Kinetic theory of inhomogeneous diffusion

This paper theoretically investigates particle diffusion in a medium where the diffusivity depends on position. We exclusively consider continuous-time, continuous-space transport and our working tool is the linear kinetic theory pertinent to guest particles in a passive host medium (Lorentz’s picture of transport). The host medium may or may not thermalize the guest particles. It may be inhomogeneous in two ways: either particle scattering features depend on position in an explicit way (geometric inhomogeneity), or they depend on position through the medium’s local temperature (thermal inhomogeneity). When the inhomogeneity is geometric, it is found that Fick’s law is valid and the particle-current equation exhibits drift without current. When the inhomogeneity is thermal, current without drift is possible, but there is no generally valid pattern for the current equation. The consistency of our results with non-equilibrium thermodynamics is brought out. The results shed light on thermodiffusion (the Ludwig-Soret effect), which often combines inhomogeneities of both kinds. Finally, a limitation of the Lorentz picture of transport in accounting for thermodiffusion is outlined.     

Thermophoresis as persistent random walk

In a simple model of a continuous random walk a particle moves in one dimension with the velocity fluctuating between +v and −v. If v is associated with the thermal velocity of a Brownian particle and allowed to be position dependent, the model accounts readily for the particle's drift along the temperature gradient and recovers basic results of the conventional thermophoresis theory.     

Brownian particles in random and quasicrystalline potentials: How they approach the equilibrium

We study the Brownian motion of an ensemble of single colloidal particles in a random square and a quasicrystalline potential when they start from non-equlibrium. For both potentials, Brownian dynamics simulations reveal a widespread subdiffusive regime before the diffusive long-time limit is reached in thermal equilibrium. We develop a random trap model based on a distribution for the depths of trapping sites that reproduces the results of the simulations in detail. Especially, it gives analytic formulas for the long-time diffusion constant and the relaxation time into the diffusive regime. Aside from detailed differences, our work demonstrates that quasicrystalline potentials can be used to mimic aspects of random potentials.     

Beyond the Navier–Stokes equations: Burnett hydrodynamics

This work is mainly concerned with the extension of hydrodynamics beyond the Navier–Stokes equations, a regime known as Burnett hydrodynamics. The derivation of the Burnett equations is considered from several theoretical approaches. In particular we discuss the Chapman–Enskog, Grad’s method, and Truesdell’s approach for solving the Boltzmann equation. Also, their derivation using the macroscopic approach given by extended thermodynamics is mentioned. The problems and successes of these equations are discussed and some alternatives proposed to improve them are mentioned. Comparisons of the predictions coming from the Burnett equations with experiments and/or simulations are given in order to have the necessary elements to give a critical assessment of their validity and usefulness.     

New problems of particle transfer in ferrocolloids: Magnetic Soret effect and thermoosmosis

The paper deals with critical reviewing of the experiments on thermodiffusion in ferrocolloids. The observed magnetic Soret effect is much stronger than that predicted theoretically. It is shown that the main reason of that is the influence of the magnetic field on mass diffusion. Besides, some measurements are affected by uncontrolled thermal and solutal magnetic convection. In porous media, when macroscopic convection is suppressed, thermodiffusion is accompanied by thermoosmosis as well as by a microconvective mass transfer induced by particle magnetophoresis on filter grains.     

General relativistic Boltzmann equation, I: Covariant treatment

This series of two articles aims at dissipating the rather dense haze existing in the present literature around the General Relativistic Boltzmann equation. In this first article, the general relativistic one-particle distribution function in phase space is defined as an average of delta functions. Thereupon, the general relativistic Boltzmann equation, to be obeyed by this function, is derived. The use of either contravariant or covariant momenta leads to different, but equivalent, forms of the equation.The results of the present article are covariant, but not manifestly covariant. The transition to a manifestly covariant treatment, on the basis of off-shell momenta, is given in the second article.     

