General Kinetic Equation Research Articles

Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1 / N in a proper thermodynamic limit N → + ∞ , where N is the number of particles. These correlations are responsible for the “collisional” evolution of the system beyond the Vlasov regime due to finite N effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard–Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size N and try to provide a coherent discussion of all the numerical results obtained for these systems.

Mechanistic model for evaluating the performance of suspended growth bioreactors for the off-gas treatment of VOCs

A mechanistic model capable to predict toluene removal in suspended growth bioreactors (SGRs) was proposed and validated. The model predicted with reasonable accuracy both toluene elimination capacities (11% of relative average error) and biomass concentration (6% of relative average error) in a Pseudomonas putida F1 culture growing in a lab-scale chemostat. Unlike previously reported mechanistic models this holistic model was capable to predict process performance under toluene, oxygen or nutrient limiting conditions. Mass transport, Henry law, and biomass yield coefficients were the most significant parameters influencing model predictions as shown by the sensitivity analysis performed. In addition, an innovative vector-based solving approach was proposed, which allowed reducing the complex systems of eight non-linear algebraic equations into a single equation and provided the dimensionless numbers necessary for bioreactor scale-up. Being based in general kinetic and mass transport equations it can be adapted to other processes where mass transport from a gaseous to an aqueous phase is involved.

A kinetic study of the strontium extraction by metallothermic reduction using submerged SrO powders injection

This work reports the results of laboratory experiments conduced to follow the kinetics of strontium recovery into the Al–Mg alloy by metallothermic reduction of SrO. The reagent was incorporated to molten alloy by the use of submerged powders injection technique. The variables analyzed were the injection time, the melt temperature and the initial magnesium content. Magnesium is added to the melt to increase the reactivity and reduce the surface tension of the molten aluminum. It was possible to increase the strontium content from 0 to 5 wt.% after 60 min of treatment. The results were fitted to a general kinetic equation, which allowed it to obtain the kinetic parameters, i.e. order of reaction and activation energy of the process. As the main mechanism of the strontium recovery process is of diffusive type, the global process rate increases as the temperature and initial amount of the magnesium increased.

A self-consistent mean field theory for diffusion in alloys

Starting from a microscopic model of the atomic transport via vacancies and interstitials in alloys, a self-consistent mean field (SCMF) kinetic theory yields the phenomenological coefficients Lij. In this theory, kinetic correlations are accounted for through a set of effective interactions within a non-equilibrium distribution function of the system. The introduction of a master equation describing the evolution with time of the distribution function and its moments leads to general self-consistent kinetic equations. The Lij of a face centered cubic alloy are calculated using the kinetic equations of Nastar (M. Nastar, Philos. Mag., 2005, 85, 3767, ref. 1) derived from a microscopic broken bond model of the vacancy jump frequency. A first approximation leads to an analytical expression of the Lij and a second approximation to a better agreement with the Monte Carlo simulations. A change of sign of the Lij is studied as a function of the microscopic parameters of the jump frequency. The Lij of a cubic centered alloy obtained for the complex diffusion mechanism of the dumbbell configuration of the interstitial (V. Barbe and M. Nastar, Philos. Mag., 2006, in press, ref. 2) are used to study the effect of an on-site rotation of the dumbbell on the transport.

Kinetic equations for reaction-subdiffusion systems: Derivation and stability analysis

We derive general kinetic equations for reacting and subdiffusing entities based on a nonlinear continuous time random walk formalism proposed by Vlad and Ross [Phys. Rev. E 66, 061908 (2002)]. Reaction and diffusion processes are separable in a typical reaction-diffusion system, and their combined influence on the evolution of the density of a species is a simple sum. Our derivation shows that this is no longer true for subdiffusive entities undergoing reactions. The strong memory effects in the transport process, i.e., the non-Markovian nature of subdiffusion, results in a nontrivial combination of reactions and spatial dispersal, which we discuss in detail. We carry out a linear stability analysis of the derived reaction-subdiffusion system to understand the effects of memory on pattern formation. We find that the Turing instability persists in the subdiffusive system. However, the memory modifies the Turing threshold and the characteristics of the band of unstable modes close to this threshold.

