Negative Diffusion Coefficient Research Articles

Transient behavior of regulated Brownian motion, I: Starting at the origin

A natural model for stochastic flow systems is regulated or reflecting Brownian motion (RBM), which is Brownian motion on the positive real line with constant negative drift and constant diffusion coefficient, modified by an impenetrable reflecting barrier at the origin. As a basis for understanding how stochastic flow systems approach steady state, this paper provides relatively simple descriptions of the moments of RBM as functions of time. In Part I attention is restricted to the case in which RBM starts at the origin; then the moment functions are increasing. After normalization by the steady-state limits, these moment c.d.f.&s (cumulative distribution functions) coincide with gamma mixtures of inverse Gaussian c.d.f.&s. The first moment c.d.f. thus coincides with the first-passage time to the origin starting in steady state with the exponential stationary distribution. From this probabilistic characterization, it follows that thekth-moment c.d.f is thek-fold convolution of the first-moment c.d.f. As a consequence, it is easy to see that the (k +1)th moment approaches its steady-state limit more slowly than thekthmoment. It is also easy to derive the asymptotic behavior ast→∞. The first two moment c.d.f.&s have completely monotone densities, supporting approximation by hyperexponential (H2)c.d.f.&s (mixtures of two exponentials). TheH2approximations provide easily comprehensible descriptions of the first two moment c.d.f.&s suitable for practical purposes. The two exponential components of theH2approximation yield simple exponential approximations in different regimes. On the other hand, numerical comparisons show that the limit related to the relaxation time does not predict the approach to steady state especially well in regions of primary interest. In Part II (Abate and Whitt (1987a)), moments of RBM with non-zero initial conditions are treated by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied here.

Chemical wave-electrical field interaction phenomena

Chemical waves in ionic media are shown to display a great variety of nonlinear phenomena. An overview of the application of bifurcation and scaling theory to the electrical field-chemical wave interaction problem is presented. It is discovered that electrical fields can induce effective negative phase diffusion coefficients in limit cycle systems and negative mode diffusion coefficients for systems near a bifurcation point. Waves are shown to exist in the system subject to an applied field when such waves do not exist in the field-free system. Entities like periodic static structures are shown to have an effective mobility and charge while fronts in some systems have an effective charge that can change sign with concentration. A host of new nonlinear equations describing these phenomena are posed. It is found that the variety of and potential to express nonlinear phenomena in reaction-transport systems is greatly enhanced by imposed or self-generated electrical fields.

Calculation of the straight and cross diffusion coefficients of a two-component mixture in a microporous zeolitic structure

A method is given which enables calculation of straight and cross diffusion coefficients of a two-component mixture adsorbed on a microporous adsorbent at constant pressure conditions. In particular, the method involves the solution of the reverse problem of calculating the matrix of diffusion coefficients on the basis of the experimental uptake curves of binary mixture adsorption. In this Paper, the method is illustrated for a binary n-hexane—ammonia mixture adsorbed on CaA type zeolite. The occurrence of a negative cross diffusion coefficient is also shown.

Void nucleation as a diffusive instability.

The dynamics of vacancies in a system with both production and annihilation mechanisms have been analyzed as to their coalescence to form voids. The modeling of the phenomena is done by introducing a vacancy-concentration field. An equation for this field is formulated which incorporates, in a simplified way, (i) vacancy diffusion, (ii) mutual pairwise interaction among vacancies, (iii) their loss due to mechanisms like recombination with interstitials and absorption at dislocations, grain boundaries, etc., and (iv) their production by incoming radiation. The recombination with interstitials is treated in two schemes. The resulting equations are analyzed for a domain, under the boundary condition that the domain boundaries are perfect sinks. Such a domain should mimic grains in materials. The equations exhibit two kinds of behaviors depending upon the magnitude of the production rate P. If P is below a certain value ${P}_{\mathrm{th}}$, the equations have steady-state solutions which are approached from the initial thermal state. When P\ensuremath{\ge}${P}_{\mathrm{th}}$, a coalescence instability occurs in time, which can be thought of as resulting from a negative diffusion coefficient. We have made theoretical estimates of ${P}_{\mathrm{th}}$ in both the schemes which are in very good agreement with numerical calculations.

