Position Probability Density Research Articles

The OU$^2$ process: Characterising dissipative confinement in noisy traps

Abstract The Ornstein-Uhlenbeck (OU) process describes the dynamics of Brownian particles in a confining harmonic potential, thereby constituting the paradigmatic model of overdamped, mean-reverting Langevin dynamics. Despite its widespread applicability, this model falls short when describing physical systems where the confining potential is itself subjected to stochastic fluctuations. However, such stochastic fluctuations generically emerge in numerous situations, including in the context of colloidal manipulation by optical tweezers, leading to inherently out-of-equilibrium trapped dynamics. To explore the consequences of stochasticity at this level, we introduce a natural extension of the OU process, in which the stiffness of the harmonic potential is itself subjected to OU-like fluctuations. We call this model the OU$^2$ process. We examine its statistical, dynamic, and thermodynamic properties through a combination of analytical and numerical methods. Importantly, we show that the probability density for the particle position presents power-law tails, in contrast to the Gaussian decay of the standard OU process. In turn, this causes the trapping behavior, extreme value statistics, first passage statistics, and entropy production of the OU$^2$ process to differ qualitatively from their standard OU counterpart. Due to the wide applicability of the standard OU process and of the proposed OU$^2$ generalisation, our study sheds light on the peculiar properties of stochastic dynamics in random potentials and lays the foundation for the refined analysis of the dynamics and thermodynamics of numerous experimental systems.

Elastic interactions compete with persistent cell motility to drive durotaxis

Many animal cells that crawl on extracellular substrates exhibit durotaxis, i.e., directed migration toward stiffer substrate regions. This has implications in several biological processes including tissue development and tumor progression. Here, we introduce a phenomenological model for single-cell durotaxis that incorporates both elastic deformation-mediated cell-substrate interactions and the stochasticity of cell migration. Our model is motivated by a key observation in an early demonstration of durotaxis: a single, contractile cell at a sharp interface between a softer and a stiffer region of an elastic substrate reorients and migrates toward the stiffer region. We model migrating cells as self-propelling, persistently motile agents that exert contractile traction forces on their elastic substrate. The resulting substrate deformations induce elastic interactions with mechanical boundaries, captured by an elastic potential. The dynamics is determined by two crucial parameters: the strength of the cellular traction-induced boundary elastic interaction (A), and the persistence of cell motility (Pe). Elastic forces and torques resulting from the potential orient cells perpendicular (parallel) to the boundary and accumulate (deplete) them at the clamped (free) boundary. Thus, a clamped boundary induces an attractive potential that drives durotaxis, while a free boundary induces a repulsive potential that prevents antidurotaxis. By quantifying the steady-state position and orientation probability densities, we show how the extent of accumulation (depletion) depends on the strength of the elastic potential and motility. We compare and contrast crawling cells with biological microswimmers and other synthetic active particles, where accumulation at confining boundaries is well known. We define metrics quantifying boundary accumulation and durotaxis, and present a phase diagram that identifies three possible regimes: durotaxis, and adurotaxis with and without motility-induced accumulation at the boundary. Overall, our model predicts how durotaxis depends on cell contractility and motility, successfully explains some previous observations, and provides testable predictions to guide future experiments.

On the Construction of Conditional Probability Densities in the Brownian and Compound Poisson Filtrations

In this paper, we construct supermartingales valued in [0,1] as solutions of an appropriate stochastic differential equation on a given reference filtration generated by either a Brownian motion or a compound Poisson process. Then, by means of the results contained in [M. Jeanblanc and S. Song, Stochastic Processes Appl. 121 (2011) 1389–1410], it is possible to construct an associated random time on some extended probability space admitting such a given supermartingale as conditional survival process and we shall check that this construction (with a particular choice of supermartingale) implies that Jacod’s equivalence hypothesis, that is, the existence of a family of strictly positive conditional probability densities for the random times with respect to the reference filtration, is satisfied. We use the components of the multiplicative decomposition of the constructed supermartingales to provide explicit expressions for the conditional probability densities of the random times on the Brownian and compound Poisson filtrations.

Dynamics of switching processes: general results and applications in intermittent active motion.

