arXiv subjects

B. Mehlig

Publications and source records attributed to B. Mehlig.

At least 109 records · Page 6Linked to original sources

arXiv2010

Clustering of exponentially separating trajectories

It might be expected that trajectories for a dynamical system which has no negative Lyapunov exponent (implying exponential growth of small separations will not cluster together. However, clustering can occur such that the density $ρ(Δx)$ of trajectories within distance $Δx$ of a reference trajectory has a power-law divergence, so that $ρ(Δx)\sim Δx^{-β}$ when $Δx$ is sufficiently small, for some $0<β<1$. We demonstrate this effect using a random map in one dimension. We find no evidence for this effect in the chaotic logistic map, and argue that the effect is harder to observe in deterministic maps.

nlin.CD↗

arXiv2009

Sample genealogies and genetic variation in populations of variable size

We consider neutral evolution of a large population subject to changes in its population size. For a population with a time-variable carrying capacity we have computed the distributions of the total branch lengths of its sample genealogies. Within the coalescent approximation we have obtained a general expression, Eq. (27), for the moments of these distributions for an arbitrary smooth dependence of the population size on time. We investigate how the frequency of population-size variations alters the distributions. This allows us to discuss their influence on the distribution of the number of mutations, and on the population homozygosity in populations with variable size.

q-bio.PE↗

arXiv2009

Multiple paternity: determining the minimum number of sires of a large brood

We describe an efficient algorithm for determining exactly the minimum number of sires consistent with the multi-locus genotypes of a mother and her progeny. We consider cases where a simple exhaustive search through all possible sets of sires is impossible in practice (because it would take too long to complete). Our algorithm for solving this combinatorial optimisation problem avoids visiting large parts of search space which would not improve the solution found so far (i.e., result in a solution with fewer number of sires). This is of particular importance when the number of allelic types in the progeny array is large and when the minimum number of sires is expected to be large. Precisely in such cases it is important to know the minimum number of sires: this number gives an exact bound on the most likely number of sires estimated by a random search algorithm in a parameter region where it may be difficult to determine whether it has converged. We apply our algorithm to data from the marine snail, Littorina saxatilis.

q-bio.PE↗

arXiv2009

Multiple regimes of diffusion

We consider the diffusion of independent particles experiencing random accelerations by a space- and time-dependent force as well as viscous damping. This model can exhibit several asymptotic behaviours, depending upon the limiting cases which are considered, some of which have been discussed in earlier work. Here we explore the full space of dimensionless parameters, and introduce an "asymptotic phase diagram" which delineates the limiting regimes.

nlin.CD↗

arXiv2009

Sequential Markov coalescent algorithms for population models with demographic structure

We analyse sequential Markov coalescent algorithms for populations with demographic structure: for a bottleneck model, a population-divergence model, and for a two-island model with migration. The sequential Markov coalescent method is an approximation to the coalescent suggested by McVean and Cardin, and Marjoram and Wall. Within this algorithm we compute, for two individuals randomly sampled from the population, the correlation between times to the most recent common ancestor and the linkage probability corresponding to two different loci with recombination rate R between them. We find that the sequential Markov coalescent method approximates the coalescent well in general in models with demographic structure. An exception is the case where individuals are sampled from populations separated by reduced gene flow. In this situation, the gene-history correlations may be significantly underestimated. We explain why this is the case.

q-bio.PE↗

arXiv2008

Shock fragmentation model for gravitational collapse

A cloud of gas collapsing under gravity will fragment. We present a new theory for this process, in which layers shocked gas fragment due to their gravitational instability. Our model explains why angular momentum does not inhibit the collapse process. The theory predicts that the fragmentation process produces objects which are significantly smaller than most stars, implying that accretion onto the fragments plays an essential role in determining the initial masses of stars. This prediction is also consistent with the hypothesis that planets can be produced by gravitational collapse.