Langevin equations with negative diffusion coefficients. A new approach to quantum optics

Stochastic differential equations with negative diffusion coefficients are introduced for applications to the analysis of open quantum systems and nonlinear optical systems. It is shown how they can be consistently employed for moment calculations. An illustration on the problem of squeezing is given.

Hydrodynamic approximation for distinct diffusion coefficients

The distinct diffusion coefficient is a measure of the coupling of the diffusive motions of two particles. It is given as the integral over a velocity cross correlation rather than the velocity self correlation that determines the self-diffusion coefficient. A hydrodynamic approximation for the distinct diffusion coefficient is proposed and then tested by comparison with data for a wide range of non-ionic binary mixtures. The hydrodynamic approximation gives negative distinct diffusion coefficients and is in qualitative agreement with most of the data. In many cases, deviations from the model results can be explained in terms of interactions which are not accurately treated by the model.

Stationary transport with partially reflecting boundary conditions. II

We consider the stationary monoenergetic transport equation (the Lorentz model) in a semiinfinite geometry with most general nonmultiplying boundary conditions at the wall x=0, accounting for absorption, specular, diffuse, and ‘‘selective’’ reflection, or combinations thereof. Performing the numerical study of a (partially) ‘‘thermalizing’’ wall, we investigate the density profile n(x), the Milne extrapolation length xM, and the space dependent diffusion coefficient D(x) as functions of the accomodation and ‘‘selection’’ coefficients. The inversed density profile (equivalent to a negative diffusion coefficient) is numerically computed. As a simple analytically solvable example, a discrete velocity model is discussed in the Appendix.

Nonlinear diffusion in spinodal decomposition: a numerical solution

The numerical solution of the one-dimensional nonlinear diffusion equation with a negative diffusion coefficient (up-hill diffusion) by a five-point approximation central difference scheme is considered. The stability criteria are discussed in detail and a numerical solution is provided for a specific case in which the time evolution of a periodic composition wave is presented with growth eventually leading to a stationary configuration. A critical comparison of the numerical solution with existing analytical solutions is shown. This leads to a simple semi-empirical growth law for studying the kinetics of spinodal decomposition in alloys.

Steady Inhomogeneous Patterns of Non-Equilibrium Superconductor –Long Time Behaviour of the µ* Model System–

Steady inhomogeneous patterns of the µ * model system for superconductor driven far from equilibrium are numerically studied. We firstly check the applicability of the reductive perturbation method to inquire into pattern formation. Then it is shown that the “global-local” concept is expedient to understand the calculated inhomogeneous spatial patterns. Further the feasibility of the abnormal (negative) diffusion in open systems is demonstrated by deriving a reduced non-linear stochastic equation with effective negative diffusion coefficient from a generalized Haken-Zwanzing model. The way of thinking developed in this paper will be useful to understand the inhomogeneous patterns in other non-equilibrium open systems.

Diffusion of a multi-component plasma

Diffusion in a weakly ionized plasma composed of negative ions as well as positive ions is examined using appropriate linearized fluid equations. For initial, electrically neutral, density perturbations of the form exp( ikx) the diffusion process is characterized by electrical non-neutrality and by three stages or time scales. For equal positive and negative ion diffusion coefficients these stages are in general (1) equilibration of the electron gas so that pressure gradient and electric forces are balanced (2) ambipolar-like diffusion of all three species, and concluding with (3) free ion diffusion. The details of the process are governed by the product kλ e (wave number times electron Debye length) and the ambient ratio of negative ion to electron number density. Numerical and analytic results for separate positive and negative diffusion coefficients show added complexity which is briefly described. These results or the more complete numerical solutions find application to the lower D region of the ionosphere.

Note on a wave-hierarchy interpretation of fluidized bed instabilities

The technological importance of fluidized beds is well known. Experi­mental observations indicate that a uniformly fluidized bed is unstable to small planar voidage disturbances that grow exponentially upwards along the bed and, under certain circumstances, attain saturation at finite amplitudes. The linearized theory is recast into a form that shares mathe­matical similarities and interpretations with ‘wave hierarchies’. In wave hierarchies, the higher-order waves, which correspond to higher-frequency waves nearer the distributor, have wave speeds that are lower than those of the lower-order waves, and the stability condition is thus violated. The lower-order waves correspond to these lower-frequency waves further along the fluidized bed. Thus voidage signals grow exponentially along the higher-order waves. Such waves will eventually be left behind and sur­passed by the faster lower-order waves along which voidage perturbations will be ‘focused’ owing to an effective negative diffusion coefficient. The peculiar double peaks in voidage perturbation waves observed by El-Kaissy and Homsy appear to be explicable by the present wave-hierarchy inter­pretation of the linearized theory.

