arXiv subjects

Timo Aspelmeier

Publications and source records attributed to Timo Aspelmeier.

15 recordsLinked to original sources

arXiv2019

Statistical Molecule Counting in Super-Resolution Fluorescence Microscopy: Towards Quantitative Nanoscopy

Super-resolution microscopy is rapidly gaining importance as an analytical tool in the life sciences. A compelling feature is the ability to label biological units of interest with fluorescent markers in living cells and to observe them with considerably higher resolution than conventional microscopy permits. The images obtained this way, however, lack an absolute intensity scale in terms of numbers of fluorophores observed. We provide an elaborate model to estimate this information from the raw data. To this end we model the entire process of photon generation in the fluorophore, their passage trough the microscope, detection and photo electron amplification in the camera, and extraction of time series from the microscopic images. At the heart of these modeling steps is a careful description of the fluorophore dynamics by a novel hidden Markov model that operates on two time scales (HTMM). Besides the fluorophore number, information about the kinetic transition rates of the fluorophore's internal states is also inferred during estimation. We comment on computational issues that arise when applying our model to simulated or measured fluorescence traces and illustrate our methodology on simulated data.

stat.AP↗

arXiv2018

Posterior analysis of $n$ in the binomial $(n,p)$ problem with both parameters unknown -- with applications to quantitative nanoscopy

Estimation of the population size $n$ from $k$ i.i.d.\ binomial observations with unknown success probability $p$ is relevant to a multitude of applications and has a long history. Without additional prior information this is a notoriously difficult task when $p$ becomes small, and the Bayesian approach becomes particularly useful. For a large class of priors, we establish posterior contraction and a Bernstein-von Mises type theorem in a setting where $p\rightarrow0$ and $n\rightarrow\infty$ as $k\to\infty$. Furthermore, we suggest a new class of Bayesian estimators for $n$ and provide a comprehensive simulation study in which we investigate their performance. To showcase the advantages of a Bayesian approach on real data, we also benchmark our estimators in a novel application from super-resolution microscopy.

math.ST↗

arXiv2015

Local Linear Convergence of the ADMM/Douglas--Rachford Algorithms without Strong Convexity and Application to Statistical Imaging

We consider the problem of minimizing the sum of a convex function and a convex function composed with an injective linear mapping. For such problems, subject to a coercivity condition at fixed points of the corresponding Picard iteration, iterates of the alternating directions method of multipliers converge locally linearly to points from which the solution to the original problem can be computed. Our proof strategy uses duality and strong metric subregularity of the Douglas--Rachford fixed point mapping. Our analysis does not require strong convexity and yields error bounds to the set of model solutions. We show in particular that convex piecewise linear-quadratic functions naturally satisfy the requirements of the theory, guaranteeing eventual linear convergence of both the Douglas--Rachford algorithm and the alternating directions method of multipliers for this class of objectives under mild assumptions on the set of fixed points. We demonstrate this result on quantitative image deconvolution and denoising with multiresolution statistical constraints.

math.OC↗

arXiv2010

Hydrodynamic Correlation Functions of a Driven Granular Fluid in Steady State

We study a homogeneously driven granular fluid of hard spheres at intermediate volume fractions and focus on time-delayed correlation functions in the stationary state. Inelastic collisions are modeled by incomplete normal restitution, allowing for efficient simulations with an event-driven algorithm. The incoherent scattering function, F_incoh(q,t), is seen to follow time-density superposition with a relaxation time that increases significantly as volume fraction increases. The statistics of particle displacements is approximately Gaussian. For the coherent scattering function S(q,omega) we compare our results to the predictions of generalized fluctuating hydrodynamics which takes into account that temperature fluctuations decay either diffusively or with a finite relaxation rate, depending on wave number and inelasticity. For sufficiently small wave number q we observe sound waves in the coherent scattering function S(q,omega) and the longitudinal current correlation function C_l(q,omega). We determine the speed of sound and the transport coefficients and compare them to the results of kinetic theory.

