Equilibrium Velocity Distribution Research Articles

Hypocoercivity and Reaction-Diffusion Limit for a Nonlinear Generation-Recombination Model

A reaction-kinetic model for a two-species gas mixture undergoing pair generation and recombination reactions is considered on a flat torus. For dominant scattering with a non-moving constant-temperature background the macroscopic limit to a reaction-diffusion system is carried out. Exponential decay to equilibrium is proven for the kinetic model by hypocoercivity estimates. This seems to be the first rigorous derivation of a nonlinear reaction-diffusion system from a kinetic model as well as the first hypocoercivity result for a nonlinear kinetic problem without smallness assumptions. The analysis profits from uniform bounds of the solution in terms of the equilibrium velocity distribution.

Non-homogeneous traffic characterization based on driver reaction and stimuli

A macroscopic model for non-homogeneous traffic is proposed based on harmonization during transitions. This model considers the lateral and forward distances between vehicles, reaction and harmonization times, and changes in velocity. Further, the equilibrium velocity distribution is characterized based on the density and travel time of real non-homogeneous traffic. The proposed and Payne–Whitham (PW) models are evaluated over a 200 m circular road using the FORCE scheme. The results obtained demonstrate that the proposed model provides a more realistic representation of non-homogeneous traffic.

A Novel Macroscopic Traffic Model based on Distance Headway

A new model is proposed to characterize changes in traffic at transitions. These changes are affected by driver response. The distance headway between vehicles is considered as it affects driver behavior. Driver response is quick with a small distance headway and slow when the distance headway is large. The variations in traffic are greater with a slow driver while traffic is smooth with a quick driver. A model is developed which characterizes traffic based on driver response and distance headway. This model is compared with the well-known and widely employed Zhang and PW models. The Zhang model characterizes driver response at transitions using an equilibrium velocity distribution and ignores distance headway and driver response. Traffic flow in the PW model is characterized using only a velocity constant. Roe decomposition is employed to evaluate the Zhang, PW, and proposed models over a 270 m circular (ring) road. Results are presented which show that Zhang model provides unrealistic results. The corresponding behavior with the proposed model has large variations in flow with a slow driver but is smooth with a quick driver. The PW model provides smooth changes in flow according to the velocity constant, but the behavior is unrealistic because it is not based on traffic physics. Doi: 10.28991/CEJ-SP2021-07-03 Full Text: PDF

Mg-mamposst: a code to test modifications of gravity with internal kinematics and lensing analyses of galaxy clusters

ABSTRACT We present an upgraded version of mg-mamposst, an extension of the mamposst (Modelling Anisotropy and Mass Profile of Spherical Observed Systems) algorithm that performs Bayesian fits of models of mass and velocity anisotropy profiles to the distribution of tracers in projected phase space, to handle modified gravity models and constrain their parameters. The new version implements two distinct types of gravity modifications, namely general chameleon and Vainshtein screening, and is further equipped with a Monte Carlo Markov chain module for an efficient parameter space exploration. The programme is complemented by the clustergen code, capable of producing mock galaxy clusters under the assumption of spherical symmetry, dynamical equilibrium, and Gaussian local velocity distribution functions as in mamposst. We demonstrate the potential of the method by analysing a set of synthetic, isolated spherically symmetric dark matter haloes, focusing on the statistical degeneracies between model parameters. Assuming the availability of additional lensing-like information, we forecast the constraints on the modified gravity parameters for the two models presented, as expected from joint lensing + internal kinematics analyses, in view of upcoming galaxy cluster surveys. In Vainshtein screening, we forecast the weak lensing effect through the estimation of the full convergence-shear profile. For chameleon screening, we constrain the allowed region in the space of the two free parameters of the model, further focusing on the $\displaystyle f(\mathcal {R})$ subclass to obtain realistic bounds on the background field $\displaystyle |f_{\mathcal {R}0}|$. Our analysis demonstrates the complementarity of internal kinematics and lensing probes for constraining modified gravity theories, and how the bounds on Vainshtein-screened theories improve through the combination of the two probes.

Averaged Boltzmann’s Kinetics for Colloidal Transport in Porous Media

AbstractDue to the stochastic nature of pore space geometry, particle velocities in a colloidal‐suspension flux are also stochastically distributed. This phenomenon is captured by Boltzmann's kinetic equation. We formulate a BGK‐form of Boltzmann's equation for particulate flow with a particle capture rate proportional to the particle speed and derive an exact method for the model's averaging. The averaged equation is of the form of a reaction‐advection‐diffusion equation with the average particle speed lower than the carrier fluid velocity. This delay, reported in numerous laboratory studies, is explained by preferential capture of fast particles. The large‐scale model coefficients of delay, dispersion, and filtration are explicitly expressed via the microscale mixing length, equilibrium velocity distribution, and filtration coefficient. The properties of the averaged coefficients are discussed, and emerging dependencies between them are presented. The derived large‐scale equation resolves two paradoxes of the traditional model for suspension‐colloidal transport in porous media. The large‐scale equation closely matches the laboratory breakthrough curves.

