Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

We consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.

PLUME-MoM-TSM 1.0.0: a volcanic column and umbrella cloud spreading model

In this paper, we present a new version of PLUME-MoM, a 1-D integral volcanic plume model based on the method of moments for the description of the polydispersity in solid particles. The model describes the steady-state dynamics of a plume in a 3-D coordinate system, and a modification of the two-size moment (TSM) method is adopted to describe changes in grain size distribution along the plume, associated with particle loss from plume margins and with particle aggregation. For this reason, the new version is named PLUME-MoM-TSM. For the first time in a plume model, the full Smoluchowski coagulation equation is solved, allowing us to quantify the formation of aggregates during the rise of the plume. In addition, PLUME-MOM-TSM allows us to model the phase change of water, which can be either magmatic, added at the vent as liquid from external sources, or incorporated through ingestion of moist atmospheric air. Finally, the code includes the possibility to simulate the initial spreading of the umbrella cloud intruding from the volcanic column into the atmosphere. A transient shallow-water system of equations models the intrusive gravity current, allowing computation of the upwind spreading. The new model is applied first to the eruption of the Calbuco volcano in southern Chile in April 2015 and then to a sensitivity analysis of the upwind spreading of the umbrella cloud to mass flow rate and meteorological conditions (wind speed and humidity). This analysis provides an analytical relationship between the upwind spreading and some observable characteristic of the volcanic column (height of the neutral buoyancy level and plume bending), which can be used to better link plume models and volcanic-ash transport and dispersion models.

Particle Size Distribution of Smoluchowski Coagulation Equation for Brownian Motion at Equilibrium

The information entropy for Smoluchowski coagulation equation is proposed based on statistical mechanics. And the normalized particle size distribution is a lognormal function at equilibrium from the principle of maximum entropy and moment constraint. The geometric mean volume and standard deviation in the distribution function are determined as simple constant. The results reveal that the assumption that algebraic mean volume be unit in self-preserving hypothesis is reasonable in some sense. Based on the present definition of information entropy, the Cercignani’s conjecture holds naturally for Smoluchowski coagulation equation. Together with the proof that the conjecture is also true for Boltzmann equation, Cercignani’s conjecture will holds for any two-body collision systems, which will benefit the understanding of Brownian motion and molecule kinematic theory, such as the stability of the dissipative system, and the mathematical theory of convergence to thermodynamic equilibrium.

The Approximation of Entropy for Smoluchowski Coagulation Equation With TEMOM

In this paper, the definition of information entropy of Smoluchowski coagulation equation for Brownian motion is introduced based on coagulation probability. The expression of entropy is the function of geometric average particle volume and standard deviation with lognormal distribution assumption. The asymptotic solution with moment method shows that the entropy is a monotone increasing function of time, which is equivalence to the entropy based on particle size distribution. the result reveals that the present definition of entropy of Smoluchowski coagulation equation are inadequate because the particle average volume at equilibrium cannot be determined from the principle of maximum entropy. This provides a basis for further exploring the global properties of Smoluchowski coagulation equation.

PLUME-MoM-TSM 1.0.0: A volcanic columns and umbrella cloud spreading model

In this paper a new version of the integral model for volcanic plumes PLUME-MoM is presented. The model describes the steady-state dynamics of a plume in a 3-D coordinate system, and the two-size moment (TSM) method is adopted to describe changes in grain-size distribution along the plume, associated with particle loss from plume margins and with particle aggregation. For this reason, the new version is named PLUME-MoM-TSM. For the first time in a plume model, the full Smoluchowski coagulation equation is solved, allowing to quantify the formation of aggregates during the rise of the plume. In addition, PLUME-MOM-TSM allows to model the phase change of water, which can be either magmatic, added at the vent as liquid from external sources, or incorporated through ingestion of moist atmospheric air. Finally, the code includes the possibility to simulate the initial spreading of the umbrella cloud intruding from the volcanic column into the atmosphere. A transient shallow water system of equations models the intrusive gravity current, allowing to compute the upwind spreading. The new model is applied first to the eruption of Calbuco volcano in southern Chile in April 2015, and then to a sensitivity analysis of the upwind spreading of the umbrella cloud to mass flow rate and meteorological conditions (wind speed and humidity). This analysis provides an analytical relationship between the upwind spreading and some characteristic of the volcanic column (height of the neutral buoyancy level and plume bending), which can be used to better link plume models and volcanic-ash transport and dispersion models.

Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel

Global weak solutions to the continuous Smoluchowski coagulation equation (SCE) are constructed for coagulation kernels featuring an algebraic singularity for small volumes and growing linearly for large volumes, thereby extending previous results obtained in Norris (1999) and Cueto Camejo & Warnecke (2015). In particular, linear growth at infinity of the coagulation kernel is included and the initial condition may have an infinite second moment. Furthermore, all weak solutions (in a suitable sense) including the ones constructed herein are shown to be mass-conserving, a property which was proved in Norris (1999) under stronger assumptions. The existence proof relies on a weak compactness method in L1 and a by-product of the analysis is that both conservative and non-conservative approximations to the SCE lead to weak solutions which are then mass-conserving.

Coagulation and universal scaling limits for critical Galton–Watson processes

The basis of this paper is the elementary observation that the n-step descendant distribution of any Galton–Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we obtain simple necessary and sufficient criteria for the convergence of scaling limits of critical Galton–Watson processes in terms of scaled family-size distributions and a natural notion of convergence of Lévy triples. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the Lévy jump measure of certain continuous-state branching processes (CSBPs) satisfies a generalized Smoluchowski equation. (This result was previously proved by Bertoin and Le Gall (2006).) Our analysis shows that the nonlinear scaling dynamics of CSBPs become linear and purely dilatational when expressed in terms of the Lévy triple associated with the branching mechanism. We prove a continuity theorem for CSBPs in terms of the associated Lévy triples, and use our scaling analysis to prove the existence of universal critical Galton–Watson processes and CSBPs analogous to Doeblin's `universal laws'. Namely, these universal processes generate all possible critical and subcritical CSBPs as subsequential scaling limits. Our convergence results rely on a natural topology for Lévy triples and a continuity theorem for Bernstein transforms (Laplace exponents) which we develop in a self-contained appendix.