Stationary Velocity Distribution Research Articles

Anomalous Transport in Three‐Dimensional Discrete Fracture Networks: Interplay Between Aperture Heterogeneity and Injection Modes

AbstractWe study how the interplay between fracture aperture heterogeneity and tracer injection mode controls fluid flow and tracer transport in three‐dimensional (3D) discrete fracture networks (DFNs). The direct 3‐D DFN simulations show that tracer injection mode has substantial effects on tracer spreading across all levels of aperture heterogeneity. The key controlling factor for effective transport is the initial Lagrangian velocity distribution, which is determined by the interplay between injection mode and aperture heterogeneity. The fundamental difference between initial Lagrangian velocity distribution and domain‐scale Eulerian velocity distribution plays a vital role in determining anomalous transport. We effectively capture the observed anomalous transport using an upscaled transport model that incorporates initial velocity distribution, stationary velocity distribution, velocity correlation length, and average advective tortuosity. With the upscaled transport model, we accurately capture the evolution of Lagrangian velocity distribution and predict longitudinal spreading in 3‐D DFN.

Collective dynamics of Stokesian swimmers with impulsive acceleration

In this paper, we investigate the characteristics of collective motion of Stokesian swimmers with impulsive acceleration (SSIA). We consider two types of intensity function, namely, the displacement-based intensity function and elapsed-time-based intensity function. The stationary velocity distribution function (VDF) of the SSIA obtained by the direct simulation Monte Carlo method is compared with the analytical approximation, which is obtained by the kinetic (mean-field) equation of the SSIA. Finally, the collective motion of the SSIA confined by 2D box is analyzed by considering Vicsek-like effects to consider the effects of binary collision on the VDF of the SSIA.

Producing suprathermal tails in the stationary velocity distribution

We revisit effective scenarios for the origin of heavy tails in stationary velocity distributions. A first analysis combines localization with diffusive acceleration. That gets realized in space plasmas to find the so-called kappa-distributions having power-law decay at high speeds. There, localization at high energy already takes place for the reversible dynamics, but becomes effective by an active diffusion in velocity space. A model for vibrating granular gases and giving rise to stretched exponential tails is also briefly discussed, where negative friction is the energizer. In all cases, the resulting nonMaxwellian velocity distributions are frenetically caused by the dependence on the speed of kinetic parameters.

The pre-breakdown solutions of electrohydrodynamic equations for liquid insulators and comparison with experiment

The system of electrohydrodynamic equations at a pre-breakdown liquid insulator is given. The influence of electric field on the partial dissociation molecules rate is taken into account. The stationary and nonstationary solutions for electric potential and hydrodynamic velocity distributions are obtained. Comparison of the experimental and theoretical results showed good agreement. The stationary velocity distribution and current-voltage characteristic are close to quadratic, as predicted by the theory.

Anomalous transport in disordered fracture networks: Spatial Markov model for dispersion with variable injection modes

We investigate tracer transport on random discrete fracture networks that are characterized by the statistics of the fracture geometry and hydraulic conductivity. While it is well known that tracer transport through fractured media can be anomalous and particle injection modes can have major impact on dispersion, the incorporation of injection modes into effective transport modeling has remained an open issue. The fundamental reason behind this challenge is that—even if the Eulerian fluid velocity is steady—the Lagrangian velocity distribution experienced by tracer particles evolves with time from its initial distribution, which is dictated by the injection mode, to a stationary velocity distribution. We quantify this evolution by a Markov model for particle velocities that are equidistantly sampled along trajectories. This stochastic approach allows for the systematic incorporation of the initial velocity distribution and quantifies the interplay between velocity distribution and spatial and temporal correlation. The proposed spatial Markov model is characterized by the initial velocity distribution, which is determined by the particle injection mode, the stationary Lagrangian velocity distribution, which is derived from the Eulerian velocity distribution, and the spatial velocity correlation length, which is related to the characteristic fracture length. This effective model leads to a time-domain random walk for the evolution of particle positions and velocities, whose joint distribution follows a Boltzmann equation. Finally, we demonstrate that the proposed model can successfully predict anomalous transport through discrete fracture networks with different levels of heterogeneity and arbitrary tracer injection modes.

Kinetic theory of vehicular traffic

We review the kinetic theory of traffic proposed by Prigogine and Herman in which the Boltzmann equation is adapted to vehicular traffic. The kinetic equation and its solution are discussed, and a novel distribution of desired velocities that is more suitable for describing real traffic conditions is analyzed. We also study the stationary velocity distribution at the transition between individual and collective flow patterns. At this transition, the distribution splits into a smoothly varying regular part, in which vehicles have nonzero velocities, and a singular one, corresponding to stopped vehicles. Computational methods for obtaining the stationary velocity distribution and the full space-time evolution of the vehicular distribution are explained.

Velocity and displacement statistics in a stochastic model of nonlinear friction showing bounded particle speed.

