Near critical catalyst reactant branching processes with controlled immigration

Near critical catalyst-reactant branching processes with controlled immigration are studied. The reactant population evolves according to a branching process whose branching rate is proportional to the total mass of the catalyst. The bulk catalyst evolution is that of a classical continuous time branching process; in addition there is a specific form of immigration. Immigration takes place exactly when the catalyst population falls below a certain threshold, in which case the population is instantaneously replenished to the threshold. Such models are motivated by problems in chemical kinetics where one wants to keep the level of a catalyst above a certain threshold in order to maintain a desired level of reaction activity. A diffusion limit theorem for the scaled processes is presented, in which the catalyst limit is described through a reflected diffusion, while the reactant limit is a diffusion with coefficients that are functions of both the reactant and the catalyst. Stochastic averaging principles under fast catalyst dynamics are established. In the case where the catalyst evolves "much faster" than the reactant, a scaling limit, in which the reactant is described through a one dimensional SDE with coefficients depending on the invariant distribution of the reflected diffusion, is obtained. Proofs rely on constrained martingale problem characterizations, Lyapunov function constructions, moment estimates that are uniform in time and the scaling parameter and occupation measure techniques.Comment: Published in at http://dx.doi.org/10.1214/12-AAP894 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Intermittency on catalysts

The present paper provides an overview of results obtained in four recent papers by the authors. These papers address the problem of intermittency for the Parabolic Anderson Model in a \emph{time-dependent random medium}, describing the evolution of a ``reactant'' in the presence of a ``catalyst''. Three examples of catalysts are considered: (1) independent simple random walks; (2) symmetric exclusion process; (3) symmetric voter model. The focus is on the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the reactant. It turns out that these exponents exhibit an interesting dependence on the dimension and on the diffusion constant.Comment: 11 pages, invited paper to appear in a Festschrift in honour of Heinrich von Weizs\"acker, on the occasion of his 60th birthday, to be published by Cambridge University Pres

Genealogy of catalytic branching models

We consider catalytic branching populations. They consist of a catalyst population evolving according to a critical binary branching process in continuous time with a constant branching rate and a reactant population with a branching rate proportional to the number of catalyst individuals alive. The reactant forms a process in random medium. We describe asymptotically the genealogy of catalytic branching populations coded as the induced forest of $\mathbb{R}$-trees using the many individuals--rapid branching continuum limit. The limiting continuum genealogical forests are then studied in detail from both the quenched and annealed points of view. The result is obtained by constructing a contour process and analyzing the appropriately rescaled version and its limit. The genealogy of the limiting forest is described by a point process. We compare geometric properties and statistics of the reactant limit forest with those of the "classical" forest.Comment: Published in at http://dx.doi.org/10.1214/08-AAP574 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Reaction-diffusion dynamics: confrontation between theory and experiment in a microfluidic reactor

We confront, quantitatively, the theoretical description of the reaction-diffusion of a second order reaction to experiment. The reaction at work is \ca/CaGreen, and the reactor is a T-shaped microchannel, 10 $\mu$m deep, 200 $\mu$m wide, and 2 cm long. The experimental measurements are compared with the two-dimensional numerical simulation of the reaction-diffusion equations. We find good agreement between theory and experiment. From this study, one may propose a method of measurement of various quantities, such as the kinetic rate of the reaction, in conditions yet inaccessible to conventional methods

Simulation of chemical reaction dynamics on an NMR quantum computer

Quantum simulation can beat current classical computers with minimally a few tens of qubits and will likely become the first practical use of a quantum computer. One promising application of quantum simulation is to attack challenging quantum chemistry problems. Here we report an experimental demonstration that a small nuclear-magnetic-resonance (NMR) quantum computer is already able to simulate the dynamics of a prototype chemical reaction. The experimental results agree well with classical simulations. We conclude that the quantum simulation of chemical reaction dynamics not computable on current classical computers is feasible in the near future.Comment: 37 pages, 7 figure

