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Boris M. Shipilevsky

Publications and source records attributed to Boris M. Shipilevsky.

8 recordsLinked to original sources
arXiv2019

Diffusion-controlled coalescence, fragmentation and collapse of $d$-dimensional $A$-particle islands in the $B$-particle sea

We present a systematic analysis of diffusion-controlled evolution and collapse of two identical spatially separated $d$-dimensional $A$-particle islands in the $B$-particle sea at propagation of the sharp reaction front $A+B\to 0$ at equal species diffusivities. We show that at a sufficiently large initial distance between the centers of islands $2\ell$ compared to their characteristic initial size and a relatively large initial ratio of concentrations island/sea the evolution dynamics of the island-sea-island system is determined unambiguously by the dimensionless parameter $Λ={\cal N}_{0}/{\cal N}_Ω$, where ${\cal N}_{0}$ is the initial particle number in the island and ${\cal N}_Ω$ is the initial number of sea particles in the volume $Ω=(2\ell)^{d}$. It is established that a) there is a $d$-dependent critical value $Λ_{\star}$ above which island coalescence occurs; b) regardless of $d$ the centers of each of the islands move towards each other along a {\it universal} trajectory merging in a united center at the $d$-dependent critical value $Λ_{s}\geqΛ_{\star}$; c) in one-dimensional systems $Λ_{\star}=Λ_{s}$, therefore at $Λ<Λ_{\star}$ each of the islands dies individually, whereas at $Λ>Λ_{\star}$ coalescence is completed by collapse of a single-centered island in the system center; d) in two- and three-dimensional systems in the range $Λ_{\star}< Λ< Λ_{s}$ coalescence is accompanied by subsequent fragmentation of a two-centered island and is completed by individual collapse of each of the islands. We discuss a detailed picture of coalescence, fragmentation and collapse of islands focusing on evolution of their shape and on behavior of the relative width of the reaction front at the final collapse stage and in the vicinity of starting coalescence and fragmentation points.

cond-mat.stat-mech↗
arXiv2019

Diffusion-controlled formation and collapse of a d-dimensional A-particle island in the B-particle sea

We consider diffusion-controlled evolution of a $d$-dimensional $A$-particle island in the $B$-particle sea at propagation of the sharp reaction front $A+B\to 0$ at equal species diffusivities. The $A$-particle island is formed by a localized (point)$A$-source with a strength $λ$ that acts for a finite time $T$. We reveal the conditions under which the island collapse time $t_{c}$ becomes much longer than the injection period $T$ (long-living island) and demonstrate that regardless of $d$ the evolution of the long-living island radius $r_{f}(t)$ is described by the universal law $ζ_{f}=r_{f}/r_{f}^{M}=\sqrt{eτ|\lnτ|}$ where $τ=t/t_{c}$ and $r_{f}^{M}$ is the maximal island expansion radius at the front turning point $t_{M}=t_{c}/e$. We find that in the long-living island regime the ratio $t_{c}/T$ changes with the increase of the injection period $T$ by the law $\propto (λ^{2}T^{2-d})^{1/d}$ i.e. increases with the increase of $T$ in the one-dimensional (1D) case, does not change with the increase of $T$ in the 2D case and decreases with the increase of $T$ in the 3D case. We derive the scaling laws for particles death in the long-living island and determine the limits of their applicability. We demonstrate also that these laws describe asymptotically the evolution of the $d$-dimensional spherical island with a uniform initial particle distribution generalizing the results obtained earlier for the quasi-one-dimensional geometry. As striking results we present a systematic analysis of the front relative width evolution for fluctuation, logarithmically modified and mean-field regimes and demonstrate that in a wide range of parameters the front remains sharp up to a narrow vicinity of the collapse point.

cond-mat.stat-mech↗
arXiv2007

Diffusion-controlled death of $A$-particle and $B$-particle islands at propagation of the sharp annihilation front $A + B \to 0$

We consider the problem of diffusion-controlled evolution of the system $A$-particle island - $B$-particle island at propagation of the sharp annihilation front $A+B\to 0$. We show that this general problem, which includes as particular cases the sea-sea and the island-sea problems, demonstrates rich dynamical behavior from self-accelerating collapse of one of the islands to synchronous exponential relaxation of the both islands. We find a universal asymptotic regime of the sharp front propagation and reveal limits of its applicability for the cases of mean-field and fluctuation fronts.