Nonlinear Markov Semigroups and Interacting Lévy Type Processes

Semigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduce and study the general class of semigroups of non-linear positivity preserving transformations on measures that is non-linear Markov or Feller semigroups. An explicit structure of generators of such groups is given in case when X is the Euclidean space R d (or more generally, a manifold) showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for Markov models of interacting particles. Well posedness results for these equations are given together with applications to interacting particles: dynamic law of large numbers and central limit theorem, the latter being new already for the standard coagulation-fragmentation models.

Municipal sludge degradation kinetic in thermophilic CSTR

AbstractThe performance of a pilot‐scale continuous‐flow stirred‐tank reactor (CSTR) treating municipal sludge under thermophilic conditions has been studied. Two pilot‐scale reactors (CSTR1 (175 L) and CSTR2 (850 L)) were operated at different hydraulic residence times (θ: 40 to 15 days). The anaerobic sludge processes are generally affected by variations in the concentration of substrate (determined as influent volatile solids, VS) and volumetric flow, both of which lead to a modification in biomass concentration and VS removal efficiency. This unsteady‐state situation is mathematically explained in terms of an autocatalytic kinetic model. The general kinetic equation in this model has been applied to experimental data obtained in CSTR1. The fit of the experimental data to the model was used to estimate kinetic parameters and the yield coefficients (μmax, α, YP/S). The estimated parameters were μmax: 0.175d−1, α: 0.358, YP/S: 0.309 m3CH4/kgVS). These parameters were subsequently used to model the substrate utilization rate and the methane generation rate in CSTR2. The model with the estimated parameters was found to provide excellent results, and is satisfactory in describing the concentration of VS and the methane generation rate in an actual digestion plant. © 2006 American Institute of Chemical Engineers AIChE J, 2006

Nonlinear Fokker–Planck–Landau integral propagator (II): transport far from equilibrium

We present a time evolving path-integral method for solving the Landau–Fokker–Planck equation to compute kinetic transport coefficients in a fully ionized plasma. The electron distribution function is advanced in time by means of the conservative short-time propagators, which we previously obtained. The validated integral operator takes into account both electron–electron and electron–ion collisions without linearizing the original Fokker–Planck collisional operator. The resulting integral formulation in velocity space is applied here to evaluate the local transport coefficients if inhomogeneities in configuration space appear. We define an effective source term through a flux particle balance in a thin slab of plasma, which leads to a nonhomogeneous Fokker–Planck equation. Hence, this new term locally models the so-called Vlasov term appearing in the general kinetic equation. Arbitrary departures from Maxwellian equilibrium can be dealt with this effective source term that preserves the positiveness of the electron distribution function, even in the runaway limit. For small perturbations of the equilibrium, the classical Spitzer and Harm transport coefficients are recovered, while a very strong reduction of the heat flux takes place for large temperature gradients, as predicted by some authors in different theories.

Anomalous behavior of LiF:Mg,Cu,P (TLD100H) irradiated with electron beams

Thermoluminescent dosimeters of LiF:Mg,Cu,P (TLD100H) were exposed to 60Co γ -ray, protons, high energy electrons and the glow curves were analyzed. We observed an unusual behavior of the TL response of dosimeters to 7 and 14 MeV electron beam. In fact the relative amplitude between two of the five peaks (2° and 3° peaks) assumes values smaller than the unity for doses up to 5 Gy and values greater than the unity for doses equal to or greater than 10 Gy. A possible explanation of this behavior is given adopting a model of the formation of the peaks 2° and 3° and examining the different ways of energy release in matter for each beam. To perform this analysis we deconvoluted the glow curve adopting general order kinetic (GOK) equation for the description of the peak shape, determining the number of peaks and their trapping parameter ( E , s , n 0 ) .