Hydrodynamic and Thermodynamic Nonequilibrium Effects around Shock Waves: Based on a Discrete Boltzmann Method.

A shock wave that is characterized by sharp physical gradients always draws the medium out of equilibrium. In this work, both hydrodynamic and thermodynamic nonequilibrium effects around the shock wave are investigated using a discrete Boltzmann model. Via Chapman–Enskog analysis, the local equilibrium and nonequilibrium velocity distribution functions in one-, two-, and three-dimensional velocity space are recovered across the shock wave. Besides, the absolute and relative deviation degrees are defined in order to describe the departure of the fluid system from the equilibrium state. The local and global nonequilibrium effects, nonorganized energy, and nonorganized energy flux are also investigated. Moreover, the impacts of the relaxation frequency, Mach number, thermal conductivity, viscosity, and the specific heat ratio on the nonequilibrium behaviours around shock waves are studied. This work is helpful for a deeper understanding of the fine structures of shock wave and nonequilibrium statistical mechanics.

Macroscopic Traffic Flow Characterization at Bottlenecks

Traffic congestion is a significant issue in urban areas. Realistic traffic flow models are crucial for understanding and mitigating congestion. Congestion occurs at bottlenecks where large changes in density occur. In this paper, a traffic flow model is proposed which characterizes traffic at the egress and ingress to bottlenecks. This model is based on driver response which includes driver reaction and traffic stimuli. Driver reaction is based on time headway and driver behavior which can be classified as sluggish, typical or aggressive. Traffic stimuli are affected by the transition width and changes in the equilibrium velocity distribution. The explicit upwind difference scheme is used to evaluate the Lighthill, Whitham, and Richards (LWR) and proposed models with a continuous injection of traffic into the system. A stability analysis of these models is given and both are evaluated over a road of length 10 km which has a bottleneck. The results obtained show that the behavior with the proposed model is more realistic than with the LWR model. This is because the LWR model cannot adequately characterize driver behavior during changes in traffic flow.

Electrically driven nematic flow in microfluidic devices containing a temperature gradient.

Fluid pumping principle has been developed utilizing the interaction, on the one hand, between the electric field E and the gradient ∇n[over ̂] of the director's field, and, on the other hand, between the ∇n[over ̂] and the temperature ∇T gradient arising in a homogeneously aligned liquid crystal (HALC) microfluidic channel. Calculations, based upon the nonlinear extension of the classical Ericksen-Leslie theory, with accounting the entropy balance equation, show that due to the coupling among the ∇T, ∇n,[over ̂] and E in the HALC microfluidic channel the horizontal flow v=v_{x}i[over ̂]=ui[over ̂] may be excited. The direction and magnitude of v is influenced both by the heat flux q across the microfluidic channel and the strength of the electric field E. The results of calculations showed that the dependence of the maximum value of the equilibrium velocity distribution |u_{max}(E/E_{th})| across the LC channel versus electric field E/E_{th} is characterized by maximum value at E/E_{th}=2.0. In the case when the electric field E≫E_{th}, the horizontal flow of the LC material completely stops and a novel mechanism of converting of the electric field in the form of the kinklike wave reorientation of the director field n[over ̂] can be excited in the LC channel.

Hypocoercivity and Fast Reaction Limit for Linear Reaction Networks with Kinetic Transport

The long time behavior of a model for a first order, weakly reversible chemical reaction network is considered, where the movement of the reacting species is described by kinetic transport. The reactions are triggered by collisions with a nonmoving background with constant temperature, determining the post-reactional equilibrium velocity distributions. Species with different particle masses are considered, with a strong separation between two groups of light and heavy particles. As an approximation, the heavy species are modeled as nonmoving. Under the assumption of at least one moving species, long time convergence is proven by hypocoercivity methods for the cases of positions in a flat torus and in whole space. In the former case the result is exponential convergence to a spatially constant equilibrium, and in the latter it is algebraic decay to zero, at the same rate as solutions of parabolic equations. This is no surprise since it is also shown that the macroscopic (or reaction dominated) behavior is governed by the diffusion equation.