Dynamics in dense suspensions of charge-stabilized colloidal particles

The dynamic behavior of charge-stabilized colloidal particles in suspension was studied by photon correlation spectroscopy with coherent X-rays (XPCS). The short-time diffusion coefficient, D(Q) , was measured for volume concentrations φ ⩽ 0.18 and compared to the free particle diffusion constant D₀ and the static structure factor S(Q) . The data show that indirect, hydrodynamic interactions are relevant for the system and hydrodynamic functions were derived. The results are in striking contrast to the predictions of the PA (pairwise-additive approximation) model, but show features typical for a hard-sphere system. The observed mobility is however considerably smaller than the one of a respective hard-sphere system. The hydrodynamic functions can be modelled quantitatively if one allows for an increased effective viscosity relative to the hard-sphere case.     

General relativistic Boltzmann equation, II: Manifestly covariant treatment

In a preceding article we presented a general relativistic treatment of the derivation of the Boltzmann equation. The four-momenta occurring in this formalism were all on-shell four-momenta, verifying the mass-shell restriction p²=m²c². Due to this restriction, the resulting Boltzmann equation, although covariant, turned out to be not manifestly covariant. In the present article we switch from mass-shell momenta to off-shell momenta, and thereby arrive at a Boltzmann equation that is manifestly covariant.     

Evidence for diffusion-induced convection in ferrofluids from small-angle neutron scattering

Isothermal convection in ferrofluids has been induced by a gradient in particle concentration antiparallel to a magnetic field gradient. The deviation of local particle concentration from its equilibrium value produced by the convective motion of the whole fluid gives rise to a corresponding spatial variation of magnetization. This variation has been observed by magnetic neutron scattering in good agreement with expectations based on flow measurements with an anemometric method.     

Forced diffusion in magnetic fluids under the influence of a strong magnetic field gradient

We have investigated the forced diffusion of magnetic nanoparticles suspended in a carrier liquid under the influence of a magnetic field gradient. A cylindrical layer of the suspension was exposed to an azimuthal magnetic field with radial gradient. The radial distribution of the concentration of magnetic particles was determined for different times. The obtained experimental data are compared with a numerical solution of the diffusion equation and good agreement has been observed.     

Nonstationary mass transfer under particle magnetophoresis in diluted ferrocolloids

Nonstationary mass transfer under nanoparticle magnetophoresis in diluted ferrocolloids is experimentally investigated. Measurements performed by using the real time holography technique indicate a difference between the concentration boundary layer parameters found in the experiment and those calculated by using the approximation of nonstationary magnetodiffusion of colloidal particles. We suppose that the final stationary concentration distribution in the boundary layer is caused by a magnetic convection. Approximative calculations of concentration magnetic convection give the mass transfer relaxation time close to the exprimentally determined one.Work is supported by the European Community, Grant ERB 3510PL92-5206     

Drift velocity in the necklace model for reptation in a two-dimensional square lattice

We report the chain dynamics in the necklace model that mimics the reptation of a chain of N particles in a two-dimensional square lattice. We focus on the drift velocity under an applied static field. The characteristics of the model allow us to determine the effects of the forces on the chains and the resulting mechanisms that affect the drift velocity. Results obtained through Monte Carlo simulations were analyzed and discussed and distinct regimes as a function of the force strength and N were identified. We found that for small total applied forces, the drift velocity scales as 1/N. When the applied force to every particle is small but the total applied force is not, the tube deforms in such a way that the drift velocity does not depend on N. Large forces, applied to every particle, can straight chains such that the distance between the chain ends increases faster than the number of particles. Also, large forces can deform the chain within the tube what is directly related to a decrease of the drift velocity.     

Three related proposals for a theoretical definition of turbulence

We pull together some developments in turbulence research to make three propositions: (1) when the steady-state solution of any transport equation produces multi-valued velocity fields, the system described is turbulent, (2) turbulent-laminar transitions are marked by the occurrence of a singularity in the derivative of velocity with respect to time, (3) the onset of turbulence in noble gases may be described quantum mechanically using the cell model of a gas, producing two testable laws describing the critical pressure of a turbulent gas.     

