Simultaneous Multiple Collisions Research Articles

Asymptotic genealogies for a class of generalized Wright–Fisher models

A class of Cannings models is studied, with population size N having a mixed multinomial offspring distribution with random success probabilities ${W_{1}},\dots ,{W_{N}}$ induced by independent and identically distributed positive random variables ${X_{1}},{X_{2}},\dots $ via ${W_{i}}:={X_{i}}/{S_{N}}$, $i\in \{1,\dots ,N\}$, where ${S_{N}}:={X_{1}}+\cdots +{X_{N}}$. The ancestral lineages are hence based on a sampling with replacement strategy from a random partition of the unit interval into N subintervals of lengths ${W_{1}},\dots ,{W_{N}}$. Convergence results for the genealogy of these Cannings models are provided under assumptions that the tail distribution of ${X_{1}}$ is regularly varying. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained by Huillet [J. Math. Biol. 68 (2014), 727–761] for the case when ${X_{1}}$ is Pareto distributed and complement those obtained by Schweinsberg [Stoch. Process. Appl. 106 (2003), 107–139] for models where sampling is performed without replacement from a supercritical branching process.

Duality and fixation in $\Xi$-Wright–Fisher processes with frequency-dependent selection

A two-types, discrete-time population model with finite, constant size is constructed, allowing for a general form of frequency-dependent selection and skewed offspring distribution. Selection is defined based on the idea that individuals first choose a (random) number of \emph{potential} parents from the previous generation and then, from the selected pool, they inherit the type of the fittest parent. The probability distribution function of the number of potential parents per individual thus parametrises entirely the selection mechanism. Using sampling- and moment-duality, weak convergence is then proved both for the allele frequency process of the selectively weak type and for the population's ancestral process. The scaling limits are, respectively, a two-types $\Xi$-Fleming-Viot jump-diffusion process with frequency-dependent selection, and a branching-coalescing process with general branching and simultaneous multiple collisions. Duality also leads to a characterisation of the probability of extinction of the selectively weak allele, in terms of the ancestral process' ergodic properties.

The genealogy of a solvable population model under selection with dynamics related to directed polymers

We consider a stochastic model describing a constant size $N$ population that may be seen as a directed polymer in random medium with $N$ sites in the transverse direction. The population dynamics is governed by a noisy traveling wave equation describing the evolution of the individual fitnesses. We show that under suitable conditions the generations are independent and the model is characterized by an extended Wright-Fisher model, in which the individual $i$ has a random fitness $\eta_i$ and the joint distribution of offspring $(\nu_1,\ldots,\nu_N)$ is given by a multinomial law with $N$ trials and probability outcomes $\eta_i$'s. We then show that the average coalescence times scales like $\log N$ and that the limit genealogical trees are governed by the Bolthausen-Sznitman coalescent, which validates the predictions by Brunet, Derrida, Mueller and Munier for this class of models. We also study the extended Wright-Fisher model, and show that, under certain conditions on $\eta_i$, the limit may be Kingman's coalescent, a coalescent with multiple collisions, or a coalescent with simultaneous multiple collisions.

The reversibility and an SPDE for the generalized Fleming–Viot processes with mutation

The (Ξ,A)-Fleming–Viot process with mutation is a probability-measure-valued process whose moment dual is similar to that of the classical Fleming–Viot process except that Kingman’s coalescent is replaced by the Ξ-coalescent, the coalescent with simultaneous multiple collisions. We first prove the existence of such a process for general mutation generator A. We then investigate its reversibility. We also study both the weak and strong uniqueness of the solution to the associated stochastic partial differential equation.

A constrained motion perspective of railway vehicles collision

Impacts, friction, and normal contact forces occur in the railway vehicles couplers. This paper presents a novel nonsmooth model of the train collision; until now, only penalized models have been approached. The train dynamics is described by an equality of measures formulated at the velocity level. The equations of motion are integrated using the Moreau time-stepping algorithm. Impulsive and normal contact forces are described by a set-valued law of Signorini type, while friction forces are described by a set-valued law of Coulomb type. The constrained forces are computed deducing a particular, simplified formulation of the Udwadia–Kalaba equations. The resulting algorithm is simple and straightforward. Both impulsive and nonimpulsive dynamics are casted in the same framework. Any feature or situation regarding train collisions may be modeled. A demonstrative application is presented. Simulations reveal nonsmooth phenomena like simultaneous multiple collisions, stick-slip, captures, and offset in the final equilibrium position.