Diffusion in a ternary system and the critical mixing point

Diffusion coefficients were measured for the ternary system water-chloroformacetic acetic acid at 25°C and at five concentrations approaching the critical mixing point. The relationship D11D22−D12D21=0 holding at the critical mixing point has been proven experimentally. By choosing water as solvent instead of acetic acid, negative main diffusion coefficients were found for chloroform, showing an interesting case of incongruent diffusion.

Nonneutral and quasi‐neutral diffusion of weakly ionized multiconstituent plasma

A theory is developed for ambipolar diffusion in a multiconstituent weakly ionized plasma. The implications of the presence of negative ions and of differing positive ion and negative ion diffusion coefficients are found. In particular, the presence of numerous negative ions is found to enhance the diffusion of the electrons. Furthermore, this theory of multiconstituent ambipolar diffusion is extended to include the nonneutral case in which the scale length of the inhomogeneity is smaller than or of the order of the electron Debye length.

The diffusive Lotka-Volterra oscillating system

The Lotka-Volterra equations are coupled with diffusion processes in homogeneous systems. The inclusion of a negative cross diffusion coefficient can result in the appearance of a stationary wave-like dissipative structure. The cross diffusion coefficients represent a deceitful relationship between the two interacting populations.

Negative ionic cross diffusion coefficients in electrolytic solutions

All existing experimental examples of oscillating homogeneous chemical reactions and dissipative structures involve ionic systems. The linear stability analysis is applied to electrolytic solutions by including diffusion potential and ionic migration effects. It is shown that negative off-diagonal diffusion coefficients can exist in electrolytic solutions, since ions diffuse interactively even in dilute solutions and are influenced by the resulting diffusion potential. The inclusion of negative off-diagonal diffusion co-efficients can increase the possibility of oscillations and spatial structures.

Measurement of negative thermal diffusion coefficients by observing the onset of thermohaline convection

Measurement of negative thermal diffusion coefficients by observing the onset of thermohaline convection

Comment on negative diffusion coefficients in quasi-linear theory

A recent paper on quasiinear theory by Klozenberg & Bernstein is commented upon. It is argued that improper treatment of the perturbed electron distribution for damped waves has led to a diffusion-type equation with a negative diffusion coefficient.

Cross terms in the thermodynamic diffusion equations for multicomponent alloys

The kinetic equations for the atom fluxes in a multicomponent alloy are derived in terms of atomic driving forces for a random alloy model. In this model, the contribution to each atom flux from the vacancy wind effect is directly proportional to the vacancy flux. The kinetic equations when written in the same form as the thermodynamic diffusion equations allow theL ik coefficients to be identified. The cross terms relating the flux of speciesi to the chemical potential gradients of other speciesk are found to be nonzero. These cross terms can have an appreciable effect on the atom fluxes and can lead to negative intrinsic diffusion coefficients even in ideal systems. Neglecting the cross terms can lead to serious inaccuracies. TheL ik in ann-component system depend on then tracer diffusion coeficients. In this respect, it still is possible to express then atom fluxes in terms of onlyn coefficients.

Contribution à la théorie des ondes glaciaires

Glacier surges may result (1) from the warming of a cold glacier, which follows an unusual snow cover, with the formation of a temperate basal layer, or (2) from the fact that two values of the sliding velocity correspond to a single value of the friction.The last results about the friction law obtained by the present author are summarized. As a rule, for increasing sliding velocities the friction goes through a maximum and next through a minimum. Thus for one value of the friction three values of the velocity, two stable ones and an unstable one, can exist.Since the derivative of the friction with respect to the sliding velocity may become negative, the coefficients c0 and D0 in the perturbation equation may become infinite (their ratio remaining finite), and next negative. By this way, when a flood wave runs down a glacier an area with a negative diffusion coefficient may appear. Then the profile of the wave is altered: it vanishes at the upper part of this area and swells beyond measure at the lower one. Factors which may limit the amplitude of this surge are pointed out.