cond-mat.stat-mech↗

arXiv2009

Partitioning of energy in highly polydisperse granular gases

A highly polydisperse granular gas is modeled by a continuous distribution of particle sizes, a, giving rise to a corresponding continuous temperature profile, T(a), which we compute approximately, generalizing previous results for binary or multicomponent mixtures. If the system is driven, it evolves towards a stationary temperature profile, which is discussed for several driving mechanisms in dependence on the variance of the size distribution. For a uniform distribution of sizes, the stationary temperature profile is nonuniform with either hot small particles (constant force driving) or hot large particles (constant velocity or constant energy driving). Polydispersity always gives rise to non-Gaussian velocity distributions. Depending on the driving mechanism the tails can be either overpopulated or underpopulated as compared to the molecular gas. The deviations are mainly due to small particles. In the case of free cooling the decay rate depends continuously on particle size, while all partial temperatures decay according to Haff's law. The analytical results are supported by event driven simulations for a large, but discrete number of species.

cond-mat.soft↗

arXiv2009

Dilute Wet Granulates: Nonequilibrium Dynamics and Structure Formation

We investigate a gas of wet granular particles, covered by a thin liquid film. The dynamic evolution is governed by two-particle interactions, which are mainly due to interfacial forces in contrast to dry granular gases. When two wet grains collide, a capillary bridge is formed and stays intact up to a certain distance of withdrawal when the bridge ruptures, dissipating a fixed amount of energy. A freely cooling system is shown to undergo a nonequillibrium dynamic phase transition from a state with mainly single particles and fast cooling to a state with growing aggregates, such that bridge rupture becomes a rare event and cooling is slow. In the early stage of cluster growth, aggregation is a self-similar process with a fractal dimension of the aggregates approximately equal to D_f ~ 2. At later times, a percolating cluster is observed which ultimately absorbs all the particles. The final cluster is compact on large length scales, but fractal with D_f ~ 2 on small length scales.

cond-mat.soft↗

arXiv2009

Cooling and aggregation in wet granulates

Wet granular materials are characterized by a defined bond energy in their particle interaction such that breaking a bond implies an irreversible loss of a fixed amount of energy. Associated with the bond energy is a nonequilibrium transition, setting in as the granular temperature falls below the bond energy. The subsequent aggregation of particles into clusters is shown to be a self-similar growth process with a cluster size distribution that obeys scaling. In the early phase of aggregation the clusters are fractals with D_f=2, for later times we observe gelation. We use simple scaling arguments to derive the temperature decay in the early and late stages of cooling and verify our results with event-driven simulations.

cond-mat.soft↗

arXiv2008

Long-time tails and cage effect in driven granular fluids

We study the velocity autocorrelation function (VACF) of a driven granular fluid in the stationary state in 3 dimensions. As the critical volume fraction of the glass transition in the corresponding elastic system is approached, we observe pronounced cage effects in the VACF as well as a strong decrease of the diffusion constant. At moderate densities the VACF is shown to decay algebraically in time (t^{-3/2}) like in a molecular fluid, as long as the driving conserves momentum locally.

cond-mat.stat-mech↗

arXiv2006

Microscopic dynamics of thin hard rods

Based on the collision rules for hard needles we derive a hydrodynamic equation that determines the coupled translational and rotational dynamics of a tagged thin rod in an ensemble of identical rods. Specifically, based on a Pseudo-Liouville operator for binary collisions between rods, the Mori-Zwanzig projection formalism is used to derive a continued fraction representation for the correlation function of the tagged particle's density, specifying its position and orientation. Truncation of the continued fraction gives rise to a generalised Enskog equation, which can be compared to the phenomenological Perrin equation for anisotropic diffusion. Only for sufficiently large density do we observe anisotropic diffusion, as indicated by an anisotropic mean square displacement, growing linearly with time. For lower densities, the Perrin equation is shown to be an insufficient hydrodynamic description for hard needles interacting via binary collisions. We compare our results to simulations and find excellent quantitative agreement for low densities and qualtitative agreement for higher densities.

cond-mat.soft↗

arXiv2001

Random walks with imperfect trapping in the decoupled-ring approximation

We investigate random walks on a lattice with imperfect traps. In one dimension, we perturbatively compute the survival probability by reducing the problem to a particle diffusing on a closed ring containing just one single trap. Numerical simulations reveal this solution, which is exact in the limit of perfect traps, to be remarkably robust with respect to a significant lowering of the trapping probability. We demonstrate that for randomly distributed traps, the long-time asymptotics of our result recovers the known stretched exponential decay. We also study an anisotropic three-dimensional version of our model, where for sufficiently large transverse diffusion the system is described by the mean-field kinetics. We discuss possible applications of some of our findings to the decay of excitons in semiconducting organic polymer materials, and emphasize the crucial influence of the spatial trap distribution on the kinetics.