Conditions for exchangeable coalescents to come down from infinity

An improved condition to come down from infinity for exchangeable coalescent processes with simultaneous multiple collisions of ancestral lineages is provided. For non-critical coalescents this leads to an improved necessary and sufficient condition for the coalescent to come down from infinity. Ananalog con- jecture for the full class of exchangeable coalescents is presented. New examples of critical coalescents are studied in detail. The results extend those obtained by J. Schweinsberg in Section 5.5 of 'Coalescents with simultaneous multiple collisions', Electron. J. Probab. 5, 1-50.

Population Genetics Models with Skewed Fertilities: A Forward and Backward Analysis

Discrete population genetics models with unequal (skewed) fertilities are considered, with an emphasis on skewed versions of Cannings models, conditional branching process models in the spirit of Karlin and McGregor, and compound Poisson models. Three particular classes of models with skewed fertilities are investigated, the Wright–Fisher model, the Dirichlet model, and the Kimura model. For each class the asymptotic behavior as the total population size N tends to infinity is investigated for power law fertilities and for geometric fertilities. This class of models can exhibit a rich variety of sub-linear or even constant effective population sizes. Therefore, the models are not necessarily in the domain of attraction of the Kingman coalescent. For a substantial range of the parameters, discrete-time coalescent processes with simultaneous multiple collisions arise in the limit.

Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson–Dirichlet coalescent

The class of coalescent processes with simultaneous multiple collisions ( Ξ -coalescents) without proper frequencies is considered. We study the asymptotic behavior of the external branch length, the total branch length and the number of mutations on the genealogical tree as the sample size n tends to infinity. The limiting random variables arising are characterized via exponential integrals of the subordinator associated with the frequency of singletons of the coalescent. The proofs are based on decompositions into external and internal branches. The asymptotics of the external branches is treated via the method of moments. The internal branches do not contribute to the limiting variables since the number C n of collisions for coalescents without proper frequencies is asymptotically negligible compared to n . The results are applied to the two-parameter Poisson–Dirichlet coalescent indicating that this particular class of coalescent processes in many respects behaves approximately as the star-shaped coalescent.

On the Speed of Coming Down from Infinity for $\Xi$-Coalescent Processes

The $\Xi$-coalescent processes were initially studied by Möhle and Sagitov (2001), and introduced by Schweinsberg (2000) in their full generality. They arise in the mathematical population genetics as the complete class of scaling limits for genealogies of Cannings' models. The $\Xi$-coalescents generalize $\Lambda$-coalescents, where now simultaneous multiple collisions of blocks are possible. The standard version starts with infinitely many blocks at time $0$, and it is said to come down from infinity if its number of blocks becomes immediately finite, almost surely. This work builds on the technique introduced recently by Berstycki, Berestycki and Limic (2009), and exhibits deterministic ``speed'' function - an almost sure small time asymptotic to the number of blocks process, for a large class of $\Xi$-coalescents that come down from infinity.

Convergence to the coalescent and its relation to the time back to the most recent common ancestor

For the class of haploid exchangeable population models with non-overlapping generations and population size $N$ it is shown that, as $N$ tends to infinity, convergence of the time-scaled ancestral process to Kingman's coalescent and convergence in distribution of the scaled times back to the most recent common ancestor (MRCA) to the corresponding times back to the MRCA of the Kingman coalescent are equivalent. Extensions of this equivalence are derived for exchangeable population models being in the domain of attraction of a coalescent process with multiple collisions. The proofs are based on the property that the total rates of a coalescent with multiple collisions already determine the distribution of the coalescent. It is finally shown that similar results cannot be obtained for the full class of exchangeable coalescents allowing for simultaneous multiple collisions of ancestral lineages, essentially because the total rates do not determine the distribution of a general exchangeable coalescent.

