sodium” (DTS), direct measurements of the velocity are performed by ultrasonic Doppler velocimetry. Differences in electric potential and the induced magnetic field are also measured to characterize the magnetohydrodynamic flow. Rotation frequencies of the inner sphere are varied between $-$30 Hz and $+$30 Hz, the magnetic Reynolds number based on measured sodium velocities and on the shell radius reaching to about 33. We have investigated the mean axisymmetric part of the flow, which consists of differential rotation. Strong super-rotation of the fluid with respect to the rotating inner sphere is directly measured. It is found that the organization of the mean flow does not change much throughout the entire range of parameters covered by our experiment. The direct measurements of zonal velocity give a nice illustration of Ferraro’s law of isorotation in the vicinity of the inner sphere, where magnetic forces dominate inertial ones. The transition from a Ferraro regime in the interior to a geostrophic regime, where inertial forces predominate, in the outer regions has been well documented. It takes place where the local Elsasser number is about 1. A quantitative agreement with nonlinear numerical simulations is obtained when keeping the same Elsasser number. The experiments also reveal a region that violates Ferraro’s law just above the inner sphere.

We solve a generalized nonautonomous nonlinear Schrödinger equation analytically by performing the Darboux transformation. The precise expressions of the soliton’s width, peak, and the trajectory of its wave center are investigated analytically, which symbolize the dynamic behavior of a nonautonomous soliton. These expressions can be conveniently and effectively applied to the management of soliton in many fields.

We propose a general strategy to stabilize the fixed points of nonlinear oscillators with augmented dynamics. By using this scheme, either the unstable fixed points of the oscillatory system or a new fixed point of the augmented system can be stabilized. The Lyapunov exponents are used to study the dynamical properties. This scheme is illustrated with a chaotic Lorenz oscillator coupled through an external linear dynamical system. The experimental demonstration of the proposed scheme to stabilize the fixed points is also presented.

Null transmission ellipsometry and resonant x-ray diffraction are employed to study the effect of enantiomeric excess (EE) on the phase behavior of antiferroelectric liquid crystal 10OTBBB1M7. Phase sequence, layer spacing, and pitch of the helical structures of the smectic-$C$${}_{\alpha }^{*}$ and smectic-$C$${}^{*}$ phases are studied as a function of temperature and EE. Upon reducing EE, a liquid-gas-type critical point of the smectic-$C$${}_{\alpha }^{*}$ to smectic-$C$${}^{*}$ transition is observed, as well as the disappearance of the smectic-$C$${}_{d4}^{*}$ and the smectic-$C$${}_{d3}^{*}$ phases. Results are analyzed in a mean-field model.

We consider a particle in one dimension submitted to amplitude and phase disorder. It can be mapped onto the complex Burgers equation, and provides a toy model for problems with interplay of interferences and disorder, such as the Nguyen-Spivak-Shklovskii model of hopping conductivity in disordered insulators and the Chalker-Coddington model for the (spin) quantum Hall effect. We also propose a direct realization in an experiment with cold atoms. The model has three distinct phases: (I) a high-temperature or weak disorder phase, (II) a pinned phase for strong amplitude disorder, and (III) a diffusive phase for strong phase disorder, but weak amplitude disorder. We compute analytically the renormalized disorder correlator, equivalent to the Burgers velocity-velocity correlator at long times. In phase III, it assumes a universal form. For strong phase disorder, interference leads to a logarithmic singularity, related to zeros of the partition sum, or poles of the complex Burgers velocity field. These results are valuable in the search for the adequate field theory for higher-dimensional systems.

The characterization of a stochastic process from its single random realization is a challenging problem for most single-particle tracking techniques which survey an individual trajectory of a tracer in a complex or viscoelastic medium. We consider two quadratic functionals of the trajectory: the time-averaged mean-square displacement (MSD) and the time-averaged squared root mean-square displacement (SRMS). For a large class of stochastic processes governed by the generalized Langevin equation with arbitrary frictional memory kernel and harmonic potential, the exact formulas for the mean and covariance of these functionals are derived. The formula for the mean value can be directly used for fitting experimental data, e.g., in optical tweezers microrheology. The formula for the variance (and covariance) allows one to estimate the intrinsic fluctuations of measured (or simulated) time-averaged MSD or SRMS for choosing the experimental setup appropriately. We show that the time-averaged SRMS has smaller fluctuations than the time-averaged MSD, in spite of much broader applications of the latter one. The theoretical results are successfully confirmed by Monte Carlo simulations of the Langevin dynamics. We conclude that the use of the time-averaged SRMS would result in a more accurate statistical analysis of individual trajectories and more reliable interpretation of experimental data.

The temporal evolution of the spatially periodic electroconvection (EC) patterns has been studied within the period of the driving ac voltage by monitoring the light intensity diffracted from the pattern. Measurements have been carried out on a variety of nematic systems, including those with negative dielectric and positive conductivity anisotropy, exhibiting “standard EC” (s-EC), those with both anisotropies negative exhibiting “nonstandard EC” (ns-EC), as well as those with the two anisotropies positive. Theoretical predictions have been confirmed for stationary s-EC and ns-EC patterns. Transitions with Hopf bifurcation have also been studied. While traveling had no effect on the temporal evolution of dielectric s-EC, traveling conductive s-EC and ns-EC patterns exhibited a substantially altered temporal behavior with a dependence on the Hopf frequency. It has also been shown that in nematics with both anisotropies positive, the pattern develops and decays within an interval much shorter than the period, even at relatively large driving frequencies.