Systems switching between different dynamical phases is a ubiquitous phenomenon. The general understanding of such a process is limited. To this end, we present a general expression that captures fluctuations of a system exhibiting a switching mechanism. Specifically, we obtain an exact expression of the Laplace-transformed characteristic function of the particle's position. Then, the characteristic function is used to compute the effective diffusion coefficient of a system performing intermittent dynamics. Furthermore, we employ two examples: (1) generalized run-and-tumble active particle, and (2) an active particle switching its dynamics between generalized active run-and-tumble motion and passive Brownian motion. In each case, explicit computations of the spatial cumulants are presented. Our findings reveal that the particle's position probability density function exhibit rich behaviours due to intermittent activity. Numerical simulations confirm our findings.

Deep learning-enhanced characterization of bubble dynamics in proton exchange membrane water electrolyzers.

The ever-increasing utility of imaging technology in proton exchange membrane water electrolyzer research raises the demand for rapid and precise image analysis. In particular, for optical video recordings, the challenge primarily lies in the large number of frames that impede the delineation of bubble dynamics with standard methods. In order to address this problem, the present study supports the automation of data analysis to facilitate swift, comprehensive, and measurable insights from captured imagery. We present a deep learning-based framework to perform high-throughput analyses of bubble dynamics using optical images of proton exchange membrane water electrolyzers. Leveraging a relatively small annotated imaging dataset of just 35 images, various configurations of the U-Net architecture were trained to perform bubble segmentation tasks. The best model achieved a precision of 95%, a recall of 78%, and an F1-score of 86% on the validation set. Subsequent to segmentation, the methodology enabled the rapid extraction of parameters such as time-resolved bubble area, size distributions, bubble position probability density, and individual bubble shape analytics. The findings underscore the potential of deep learning to enhance the analysis of polymer electrolyte membrane water electrolyzer imaging, offering a path toward more efficient and informative evaluations in electrochemical research.

Random walk on spheres method for solving anisotropic transient diffusion problems and flux calculations

Abstract The Random Walk on Spheres (RWS) algorithm for solving anisotropic transient diffusion problems based on a stochastic learning procedure for calculation of the exit position of the anisotropic diffusion process on a sphere is developed. Direct generalization of the Random Walk on Spheres method to anisotropic diffusion equations is not possible, therefore, we have numerically calculated the probability density for the exit position on a sphere. The first passage time is then represented explicitly. The method can easily be implemented to solve diffusion problems with spatially varying diffusion coefficients for complicated three-dimensional domains. Particle tracking algorithm is highly efficient for calculation of fluxes to boundaries. We apply the developed algorithm for solving an exciton transport in a semiconductor material with a threading dislocation where the measured functions are the exciton fluxes to the semiconductor’s substrate and on the dislocation surface.

A random error estimation method for satellites being launched into orbit based on ellipsoid theory

Accurate and fast random error estimation is essential to improve satellite control. In this paper, the random error of the satellite being launched into orbit is explored. According to the uncertainty matrix of six satellite orbit parameters associated with the project, the uncertainty of the orbit position in the three axes of the geocentric inertial frame of reference is calculated via the propagation rate of the covariance matrix. Then, the random error probability density of the satellite position is constructed for normal distribution. The surface of the error with the same probability density is approximated as an ellipsoid, the size is deduced along with the ellipsoid's shape and direction, and the satellite's spatial position probability is obtained. Monte Carlo method and STK software were employed to simulate low-earth orbit micro-nano satellite launching into the same orbital position. It is confirmed that the results obtained via the theoretical model are in line with practice. Therefore, the ellipsoid theory proposed in this paper can be used to describe the satellite random position error.