Enskog’s kinetic theory of dense gases for chemically reacting binary mixtures, II: Light scattering and sound propagation

Enskog’s kinetic theory for a symmetric moderately dense reaction A+A?B+B is used to determine Fick’s and Fourier’s law. The transport coefficients of diffusion, thermal-diffusion rate and thermal conductivity are represented graphically for endothermic and exothermic reactions and are analyzed as a function of the activation energy and of the density of the mixture. The Onsager reciprocity relations are numerically investigated and verified. The problems related to sound propagation and light scattering are investigated for such a mixture and it is shown that the influence of chemical reactions on phase velocity, attenuation coefficient and light scattering spectra is more pronounced for rarefied gases although there is a considerable change in these quantities as the mixture becomes denser.     

Results for a fractional diffusion equation with a nonlocal term in spherical symmetry

We obtain time dependent solutions for a fractional diffusion equation containing a nonlocal term by considering the spherical symmetry and using the Green function approach. The nonlocal term incorporated in the diffusion equation may also be related to the spatial and time fractional derivative and introduces different regimes of spreading of the solution with the time evolution. In addition, a rich class of anomalous diffusion processes may be described from the results obtained here.     

On a unified theory of cold dark matter halos based on collisionless Boltzmann-Poisson polytropes

Collisionless particles in dark matter halos constitute ideal systems for applying the theory of collisionless Boltzmann-Poisson (BP) polytropes. This analysis provides a powerful complementary method for studying galactic halos. A comparison of the results obtained here and the Navarro, Frenk and White (NFW), Isothermal and Burkert profiles is shown. We obtain very good agreement with NFW profiles with errors of the order of 3%. The best polytropic profile for a finite mass halo is close to the Plummer/Schuster profile. We can explore in detail the central region and find the inner profile that complements the NFW profile. A simple formula for the inner region to which the NFW should converge for R→0 could be ρ=ρ₀(1−(R/C)²)^4.7, ρ₀ and C being constants. Boltzmann-Poisson polytropes provide a theoretical approach fully compatible with the universal NFW profiles at intermediate radii and complementing them at low radii: they permit the determination of the density profile in the inner kiloparsec inaccessible to N-body simulations because of resolution limits.     

Dissociative attachment of an electron to a molecule: kinetic theory

Test particles interact with a medium by means of a bimolecular reversible chemical reaction. Two species are assumed to be much more numerous so that they are distributed according to fixed distributions: Maxwellians and Dirac's deltas. Equilibrium and its stability are investigated in the first case. For the second case, a system is constructed, in view of an approximate solution.     

Single-particle thermal diffusion of charged colloids: Double-layer theory in a temperature gradient

The double-layer contribution to the single-particle thermal diffusion coefficient of charged, spherical colloids with arbitrary double-layer thickness is calculated and compared to experiments. The calculation is based on an extension of the Debye-Hückel theory for the double-layer structure that includes a small temperature gradient. There are three forces that constitute the total thermophoretic force on a charged colloidal sphere due to the presence of its double layer: i) the force F _W that results from the temperature dependence of the internal electrostatic energy W of the double layer, ii) the electric force F _el with which the temperature-induced non-spherically symmetric double-layer potential acts on the surface charges of the colloidal sphere and iii) the solvent-friction force F _sol on the surface of the colloidal sphere due to the solvent flow that is induced in the double layer because of its asymmetry. The force F _W will be shown to reproduce predictions based on irreversible-thermodynamics considerations. The other two forces F _el and F _sol depend on the details of the temperature-gradient-induced asymmetry of the double-layer structure which cannot be included in an irreversible-thermodynamics treatment. Explicit expressions for the thermal diffusion coefficient are derived for arbitrary double-layer thickness, which complement the irreversible-thermodynamics result through the inclusion of the thermophoretic velocity resulting from the electric- and solvent-friction force.