Wave turbulence in incompressible Hall magnetohydrodynamics

We investigate the steepening of the magnetic fluctuation power law spectra observed in the inner solar wind for frequencies higher than 0.5 Hz. This high frequency part of the spectrum may be attributed to dispersive nonlinear processes. In that context, the long-time behavior of weakly interacting waves is examined in the framework of 3D incompressible Hall MHD turbulence. The Hall term added to the standard MHD equations makes the Alfv\'en waves dispersive and circularly polarized. We introduce the generalized Els\"asser variables and, using a complex helicity decomposition, we derive for three-wave interaction processes the general wave kinetic equations; they describe the nonlinear dynamics of Alfv\'en, whistler and ion cyclotron wave turbulence in the presence of a strong uniform magnetic field $B_0 \ep$. Hall MHD turbulence is characterized by anisotropies of different strength. We show that electron and standard MHD turbulence can be seen as two frequency limits of the present theory but the standard MHD limit is singular; additionally, we analyze in detail the ion MHD turbulence limit. Exact power law solutions of the master wave kinetic equations are given in the small and large scale limits for which we have, respectively, the total energy spectra $E(\kpn,\kpa) \sim \kpn^{-5/2} |\kpa|^{-1/2}$ and $E(\kpn,\kpa) \sim \kpn^{-2}$. An anisotropic phenomenology is developed to describe continuously the different scaling laws of the energy spectrum; one predicts $E(\kpn,\kpa) \sim \kpn^{-2} |\kpa|^{-1/2} (1+\kpn^2d_i^2)^{-1/4}$. Nonlocal interactions between Alfv\'en, whistler and ion cyclotron waves are investigated; a non trivial dynamics exists only when a discrepancy from the equipartition between the large scale kinetic and magnetic energies happens.

Glow curve analysis of TLD-100H irradiated with radiation of different LET: Comparison between two theoretical method

In the present paper we show the result of study on the thermoluminescent signal of LiF:Mg,Cu,P (TLD-100H) after irradiation with beams of different LET (photons, electrons and protons). Through the analysis of the glow curve it is possible to obtain information on the LET of the radiation beam. To perform this analysis we have deconvoluted the glow curves adopting general order kinetic equation for the description of the peak shape. In addition a numerical simulation has been performed to find a connection among the physical parameters of the model (OTOR) and the parameters of the empirical expression used in the analysis of the experimental data (GOK). The methodology indicate that the kinetic order b of the glow peak depends on the LET.

Kinetic description of avalanching systems

Avalanching systems are treated analytically using the renormalization group (in the self-organized-criticality regime) or mean-field approximation, respectively. The latter describes the state in terms of the mean number of active and passive sites, without addressing the inhomogeneity in their distribution. This paper goes one step further by proposing a kinetic description of avalanching systems making use of the distribution function for clusters of active sites. We illustrate an application of the kinetic formalism to a model proposed for the description of the avalanching processes in the reconnecting current sheet of the Earth's magnetosphere. A description of avalanching systems is proposed that makes use of the distribution function for clusters of active sites. A general kinetic equation is derived that describes the temporal evolution of the distribution function, in terms of growth and shrinking probabilities. The distribution of clusters is derived for the stationary regime, for a quite general class of avalanching systems or arbitrary dimensionality. The approach, including the probability calculation, is illustrated by an application of the kinetic description to the recently proposed burning model.

Pair Winds in Schwarzschild Spacetime with Application to Hot Bare Strange Stars

We consider a time-dependent, spherically outflowing wind in Schwarzschild spacetime consisting of electron-positron pairs and photons. Without assuming thermal equilibrium, we account for the microphysics, including two-body processes (ee ? ee, ?e ? ?e, and e+e- ? ??) and their radiative three-body variants (ee ? ee?, ?e ? ?e?, and e+e- ? ???). We present a finite-difference scheme for solving the general relativistic kinetic Boltzmann equations for pairs and photons. We apply this to the concrete example of a wind from a hot bare strange star, predicted to be a powerful source of hard X-ray photons and e? pairs created by the Coulomb barrier at the quark surface. We study the kinetics of the wind particles and the emerging emission in photons and pairs for stationary winds with total luminosities in the range 1034-1039 ergs?s-1 for different values of the injected photon-to-pair ratio. The wind parameters?such as the mean optical depth for photons, rates of particle number and energy outflows, bulk velocity, and number density of the pair plasma?are presented as functions of the distance from the stellar surface, as well as characteristics of the emergent radiation. We find that photons dominate in the emerging emission and that the emerging photon spectrum is rather hard and differs substantially from the thermal spectrum expected from a neutron star with the same luminosity. This might help distinguish the putative bare strange stars from neutron stars.