Stochastic Quantum Probability and Subatomic Particle Experiment Design

This article is the third of a trilogy of articles on the nature of probability in quantum mechanics. The first article [1] began by noting that superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into the position probability density of an atomic particle, which happens nowhere else in probability theory. It went on to also note that there is an unexplained coincidence in quantum mechanics in that the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables and went on to examine whether there could be an archetypical variable, in the Platonic sense of true form, behind quantum probability that would reconcile quantum probability with classic probability. This examination found such a variable, which can encompass both local and nonlocal quantum events. The second article [2] provided evidence that quantum probability has such a stochastic nature. This evidence was based on the number of electrons that need to be sent through a two-slit interferometer to gain a clear pattern of self-interference, which when compared with the number that would be expected to be sufficient in order for the position probability distribution of the self-interference wavefunction to take clear shape suggests that there is more variability present than that described by the formulation of quantum mechanics, which implies the presence of an underlying and as yet unrecognized physical process. This final article completes the trilogy by considering how a key aspect of experimental design would be affected by the increased variability that would be present if quantum probability is itself stochastic in the manner suggested in the previous two articles.

Close encounters of the sticky kind: Brownian motion at absorbing boundaries.

Encounter-based models of diffusion provide a probabilistic framework for analyzing the effects of a partially absorbing reactive surface, in which the probability of absorption depends upon the amount of surface-particle contact time. In this paper we develop a class of encounter-based models that deal with absorption at sticky boundaries. Sticky boundaries occur in a diverse range of applications, including cell biology, colloidal physics, finance, and human crowd dynamics. They also naturally arise in active matter, where confined active particles tend to spontaneously accumulate at boundaries even in the absence of any particle-particle interactions. We begin by constructing a one-dimensional encounter-based model of sticky Brownian motion (BM), which is based on the zero-range limit of nonsticky BM with a short-range attractive potential well near the origin. In this limit, the boundary-contact time is given by the amount of time (occupation time) that the particle spends at the origin. We calculate the joint probability density or propagator for the particle position and the occupation time, and then identify an absorption event as the first time that the occupation time crosses a randomly generated threshold. We illustrate the theory by considering diffusion in a finite interval with a partially absorbing sticky boundary at one end. We show how various quantities, such as the mean first passage time (MFPT) for single-particle absorption and the relaxation to steady state at the multiparticle level, depend on moments of the random threshold distribution. Finally, we determine how sticky BM can be obtained by taking a particular diffusion limit of a sticky run-and-tumble particle (RTP).

Encounter-based model of a run-and-tumble particle II: absorption at sticky boundaries

In this paper we develop an encounter-based model of a run-and-tumble particle (RTP) confined to a finite interval with partially absorbing, sticky boundaries at both ends. We assume that the particle switches between two constant velocity states ±v at a rate α. Whenever the particle hits a boundary, it becomes stuck by pushing on the boundary until either a tumble event reverses the swimming direction or it is permanently absorbed. We formulate the absorption process by identifying the first passage time (FPT) for absorption with the event that the time A(t) spent attached to either wall up to time t (the occupation time) crosses some random threshold . Taking to be an exponential distribution, , we show that the joint probability density for particle position X(t) and velocity state satisfies a well-defined boundary value problem (BVP) with κ 0 representing a constant absorption rate. The solution of this BVP determines the so-called occupation time propagator, which is the joint probability density for the triplet . The propagator is then used to incorporate more general models of absorption based on non-exponential (non-Markovian) distributions . We illustrate the theory by calculating the mean FPT and splitting probabilities for absorption. We also show how our previous results for partially absorbing, non-sticky boundaries can be recovered in an appropriate limit. Absorption now depends on the number of collisions of the RTP with the boundary. Finally, we extend the theory by taking absorption to depend on the individual occupation times at the two ends.

Two New Methods in Stochastic Electrodynamics for Analyzing the Simple Harmonic Oscillator and Possible Extension to Hydrogen

The position probability density function is calculated for a classical electric dipole harmonic oscillator bathed in zero-point plus Planckian electromagnetic fields, as considered in the physical theory of stochastic electrodynamics (SED). The calculations are carried out via two new methods. They start from a general probability density expression involving the formal integration over all probabilistic values of the Fourier coefficients describing the stochastic radiation fields. The first approach explicitly carries out all these integrations; the second approach shows that this general probability density expression satisfies a partial differential equation that is readily solved. After carrying out these two fairly long analyses and contrasting them, some examples are provided for extending this approach to quantities other than position, such as the joint probability density distribution for positions at different times, and for position and momentum. This article concludes by discussing the application of this general probability density expression to a system of great interest in SED, namely, the classical model of hydrogen.

