Phase transitions in systems with aggregation and shattering

Krapivsky, P. L.; Otieno, W.; Brilliantov, Nikolai V.

doi:10.1103/physreve.96.042138

Cited by 16 publications

(10 citation statements)

References 29 publications

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“…where F ij quantifies the shattering rate. Models with shattering exhibit interesting behaviors including dynamical phase transitions [15]. It has been shown [10] that more general fragmentation models with a large number of fragments yield qualitatively similar size distribution provided the small-size debris strongly dominates over the large size ones [10].…”

Section: B Aggregation With Fragmentationmentioning

confidence: 99%

Steady oscillations in aggregation-fragmentation processes

et al. 2018

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We report surprising steady oscillations in aggregation-fragmentation processes. Oscillating solutions are observed for the class of aggregation kernels K_{i,j}=i^{ν}j^{μ}+j^{ν}i^{μ} homogeneous in masses i and j of merging clusters and fragmentation kernels, F_{ij}=λK_{ij}, with parameter λ quantifying the intensity of the disruptive impacts. We assume a complete decomposition (shattering) of colliding partners into monomers. We show that an assumption of a steady-state distribution of cluster sizes, compatible with governing equations, yields a power law with an exponential cutoff. This prediction agrees with simulation results when θ≡ν-μ<1. For θ=ν-μ>1, however, the densities exhibit an oscillatory behavior. While these oscillations decay for not very small λ, they become steady if θ is close to 2 and λ is very small. Simulation results lead to a conjecture that for θ<1 the system has a stable fixed point, corresponding to the steady-state density distribution, while for any θ>1 there exists a critical value λ_{c}, such that for λ<λ_{c}, the system has an attracting limit cycle. This is rather striking for a closed system of Smoluchowski-like equations, lacking any sinks and sources of mass.

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Section: B Aggregation With Fragmentationmentioning

confidence: 99%

Steady oscillations in aggregation-fragmentation processes

et al. 2018

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“…For example, there exist a class of 'jammed state' corresponding to possible non-uniqueness of solutions of the steady-state problem. This state has been observed for much simpler models by [36] or [37] basing on fact that some concentrations of clusters can become zeros and freeze the aggregation process without further relaxation of particle size distributions.…”

Section: Theoretical Analysismentioning

confidence: 77%

Oscillating stationary distributions of nanoclusters in an open system

Matveev

Sorokin

Смирнов³

et al. 2020

Mathematical and Computer Modelling of Dynamical Systems

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Steady-state oscillations of nanoparticle populations in the system of colliding monomers and seed-clusters are observed for the range of the seed-cluster source with diffusion and ballistic collision kernels. The dynamics of nanoparticles in this system is driven by monomer-cluster and cluster-cluster irreversible aggregation and described in terms of the number of primary monomers per nanoparticle based on solving the population balance equations as described by the classical system of Smoluchowski equations. The oscillations of particles' concentrations arise with growing power of the source of seed-clusters and can remain visible for several orders of magnitute of particle sizes k. For the case of constant kinetic coefficients the novel semi-analytial solution of the utilized aggregation model is found and results of numerical simulations with use of up to 2 20 non-linear kinetic equations agree excellently with proposed theory.

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“…An analogous behavior is known to occur in fragmentation in the form of a ghost phase of clusters with zero mass (“dust”) that contain a finite fraction of the total mass [ 5 , 6 ]. This process of “shattering” has evoked analogies to phase transitions and motivated numerous investigations of the kinetic equations of fragmentation and the conditions that may result to this unique behavior [ 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 ].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Theory of Discrete Binary Fragmentation—Kinetics and Thermodynamics

Matsoukas

2022

Entropy

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We formulate binary fragmentation as a discrete stochastic process in which an integer mass k splits into two integer fragments j, k−j, with rate proportional to the fragmentation kernel Fj,k−j. We construct the ensemble of all distributions that can form in fixed number of steps from initial mass M and obtain their probabilities in terms of the fragmentation kernel. We obtain its partition function, the mean distribution and its evolution in time, and determine its stability using standard thermodynamic tools. We show that shattering is a phase transition that takes place when the stability conditions of the partition function are violated. We further discuss the close analogy between shattering and gelation, and between fragmentation and aggregation in general.

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Phase transitions in systems with aggregation and shattering

Cited by 16 publications

References 29 publications

Steady oscillations in aggregation-fragmentation processes

Steady oscillations in aggregation-fragmentation processes

Oscillating stationary distributions of nanoclusters in an open system

Stochastic Theory of Discrete Binary Fragmentation—Kinetics and Thermodynamics

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