Evolution of a Keplerian disk of colliding and fragmenting particles: a kinetic model with application to the Edgeworth–Kuiper belt

We present a kinetic model of a disk of solid particles, orbiting a primary and experiencing inelastic collisions. In distinction to other collisional models that use a 2D (mass–semimajor axis) binning and perform a separate analysis of the velocity (eccentricity, inclination) evolution, we choose mass and orbital elements as independent variables of a phase space. The distribution function in this space contains full information on the combined mass, spatial, and velocity distributions of particles. A general kinetic equation for the distribution function is derived, valid for any set of orbital elements and for any collisional outcome, specified by a single kernel function. The first implementation of the model utilizes a 3D phase space (mass–semimajor axis–eccentricity) and involves averages over the inclination and all angular elements. We assume collisions to be destructive, simulate them with available material- and size-dependent scaling laws, and include collisional damping. A closed set of kinetic equations for a mass–semimajor axis–eccentricity distribution is written and transformation rules to usual mass and spatial distributions of the disk material are obtained. The kinetic “core” of our approach is generic. It is possible to add inclination as an additional phase space variable, to include cratering collisions and agglomeration, dynamical friction and viscous stirring, gravity of large perturbers, drag forces, and other effects into the model. As a specific application, we address the collisional evolution of the classical population in the Edgeworth–Kuiper belt (EKB). We run the model for different initial disk's masses and radial profiles and different impact strengths of objects. Our results for the size distribution, collisional timescales, and mass loss are in agreement with previous studies. In particular, collisional evolution is found to be most substantial in the inner part of the EKB, where the separation size between the survivors over EKB's age and fragments of earlier collisions lies between a few and several tens of km. The size distribution in the EKB is not a single Dohnanyi-type power law, reflecting the size dependence of the critical specific energy in both strength and gravity regimes. The net mass loss rate of an evolved disk is nearly constant and is dominated by disruption of larger objects. Finally, assuming an initially uniform distribution of orbital eccentricities, we show that an evolved disk contains more objects in orbits with intermediate eccentricities than in nearly circular or more eccentric orbits. This property holds for objects of any size and is explained in terms of collisional probabilities. The effect should modulate the eccentricity distribution shaped by dynamical mechanisms, such as resonances and truncation of perihelia by Neptune.

The EIRENE and B2-EIRENE Codes

The EIRENE neutral gas transport Monte Carlo code has been developed initially for TEXTOR since the early 1980s. It is currently applied worldwide in most fusion laboratories for a large variety of different purposes. The main goal of code development was to provide a tool to investigate neutral gas transport in magnetically confined plasmas. But, due to its flexibility, it also can be used to solve more general linear kinetic transport equations by applying a stochastic rather than a numerical or analytical method of solution. Major applications of EIRENE are in connection with plasma fluid codes, in particular with the various versions of the B2 two-dimensional plasma edge fluid code. The combined code package B2-EIRENE was developed, again initially for TEXTOR applications, in the late 1980s. It too has become a standard tool in plasma edge science. It is currently mainly used for divertor configurations, such as by the ITER central team, to assist the design of the ITER divertor. Both the EIRENE and B2-EIRENE concepts are introduced and illustrated with sample applications.