astro-ph↗

arXiv2008

Variable-range Projection Model for Turbulence-driven Collisions

We discuss the probability distribution of relative speed $ΔV$ of inertial particles suspended in a highly turbulent gas when the Stokes numbers, a dimensionless measure of their inertia, is large. We identify a mechanism giving rise to the distribution $P(ΔV) \sim \exp(-C |ΔV|^{4/3})$ (for some constant $C$). Our conclusions are supported by numerical simulations and the analytical solution of a model equation of motion. The results determine the rate of collisions between suspended particles. They are relevant to the hypothesised mechanism for formation of planets by aggregation of dust particles in circumstellar nebula.

physics.flu-dyn↗

arXiv2008

Collisions of particles advected in random flows

We consider collisions of particles advected in a fluid. As already pointed out by Smoluchowski [Z. f. physik. Chemie XCII, 129-168, (1917)], macroscopic motion of the fluid can significantly enhance the frequency of collisions between the suspended particles. This effect was invoked by Saffman and Turner [J. Fluid Mech. 1, 16-30, (1956)] to estimate collision rates of small water droplets in turbulent rain clouds, the macroscopic motion being caused by turbulence. Here we show that the Saffman-Turner theory is unsatisfactory because it describes an initial transient only. The reason for this failure is that the local flow in the vicinity of a particle is treated as if it were a steady hyperbolic flow, whereas in reality it must fluctuate. We derive exact expressions for the steady-state collision rate for particles suspended in rapidly fluctuating random flows and compute how this steady state is approached. For incompressible flows, the Saffman-Turner expression is an upper bound.

nlin.CD↗

arXiv2007

Stokes trapping and planet formation

It is believed that planets are formed by aggregation of dust particles suspended in the turbulent gas forming accretion disks around developing stars. We describe a mechanism, termed 'Stokes trapping', by which turbulence limits the growth of aggregates of dust particles, so that their Stokes number (defined as the ratio of the damping time of the particles to the Kolmogorov dissipation timescale) remains close to unity. We discuss possible mechanisms for avoiding this barrier to further growth. None of these is found to be satisfactory and we introduce a new theory which does not involve the growth of small clusters of dust grains.

astro-ph↗

arXiv2007

Stick-Breaking Model for Variable-Range Hopping

We consider the optimal conduction path of the one-dimensional variable-range hopping problem. We describe a hierarchical procedure for constructing the path which is in excellent agreement with numerical results obtained from a percolation approach. The advantage of the hierarchical construction is that it is easier to analyse. We show that the distribution of hopping lengths is well approximated by a model for the repeated breaking of a stick at its weakest point, until the fragments are too strong to be broken.

cond-mat.dis-nn↗

arXiv2007

An accurate model for genetic hitch-hiking

We suggest a simple deterministic approximation for the growth of the favoured-allele frequency during a selective sweep. Using this approximation we introduce an accurate model for genetic hitch-hiking. Only when Ns < 10 (N is the population size and s denotes the selection coefficient), are discrepancies between our approximation and direct numerical simulations of a Moran model noticeable. Our model describes the gene genealogies of a contiguous segment of neutral loci close to the selected one, and it does not assume that the selective sweep happens instantaneously. This enables us to compute SNP distributions on the neutral segment without bias.

q-bio.PE↗

arXiv2007

Precise asymptotics for a variable-range hopping model

For a system of localised electron states the DC conductivity vanishes at zero temperature, but localised electrons can conduct at finite temperature. Mott gave a theory for the low-temperature conductivity in terms of a variable-range hopping model, which is hard to analyse. Here we give precise asymptotic results for a modified variable-range hopping model proposed by S. Alexander [Phys. Rev. B 26, 2956 (1982)].