cond-mat.stat-mech↗

arXiv2000

Stress Relaxation of Near-Critical Gels

The time-dependent stress relaxation for a Rouse model of a crosslinked polymer melt is completely determined by the spectrum of eigenvalues of the connectivity matrix. The latter has been computed analytically for a mean-field distribution of crosslinks. It shows a Lifshitz tail for small eigenvalues and all concentrations below the percolation threshold, giving rise to a stretched exponential decay of the stress relaxation function in the sol phase. At the critical point the density of states is finite for small eigenvalues, resulting in a logarithmic divergence of the viscosity and an algebraic decay of the stress relaxation function. Numerical diagonalization of the connectivity matrix supports the analytical findings and has furthermore been applied to cluster statistics corresponding to random bond percolation in two and three dimensions.

cond-mat.soft↗

arXiv2000

Free cooling of particles with rotational degrees of freedom

Free cooling of granular materials is analyzed on the basis of a pseudo-Liouville operator. Exchange of translational and rotational energy requires surface roughness for spherical grains, but occurs for non-spherical grains, like needles, even if they are perfectly smooth. Based on the assumption of a homogeneous cooling state, we derive an approximate analytical theory. It predicts that cooling of both rough spheres and smooth needles proceeds in two stages: An exponentially fast decay to a state with stationary ratio of translational and rotational energy and a subsequent algebraic decay of the total energy. These results are confirmed by simulations for large systems of moderate density. For higher densities, we observe deviations from the homogeneous state as well as large-scale structures in the velocity field. We study non-Gaussian distributions of the momenta perturbatively and observe a breakdown of the expansion for particular values of surface roughness and normal restitution.

cond-mat.stat-mech↗

arXiv1999

Universal aspects of vacancy-mediated disordering dynamics: the effect of external fields

We investigate the disordering of an initially phase-segregated binary alloy, due to a highly mobile defect which couples to an electric or gravitational field. Using both mean-field and Monte Carlo methods, we show that the late stages of this process exhibit dynamic scaling, characterized by a set of exponents and scaling functions. A new scaling variable emerges, associated with the field. While the scaling functions carry information about the field and the boundary conditions, the exponents are universal. They can be computed analytically, in excellent agreement with simulation results.

cond-mat.stat-mech↗

arXiv1999

Granular cooling of hard needles

We have developed a kinetic theory of hard needles undergoing binary collisions with loss of energy due to normal and tangential restitution. In addition, we have simulated many particle systems of granular hard needles. The theory, based on the assumption of a homogeneous cooling state, predicts that granular cooling of the needles proceeds in two stages: An exponential decay of the initial configuration to a state where translational and rotational energies take on a time independent ratio (not necessarily unity), followed by an algebraic decay of the total kinetic energy $\sim t^{-2}$. The simulations support the theory very well for low and moderate densities. For higher densities, we have observed the onset of the formation of clusters and shear bands.

cond-mat.stat-mech↗

arXiv1997

Cooling dynamics of a dilute gas of inelastic rods: a many particle simulation

We present results of simulations for a dilute gas of inelastically colliding particles. Collisions are modelled as a stochastic process, which on average decreases the translational energy (cooling), but allows for fluctuations in the transfer of energy to internal vibrations. We show that these fluctuations are strong enough to suppress inelastic collapse. This allows us to study large systems for long times in the truely inelastic regime. During the cooling stage we observe complex cluster dynamics, as large clusters of particles form, collide and merge or dissolve. Typical clusters are found to survive long enough to establish local equilibrium within a cluster, but not among different clusters. We extend the model to include net dissipation of energy by damping of the internal vibrations. Inelatic collapse is avoided also in this case but in contrast to the conservative system the translational energy decays according to the mean field scaling law, E(t)\propto t^{-2}, for asymptotically long times.

cond-mat.stat-mech↗