Analytical asymptotic solutions of nA+mB→C reaction-diffusion equations in two-layer systems: A general study

Large time evolution of concentration profiles is studied analytically for reaction-diffusion systems where the reactants A and B are each initially separately contained in two immiscible solutions and react upon contact and transfer across the interface according to a general nA+mB-->C reaction scheme. This study generalizes to immiscible two-layer systems the large time analytical asymptotic limits of concentrations derived by Koza [J. Stat. Phys. 85, 179 (1996)] for miscible fluids and for reaction rates of the form A;{n}B;{m} with arbitrary diffusion coefficients and homogeneous initial concentrations. In addition to a dependence on the parameters already characterizing the miscible case, the asymptotic concentration profiles in immiscible systems depend now also on the partition coefficients of the chemical species between the two solution layers and on the ratio of diffusion coefficients of a given species in the two fluids. The miscible time scalings are found to remain valid for the immiscible fluids case. However, for immiscible systems, the reaction front speed is enhanced by increasing the stoichiometry of the invading species over that of the species being invaded. The direction of the front propagation is found to depend on the diffusion coefficient of the invading species in its initial fluid but not on its value in the invading fluid. Hence, a reaction front in immiscible fluids can travel in the opposite direction to the reaction front formed in miscible fluids for a range of parameter values. The value of the invading species partition coefficient affects the magnitude of the front speed but it cannot alter the direction of the front. For sufficiently large times, the total amount of product produced in time is independent of the rate of the reaction. The centre of mass of the product can move in the opposite direction to the center of mass of the reaction rate.Journal Articleinfo:eu-repo/semantics/publishe

Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions

We consider the $d=1$ nonlinear Fokker-Planck-like equation with fractional derivatives $\frac{\partial}{\partial t}P(x,t)=D \frac{\partial^{\gamma}}{\partial x^{\gamma}}[P(x,t) ]^{\nu}$. Exact time-dependent solutions are found for $\nu = \frac{2-\gamma}{1+ \gamma}$ ($-\infty<\gamma \leq 2$). By considering the long-distance {\it asymptotic} behavior of these solutions, a connection is established, namely $q=\frac{\gamma+3}{\gamma+1}$ ($0<\gamma \le 2$), with the solutions optimizing the nonextensive entropy characterized by index $q$ . Interestingly enough, this relation coincides with the one already known for L\'evy-like superdiffusion (i.e., $\nu=1$ and $0<\gamma \le 2$). Finally, for $(\gamma,\nu)=(2, 0)$ we obtain $q=5/3$ which differs from the value $q=2$ corresponding to the $\gamma=2$ solutions available in the literature ($\nu<1$ porous medium equation), thus exhibiting nonuniform convergence.Comment: 3 figure

Reaction-diffusion systems and nonlinear waves

The authors investigate the solution of a nonlinear reaction-diffusion equation connected with nonlinear waves. The equation discussed is more general than the one discussed recently by Manne, Hurd, and Kenkre (2000). The results are presented in a compact and elegant form in terms of Mittag-Leffler functions and generalized Mittag-Leffler functions, which are suitable for numerical computation. The importance of the derived results lies in the fact that numerous results on fractional reaction, fractional diffusion, anomalous diffusion problems, and fractional telegraph equations scattered in the literature can be derived, as special cases, of the results investigated in this article.Comment: LaTeX, 16 pages, corrected typo

Anomalous diffusion with absorption: Exact time-dependent solutions

Recently, analytical solutions of a nonlinear Fokker-Planck equation describing anomalous diffusion with an external linear force were found using a non extensive thermostatistical Ansatz. We have extended these solutions to the case when an homogeneous absorption process is also present. Some peculiar aspects of the interrelation between the deterministic force, the nonlinear diffusion and the absorption process are discussed.Comment: RevTex, 16 pgs, 4 figures. Accepted in Physical Review

Rescaling of diffusion coefficients in two-time scale chemical systems

We study reaction-diffusion systems which involve processes that occur on different time scales. In particular, we apply a multiscale analysis to obtain a reduced description of the slow dynamics. Under certain assumptions this reduction yields a new set of reaction-diffusion equations with rescaled diffusion coefficients. We analyze the Selkov model &#91;E. E. Selkov, Eur. J. Biochem. 4, 79 (1968)&#93; and the ferrocyanide-iodide-sulfite reaction &#91;E. C. Edblom et al., J. Am. Chem. Soc. 108, 2826 (1986)&#93; to determine whether the rescaling in this case may account for the difference of diffusivities that the formation of certain types of patterns requires. © 2000 American Institute of Physics.Fil:Strier, D.E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.Fil:Dawson, S.P. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

Exact expression for the diffusion propagator in the time-dependent anharmonic potential: V(#chi#,t) = 1on2a(t)#chi#&quot;2 + bln#chi#

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