arXiv subjects

Julian Kappler

Publications and source records attributed to Julian Kappler.

14 recordsLinked to original sources

arXiv2024

Short-time Fokker-Planck propagator beyond the Gaussian approximation

We present a perturbation approach to calculate the short-time propagator, or transition density, of the one-dimensional Fokker-Planck equation, to in principle arbitrary order in the time increment. Our approach preserves probability exactly and allows us to evaluate expectation values of analytical observables to in principle arbitrary accuracy; to showcase this, we derive perturbation expansions for the moments of the spatial increment, the finite-time Kramers-Moyal coefficients, and the mean medium entropy production rate. For an explicit multiplicative-noise system with available analytical solution, we validate all our perturbative results. Throughout, we compare our perturbative results to those obtained from the widely used Gaussian approximation of the short-time propagator; we demonstrate that this Gaussian propagator leads to errors that can be many orders of magnitude larger than those resulting from our perturbation approach. Potential applications of our results include parametrizing diffusive stochastic dynamics from observed time series, and sampling path spaces of stochastic trajectories numerically.

cond-mat.stat-mech↗

arXiv2024

Sojourn probabilities in tubes and pathwise irreversibility for Itô processes

The sojourn probability of an Itô diffusion process, i.e. its probability to remain in the tubular neighborhood of a smooth path, is a central quantity in the study of path probabilities. For $N$-dimensional Itô processes with state-dependent full-rank diffusion tensor, we derive a general expression for the sojourn probability in tubes whose radii are small but finite, and fixed by the metric of the ambient Euclidean space. The central quantity in our study is the exit rate at which trajectories leave the tube for the first time. This has an interpretation as a Lagrangian and can be measured directly in experiment, unlike previously defined sojourn probabilities which depend on prior knowledge of the state-dependent diffusivity. We find that while in the limit of vanishing tube radius the ratio of sojourn probabilities for a pair of distinct paths is in general divergent, the same for a path and its time-reversal is always convergent and finite. This provides a pathwise definition of irreversibility for Itô processes that is agnostic to the state-dependence of the diffusivity. For one-dimensional systems we derive an explicit expression for our Lagrangian in terms of the drift and diffusivity, and find that our result differs from previously reported multiplicative-noise Lagrangians. We confirm our result by comparing to numerical simulations, and relate our theory to the Stratonovich Lagrangian for multiplicative noise. For one-dimensional systems, we discuss under which conditions the vanishing-radius limiting ratio of sojourn probabilities for a pair of forward and backward paths recovers the established pathwise entropy production. Finally, we demonstrate for our one-dimensional example system that the most probable tube for a barrier crossing depends sensitively on the tube radius, and hence on the tolerated amount of fluctuations around the smooth reference path.

cond-mat.stat-mech↗

arXiv2024

Resolution dependence of most probable pathways with state-dependent diffusivity

Recent experiments have probed the relative likelihoods of trajectories in stochastic systems by observing survival probabilities within a tube of radius $R$ in spacetime. We measure such probabilities here for a colloidal particle in a corrugated channel, corresponding to a bistable potential with state-dependent diffusivity. In contrast to previous findings for state-independent noise, we find that the most probable pathway changes qualitatively as the tube radius $R$ is altered. We explain this by computing the survival probabilities predicted by overdamped Langevin dynamics. At high enough resolution (small enough $R$), survival probabilities depend solely on diffusivity variations, independent of deterministic forces; finite $R$ corrections yield a generalization of the Onsager-Machlup action. As corollary, ratios of survival probabilities are singular as $R \to 0$, but become regular, and described by the classical Onsager-Machlup action, only in the special case of state-independent noise.

cond-mat.stat-mech↗

arXiv2022

Diffusivity dependence of the transition path ensemble

Transition pathways of stochastic dynamical systems are typically approximated by instantons. Here we show, using a dynamical system containing two competing pathways, that at low-to-intermediate temperatures, instantons can fail to capture the most likely transition pathways. We construct an approximation which includes fluctuations around the instanton and, by comparing with the results of an accurate and efficient path-space Monte Carlo sampling method, find this approximation to hold for a wide range of temperatures. Our work delimits the applicability of large deviation theory and provides methods to probe these limits numerically.