Unsteady Plasma Flow Near an Oscillating Rigid Plane Plate Under the Influence of an Unsteady Nonlinear External Magnetic Field

In an irreversible thermodynamics framework, Stokes’ second problem was examined for unsteady oscillating flow. The objective was to apply that for a plasma near an illimitable oscillating rigid plane plate under the influence of an unsteady nonlinear applied magnetic field. The Bhatnagar-Gross-Krook (BGK) pattern of the Boltzmann kinetic equation supplemented by Maxwell’s equations was investigated. The method of moments was applied with a two-sided distribution function. The exact traveling wave solution was obtained for a system consisting of four non-homogeneous partial differential equations. The velocity, shear stress, viscosity coefficient, generated electric field, applied nonlinear magnetic field, polarization, gyro-radius, and gyro-frequency were calculated. Furthermore, the distinction between the equilibrium velocity distribution function and the perturbed distribution functions was theoretically clarified at distinct time values. The advantage of the Boltzmann equation permitted us to consider irreversible non-equilibrium thermodynamics principles. For that purpose, the calculated distribution functions should be used in the formulae of entropy, entropy flux, thermodynamic forces, and kinetic coefficient. From the analysis of the results, it is found that Boltzmann’s H-theorem, thermodynamics laws, and Le Chatelier’s principle were consistent with our model for the whole system. The distinct contributions of the forces exerted on the system modified its internal energies; they were expressed via the Gibbs formula. The results demonstrated that the proposed model is capable of describing the behavior of plasma helium gas in the upper atmosphere ionic belts. Based on the analytical calculations, 3D-Graphics illustrating the physical quantities were drawn to predict their conduct, and the results are deeply discussed.

A macroscopic traffic model for heterogeneous flow

A new macroscopic heterogeneous traffic flow model is proposed in this paper. Driver anticipation is developed using analogies from Darcy's law and is based on the time and lateral distance headways. An equilibrium velocity distribution for heterogeneous traffic based on traffic alignment is employed. The performance of the Extended Speed Gradient (ESG), Zhang and proposed models is compared for the same traffic conditions using the Roe decomposition technique. The results obtained show the significance of the time and lateral distance headways on the evolution of traffic flow. Further, the proposed model provides a more realistic characterization of traffic flow than the ESG and Zhang models.

Relativistic Poisson-Kac processes and equilibrium Jüttner distribution

It is shown that relativistic Ornstein-Uhlenbeck processes driven by stochastic perturbations possessing finite propagation velocity, and specifically by Poisson-Kac processes, are consistent with the known equilibrium velocity distribution of a relativistic gas (Jüttner distribution). Some observations on the boundedness of acceleration are addressed from elementary quantum principles (quantization of the action variable).

A Macroscopic Traffic Model Based on Anticipation

In this paper, a new traffic model is presented which considers spatial changes in traffic density to characterize the traffic flow during transitions. The commonly employed Zhang model adjusts the traffic according to the gradient of the equilibrium velocity distribution during transitions regardless of the interaction between vehicles, maximum road density (capacity), or driver sensitivity. As a consequence, the resulting speed and flow can be unrealistic. With the proposed model, the speed and traffic flow during transitions are a function of the headway, traffic density, and velocity profile. Results are presented for discontinuous traffic densities caused by a bottleneck along a circular road which show that this model eliminates the unrealistic behavior of the Zhang model.

Analytic slowing-down distributions as modified by turbulent transport

The effect of electrostatic microturbulence on fast particles rapidly decreases at high energy, but can be significant at moderate energy. Previous studies found that, in addition to changes in the energetic particle density, this results in non-trivial changes to the equilibrium velocity distribution. These effects have implications for plasma heating and the stability of Alfvén eigenmodes, but make multiscale simulations much more difficult without further approximations. Here, several related analytic model distribution functions are derived from first principles. A single dimensionless parameter characterizes the relative strength of turbulence relative to collisions, and this parameter appears as an exponent in the model distribution functions. Even the most simple of these models reproduces key features of the numerical phase-space transport solution and provides a useful a priori heuristic for determining how strong the effect of turbulence is on the redistribution of energetic particles in toroidal plasmas.

Study on failure mechanism of Cu-polyethylene-Cu sandwich structure by molecular dynamics simulation

The tensile failure mechanism of Cu-Polyethylene (PE)-Cu (CPC) sandwich structure was clarified by molecular dynamics (MD) simulations subjected to a uniaxial tensile loading at microscopic scale. The sensitivity analysis of parameters such as mixing rules in describing the interaction between the wall (Cu) and the sandwich layer (PE), model size, relaxation time for equilibrium and initial velocity distribution was carried out to verify the rationality of modeling. The evolutions of stress-strain relationship and each potential energy component were provided to describe the failure process of the structure. The peak of non-bond energy shows a delay compared to the yield point in stress-strain curve, which coincides with the local maximum point of the trans-fraction curve of dihedral angles. After that, an inflexion appeared in the trans-fraction curve indicates an energy transport process, which corresponds with the slope change of the stress-strain curve. It is assumed that the dihedral distribution plays a crucial role in the damage process of CPC structure. In addition, the temperature field and the density profile were adopted to predict the position of damage initiation, which was confirmed by the microstructure evolution. The intrinsic thickness-dependence of CPC was explored by taking the coupling effect of bridging and entanglement into account, which is in reverse proportion with the yield strength of CPC.