On the number of segregating sites for populations with large family sizes

We present recursions for the total number,Sn, of mutations in a sample ofnindividuals, when the underlying genealogical tree of the sample is modelled by a coalescent process with mutation rater>0. The coalescent is allowed to have simultaneous multiple collisions of ancestral lineages, which corresponds to the existence of large families in the underlying population model. For the subclass of Λ-coalescent processes allowing for multiple collisions, such that the measure Λ(dx)/xis finite, we prove thatSn/(nr) converges in distribution to a limiting variable,S, characterized via an exponential integral of a certain subordinator. When the measure Λ(dx)/x2is finite, the distribution ofScoincides with the stationary distribution of an autoregressive process of order 1 and is uniquely determined via a stochastic fixed-point equation of the formwith specific independent random coefficientsAandB. Examples are presented in which explicit representations for (the density of)Sare available. We conjecture thatSn/E(Sn)→1 in probability if the measure Λ(dx)/xis infinite.

On the number of segregating sites for populations with large family sizes

We present recursions for the total number,Sn, of mutations in a sample ofnindividuals, when the underlying genealogical tree of the sample is modelled by a coalescent process with mutation rater>0. The coalescent is allowed to have simultaneous multiple collisions of ancestral lineages, which corresponds to the existence of large families in the underlying population model. For the subclass of Λ-coalescent processes allowing for multiple collisions, such that the measure Λ(dx)/xis finite, we prove thatSn/(nr) converges in distribution to a limiting variable,S, characterized via an exponential integral of a certain subordinator. When the measure Λ(dx)/x2is finite, the distribution ofScoincides with the stationary distribution of an autoregressive process of order 1 and is uniquely determined via a stochastic fixed-point equation of the formwith specific independent random coefficientsAandB. Examples are presented in which explicit representations for (the density of)Sare available. We conjecture thatSn/E(Sn)→1 in probability if the measure Λ(dx)/xis infinite.

A coalescent model for the effect of advantageous mutations on the genealogy of a population

When an advantageous mutation occurs in a population, the favorable allele may spread to the entire population in a short time, an event known as a selective sweep. As a result, when we sample n individuals from a population and trace their ancestral lines backwards in time, many lineages may coalesce almost instantaneously at the time of a selective sweep. We show that as the population size goes to infinity, this process converges to a coalescent process called a coalescent with multiple collisions. A better approximation for finite populations can be obtained using a coalescent with simultaneous multiple collisions. We also show how these coalescent approximations can be used to get insight into how beneficial mutations affect the behavior of statistics that have been used to detect departures from the usual Kingman's coalescent.

Monte Carlo simulation of charged particle diffusion through an ionized gas: contributions of simultaneous multiple collisions

When a charged particle moves through fully or partially ionized plasmas, energy loss and momentum transfer occur through various kinds of elastic and inelastic collisions. In ordinary gases, composed primarily of neutral particles, the interactions are mainly close binary collisions, being the impact parameter well shorter than the mean free path. In a gas composed of charged particles, neglecting simultaneous multiple collisions can be a very poor approximation because the electromagnetic forces decrease much more slowly with distance than do the forces between neutral atoms. Here we present a Monte Carlo code, where the effects of large impact parameter multiple scattering are considered.

Coalescent processes obtained from supercritical Galton–Watson processes

Consider a population model in which there are N individuals in each generation. One can obtain a coalescent tree by sampling n individuals from the current generation and following their ancestral lines backwards in time. It is well-known that under certain conditions on the joint distribution of the family sizes, one gets a limiting coalescent process as N→∞ after a suitable rescaling. Here we consider a model in which the numbers of offspring for the individuals are independent, but in each generation only N of the offspring are chosen at random for survival. We assume further that if X is the number of offspring of an individual, then P( X⩾ k)∼ Ck − a for some a>0 and C>0. We show that, depending on the value of a, the limit may be Kingman's coalescent, in which each pair of ancestral lines merges at rate one, a coalescent with multiple collisions, or a coalescent with simultaneous multiple collisions.

