Gene flow across geographical barriers - scaling limits of random walks with obstacles

In this paper, we study the scaling limit of a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. If the probability of crossing the barrier scales as $1/\sqrt{n}$ as we rescale space by $\sqrt{n}$ and time by $n$, we obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at the origin reaches an independent exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at the origin reaches another exponential level, and so on. We give a martingale problem characterisation of this process as well as another construction and an explicit formula for its transition density. This result has applications in the field of population genetics where such a random walk is used to trace the position of one's ancestor in the past in the presence of a barrier to gene flow.Comment: Stochastic Processes and Their Applications, in pres

Experimental constrains on shear-induced crystal breakage in magmas

International audienceCrystal breakage occurs along margins of conduit walls and basal zones of lava flows. It is usually interpreted as flow-related textures developed at large finite strains and strains rates. We have investigated the grain size and shape distributions in an experimentally deformed crystal-melt suspension in order to constrain the temperature T, the strain γ and the strain rate γr ranges of the crystal breakage process. The starting crystal-melt suspension is composed of a haplogranitic melt with 54 vol% alumina crystals. Torsion experiments were performed in a gas medium Paterson apparatus at 300 MPa confining pressure and subsolidus temperatures. Crystal size distribution and aspect ratio of alumina grains were measured on polished sections normal to the shear direction, i.e. from the centre to the rim of the deformed cylinders. A first minor occurrence of crystal breakage is evidenced in all experiments and low strains. It is related to intense stress localisation at some grain contacts in the initially connected solid framework. A second intense and penetrative crystal breakage process is observed for T≤ 550°C and γr > 6.2x10-4 s- 1. The evolution of the size distribution as a function of finite strain and the reduced aspect ratios of preserved largest crystals in intensely strained zones support that breakage occurs by abrasion of the larger crystals. This abrasion can be attributed to the partial stress propagation over both the melt and partially isolated crystals under visco-elastic conditions. Mechanical data show a transition from slight shear softening at low strain rates and highest temperatures to strain hardening for experiments that produced penetrative crystal breakage. The crystal-melt suspension exhibits a shear thinning behaviour with a stress exponent larger than 2.06 over the explored strain rate and temperature domain for the experiments without intensive crystal breakage. Our results are applicable to the interpretation of the crystal breakage often observed at the base of lava flows, in domes, and near conduit walls. This experimental reproduction of a process observed in nature is important because the controls of stress-induced breakage we quantified are also key parameters governing magma transport

Settling and compaction of olivine in basaltic magmas: an experimental study on the time scales of cumulate formation

A series of centrifuge-assisted settling experiments of 30 vol% olivine in 70 vol% basaltic melt was conducted to elucidate the formation mechanisms and time scales of gravitational cumulates. The settling experiments were performed in a centrifuging piston cylinder at 200-1,500g, 1,270-1,280°C, and 0.8-1.1GPa on previously annealed and texturally equilibrated samples. The mechanical settling of the dense olivine suspension occurs at about 1/6 the speed of simple Stokes settling, resulting in a sedimentation exponent n=4.1(6) in agreement with predictions from analogue systems. The porosity (φ m ) of the orthocumulate resulting from gravitational settling of crystals is about 54% and formation times of olivine orthocumulates result to 0.1-10mday−1 (for an initial crystal content of the melt of 1-5% and grain sizes of 2-10mm). After mechanical settling, olivine grains rest on each other, and further compaction occurs through pressure dissolution at grain contacts, olivine reprecipitation where olivine is in contact with melt, and concomitant expulsion of excess liquid from the cumulate layer. With centrifugation at 400g for 50h, porosities as low as 30.3 vol% were achieved. The olivine content at the bottom of the gravitational cumulate is 1−φm~log(Δρ·h·a·t), where Δρ is the density difference between crystals and melt, h the crystal layer thickness, a the acceleration, and t the time of centrifuging. Compaction is hence proportional to effective stress integrated over time indicating that pressure dissolution is the dominant mechanism for chemical compaction. The compaction limit, that is the lowermost porosity to be reached by this mechanism, is calculated by equating the lithostatic and hydraulic pressure gradients in the cumulate and results to 3-5% porosity for the experiments. Crystal size distribution curves and a growth exponent n of 3.1(3) indicate that diffusion-controlled Ostwald ripening is the dominant crystal growth mechanism. The above relationship, combined with a linear scaling for grain size as appropriate for reaction-controlled pressure solution creep, allows calculation of formation times of adcumulates. If chemical compaction is dissolution-reprecipitation limited, then single layers of natural olivine adcumulates of ½ m thickness with 70-75 vol% olivine at the base (as observed in the Rhum layered intrusion) would have typical formation times of 0.4-3years for grain sizes of 2-10mm. This time scale compares favourably with characteristic cooling times of sills. If a greater than20-m-thick series of cumulate layers pressurizes a base layer with the porosity still filled by a melt, then compaction proceeds to the compaction limit within a few years. It can thus be expected that in layered mafic intrusions where cumulates are continuously deposited from a large magma chamber and which characteristic cooling times of more than decades, a compaction zone of several tens of metres forms with adcumulates only maintaining porosities in the order of 5%. In conclusion, gravitational settling and gravitation-driven chemical compaction are feasible cumulate-forming processes for dense mafic minerals in basaltic magmas and in particular in large layered intrusion