A new low-cost biofilm carrier for the treatment of municipal wastewater in a moving bed reactor

The Moving Bed Biofilm Reactor has proven to be an efficient system in wastewater treatment and has become a viable solution for small treatment plants. The main objective of this research was to analyse the performance of a moving bed reactor using low-cost local material when fed with municipal wastewater. A pilot reactor with a total volume of 900 litres was built and it was fed continuously with municipal wastewater. The operation of the system was adjusted to six different organic loading rates. The biofilm carrier was polyethylene tubing with internal diameter of 1.1 cm, cut into pieces of 1.2 cm. The tested material offered a specific surface area of 590 m2/m3. Air was provided with a fine-bubble diffuser. The main results show that the reactor performance was stable and predictable. The COD removal confidently behaves according to a general hyperbolic kinetic equation. The maximal total COD removal attained was 81%. Nitrification was observed only for organic loads with values under 5.7 gCOD/m2 x d. Good adherence of the microorganisms was observed for the applied organic loading rates. After several months of operation, the material showed no signs of abrasion or deformation. The sludge production behaved linearly with the organic load reaching 979 gTSS/d with the highest organic load of 35.7 gCOD/m2 x d. The amount of microorganisms attached to the carrier increased with the organic load tending to an asymptotical maximal value of 17.3 g/m2 (as dry solids). Mean cellular retention times from 2.0 to 23.1 days were determined.

A hybrid kinetic–fluid model for solving the Vlasov–BGK equation

Our purpose is to derive a model for charged particles which combines a kinetic description of the fast particles with a fluid description of the slow ones. In a previous paper, a similar model was derived from a kinetic BGK equation that uses a constant relaxation time and does not include the effect of an electric field. In this paper, we consider a more general kinetic equation including an electric field and a varying relaxation time. Fast particles will be described through a collisional kinetic equation of Vlasov–BGK type while thermal particles will be modeled by a hydrodynamic model. Then, we construct a numerical scheme for this model and validate the approach by presenting various numerical tests.

Kinetic model of the evaporation process of benzylbutyl phthalate from plasticized poly(vinyl chloride)

The kinetic model of the physical process of evaporation of plasticizer from plasticized PVC foils was developed from the results of isothermal thermogravimetric investigation of evaporation of benzyl-butyl phthalate in the temperature range 120–150 °C under nitrogen flow. The kinetic parameters were estimated by integral method of analysis. Mathematical modeling of the kinetic of plasticizers evaporation was performed on the basis of function c= f( T, t) and kinetic equation of evaporation −d c/d t= f( T, c 0, c( t)). The developed mathematical model was described by the general kinetic equation − dc/ dt=(k r exp(k pT)c 0+k qT+k w)c(t) . The differential quotients δ(−d c/d t)/ δT= f( T, c 0, c( t))= f( T, c 0, t) and δ(−d c/d t)/ δc 0= f( T, c( t))= f( T, c 0, t) were performed, and mathematical definition of the changes of the evaporation rate constant with the change of temperature and the change of the initial plasticized concentration were discussed.

Kinetic study of poly- N-methylpyrrole electrogeneration

In this kinetic study ex situ microgravimetry, chronoamperometry and cyclic voltammetry were used to determine the orders of reaction with respect to the electrolyte, the monomer and the general kinetic equation in N-methylpyrrole electropolymerization in an acetonitrile medium with sodium perchlorate as dopant. Physical properties such as charge productivity P Q , charge storage efficiency C Q , and doping level y were calculated and their variations were analyzed. Data from the kinetic study suggest that the influence of the electrolyte on the electropolymerization mechanism is considerable and is directly related to its concentration.

Electron acceleration by a stochastic electric field in the atmospheric layer

Electron acceleration by a stochastic (in space) electric field is considered in the atmosphere taking into account limited scales of the acceleration layer. The general kinetic equation for the isotropic part of the electron distribution function is thereby derived, which includes runaway effects and influence of a stochastic electric field on spatial diffusion. A tail of accelerated electrons in the distribution function appears when the electric field intensity overtakes the threshold value, which is close to the threshold in a constant electric field. Stationary solutions of the kinetic equation in a finite layer are analyzed numerically in the presence of source of energetic electrons. These solutions are discussed in connection with γ- and x-ray emissions observed inside thunderclouds.