cond-mat.stat-mech↗

arXiv2022

Barrier-crossing times for different non-Markovian friction in well and barrier $-$ A numerical study

We introduce a generalized Langevin model system for different non-Markovian effects in the well and barrier regions of a potential, and use it to numerically study the dependence of the barrier-crossing time. In the appropriate limits, our model interpolates between the theoretical barrier-crossing-time predictions by Grote and Hynes (GH), as well as by Pollak et al., which for a single barrier memory time can differ by several orders of magnitude. Our model furthermore allows to test an analytic rate theory for space-inhomogeneous memory, which disagrees with our numerical results in the long well-memory regime. In this regime, we find that short barrier memory decreases the barrier-crossing time as compared to long barrier memory. This is in contrast with the short well-memory regime, where both our numerical results and GH theory predict an acceleration of the barrier crossing time with increasing barrier memory time. Both effects, the `Markovian-barrier acceleration' and GH `non-Markovian-barrier acceleration' can be understood from a committor analysis. Our model combines finite relaxation times of orthogonal degrees of freedom with a space-inhomogeneous coupling to such degrees, and represents a step towards more realistic modeling of physical reaction coordinates.

physics.chem-ph↗

arXiv2022

Measurement of irreversibility and entropy production via the tubular ensemble

The appealing theoretical measure of irreversibility in a stochastic process, as the ratio of the probabilities of a trajectory and its time reversal, cannot be accessed directly in experiment since the probability of a single trajectory is zero. We regularize this definition by considering, instead, the limiting ratio of probabilities for trajectories to remain in the tubular neighborhood of a smooth path and its time reversal. The resulting pathwise medium entropy production agrees with the formal expression from stochastic thermodynamics, and can be obtained from measurable tube probabilities. Estimating the latter from numerically sampled trajectories for Langevin dynamics yields excellent agreement with theory. By combining our measurement of pathwise entropy production with a Markov Chain Monte Carlo algorithm, we infer the entropy-production distribution for a transition path ensemble directly from short recorded trajectories. Our work enables the measurement of irreversibility along individual paths, and path ensembles, in a model-free manner.

cond-mat.stat-mech↗

arXiv2022

Time-dependent friction effects on vibrational infrared frequencies and line shapes of liquid water

From ab initio simulations of liquid water, the time-dependent friction functions and time-averaged non-linear effective bond potentials for the OH stretch and HOH bend vibrations are extracted. The obtained friction exhibits adiabatic contributions at and below the vibrational time scales, but also much slower non-adiabatic contributions, reflecting homogeneous and inhomogeneous line broadening mechanisms, respectively. Compared to the gas phase, hydration softens both stretch and bend potentials, which by itself would lead to a red-shift of the corresponding vibrational bands. In contrast, non-adiabatic friction contributions cause a spectral blue shift. For the stretch mode, the potential effect dominates and thus a significant red shift when going from gas to the liquid phase results. For the bend mode, potential and non-adiabatic friction effects are of comparable magnitude, so that a slight blue shift results, in agreement with well-known but puzzling experimental findings. The observed line broadening is shown to be roughly equally caused by adiabatic and non-adiabatic friction contributions for both, the stretch and bend modes in liquid water. Thus, the understanding of infrared vibrational frequencies and line shapes is considerably advanced by the quantitative analysis of the time-dependent friction that acts on vibrational modes in liquids.

physics.chem-ph↗

arXiv2021

Efficient Bayesian inference of fully stochastic epidemiological models with applications to COVID-19