Coalescents with Simultaneous Multiple Collisions

We study a family of coalescent processes that undergo ``simultaneous multiple collisions,'' meaning that many clusters of particles can merge into a single cluster at one time, and many such mergers can occur simultaneously. This family of processes, which we obtain from simple assumptions about the rates of different types of mergers, essentially coincides with a family of processes that Mohle and Sagitov obtain as a limit of scaled ancestral processes in a population model with exchangeable family sizes. We characterize the possible merger rates in terms of a single measure, show how these coalescents can be constructed from a Poisson process, and discuss some basic properties of these processes. This work generalizes some work of Pitman, who provides similar analysis for a family of coalescent processes in which many clusters can coalesce into a single cluster, but almost surely no two such mergers occur simultaneously.

Amolecular dynamics simulation of collisional excitation mechanisms in Al

A modified version of the SPUT2 molecular dynamics sputtering code was used to reinvestigate core excitation in Al atoms following bombardment with 1–5 keV Ar + ions. For all bombarding energies, asymmetric collisions between the incoming ion and target atoms yielded smaller minimum distances-of-closest-approach between the collision partners for hard collisions than did symmetric collisions between pairs of target atoms. Simple critical distance-of-closest-approach models were used to estimate core excitation for both asymmetric and symmetric collisions. A single value of R c (0.367 Å) was used for asymmetric Ar-Al collisions, while two choices of R c were used for symmetric Al-Al collisions (0.442 and 0.530 Å). With the smaller R c value for Al-Al collisions, we find that core excitation proceeds predominantly by asymmetric collisions at all bombarding energies above threshold. At 5 keV bombarding energy the percentage of sputtered, core-excited atoms originating from asymmetric collisions ranged from 89 to 95% depending on the incident direction of the projectile. With the larger R c value, core excitation proceeds predominantly by asymmetric collisions at bombarding energies above approximately 3 keV; and at 5 keV asymmetric collisions accounted for ∼ 60 to ∼ 84% of sputtered, core-excited atoms. Lifetime corrections and corrections for Auger neutralization near the target surface had little effect on the ratio of asymmetric to symmetric collisions responsible for atomic-like Auger emission. These simulation results suggest that simultaneous multiple collisions are very important in the initial energy- and momentum-transfer stage which initiates the cascade.

Simulation of cluster impacts on metallic surfaces.

Energy, number-density, and confinement-time properties of 1-keV/atom ${\mathrm{Al}}_{32}$ and ${\mathrm{Al}}_{63}$ cluster impacts on Al and Au targets have been investigated by molecular-dynamics simulation. Inertial confinement of the incoming clusters leads to number-density increases of \ensuremath{\sim}2 for very short time periods (20 fs) in the primary impact zone. Simultaneous and near-simultaneous multiple collisions occasionally increase the potential energy of a particle to more than twice that allowed in isolated two-body collisions. Up to 12% of the kinetic energy of the incoming cluster is transferred to atoms which eject early in the collision cascade.

Experimental investigations on electron interaction

Certain observations in low-pressure gas discharges and in magnetrons appear to indicate an electron interaction effect several orders of magnitude in excess of the theoretical values based on binary encounters only. As the Coulomb interaction in low-pressure plasmas and in electron clouds usually extends over a range which contains at least several thousand electrons, it was not clear whether these observations indicated a breakdown of the theory, or some new effects, other than electron interaction. New experiments were therefore designed to establish an upper limit for the magnitude of the electron interaction effect. Both plasmas in thermal equilibrium and pure electronic space charges were investigated. The basic experimental method in each case was shooting a very carefully collimated electron beam through the charge cloud, and measuring the scattering by direct observation on a fluorescent screen. The apparatus was designed to resolve angles down to 10 -4 . The density of the equilibrium plasma was inferred from capacity measurements on a thin wire probe in the plasma. The density of the electron cloud was determined by measuring the electronoptical effects produced. It is concluded that both for the plasma as for the electron cloud the scattering is less than five times the value predicted by the theories neglecting simultaneous multiple collisions. The interaction is therefore so small that it cannot be expected to account for the observed anomalies either in low-pressure gas discharges or in other electronic devices.