Analytical formulas for the probability density of aircraft positions during flights in the piloting area

Analytical formulas for the probability densities of possible aircraft positions during flights in the piloting area are presented. Two-way exponential and normal navigation deviations of the aircraft from the selected track in the lateral direction with respect to the direction of movement in the piloting area are considered. Keywords aircraft, probability density of aircraft position, Gaussian distribution, DE distribution

Encounter-based model of a run-and-tumble particle

In this paper we extend the encounter-based model of diffusion-mediated surface absorption to the case of an unbiased run-and-tumble particle (RTP) confined to a finite interval [0, L] and switching between two constant velocity states ±v at a rate α. The encounter-based formalism is motivated by the observation that various surface-based reactions are better modeled in terms of a reactivity that is a function of the amount of time that a particle spends in a neighborhood of an absorbing surface, which is specified by a functional known as the boundary local time. The effects of surface reactions are taken into account by identifying the first passage time (FPT) for absorption with the event that the local time crosses some random threshold . In the case of a Brownian particle, the local time ℓ(t) is a continuous non-decreasing function of the time t. Taking to be an exponential distribution, , is equivalent to imposing a Robin boundary condition with a constant rate of absorption κ 0. One major difference in the encounter-based model of an RTP is that the boundary local time ℓ(t) is a now a discrete random variable that counts the number of collisions of the RTP with the boundary. Given this modification, we show that in the case of a geometric distribution Ψ(ℓ) = z ℓ , z = 1/(1 + κ 0/v), we recover the RTP analog of the Robin boundary condition. This allows us to solve the boundary value problem (BVP) for the joint probability density for particle position and the local time, and thus incorporate more general models of absorption based on non-geometric distributions Ψ(ℓ). We illustrate the theory by calculating the mean FPT (MFPT) for absorption at x = L given a totally reflecting boundary at x = 0. We also determine the splitting probability for absorption at x = L when the boundary at x = 0 is totally absorbing.

On the scalar turbulent/turbulent interface of axisymmetric jets

The effect of a zero-mean-flow turbulent ambient on the geometry of the turbulent/turbulent interface (TTI) of an axisymmetric jet is investigated and compared with the traditional turbulent/non-turbulent interface (TNTI). The ambient turbulence is generated using a random jet array. Orthogonal cross-sections of the jet subjected to different levels of ambient turbulence intensities and length scales are captured using planar laser-induced fluorescence. The enhanced radial transport of concentration from the jet core towards the interface and the increased scalar fluctuations within the jet caused by the ambient turbulence result in steeper mean and root-mean-square scalar conditional jumps across the TTI layer compared with those of the TNTI, respectively. When compared with the quiescent ambient, the mean effect of background turbulence is to stretch and corrugate the interface surface area as evident from the wider probability density of the interface radial position, the lowered occurrence of zero-curvature surface elements, the greater misalignment between the radial and normal unit vectors of the interface, the increased tortuosity and the increased magnitude of the fractal exponent.

Analytical Investigations into Anomalous Diffusion Driven by Stress Redistribution Events: Consequences of Lévy Flights

This research is concerned with developing a generalised diffusion equation capable of describing diffusion processes driven by underlying stress-redistributing type events. The work utilises the development of an appropriate continuous time random walk framework as a foundation to consider a new generalised diffusion equation. While previous work has explored the resulting generalised diffusion equation for jump-timings motivated by stick-slip physics, here non-Gaussian probability distributions of the jump displacements are also considered, specifically Lévy flights. This work illuminates several features of the analytic solution to such a generalised diffusion equation using several known properties of the Fox H function. Specifically demonstrated are the temporal behaviour of the resulting position probability density function, and its normalisation. The reduction of the proposed form to expected known solutions upon the insertion of simplifying parameter values, as well as a demonstration of asymptotic behaviours, is undertaken to add confidence to the validity of this equation. This work describes the analytical solution of such a generalised diffusion equation for the first time, and additionally demonstrates the capacity of the Fox H function and its properties in solving and studying generalised Fokker–Planck equations.

Narrow capture problem: An encounter-based approach to partially reactive targets.