Diffusion of colloidal particles in a complex environment such as polymer networks or biological cells is a topic of high complexity with significant biological and medical relevance. In such situations, the interaction between the surroundings and the particle motion has to be taken into account. We analyze a simplified diffusion model that includes some aspects of a complex environment in the framework of a nonlinear friction process: at low particle speeds, friction grows linearly with the particle velocity as for regular viscous friction; it grows more than linearly at higher particle speeds; finally, at a maximum of the possible particle speed, the friction diverges. In addition to bare diffusion, we study the influence of a constant drift force acting on the diffusing particle. While the corresponding stationary velocity distributions can be derived analytically, the displacement statistics generally must be determined numerically. However, as a benefit of our model, analytical progress can be made in one case of a special maximum particle speed. The effect of a drift force in this case is analytically determined by perturbation theory. It will be interesting in the future to compare our results to real experimental systems. One realization could be magnetic colloidal particles diffusing through a shear-thickening environment such as starch suspensions, possibly exposed to an external magnetic field gradient.

One specific velocity color mapping using optical coherence tomography.

Depth resolved coherence gating along with Doppler shift detection of the carrier frequency is used for one predetermined velocity mapping in different flows. Bidirectional rapid scanning optical delay of optical coherence tomography system is applied in the reference arm. Tilted capillary entry is used as a hydrodynamic phantom to model a sign-variable flow with complex geometry. Structural and one specific velocity images are obtained from the scanning interferometer signal processing in the frequency domain using analog and digital filtering. A standard structural image is decomposed into three parts: stationary object, and positive and negative velocity distributions. The latter two show equivelocity maps of the flow. The final image is represented as the complexation of the three.

Stationary electron velocity distribution function in crossed electric and magnetic fields with collisions

Analytical studies and numerical simulations show that the electron velocity distribution function in a Hall thruster discharge with crossed electric and magnetic fields is not Maxwellian. This is due to the fact that the mean free path between collisions is greater than both the Larmor radius and the characteristic dimensions of the discharge channel. However in numerical models of Hall thrusters, a hydrodynamic approach is often used to describe the electron dynamics, because discharge simulation in a fully kinetic approach requires large computing resources and is time consuming. A more accurate modeling of the electron flow in the hydrodynamic approximation requires taking into account the non-Maxwellian character of the distribution function and finding its moments, an approach that reflects the properties of electrons drifting in crossed electric and magnetic fields better than the commonly used Euler or Navier-Stokes approximations. In the present paper, an expression for the electron velocity distribution function in rarefied spatially homogeneous stationary plasma with crossed electric and magnetic fields and predominance of collisions with heavy particles is derived in the relaxation approximation. The main moments of the distribution function including longitudinal and transversal temperatures, the components of the viscous stress tensor, and of the heat flux vector are calculated. Distinctive features of the hydrodynamic description of electrons with a strongly non-equilibrium distribution function and the prospects for further development of the proposed approach for calculating the distribution function in spatially inhomogeneous plasma are discussed.

Active Brownian particles

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.

Relaxation of ideal classical particles in a one‐dimensional box

AbstractWe study the deterministic dynamics of non‐interacting classical gas particles confined to a one‐dimensional box as a pedagogical toy model for the relaxation of the Boltzmann distribution towards equilibrium. Hard container walls alone induce a uniform distribution of the gas particles at large times. For the relaxation of the velocity distribution we model the dynamical walls by independent scatterers. The Markov property guarantees a stationary but not necessarily thermal velocity distribution for the gas particles at large times. We identify the conditions for physical walls where the stationary velocity distribution is the Maxwell distribution. For our numerical simulation we represent the wall particles by independent harmonic oscillators. The corresponding dynamical map for oscillators with a fixed phase (Fermi–Ulam accelerator) is chaotic for mesoscopic box dimensions.

Relation between cooperative molecular motors and active Brownian particles

Active Brownian particles (ABPs), obeying a nonlinear Langevin equation with speed-dependent drift and noise amplitude, are well-known models used to describe self-propelled motion in biology. In this paper we study a model describing the stochastic dynamics of a group of coupled molecular motors (CMMs). Using two independent numerical methods, one based on the stationary velocity distribution of the motors and the other one on the local increments (also known as the Kramers-Moyal coefficients) of the velocity, we establish a connection between the CMM and the ABP models. The parameters extracted for the ABP via the two methods show good agreement for both symmetric and asymmetric cases and are independent of N, the number of motors, provided that N is not too small. This indicates that one can indeed describe the CMM problem with a simpler ABP model. However, the power spectrum of velocity fluctuations in the CMM model reveals a peak at a finite frequency, a peak which is absent in the velocity spectrum of the ABP model. This implies richer dynamic features of the CMM model which cannot be captured by an ABP model.

Time parametrization and stationary distributions in a relativistic gas

In this paper we consider the effect of different time parametrizations on the stationary velocity distribution function for a relativistic gas. We clarify the distinction between two such distributions, namely, the Jüttner and the Modified Jüttner distributions. Using a recently proposed model of a relativistic gas, we show that the obtained results for the proper-time averaging does not lead to the Modified Jüttner distribution (as recently conjectured), but introduces a Lorentz factor γ(v) (i.e., energy) to the well-known Jüttner function which results from observer-time averaging. These two modifications (i.e., Modified Jüttner function or Jüttner divided by energy) are identical in the rest frame; however, their distinction comes to light when one considers a moving frame. We obtain results for rest frame as well as moving frame in order to support our claim. We finally conclude that reparametrizing time simply rescales stationary (Jüttner) distribution function according to a well-defined frame-independent relation.

