We deal with a generalized statistical description of nonequilibrium complex systems based on least biased distributions given some prior information. A maximum entropy principle is introduced that allows for the determination of the distribution of the fluctuating intensive parameter $\beta$ of a superstatistical system, given certain constraints on the complex system under consideration. We apply the theory to three examples: the superstatistical quantum-mechanical harmonic oscillator, the superstatistical classical ideal gas, and velocity time series as measured in a turbulent Taylor-Couette flow.

The relationship between the microscopic arrangement of molecules in a supercooled liquid and its slow dynamics at low temperature near glass transition is studied by molecular dynamics simulations. A Lennard-Jones liquid with polydispersity in size and mass of constituent particles is chosen as the model system. Our studies reveal that the local structure (that varies with polydispersity) plays a crucial role both in the slowing down of dynamics and in the growth of the dynamic heterogeneities, besides determining the glass forming ability of the system. Increasing polydispersity at fixed volume fraction is found to suppress the rate of growth of dynamic correlations, as detected by the growth in the peak of the nonlinear density response function, ${\chi }_4\left(t\right)$. The growth in dynamical correlation is manifested in a stronger than usual breakdown of Stokes-Einstein relation at lower polydispersity at low temperatures and also leads to a decrease in the fragility of the system with polydispersity. We show that the suppression of the rate of growth of the dynamic heterogeneity can be attributed to the loss of structural correlations (as measured by the structure factor and the local bond orientational order) with polydispersity. While a critical polydispersity is required to avoid crystallization, we find that a further increase in polydispersity lowers the glass forming ability.

The molecular dynamics of 4-$n$-octyl-$4^{\prime }$-cyanobiphenyl (8CB) confined inside the pores of a series of AlMCM-41 samples with the same structure, constant composition $(\mathrm{Si}∕\mathrm{Al}=14.7)$ but different pore sizes (diameter between 2.3 and $4.6\mathrm{nm}$) was investigated by broadband dielectric spectroscopy $({10}^{-2}–{10}^9\mathrm{Hz})$ in a large temperature interval. Two relaxation processes are observed: one has a bulklike behavior and is assigned to the 8CB in the pore center. The relaxation time of the second relaxation process is essentially slower than that of the former one and this process is related to the dynamics of molecules in a surface layer with a paranematic order. Both relaxation processes are specifically influenced by the interaction of the molecules with the surface and by the confinement. Above the clearing temperature the temperature dependence of the relaxation rate of the bulklike process obeys the Vogel-Fulcher-Tammann (VFT) law. The Vogel temperature increases with decreasing pore size. This is explained by increasing influence of paranematic potential of the surface layer with decreasing pore size. The temperature dependence of the relaxation rate of the surface layer follows also the VFT formula and the Vogel temperature decreases with decreasing pore size. This temperature dependence is controlled by both the interaction of the 8CB molecules with the surface via hydrogen bonding and by spatial confinement effects. To discriminate between both effects the data for the surface layer of 8CB confined to the molecular sieves are compared with results concerning 8CB adsorbed as a quasimonolayer on the surface of silica spheres of aerosil. On this basis a confinement parameter is defined and discussed.

The range of conduction delays between connected neuronal populations is often modeled as a single discrete delay, assumed to be an effective value averaging over all fiber velocities. This paper shows the effects of distributed delays on signal propagation. A distribution acts as a linear filter, imposing an upper frequency cutoff that is inversely proportional to the delay width. Distributed thalamocortical and corticothalamic delays are incorporated into a physiologically based mean-field model of the cortex and thalamus to illustrate their effects on the electroencephalogram (EEG). The power spectrum is acutely sensitive to the width of the thalamocortical delay distribution, and more so than the corticothalamic distribution, because all input signals must travel along the thalamocortical pathway. This imposes a cutoff frequency above which the spectrum is overly damped. The positions of spectral peaks in the resting EEG depend primarily on the distribution mean, with only weak dependences on distribution width. Increasing distribution width increases the stability of fixed point solutions. A single discrete delay successfully approximates a distribution for frequencies below a cutoff that is inversely proportional to the delay width, provided that other model parameters are moderately adjusted. A pair of discrete delays together having the same mean, variance, and skewness as the distribution approximates the distribution over the same frequency range without needing parameter adjustment. Delay distributions with large fractional widths are well approximated by low-order differential equations.