cond-mat.stat-mech↗
arXiv2006

Annihilation Catastrophe: From Formation to Universal Explosion

I present a systematic analysis of formation of the universal annihilation catastrophe which develops in an open system, where species $A$ and $B$ diffuse from the bulk of restricted medium and die on its surface (desorb) by the reaction $A + B \to 0$. This phenomenon arises in the diffusion-controlled limit as a result of self-organizing explosive growth (drop) of the surface concentrations of, respectively, slow and fast particles ({\it concentration explosion}) and manifests itself in the form of an abrupt singular jump of the desorption flux relaxation rate. As striking results I find the dependences of time and amplitude of the catastrophe on the initial particle number, and answer the basic questions of when and how universality is achieved.

cond-mat.stat-mech↗
arXiv2003

Diffusion-controlled annihilation $A + B \to 0$: The growth of an $A$ particle island from a localized $A$-source in the $B$ particle sea

We present the growth dynamics of an island of particles $A$ injected from a localized $A$-source into the sea of particles $B$ and dying in the course of diffusion-controlled annihilation $A+B\to 0$. We show that in the 1d case the island unlimitedly grows at any source strength $Λ$, and the dynamics of its growth {\it does not depend} asymptotically on the diffusivity of $B$ particles. In the 3d case the island grows only at $Λ> Λ_{c}$, achieving asymptotically a stationary state ({\it static island}). In the marginal 2d case the island unlimitedly grows at any $Λ$ but at $Λ< Λ_{*}$ the time of its formation becomes exponentially large. For all the cases the numbers of surviving and dying $A$ particles are calculated, and the scaling of the reaction zone is derived.

cond-mat.stat-mech↗
arXiv2003

Diffusion-controlled annihilation $A + B \to 0$ with initially separated reactants: The death of an $A$ particle island in the $B$ particle sea

We consider the diffusion-controlled annihilation dynamics $A+B\to 0$ with equal species diffusivities in the system where an island of particles $A$ is surrounded by the uniform sea of particles $B$. We show that once the initial number of particles in the island is large enough, then at any system's dimensionality $d$ the death of the majority of particles occurs in the {\it universal scaling regime} within which $\approx 4/5$ of the particles die at the island expansion stage and the remaining $\approx 1/5$ at the stage of its subsequent contraction. In the quasistatic approximation the scaling of the reaction zone has been obtained for the cases of mean-field ($d \geq d_{c}$) and fluctuation ($d < d_{c}$) dynamics of the front.

cond-mat.stat-mech↗
arXiv2002

Dynamics of the reaction-diffusion system $A + B \to 0 $ with input of particles

We study dynamics of filling of an initially empty finite medium by diffusing particles $A$ and $B$, which arise on the surface upon dissociation of $AB$ molecules, impinging on it with a fixed flux density $I$, and desorb from it by the reaction $A + B\to AB\to 0$. We show that once the bulk diffusivities differ ($p=D_{A}/D_{B}<1$), there exists a critical flux density $I_{c}(p)$, above which the relaxation dynamics to the steady state is qualitatively changed: on time dependencies of $c_{As}/c_{e}$ ($c_{e}$ being the steady state concentration at $t\to \infty$) a maximum appears, the amplitude of which grows both with $I$ and with $D_{B}/D_{A}$ ratio. In the diffusion-controlled limit $I \gg I_{c}$ at $p \ll 1$ the reaction "selects" the {\it universal laws} for the particles number growth ${\cal N}_{A}={\cal N}_{B}\propto t^{1/4}$ and the evolution of the surface concentrations $c_{As}\propto t^{-1/4},c_{Bs}\propto t^{1/4}$, which are approached by one of the {\it two characteristic regimes} with the corresponding hierarchy of the intermediate power-law asymptotics. In the first of these $c_{As}$ goes through a comparatively {\it sharp} max$(c_{As}/c_{e})\propto I^{1/6}$, the amplitude of which is $p$-independent, in the second one $c_{As}$ goes through a {\itplateau-like} max$(c_{As}/c_{e})\propto p^{-1/4}$, the amplitude of which is $I$-independent. We demonstrate that on the main filling stage the evolution of the ${\cal N}(t)/{\cal N}_{e}, c_{As}(t)/c_{e},$ and $c_{Bs}(t)/c_{e}$ trajectories with changing $p$ or $J$ between the limiting regimes is unambiguously defined by the value of the scaling parameter ${\cal K}=p^{3/2}J$ ($J$ being the reduced flux density) and is described by the set of {\it scaling laws}, which we study in detail analytically and numerically.

cond-mat.stat-mech↗