Rectification of motion in nonlinear media with asymmetric random drive

We consider moving particles in media with nonlinear friction and drive them by an asymmetric dichotomic Markov process. Due to different energy dissipations, during the forward and backward stroke, we obtain a mean non-vanishing directed flow of the particles. Starting with the stationary velocity distribution, we calculate the stationary current of particles, the variance and the relative variance in dependence on the degree of nonlinearity of the friction, on the asymmetry and for different strengths of friction. In two dimensions the particle performs diffusional motion, if in addition the direction of the asymmetric drive changes stochastically.

Granular rough sphere in a low-density thermal bath

We study the stationary state of a rough granular sphere immersed in a thermal bath composed of point particles. When the centre of mass of the sphere is fixed the stationary angular velocity distribution is shown to be Gaussian with an effective temperature lower than that of the bath. For a freely moving rough sphere coupled to the thermostat via inelastic collisions we find a condition under which the joint distribution of the translational and rotational velocities is a product of Gaussian distributions with the same effective temperature. In this rather unexpected case we derive a formula for the stationary energy flow from the thermostat to the sphere in accordance with the Fourier law.

Bucket brigades with worker learning

The dynamics and throughput of a bucket brigade production system is studied when workers’ speeds increase due to learning. It is shown that, if the rules of the bucket brigade system allow a re-ordering of its workers then the bucket brigade production system is very robust and will typically rebalance to a self-organizing optimal production arrangement. As workers learn only those parts of the production line that they work on, the stationary velocity distribution for the workers of a stable bucket brigade is non-uniform over the production line. Hence, depending on the initial placement of the workers, there are many different stationary velocity distributions. It is shown that all the stationary distributions lead to the same throughput.

Stationary state of a heated granular gas: Fate of the usual H-functional

We consider the characterization of the nonequilibrium stationary state of a randomly driven granular gas in terms of an entropy-production-based variational formulation. Enforcing spatial homogeneity, we first consider the temporal stability of the stationary state reached after a transient. In connection, two heuristic albeit physically motivated candidates for the nonequilibrium entropy production are put forward. It turns out that none of them displays an extremum for the stationary velocity distribution selected by the dynamics. Finally, the relevance of the relative Kullbach entropy is discussed.

Dynamics of a tracer granular particle as a nonequilibrium Markov process

The dynamics of a tracer particle in a stationary driven granular gas is investigated. We show how to transform the linear Boltzmann equation, describing the dynamics of the tracer into a master equation for a continuous Markov process. The transition rates depend on the stationary velocity distribution of the gas. When the gas has a Gaussian velocity probability distribution function (PDF), the stationary velocity PDF of the tracer is Gaussian with a lower temperature and satisfies detailed balance for any value of the restitution coefficient alpha. As soon as the velocity PDF of the gas departs from the Gaussian form, detailed balance is violated. This nonequilibrium state can be characterized in terms of a Lebowitz-Spohn action functional W(tau) defined over trajectories of time duration tau. We discuss the properties of this functional and of a similar functional W(tau), which differs from the first for a term that is nonextensive in time. On the one hand, we show that in numerical experiments (i.e., at finite times tau), the two functionals have different fluctuations and W always satisfies an Evans-Searles-like symmetry. On the other hand, we cannot observe the verification of the Lebowitz-Spohn-Gallavotti-Cohen (LS-GC) relation, which is expected for W(tau) at very large times tau. We give an argument for the possible failure of the LS-GC relation in this situation. We also suggest practical recipes for measuring W(tau) and W(tau) in experiments.

Stationary States and Energy Cascades in Inelastic Gases

We find a general class of nontrivial stationary states in inelastic gases where, due to dissipation, energy is transferred from large velocity scales to small velocity scales. These steady states exist for arbitrary collision rules and arbitrary dimension. Their signature is a stationary velocity distribution f(v) with an algebraic high-energy tail, f(v) approximately v(-sigma). The exponent sigma is obtained analytically and it varies continuously with the spatial dimension, the homogeneity index characterizing the collision rate, and the restitution coefficient. We observe these stationary states in numerical simulations in which energy is injected into the system by infrequently boosting particles to high velocities. We propose that these states may be realized experimentally in driven granular systems.

Electro- and Magnetostatic Fields in Media with a Nonuniform Velocity

A motion-induced magnetic or electric field is calculated by the integral method in the first order of smallness in the ratio of the motion velocity to the speed of light for homogenous media with an arbitrary stationary velocity distribution that are placed in static electric or magnetic fields. For the case of rotation of a sphere, the validity of the results is corroborated by comparing with a solution obtained by joining the fields in the moving and quiescent parts of the medium. Estimates suggest the feasibility of experimentally observing this effect of continuum electrodynamics.