Generalized Boltzmann-Type Equations for Aggregation in Gases

The coalescence and fragmentation of particles in a dispersion system are investigated by applying kinetic theory methods, namely, by generalizing the Boltzmann kinetic equation to coalescence and fragmentation processes. Dynamic equations for the particle concentrations in the system are derived using the kinetic equations of motion. For particle coalescence and fragmentation, equations for the particle momentum, coordinate, and mass distribution functions are obtained and the coalescence and fragmentation coefficients are calculated. The equilibrium mass and velocity distribution functions of the particles in the dispersion system are found in the approximation of an active terminal group (Becker–Doring-type equation). The transition to a continuum description is performed.

Isotropic pressure model in the presence of coarse-graining scale Alfvén waves and its consequence for modulational instability

A scalar pressure model of solar wind plasma in the presence of Alfvén waves at the coarse-graining scale is discussed. Using a simple sub-grid-scale model, it is shown that the scalar pressure model can be applied to describe the background pressure, because the off-diagonal components of the sub-grid-scale stress tensor disappear in the presence of one-dimensional, sub-grid scale Alfvén waves. Local equilibrium velocity distribution functions corresponding to the scalar pressure including wave dynamics (apparent temperature) are also presented. The specific heat ratio of the apparent temperature is given by the Wentzel–Kramers–Brillouin approximation of the Alfvén waves. The consequence of the local equilibrium velocity distribution functions in the coarse-graining scale model is discussed by using a nonlinear equation set describing envelope-modulated nonlinear Alfvén waves. It is recognized that without sufficient separation between the scale of waves and the scale of integration, artificial changes of the growth rates appear in the solutions of linear growth rates owing to the finite length of the filter function.

Note on one-fluid modeling of low-frequency Alfvénic fluctuations in a solar wind plasma with multi-ion components

A simple point of view that non-zero Alfvén ratio (residual energy) appears as a consequence of one-fluid modeling of uni-directional Alfvén waves in a solar wind plasma is presented. Since relative speeds among ions are incorporated into the one-fluid model as a pressure anisotropy, the Alfvén ratio can be finite due to the decrease in the phase velocity. It is shown that a proton beam component typically found in the solar wind plasma can contribute to generating non-zero Alfvén ratio observed in the solar wind plasma. Local equilibrium velocity distribution functions of each ion component are also discussed by using maximum entropy principle.

Revised lattice Boltzmann model for traffic flow with equilibrium traffic pressure

A revised lattice Boltzmann model concerning the equilibrium traffic pressure is proposed in this study to tackle the phase transition phenomena of traffic flow system. The traditional lattice Boltzmann model has limitation to investigate the complex traffic phase transitions due to its difficulty for modeling the equilibrium velocity distribution. Concerning this drawback, the equilibrium traffic pressure is taken into account to derive the equilibrium velocity distribution in the revised lattice Boltzmann model. In the proposed model, a three-dimensional velocity-space is assumed to determine the equilibrium velocity distribution functions and an alternative, new derivative approach is introduced to deduct the macroscopic equations with the first-order accuracy level from the lattice Boltzmann model. Based on the linear stability theory, the stability conditions of the corresponding macroscopic equations can be obtained. The outputs indicate that the stability curve is divided into three regions, i.e., the stable region, the neutral stability region, and the unstable region. In the stable region, small disturbance appears in the initial uniform flow and will vanish after long term evolution, while in the unstable region, the disturbance will be enlarged and finally leads to the traffic system entering the congested state. In the neutral stability region, small disturbance does not vanish with time and maintains its amplitude in the traffic system. Conclusively, the stability of traffic system is found to be enhanced as the equilibrium traffic pressure increases. Finally, the numerical outputs of the proposed model are found to be consistent with the recognized, theoretical results.

Equilibrium of a Brownian particle in a damping induced inhomogeneous space: an alternative approach

We look at the equilibrium of a Brownian particle in an inhomogeneous space following the alternative approach proposed in (Bhattacharyay 2013 Physica A 392 4265). We consider a coordinate dependent damping resulting in a stochastic problem with multiplicative noise. We convert this stochastic problem with multiplicative noise to additive noise form using our alternative approach. The resulting equilibrium velocity distribution is locally Maxwellian with a zero mean. The position distribution is of a modified Boltzmann form where that of a free particle is a function of damping constant.