Simulating the Holtsmark distribution by the Monte Carlo method

Kozlitin, I. A.

doi:10.1134/s207004821101008x

Cited by 4 publications

(3 citation statements)

References 2 publications

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“…This expression, frequently used in numerical studies in the cosmological context (see Pietronero et al 2002;Bottaccio et al 2002, and references therein) still bears the same problems of its full integral form, being non-normalizable and with divergent standard deviation. Typically, in order to avoid a diverging cumulative distribution and diverging energy density of the fluctuating field (Kozlitin 2011), when sampling H(F) in numerical schemes one is forced to fix bona fide cut-offs at large and small F.…”

Section: Methodsmentioning

confidence: 99%

Taking apart the dynamical clock

Pasquato

Cintio

2020

A&A

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Context. In globular clusters (GCs), blue straggler stars (BSS) are heavier than the average star, so dynamical friction strongly affects them. The radial distribution of BSS, normalized to a reference population, appears bimodal in a fraction of Galactic GCs, with a density peak in the core, a prominent zone of avoidance at intermediate radii, and again higher density in the outskirts. The zone of avoidance appears to be located at larger radii the more relaxed the host cluster, acting as a sort of dynamical clock. Aims. We use a new method to compute the evolution of the BSS radial distribution under dynamical friction and diffusion. Methods. We evolve our BSS in the mean cluster potential under dynamical friction plus a random fluctuating force, solving the Langevin equation with the Mannella quasi symplectic scheme. This is a new simulation method that is much faster and simpler than direct N-body codes, but retains their main feature: diffusion powered by strong, if infrequent, kicks. Results. We compute the radial distribution of initially unsegregated BSS normalized to a reference population as a function of time. We trace the evolution of its minimum, corresponding to the zone of avoidance. We compare the evolution under kicks extracted from a Gaussian distribution to that obtained using a Holtsmark distribution. The latter is a fat-tailed distribution which correctly models the effects of close gravitational encounters. We find that the zone of avoidance moves outwards over time, as expected based on observations, only when using the Holtsmark distribution. Thus, the correct representation of near encounters is crucial to reproduce the dynamics of the system. Conclusions. We confirm and extend earlier results that showed how the dynamical clock indicator depends on dynamical friction and on effective diffusion powered by dynamical encounters. We demonstrated the high sensitivity of the clock to the details of the mechanism underlying diffusion, which may explain the difficulties in reproducing the motion of the zone of avoidance across different simulation methods.

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Section: Methodsmentioning

confidence: 99%

Taking apart the dynamical clock

Pasquato

Cintio

2020

A&A

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“…The latter expression, frequently used in numerical studies in the cosmological context (see Pietronero et al 2002;Bottaccio et al 2002, and references therein) still bears the same problems of its full integral form, being non-normalizable and with divergent standard deviation. Typically, in order to avoid a diverging cumulative distribution and diverging energy density of the fluctuating field (Kozlitin 2011), when sampling H(F) in numerical schemes one is forced to fix bona fide cut-offs at large and and small F.…”

Section: Methodsmentioning

confidence: 99%

Taking apart the dynamical clock. Fat-tailed dynamical kicks shape the blue-straggler star bimodality

Pasquato,

Di Cintio

2020

Preprint

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Context. In globular clusters (GCs), blue straggler stars (BSS) are heavier than the average star, so dynamical friction strongly affects them. The radial distribution of BSS, normalized to a reference population, appears bimodal in a fraction of Galactic GCs, with a density peak in the core, a prominent zone of avoidance at intermediate radii, and again higher density in the outskirts. The zone of avoidance appears to be located at larger radii the more relaxed the host cluster, acting as a sort of dynamical clock. Aims. We use a new method to compute the evolution of the BSS radial distribution under dynamical friction and diffusion. Methods. We evolve our BSS in the mean cluster potential under dynamical friction plus a random fluctuating force, solving the Langevin equation with the Mannella quasi symplectic scheme. This amounts to a new simulation method which is much faster and simpler than direct N-body codes but retains their main feature: diffusion powered by strong, if infrequent, kicks. Results. We compute the radial distribution of initially unsegregated BSS normalized to a reference population as a function of time. We trace the evolution of its minimum, corresponding to the zone of avoidance. We compare the evolution under kicks extracted from a Gaussian distribution to that obtained using a Holtsmark distribution. The latter is a fat tailed distribution which correctly models the effects of close gravitational encounters. We find that the zone of avoidance moves outwards over time, as expected based on observations, only when using the Holtsmark distribution. Thus the correct representation of near encounters is crucial to reproduce the dynamics of the system. Conclusions. We confirm and extend earlier results that showed how the dynamical clock indicator depends both on dynamical friction and effective diffusion powered by dynamical encounters. We demonstrated the high sensitivity of the clock to the details of the mechanism underlying diffusion, which may explain the difficulties in reproducing the motion of the zone of avoidance across different simulation methods.

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“…It is often believed that it is not possible to obtain a rather compact expression of S(β) (or H(β)) in terms of known functions (see for instance page 183 of the book "Atomic Physics in Hot Plasmas", by D. Salzmann: "There is no analytical solution for the integral in terms of known elementary or special functions" [4]). There are many interesting works about the numerical computation of the Holtsmark distribution, for instance using Monte Carlo methods [16] or rational-fraction approximations [17]. The peak value of S(β) is…”

Section: Introductionmentioning

confidence: 99%

Expression of the Holtsmark function in terms of hypergeometric $$_{2}F_2$$ and Airy Bi functions

Pain

2020

Eur. Phys. J. Plus

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The Holtsmark distribution has applications in plasma physics, for the electric-microfield distribution involved in spectral line shapes for instance, as well as in astrophysics for the distribution of gravitating bodies. It is one of the few examples of a stable distribution for which a closed-form expression of the probability density function is known. However, the latter is not expressible in terms of elementary functions. In the present work, we mention that the Holtsmark probability density function can be expressed in terms of hypergeometric function 2F2 and of Airy function of the second kind Bi and its derivative. The new formula is simpler than the one proposed by Lee involving 2F3 and 3F4 hypergeometric functions.

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Simulating the Holtsmark distribution by the Monte Carlo method

Cited by 4 publications

References 2 publications

Taking apart the dynamical clock

Taking apart the dynamical clock

Taking apart the dynamical clock. Fat-tailed dynamical kicks shape the blue-straggler star bimodality

Expression of the Holtsmark function in terms of hypergeometric $$_{2}F_2$$ and Airy Bi functions

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