This Brief Report examines Levy motion in a slit pore with sticky boundaries, i.e., boundaries that absorb particles for a random amount of time. A set of equations is developed that can explicitly be solved for mean travel distance to a plane for a particle released from the origin and can iteratively be used to compute mean first-passage time (MFPT). Results from the theory compare favorably with Monte Carlo simulations.

We describe the Brownian demixing of sedimenting suspensions, a recently discovered phenomenon in which increases in the thermal energy can destabilize a system of orientable particles subjected to a torque to fluctuations in concentration. Through use of Brownian dynamics simulation and a mean-field analysis, we demonstrate that demixing occurs in a model system composed of slender rigid rods sedimenting between no-slip walls. Additionally, we describe the effects of wall separation distance on suspension stability.

Wang-Landau sampling (WLS) of large systems requires dividing the energy range into “windows” and joining the results of simulations in each window. The resulting density of states (and associated thermodynamic functions) is shown to suffer from boundary effects in simulations of lattice polymers and the five-state Potts model. Here, we implement WLS using adaptive windows. Instead of defining fixed energy windows (or windows in the energy-magnetization plane for the Potts model), the boundary positions depend on the set of energy values on which the histogram is flat at a given stage of the simulation. Shifting the windows each time the modification factor $f$ is reduced, we eliminate border effects that arise in simulations using fixed windows. Adaptive windows extend significantly the range of system sizes that may be studied reliably using WLS.

We study the phase transition of a nonequilibrium statistical-mechanical model, in which two degrees of freedom with different time scales separated from each other touch their own heat bath. A general condition for finding anomalous negative latent heat recently discovered is derived from a thermodynamic argument. As a specific example, the phase diagram of a spin–lattice-gas model is studied based on a mean-field analysis with the replica method. While configurational variables are spin and particle in this model, it is found that the negative latent heat appears in a parameter region of the model, irrespective of the order of their time scale. Qualitative differences in the phase diagram are also discussed.

The motion of driven interfaces in random media at finite temperature $T$ and small external force $F$ is usually described by a linear displacement $h_G\left(t\right)\sim V(F,T)t$ at large times, where the velocity vanishes according to the creep formula as $V(F,T)\sim e^{-K\left(T\right)∕F^{\mu }}$ for $F\to 0$. In this paper, we question this picture on the specific example of the directed polymer in a two-dimensional random medium. We have recently shown [C. Monthus and T. Garel, J. Phys. A 41, 255002 (2008)] that its dynamics for $F=0$ can be analyzed in terms of a strong disorder renormalization procedure, where the distribution of renormalized barriers flows towards some “infinite disorder fixed point.” In the present paper, we obtain that for small $F$, this “infinite disorder fixed point” becomes a "strong disorder fixed point" with an exponential distribution of renormalized barriers. The corresponding distribution of trapping times then only decays as a power law $P\left(\tau \right)\sim 1∕{\tau }^{1+\alpha }$, where the exponent $\alpha (F,T)$ vanishes as $\alpha (F,T)\propto F^{\mu }$ as $F\to 0$. Our conclusion is that in the small force region $\alpha (F,T)<1$, the divergence of the averaged trapping time $\overline{\tau }=+\infty$ induces strong non-self-averaging effects that invalidate the usual creep formula obtained by replacing all trapping times by the typical value. We find instead that the motion is only sublinearly in time $h_G\left(t\right)\sim t^{\alpha (F,T)}$, i.e., the asymptotic velocity vanishes $V=0$. This analysis is confirmed by numerical simulations of a directed polymer with a metric constraint driven in a traps landscape. We moreover obtain that the roughness exponent, which is governed by the equilibrium value ${\zeta }_{\mathrm{eq}}=2∕3$ up to some large scale, becomes equal to $\zeta =1$ at the largest scales.