Epidemiological forecasts are beset by uncertainties about the underlying epidemiological processes, and the surveillance process through which data are acquired. We present a Bayesian inference methodology that quantifies these uncertainties, for epidemics that are modelled by (possibly) non-stationary, continuous-time, Markov population processes. The efficiency of the method derives from a functional central limit theorem approximation of the likelihood, valid for large populations. We demonstrate the methodology by analysing the early stages of the COVID-19 pandemic in the UK, based on age-structured data for the number of deaths. This includes maximum a posteriori estimates, MCMC sampling of the posterior, computation of the model evidence, and the determination of parameter sensitivities via the Fisher information matrix. Our methodology is implemented in PyRoss, an open-source platform for analysis of epidemiological compartment models.

stat.ME↗

arXiv2021

Experimental measurement of relative path probabilities and stochastic actions

For diffusive stochastic dynamics, the probability to observe any individual trajectory is vanishingly small, making it unclear how to experimentally validate theoretical results for ratios of path probabilities. We provide the missing link between theory and experiment, by establishing a protocol to extract ratios of path probabilities from measured time series. For experiments on a single colloidal particle in a microchannel, we extract both ratios of path probabilities, and the most probable path for a barrier crossing, and find excellent agreement with independently calculated predictions based on the Onsager-Machlup stochastic action. Our experimental results at room temperature are found to be inconsistent with the low-noise Freidlin-Wentzell stochastic action, and we discuss under which circumstances the latter is expected to describe the most probable path. Furthermore, while the experimentally accessible ratio of path probabilities is uniquely determined, the formal path-integral action is known to depend on the time-discretization scheme used for deriving it; we reconcile these two seemingly contradictory facts by careful analysis of the time-slicing derivation of the path integral. Our experimental protocol enables us to probe probability distributions on path space, and allows us to relate theoretical single-trajectory results to measurement.

cond-mat.stat-mech↗

arXiv2020

Stochastic action for tubes: Connecting path probabilities to measurement

The trajectories of diffusion processes are continuous but non-differentiable, and each occurs with vanishing probability. This introduces a gap between theory, where path probabilities are used in many contexts, and experiment, where only events with non-zero probability are measurable. Here we bridge this gap by considering the probability of diffusive trajectories to remain within a tube of small but finite radius around a smooth path. This probability can be measured in experiment, via the rate at which trajectories exit the tube for the first time, thereby establishing a link between path probabilities and physical observables. Considering $N$-dimensional overdamped Langevin dynamics, we show that the tube probability can be obtained theoretically from the solution of the Fokker-Planck equation. Expressing the resulting exit rate as a functional of the path and ordering it as a power series in the tube radius, we identify the zeroth-order term as the Onsager-Machlup stochastic action, thereby elevating it from a mathematical construct to a physical observable. The higher-order terms reveal, for the first time, the form of the finite-radius contributions which account for fluctuations around the path. To demonstrate the experimental relevance of this action functional for tubes, we numerically sample trajectories of Brownian motion in a double-well potential, compute their exit rate, and show an excellent agreement with our analytical results. Our work shows that smooth tubes are surrogates for non-differentiable diffusive trajectories, and provide a direct way of comparing theoretical results on single trajectories, such as path-wise definitions of irreversibility, to measurement.

cond-mat.stat-mech↗

arXiv2020

Inference, prediction and optimization of non-pharmaceutical interventions using compartment models: the PyRoss library

PyRoss is an open-source Python library that offers an integrated platform for inference, prediction and optimisation of NPIs in age- and contact-structured epidemiological compartment models. This report outlines the rationale and functionality of the PyRoss library, with various illustrations and examples focusing on well-mixed, age-structured populations. The PyRoss library supports arbitrary structured models formulated stochastically (as master equations) or deterministically (as ODEs) and allows mid-run transitioning from one to the other. By supporting additional compartmental subdivision ad libitum, PyRoss can emulate time-since-infection models and allows medical stages such as hospitalization or quarantine to be modelled and forecast. The PyRoss library enables fitting to epidemiological data, as available, using Bayesian parameter inference, so that competing models can be weighed by their evidence. PyRoss allows fully Bayesian forecasts of the impact of idealized NPIs by convolving uncertainties arising from epidemiological data, model choice, parameters, and intrinsic stochasticity. Algorithms to optimize time-dependent NPI scenarios against user-defined cost functions are included. PyRoss's current age-structured compartment framework for well-mixed populations will in future reports be extended to include compartments structured by location, occupation, use of travel networks and other attributes relevant to assessing disease spread and the impact of NPIs. We argue that such compartment models, by allowing social data of arbitrary granularity to be combined with Bayesian parameter estimation for poorly-known disease variables, could enable more powerful and robust prediction than other approaches to detailed epidemic modelling. We invite others to use the PyRoss library for research to address today's COVID-19 crisis, and to plan for future pandemics.