Reentrant phase diagram and pH effects in cross-linked gelatin gels

Experimental results have shown that the kinetics of bond formation in chemical crosslinking of gelatin solutions is strongly affected not only by gelatin and reactant concentrations but also by the solution pH. We present an extended numerical investigation of the phase diagram and of the kinetics of bond formation as a function of the pH, via Monte Carlo simulations of a lattice model for gelatin chains and reactant agent in solution. We find a reentrant phase diagram, namely gelation can be hindered either by loop formation, at low reactant concentrations, or by saturation of active sites of the chains via formation of single bonds with crosslinkers, at high reactant concentrations. The ratio of the characteristic times for the formation of the first and of the second bond between the crosslinker and an active site of a chain is found to depend on the reactant reactivity, in good agreement with experimental data.Comment: 8 pages, 8 figure

Isomerization dynamics of a buckled nanobeam

We analyze the dynamics of a model of a nanobeam under compression. The model is a two mode truncation of the Euler-Bernoulli beam equation subject to compressive stress. We consider parameter regimes where the first mode is unstable and the second mode can be either stable or unstable, and the remaining modes (neglected) are always stable. Material parameters used correspond to silicon. The two mode model Hamiltonian is the sum of a (diagonal) kinetic energy term and a potential energy term. The form of the potential energy function suggests an analogy with isomerisation reactions in chemistry. We therefore study the dynamics of the buckled beam using the conceptual framework established for the theory of isomerisation reactions. When the second mode is stable the potential energy surface has an index one saddle and when the second mode is unstable the potential energy surface has an index two saddle and two index one saddles. Symmetry of the system allows us to construct a phase space dividing surface between the two "isomers" (buckled states). The energy range is sufficiently wide that we can treat the effects of the index one and index two saddles in a unified fashion. We have computed reactive fluxes, mean gap times and reactant phase space volumes for three stress values at several different energies. In all cases the phase space volume swept out by isomerizing trajectories is considerably less than the reactant density of states, proving that the dynamics is highly nonergodic. The associated gap time distributions consist of one or more `pulses' of trajectories. Computation of the reactive flux correlation function shows no sign of a plateau region; rather, the flux exhibits oscillatory decay, indicating that, for the 2-mode model in the physical regime considered, a rate constant for isomerization does not exist.Comment: 42 pages, 6 figure

An Analytical Construction of the SRB Measures for Baker-type Maps

For a class of dynamical systems, called the axiom-A systems, Sinai, Ruelle and Bowen showed the existence of an invariant measure (SRB measure) weakly attracting the temporal average of any initial distribution that is absolutely continuous with respect to the Lebesgue measure. Recently, the SRB measures were found to be related to the nonequilibrium stationary state distribution functions for thermostated or open systems. Inspite of the importance of these SRB measures, it is difficult to handle them analytically because they are often singular functions. In this article, for three kinds of Baker-type maps, the SRB measures are analytically constructed with the aid of a functional equation, which was proposed by de Rham in order to deal with a class of singular functions. We first briefly review the properties of singular functions including those of de Rham. Then, the Baker-type maps are described, one of which is non-conservative but time reversible, the second has a Cantor-like invariant set, and the third is a model of a simple chemical reaction $R \leftrightarrow I \leftrightarrow P$. For the second example, the cases with and without escape are considered. For the last example, we consider the reaction processes in a closed system and in an open system under a flux boundary condition. In all cases, we show that the evolution equation of the distribution functions partially integrated over the unstable direction is very similar to de Rham's functional equation and, employing this analogy, we explicitly construct the SRB measures.Comment: 53 pages, 10 figures, to appear in CHAO

Semiclassical transmission across transition states

It is shown that the probability of quantum-mechanical transmission across a phase space bottleneck can be compactly approximated using an operator derived from a complex Poincar\'e return map. This result uniformly incorporates tunnelling effects with classically-allowed transmission and generalises a result previously derived for a classically small region of phase space.Comment: To appear in Nonlinearit