cond-mat.dis-nn↗

arXiv2007

Advective collisions

Small particles advected in a fluid can collide (and therefore aggregate) due to the stretching or shearing of fluid elements. This effect is usually discussed in terms of a theory due to Saffman and Turner [J. Fluid Mech., 1, 16-30, (1956)]. We show that in complex or random flows the Saffman-Turner theory for the collision rate describes only an initial transient (which we evaluate exactly). We obtain precise expressions for the steady-state collision rate for flows with small Kubo number, including the influence of fractal clustering on the collision rate for compressible flows. For incompressible turbulent flows, where the Kubo number is of order unity, the Saffman-Turner theory is an upper bound.

nlin.CD↗

arXiv2006

Unmixing in Random Flows

We consider particles suspended in a randomly stirred or turbulent fluid. When effects of the inertia of the particles are significant, an initially uniform scatter of particles can cluster together. We analyse this 'unmixing' effect by calculating the Lyapunov exponents for dense particles suspended in such a random three-dimensional flow, concentrating on the limit where the viscous damping rate is small compared to the inverse correlation time of the random flow (that is, the regime of large Stokes number). In this limit Lyapunov exponents are obtained as a power series in a parameter which is a dimensionless measure of the inertia. We report results for the first seven orders. The perturbation series is divergent, but we obtain accurate results from a Pade-Borel summation. We deduce that particles can cluster onto a fractal set and show that its dimension is in satisfactory agreement with previously reported in simulations of turbulent Navier-Stokes flows. We also investigate the rate of formation of caustics in the particle flow.

nlin.CD↗

arXiv2006

Generalised Ornstein-Uhlenbeck processes

We solve a physically significant extension of a classic problem in the theory of diffusion, namely the Ornstein-Uhlenbeck process [G. E. Ornstein and L. S. Uhlenbeck, Phys. Rev. 36, 823, (1930)]. Our generalised Ornstein-Uhlenbeck systems include a force which depends upon the position of the particle, as well as upon time. They exhibit anomalous diffusion at short times, and non-Maxwellian velocity distributions in equilibrium. Two approaches are used. Some statistics are obtained from a closed-form expression for the propagator of the Fokker-Planck equation for the case where the particle is initially at rest. In the general case we use spectral decomposition of a Fokker-Planck equation, employing nonlinear creation and annihilation operators to generate the spectrum which consists of two staggered ladders.

cond-mat.stat-mech↗

arXiv2005

Staggered Ladder Spectra

We exactly solve a Fokker-Planck equation by determining its eigenvalues and eigenfunctions: we construct nonlinear second-order differential operators which act as raising and lowering operators, generating ladder spectra for the odd and even parity states. These are staggered: the odd-even separation differs from even-odd. The Fokker-Planck equation describes, in the limit of weak damping, a generalised Ornstein-Uhlenbeck process where the random force depends upon position as well as time. Our exact solution exhibits anomalous diffusion at short times and a stationary non-Maxwellian momentum distribution.

cond-mat.soft↗

arXiv2005

Clustering in mixing flows

We calculate the Lyapunov exponents for particles suspended in a random three-dimensional flow, concentrating on the limit where the viscous damping rate is small compared to the inverse correlation time. In this limit Lyapunov exponents are obtained as a power series in epsilon, a dimensionless measure of the particle inertia. Although the perturbation generates an asymptotic series, we obtain accurate results from a Pade-Borel summation. Our results prove that particles suspended in an incompressible random mixing flow can show pronounced clustering when the Stokes number is large and we characterise two distinct clustering effects which occur in that limit.

cond-mat.dis-nn↗

arXiv2004

Caustics in turbulent aerosols

Networks of caustics can occur in the distribution of particles suspended in a randomly moving gas. These can facilitate coagulation of particles by bringing them into close proximity, even in cases where the trajectories do not coalesce. We show that the long-time morphology of these caustic patterns is determined by the Lyapunov exponents lambda_1, lambda_2 of the suspended particles, as well as the rate J at which particles encounter caustics. We develop a theory determining the quantities J, lambda_1, lambda_2 from the statistical properties of the gas flow, in the limit of short correlation times.

cond-mat.stat-mech↗