Zakharov’s transformation in the problem of galaxy mass distribution function

Конторович, В. М.

doi:10.1016/s0167-2789(01)00200-7

Cited by 12 publications

(12 citation statements)

References 12 publications

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“…We have found that V. M. Kontorovich has recently applied the Zakharov transformation to aggregation problems [22] for a class of kernels arising from the study of galactic mergers in astrophysics.…”

Section: Discussionmentioning

confidence: 99%

“…If we continue to add mass to the system at a constant rate J 0 then we know that the final steady state is given by the Kolmogorov spectrum Eq. (22). When the total mass contained in this solution is finite then mass conservation must be violated at some time since the total mass supplied to the system grows linearly in time.…”

Section: Finite and Infinite Capacity Cases-gelling And Nongellingmentioning

confidence: 99%

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

Connaughton

Rajesh²,

Zaboronski³

2004

Phys. Rev. E

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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. We derive a "locality criterion," expressed in terms of the asymptotic properties of the kernel, that must be satisfied in order for the Kolmogorov spectrum to be an admissible solution. Whether a given kernel leads to a gelation transition or not can be determined by computing the mass capacity of the Kolmogorov spectrum. As an example, we compute the exact stationary state for the family of kernels, K(zeta) ( m(1), m(2) )= ( m(1) m(2) )(zeta/2) which includes both gelling and nongelling cases, reproducing the known solution in the case zeta=0. Surprisingly, the Kolmogorov constant is the same for all kernels in this family.

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Section: Discussionmentioning

confidence: 99%

Section: Finite and Infinite Capacity Cases-gelling And Nongellingmentioning

confidence: 99%

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

Connaughton

Rajesh²,

Zaboronski³

2004

Phys. Rev. E

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“…This can occur in practice. Examples include coagulation of ice in planetary rings [4], gravitational clustering [10], and droplet sedimentation in static fluids [11].…”

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confidence: 99%

Collective Oscillations in Irreversible Coagulation Driven by Monomer Inputs and Large-Cluster Outputs

et al. 2012

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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. This universality is lost, however, if the aggregation rate between large and small clusters increases sufficiently steeply as a function of cluster sizes. We identify a transition to a regime in which the stationary state vanishes as the cutoff grows. This nonuniversal stationary state becomes unstable as the cutoff is increased. It undergoes a Hopf bifurcation after which the stationary state is replaced by persistent and periodic collective oscillations. These oscillations, which bear some similarities to relaxation oscillations in excitable media, carry pulses of mass through the space of cluster sizes such that the average mass flux through any cluster size remains constant. Universality is partially restored in the sense that the scaling of the period and amplitude of oscillation is inherited from the dynamical scaling exponents of the universal regime.

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“…The stationary state of this equation is best studied using the method of Zakharov transformations as detailed in [10,26] …”

Section: Renormalised Smoluchowski Equationmentioning

confidence: 99%

“…arguments. Strictly speaking, the Zakharov transformation technology of [10,26] only works for homogeneous kernels which leaves us with the question of what to do with the logarithm in eq. (87).…”

Section: Renormalised Smoluchowski Equationmentioning

confidence: 99%

Cluster–cluster aggregation as an analogue of a turbulent cascade: Kolmogorov phenomenology, scaling laws and the breakdown of self-similarity

Connaughton¹,

Rajesh²,

Zaboronski³

2006

Physica D: Nonlinear Phenomena

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We present a detailed study of the statistical properties of a system of diffusing aggregating particles in the presence of a steady source of monomers. We emphasise the case of low spatial dimensions where strong diffusive fluctuations invalidate the mean-field description provided by standard Smoluchowski kinetic theory. The presence of a source of monomers allows the system to reach a statistically stationary state at large times. This state is characterised by a constant flux of mass directed from small to large masses. It therefore admits a phenomenological description based on the assumption of self-similarity and constant mass flux analogous to the Kolmogorov's 1941 theory of turbulence. Unlike turbulence, the aggregation problem is analytically tractable using powerful methods of statistical field theory. We explain in detail how these methods should be adapted to study the far-from-equilibrium, flux-dominated states characteristic of turbulent systems. We consider multipoint correlation functions of the mass density. By an exact evaluation of the scaling exponents for the one and two-point correlation functions, we show that the assumption of self-similiarity breaks down at large masses for spatial dimensions, d ≤ 2. We calculate nonrigourously the exponents of the higher order correlation functions as an ǫ-expansion where ǫ = 2−d. We show that the mass distribution exhibits non-trivial multiscaling. An analogy can be drawn with the case of hydrodynamic turbulence. The physical origin of this multiscaling is traced to the presence of strong correlations between particles participating in large mass aggregation events. These correlations stem from the recurrence of diffusion processes in d ≤ 2. The analytic methods developed here will have more general applicability beyond the study of this specific problem.

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Zakharov’s transformation in the problem of galaxy mass distribution function

Cited by 12 publications

References 12 publications

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

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

Collective Oscillations in Irreversible Coagulation Driven by Monomer Inputs and Large-Cluster Outputs

Cluster–cluster aggregation as an analogue of a turbulent cascade: Kolmogorov phenomenology, scaling laws and the breakdown of self-similarity

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