We present the results of numerical simulations of the fluctuations of the electrostatic potential and electric field inside cavities created in the fluid of dipolar hard spheres. We found that the thermodynamics of polar solvation changes dramatically when the cavity size becomes about 4–5 times larger than the size of the liquid particle. The range of small cavities can be reasonably understood within the framework of current solvation models. On the contrary, the regime of large cavities is characterized by a significant softening of the cavity interface resulting in a decay of the fluctuation variances with the cavity size much faster than anticipated by both the continuum electrostatics and microscopic theories. For instance, the variance of the electrostatic potential at the cavity center decays with the cavity radius $R_0$ approximately as $1∕R_0^{4-6}$ instead of the $1∕R_0$ scaling expected from the standard electrostatics. Our results suggest that cores of nonpolar molecular assemblies in polar liquids lose solvation strength much faster than is traditionally anticipated.

We study macroscopic aspects of crystal surface relaxation in $2+1$ dimensions by accounting for near-equilibrium kinetics of transparent steps at the nanoscale. For slowly varying step geometries, we show that step permeability can simply renormalize a parameter in a known relation between the large-scale surface flux and the step chemical potential. This leads to a nonlinear fourth-order partial differential equation for the surface height profile.

In order to study the resonance spectra of chaotic cavities subject to some damping (which can be due to absorption or partial reflection at the boundaries), we use a model of damped quantum maps. In the high-frequency limit, the distribution of (quantum) decay rates is shown to cluster near a “typical” value, which is larger than the classical decay rate of the corresponding damped ray dynamics. The speed of this clustering may be quite slow, which could explain why it has not been detected in previous numerical data.

Particles suspended in a film flow can either settle out of the flow, remain well mixed, or even advance faster than the fluid, accumulating at the moving contact line. Recent experiments by Zhou et al. [Phys. Rev. Lett. 94, 117803 (2005)] have demonstrated that these three settling behaviors can be achieved by control of the average particle concentration $\varphi$ and inclination angle $\theta$. This work presents a theory for determining the settling behavior in the Stokes regime by calculating the depth profile of $\varphi$ and the depth-averaged velocities of the liquid and particle phases. It is found that shear-induced particle fluxes can lead to an inversely stratified flow, in which the particles move on average faster than the liquid. The theory is directly compared to Zhou et al.’s experimental data, and the implications of stratification for lubrication-type models are also discussed.

This work considers the stability of an electro-osmotic microchannel flow with streamwise electrical conductivity gradients, a configuration common in microfluidic applications such as field amplified sample stacking. Previous work on such flows has focused on how streamwise conductivity gradients set a nonuniform electro-osmotic velocity which results in dispersion of the conductivity field. However, it has been known for many years that electric fields can couple with conductivity gradients to generate unstable flows. This work demonstrates that at high electric fields such an electrohydrodynamic instability arises in this configuration and the basic mechanisms are explored through numerical simulations. The instability is unique in that the nonuniform electro-osmotic flow sets the shape of the underlying conductivity field in a way that makes it susceptible to instability. While nonuniform electro-osmotic flow sets the stage, the instability is ultimately the result of electric body forces due to slight departure from electroneutrality in the fluid bulk. A simple stability map is created where two dimensionless numbers can predict system stability reasonably well, even though the system formally depends on six dimensionless groups.

Lundgren’s vortex model for the intermittent fine structure of high-Reynolds-number turbulence is applied to the Navier-Stokes $\alpha \beta$ equations and specialized to the Navier-Stokes $\alpha$ equations. The Navier-Stokes $\alpha \beta$ equations involve dispersive and dissipative length scales $\alpha$ and $\beta$, respectively. Setting $\beta$ equal to $\alpha$ reduces the Navier-Stokes $\alpha \beta$ equations to the Navier-Stokes $\alpha$ equations. For the Navier-Stokes $\alpha$ equations, the energy spectrum is found to obey Kolmogorov’s $-5∕3$ law in a range of wave numbers identical to that determined by Lundgren for the Navier-Stokes equations. For the Navier-Stokes $\alpha \beta$ equations, Kolmogorov’s $-5∕3$ law is also recovered. However, granted that $\beta <\alpha$, the range of wave numbers for which this law holds is extended by a factor of $\alpha ∕\beta$. This suggests that simulations based on the Navier-Stokes $\alpha \beta$ equations may have the potential to resolve features smaller than those obtainable using the Navier-Stokes $\alpha$ equations.

