Exact Calculation of the Probabilities of Rare Events in Cluster-Cluster Aggregation.

We develop an action formalism to calculate probabilities of rare events in cluster-cluster aggregation for arbitrary collision kernels and establish a pathwise large deviation principle with total mass being the rate. As an application, the rate function for the number of surviving particles as well as the optimal evolution trajectory are calculated exactly for the constant, sum, and product kernels. For the product kernel, we argue that the second derivative of the rate function has a discontinuity.

Sample-path large deviations for stochastic evolutions driven by the square of a Gaussian process.

Recently, a number of physical models have emerged described by a random process with increments given by a quadratic form of a fast Gaussian process. We find that the rate function which describes sample-path large deviations for such a process can be computed from the large domain size asymptotic of a certain Fredholm determinant. The latter can be evaluated analytically using a theorem of Widom which generalizes the celebrated Szegő-Kac formula to the multidimensional case.

Stationary mass distribution and nonlocality in models of coalescence and shattering.

We study the asymptotic properties of the steady state mass distribution for a class of collision kernels in an aggregation-shattering model in the limit of small shattering probabilities. It is shown that the exponents characterizing the large and small mass asymptotic behavior of the mass distribution depend on whether the collision kernel is local (the aggregation mass flux is essentially generated by collisions between particles of similar masses) or nonlocal (collision between particles of widely different masses give the main contribution to the mass flux). We show that the nonlocal regime is further divided into two subregimes corresponding to weak and strong nonlocality.

Collective oscillations in irreversible coagulation driven by monomer inputs and large-cluster outputs.

We describe collective oscillatory behavior in the kinetics of irreversible coagulation with a constant input of monomers and removal of large clusters. For a broad class of collision rates, this system reaches a nonequilibrium stationary state at large times and the cluster size distribution tends to a universal form characterized by a constant flux of mass through the space of cluster sizes. Universality, in this context, means that the stationary state becomes independent of the cutoff as the cutoff grows.

Instantaneous gelation in Smoluchowski's coagulation equation revisited.

We study the solutions of the Smoluchowski coagulation equation with a regularization term which removes clusters from the system when their mass exceeds a specified cutoff size, M. We focus primarily on collision kernels which would exhibit an instantaneous gelation transition in the absence of any regularization. Numerical simulations demonstrate that for such kernels with monodisperse initial data, the regularized gelation time decreases as M increases, consistent with the expectation that the gelation time is zero in the unregularized system.

Constant flux relation for diffusion-limited cluster-cluster aggregation.

In a nonequilibrium system, a constant flux relation (CFR) expresses the fact that a constant flux of a conserved quantity exactly determines the scaling of the particular correlation function linked to the flux of that conserved quantity. This is true regardless of whether mean-field theory is applicable or not. We focus on cluster-cluster aggregation and discuss the consequences of mass conservation for the steady state of aggregation models with a monomer source in the diffusion-limited regime.

Constant flux relation for driven dissipative systems.

Conservation laws constrain the stationary state statistics of driven dissipative systems because the average flux of a conserved quantity between driving and dissipation scales should be constant. This requirement leads to a universal scaling law for flux-measuring correlation functions, which generalizes the 4/5th law of Navier-Stokes turbulence. We demonstrate the utility of this simple idea by deriving new exact scaling relations for models of aggregating particle systems in the fluctuation-dominated regime and for energy and wave action cascades in models of strong wave turbulence.

Multiscaling of correlation functions in single species reaction-diffusion systems.

We derive the multi-scaling of probability distributions of multi-particle configurations for the binary reaction-diffusion system in A + A --> O in d < or =2 and for the ternary system in 3A --> O in d=1. For the binary reaction we find that the probability Pt(N, Delta V) of finding N particles in a fixed volume element Delta V at time t decays in the limit of large time as (ln t/t)N(ln t)-N(N-1)/2 for d=2 and t-Nd/2 t-N(N-1)epsilon/4+O(epsilon2) for d<2. Here epsilon=2-d.

Breakdown of Kolmogorov scaling in models of cluster aggregation.

We describe a model of cluster aggregation with a source which provides a rare example of an analytically tractable turbulent system. The steady state is characterized by a constant mass flux from small masses to large. Thus it can be studied using a phenomenological theory, inspired by Kolmogorov's 1941 theory, which assumes constant flux and self-similarity.

Survival probability of a diffusing test particle in a system of coagulating and annihilating random walkers.

We calculate the survival probability of a diffusing test particle in an environment of diffusing particles that undergo coagulation at rate lambda(c) and annihilation at rate lambda(a) . The test particle is annihilated at rate lambda(') on coming into contact with the other particles. The survival probability decays algebraically with time as t(-theta;) .

Stationary Kolmogorov solutions of the Smoluchowski aggregation equation with a source term.

In this paper we show how the method of Zakharov transformations may be used to analyze the stationary solutions of the Smoluchowski aggregation equation with a source term for arbitrary homogeneous coagulation kernel. The resulting power-law mass distributions are of Kolmogorov type in the sense that they carry a constant flux of mass from small masses to large. They are valid for masses much larger than the characteristic mass of the source.

Persistence properties of a system of coagulating and annihilating random walkers.

We study a d-dimensional system of diffusing particles that on contact either annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1). In one dimension, the system models the zero-temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m-1) coagulations.

Kang-Redner small-mass anomaly in cluster-cluster aggregation.

The large-time small-mass asymptotic behavior of the average mass distribution P(m,t) is studied in a d-dimensional system of diffusing aggregating particles for 1< or =d< or =2. By means of both a renormalization group computation as well as a direct resummation of leading terms in the small-reaction-rate expansion of the average mass distribution, it is shown that P(m,t) approximately (1/t(d))(m(1/d)/sqrt[t])(e(KR)) for m< View Article and Find Full Text PDF