An extension of the Ising-Curie-Weiss model of self-organized criticality with a threshold on the interaction range

In arXiv:1301.6911, Cerf and Gorny constructed a model of self-organized criticality, by introducing an automatic control of the temperature parameter in the generalized Ising Curie-Weiss model. The fluctuations of the magnetization of this spin model are of order $n^{3/4}$ with a limiting law of the form $C\exp(-x^4)$, as in the critical regime of the Curie-Weiss model. In this article, we build upon this model by replacing the mean-field interaction with a one-dimensional interaction with a certain range $r_n$ which varies as a function of the number $n$ of particles. In the Gaussian case, we show that the self-critical behaviour observed in the mean-field case extends to interaction ranges $r_n\gg n^{3/4}$ and we show that this threshold is sharp, with different fluctuations when the interaction range is of order of $n^{3/4}$ or smaller than $n^{3/4}$.Comment: 24 pages. This new simplified version merges this article with arXiv:2110.0794

Isolation by distance patterns arising from short range and long range dispersal -- a forwards in time approach

In this paper, we consider a mathematical model for the evolution of neutral genetic diversity in a spatial continuum including mutations, genetic drift and either short range or long range dispersal. The model we consider is the spatial $\Lambda$-Fleming-Viot process introduced by Barton, Etheridge and V\'eber, which describes the state of the population at any time by a measure on $\mathbb{R}^d \times [0,1]$, where $\mathbb{R}^d$ is the geographical space and $[0,1]$ is the space of genetic types. In both cases (short range and long range dispersal), we prove a functional central limit theorem for the process as the population density becomes large and under some space-time rescaling. We then deduce from these two central limit theorems a formula for the asymptotic probability of identity of two individuals picked at random from two given spatial locations. In the case of short range dispersal, we recover the classical Wright-Mal\'ecot formula, which is widely used in demographic inference for spatially structured populations. In the case of long range dispersal, however, our formula appears to be new, and could open the way for a better appraisal of long range dispersal in inference methods

An extension of the Ising-Curie-Weiss model of self-organized criticality with long range interactions

In [CG16], Cerf and Gorny constructed a model of self-organized criticality, by introducing an automatic control of the temperature parameter in the generalized Ising Curie-Weiss model. In this article, we build upon this model by replacing the mean-field interaction of [CG16] with a one-dimensional interaction with a certain range d(n) which varies as a function of the number n of particles. In the Gaussian case, we show that, for a very long range of interaction (d(n) of order n), the model exhibits the same behaviour as in the mean-field case, whereas in the case of a nearest neighbour interaction (d(n) = 1), the behaviour highlighted by Cerf and Gorny breaks out.Comment: Article merged with arXiv:2110.07949 in a new simplified version which gathers the results of both article

A planar Ising model of self-organized criticality

We consider the planar Ising model in a finite square box and we replace the temperature parameter with a function depending on the magnetization. This creates a feedback from the spin configuration onto the parameter, which drives the system towards the critical point. Using the finite-size scaling results of arXiv:0811.4507, we show that, when the size of the box grows to infinity, the temperature concentrates around the critical temperature of the planar Ising model on the square lattice

Centrifuge experiments with magmatic systems : from melt segregation to pluton emplacement