q-bio.PE↗

arXiv2020

Molecular interpretation of the non-Newtonian viscoelastic behavior of liquid water at high frequencies

Using classical as well as ab-initio molecular dynamics simulations, we calculate the frequency-dependent shear viscosity of pure water and water-glycerol mixtures. In agreement with recent experiments, we find deviations from Newtonian-fluid behavior in the THz regime. Based on an extension of the Maxwell model, we introduce a viscoelastic model to describe the observed viscosity spectrum of pure water. We find four relaxation modes in the spectrum which we attribute to i) hydrogen-bond network topology changes, ii) hydrogen-bond stretch vibrations of water pairs, iii) collective vibrations of water molecule triplets, and iv) librational excitations of individual water molecules. Our model quantitatively describes the viscoelastic response of liquid water on short timescales, where the hydrodynamic description via a Newtonian-fluid model breaks down.

physics.flu-dyn↗

arXiv2019

Non-Markovian barrier crossing with two-time-scale memory is dominated by the faster memory component

We investigate non-Markovian barrier-crossing kinetics of a massive particle in one dimension in the presence of a memory function that is the sum of two exponentials with different memory times $τ_1$ and $τ_2$. Our Langevin simulations for the special case where both exponentials contribute equally to the total friction show that the barrier crossing time becomes independent of the longer memory time if at least one of the two memory times is larger than the intrinsic diffusion time. When we associate memory effects with coupled degrees of freedom that are orthogonal to a one-dimensional reaction coordinate, this counterintuitive result shows that the faster orthogonal degrees of freedom dominate barrier-crossing kinetics in the non-Markovian limit and that the slower orthogonal degrees become negligible, quite contrary to the standard time-scale separation assumption and with important consequences for the proper setup of coarse-graining procedures in the non-Markovian case. By asymptotic matching and symmetry arguments, we construct a crossover formula for the barrier crossing time that is valid for general multi-exponential memory kernels. This formula can be used to estimate barrier-crossing times for general memory functions for high friction, i.e. in the overdamped regime, as well as for low friction, i.e. in the inertial regime. Typical examples where our results are important include protein folding in the high-friction limit and chemical reactions such as proton-transfer reactions in the low-friction limit.

physics.chem-ph↗

arXiv2017

Nonlinear fractional waves at elastic interfaces

We derive the nonlinear fractional surface wave equation that governs compression waves at an interface that is coupled to a viscous bulk medium. The fractional character of the differential equation comes from the fact that the effective thickness of the bulk layer that is coupled to the interface is frequency dependent. The nonlinearity arises from the nonlinear dependence of the interface compressibility on the local compression, which is obtained from experimental measurements and reflects a phase transition at the interface. Numerical solutions of our nonlinear fractional theory reproduce several experimental key features of surface waves in phospholipid monolayers at the air-water interface without freely adjustable fitting parameters. In particular, the propagation length of the surface wave abruptly increases at a threshold excitation amplitude. The wave velocity is found to be of the order of 40 cm/s both in experiments and theory and slightly increases as a function of the excitation amplitude. Nonlinear acoustic switching effects in membranes are thus shown to arise purely based on intrinsic membrane properties, namely the presence of compressibility nonlinearities that accompany phase transitions at the interface.

physics.flu-dyn↗