Accurate dispersion relations are fundamental in the evaluation of many astrophysical processes such as, for example, heat loss by neutrino emission. In this paper, the parallel propagating wave modes of a magnetized, degenerate, electron gas are evaluated exactly using the $S$-matrix form of the linear response tensor (for a relativistic quantum magnetized electron gas) and the vertex formalism. The modes are then discussed in terms of the dissipative regions in which they occur. In addition to analogous modes to the unmagnetized case, there also exist doublet modes due to the logarithmic singularites that occur in the dissipative regimes of both the longitudinal and transverse dispersion relations and modes above the gyromagnetic thresholds for one of the transverse components. These additional modes only appear over a restricted range of wave numbers and are unique to the magnetized case.

We consider the random input problem for a nonlinear system modeled by the integrable one-dimensional self-focusing nonlinear Schrödinger equation (NLSE). We concentrate on the properties obtained from the direct scattering problem associated with the NLSE. We discuss some general issues regarding soliton creation from random input. We also study the averaged spectral density of random quasilinear waves generated in the NLSE channel for two models of the disordered input field profile. The first model is symmetric complex Gaussian white noise and the second one is a real dichotomous (telegraph) process. For the former model, the closed-form expression for the averaged spectral density is obtained, while for the dichotomous real input we present the small noise perturbative expansion for the same quantity. In the case of the dichotomous input, we also obtain the distribution of minimal pulse width required for a soliton generation. The obtained results can be applied to a multitude of problems including random nonlinear Fraunhoffer diffraction, transmission properties of randomly apodized long period Fiber Bragg gratings, and the propagation of incoherent pulses in optical fibers.

During steady-state water evaporation, when the vapor phase is heated electrically, the temperature on the vapor side of the interface has been reported to be as much as $27.83°\mathrm{C}$ greater than that on the liquid side. The reported interfacial temperatures were measured with thermocouple beads that were less than $50\mu \mathrm{m}$ in diameter and centered $35\mu \mathrm{m}$ from the interface in each phase. We examine the reliability of these measurements by using them with a theory of kinetics to predict the interfacial-liquid temperature. The predicted temperature discontinuities are found to be in agreement with those measured up to a temperature discontinuity of $15.69°\mathrm{C}$, but larger discontinuities cannot be confirmed because of uncertainties in the vapor-phase pressure measurements. The theory of kinetics used in the analysis includes molecular phonons in the expression for the evaporation flux. We show it is essential to include these terms if the theory is to be used to predict the temperature discontinuities.

The existence probability $E_p$ and the percolation probability $P$ of bond percolation on rectangular domains with different aspect ratios $R$ are studied via the mapping functions between systems with different aspect ratios. The superscaling behavior of $E_p$ and $P$ for such systems with exponents $a$ and $b$, respectively, found by Watanabe et al. [Phys. Rev. Lett. 93, 190601 (2004)] can be understood from the lower-order approximation of the mapping functions $f_R$ and $g_R$ for $E_p$ and $P$, respectively; the exponents $a$ and $b$ can be obtained from numerically determined mapping functions $f_R$ and $g_R$, respectively.

A theoretical model for self-dynamic response is developed using vibration-transit theory, and is applied to liquid sodium at all wave vectors $q$ from the hydrodynamic regime to the free particle limit. In this theory the zeroth-order Hamiltonian describes the vibrational motion in a single random valley harmonically extended to infinity. This Hamiltonian is tractable, is evaluated a priori for monatomic liquids, and the same Hamiltonian (the same set of eigenvalues and eigenvectors) is used for equilibrium and nonequilibrium theory. Here, for the self-intermediate scattering function $F^s(q,t)$, we find the vibrational contribution is in near perfect agreement with molecular dynamics (MD) through short and intermediate times, at all $q$. This is direct confirmation that normal mode vibrational correlations are present in the motion of the liquid state. The primary transit effect is the diffusive motion of the vibrational equilibrium positions, as the liquid transits rapidly among random valleys. This motion is modeled as a standard random walk, and the resulting theoretical $F^s(q,t)$ is in excellent agreement with MD results at all $q$ and $t$. In the limit $q\to \infty$, the theory automatically exhibits the correct approach to the free-particle limit. Also, in the limit $q\to 0$, the hydrodynamic limit emerges as well. In contrast to the benchmark theories of generalized hydrodynamics and mode coupling, the present theory is near a priori, while achieving modestly better accuracy. Therefore, in our view, it constitutes an improvement over the traditional theories.