In this thesis, laboratory investigations have been conducted to investigate several processes occurring during the melt segregation (crystal settling and compaction processes), as well as during emplacement of plutons. With the help of three different sets of centrifuge experiments rates of these three magmatic processes have been evaluated. In the first series of the centrifuge experiments, the diapiric ascent of buoyant material from two source layers at different depths was studied. Through five models, the hypothesis of ascending diapirs was tested and it was demonstrated whether a rising diapir ascends straight upward or if its ascent might be deviated by another buoyant, softer – and consequently easier to travel through – layer which is located within the overburden strata. We were interested under which conditions they can be formed. For this purpose we placed perturbations on top of both the buoyant layers; either with a set-off of both the protrusions (for three of these experiments), or with both protrusion sitting directly on top of each other (for one of the experiments). In the first experiment, we omitted the perturbations, to test which pathways diapirs take which grow from natural Rayleigh-Taylor instabilities. Three others experiments differed in the viscosity contrast between the overburden and the buoyant material. Through the experimental runs, the effects of different overburden viscosities and perturbation positions on the number of the diapirs were observed. The modeling results show that two diapirs rising from the offset perturbations do not take the same pathway through the overburden layer. Rather, each diapir takes a different pathway, with the deeper diapir piercing through its overburden while rising, regardless if it was a buoyant layer or denser overburden layers. However, when the two perturbations were situated directly above each other in the different PDMS layers, this resulted in the formation of one big diapir rather than several smaller ones, and the overburden layer was less deformed than with offset perturbations. Diapiric structures as those derived from the models without perturbation and where the perturbation are offset occur within Great Kavir Basin (Iran), where numerous salt diapirs grew from several salt horizons, which show a similar spatial distribution. The resulting structure observed in the model where the two perturbations situated directly above each other, is close to what is observed in composite batholiths such as the Flasergranitoid Zone within the Bergsträßer Odenwald Crystalline Complex (Germany). The second series of models were aimed to study crystal settling within a magma. For this purpose experiments with an artificial magma of 30 vol% olivine in 70 vol% basaltic melt were conducted to elucidate the formation mechanisms and time scales of gravitational cumulates. Through the experiments, two physical processes have been observed: (i) purely mechanical compaction, and (ii) chemical compaction induced by dissolution and re-precipitation of settled crystals. The results reveals that the mechanical settling of the dense olivine suspension occurs at about 1/6 the speed of simple Stokes settling, and a sedimentation exponent n of 4.1 is found. Evidences of chemical compaction induced by dissolution and re-precipitation of settled crystals have been highlighted by a detailed analysis of the fine structure of olivine grain boundaries. This last has revealed (1) the presence of Ca, which is characteristic only for MORB-melt, at the interface of two adjacent Ol-grains even when no melt is present; (2) a not fully crystallized boundary layer between two adjacent olivine grains. The crystal size distribution curves and the grain size growth exponent n ~3.6 indicate that diffusion controlled Ostwald ripening is the dominant crystal growth mechanism in concentrated magmatic suspensions. Finally, the formation times in natural olivine adcumulates have been calculated. The last series of centrifuge experiments deals with the crystal-melt settling-floating mechanism in a system composed of natural two pyroxene gabbro. The results have revealed a vertical evolution of the major and trace elements in the melt phase. Then, a numerical modelling of the sedimentation process of the crystals has been made in order to describe the compaction evolution with time. In comparing the numerical simulation with the centrifuge modelling, the stratification of the compacted layer in the runs is reproduced in numerical models. Moreover, on the base of the numerical and centrifuge modelling, a sedimentation exponent describing a deviation of settling in concentrated suspensions from Stokes sedimentation has been evaluated. Finally, the numerical simulation is applied to the Muskox intrusion to estimate the formation time and the melt fraction evolution in using the hindered sedimentation model calculations.Schmelzen und Magma (Schmelze mit darin enthaltenen Feststoffen) müssen sowohl innerhalb ihrer Schmelzregion als auch aus dieser heraus transportiert werden, bevor sie schließlich abgelagert werden. Der Transportvorgang geschieht in zwei Größenskalen: Segregation, die gekennzeichnet ist durch kleinskalige Bewegung der Schmelze im Zentimeter- bis Dezimeterbereich und vor allem im Quellbereich stattfindet, sowie langskaliger Aufstieg im Kilometerbereich durch die kontinentale Kruste bis zu dem Ort der abschließenden Platznahme. Diese Arbeit untersucht anhand von drei Versuchsreihen an Zentrifugen verschiedene Prozesse, die während der Schmelzsegregation (Absinken von Kristallen und Kompaktions-Prozesse), des Aufstiegs und der Plutonplatznahme stattfinden. Unter Verwendung von drei verschiedenen Zentrifugen habe ich versucht, diese Mechanismen genauer zu verstehen

Central limit theorems describing isolation by distance under various forms of power-law dispersal

In this paper, we uncover new asymptotic isolation by distance patterns occurring under long-range dispersal of offspring. We extend a recent work of the first author, in which this information was obtained from forwards-in-time dynamics using a novel stochastic partial differential equations approach for spatial $\Lambda$-Fleming-Viot models. The latter were introduced by Barton, Etheridge and V\'eber as a framework to model the evolution of the genetic composition of a spatially structured population. Reproduction takes place through extinction-recolonisation events driven by a Poisson point process. During an event, in certain ball-shaped areas, a parent is sampled and a proportion of the population is replaced. We generalize the previous approach of the first author by allowing the area from which a parent is sampled during events to differ from the area in which offspring are dispersed, and the radii of these regions follow power-law distributions. In particular, while in previous works the motion of ancestral lineages and coalescence behaviour were closely linked, we demonstrate that local and non-local coalescence is possible for ancestral lineages governed by both fractional and standard Laplacians.Comment: 46 pages, 3 Figure