A general topic of current interest is the analysis of diffusion problems in singularly perturbed domains with small interior targets or traps (the narrow capture problem). One major application is to intracellular diffusion, where the targets typically represent some form of reactive biochemical substrate. Most studies of the narrow capture problem treat the target boundaries as totally absorbing (Dirichlet), that is, the chemical reaction occurs immediately on first encounter between particle and target surface. In this paper, we analyze the three-dimensional narrow capture problem in the more realistic case of partially reactive target boundaries. We begin by considering classical Robin boundary conditions. Matching inner and outer solutions of the single-particle probability density, we derive an asymptotic expansion of the Laplace transformed flux into each reactive surface in powers of ε, where ερ is a given target size. In turn, the fluxes determine the splitting probabilities for target absorption. We then extend our analysis to more general types of reactive targets by combining matched asymptotic analysis with an encounter-based formulation of diffusion-mediated surface reactions. That is, we derive an asymptotic expansion of the joint probability density for particle position and the so-called boundary local time, which characterizes the amount of time that a Brownian particle spends in the neighborhood of a point on a totally reflecting boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. Robin boundary conditions are recovered in the special case of an exponential law for the stopping local times. Finally, we illustrate the theory by exploring how the leading-order contributions to the splitting probabilities depend on the choice of surface reactions. In particular, we show that there is an effective renormalization of the target radius of the form ρ→ρ-Ψ[over ̃](1/ρ), where Ψ[over ̃] is the Laplace transform of the stopping local time distribution.

Superdiffusion in self-reinforcing run-and-tumble model with rests.

This paper introduces a run-and-tumble model with self-reinforcing directionality and rests. We derive a single governing hyperbolic partial differential equationfor the probability density of random-walk position, from which we obtain the second moment in the long-time limit. We find the criteria for the transition between superdiffusion and diffusion caused by the addition of a rest state. The emergence of superdiffusion depends on both the parameter representing the strength of self-reinforcement and the ratio between mean running and resting times. The mean running time must be at least 2/3 of the mean resting time for superdiffusion to be possible. Monte Carlo simulations validate this theoretical result. This work demonstrates the possibility of extending the telegrapher's (or Cattaneo) equationby adding self-reinforcing directionality so that superdiffusion occurs even when rests are introduced.

Short-Term Collision Probability Algorithm for Parallelepiped-Shaped Satellites

The probability of collision of two solid satellites, one of which is a parallelepiped and the other is a parallelepiped or a small sphere, is computed under the assumptions of short-term encounter. The probability is given by the integral of the relative position probability density over a region of the collision plane. This region is the Minkowski sum of the projections of the objects’ shapes onto the collision plane. The integration is the only step that cannot be done analytically. When the position probability densities are Gaussians, a very fast and accurate algorithm found in the literature can be applied after several transformations have been performed to the integration region. A comparison against several current methods has been included. When a parallelepiped is involved, the presented method is advantageous with respect to the existing ones.

Signal preservation of exomoon transits during light curve folding

In the search for moons around extrasolar planets (exomoons), astronomers are confronted with a stunning observation. Although 3400 of the 4500 exoplanets were discovered with the transit method and although there are well over 25 times as many moons than planets known in the Solar System (two of which are larger than Mercury), no exomoon has been discovered to date. In the search for exoplanet transits, stellar light curves are usually phase-folded over a range of trial epochs and periods. This approach, however, is not applicable in a straightforward manner to exomoons. Planet-moon transits either have to be modeled in great detail (including their orbital dynamics, mutual eclipses, etc.), which is computationally expensive, or key simplifications have to be assumed in the modeling. One such simplification is to search for moon transits outside of the planetary transits. The question we address in this report is how much in-transit data of an exomoon remains uncontaminated by the near-simultaneous transits of its host planet. We develop an analytical framework based on the probability density of the sky-projected apparent position of an exomoon relative to its planet and test our results with a numerical planet-moon transit simulator. For exomoons with planet-moon orbital separations similar to the Galilean moons, we find that only a small fraction of their in-transit data is uncontaminated by planetary transits: 14% for Io, 20% for Europa, 42% for Ganymede, and 73% for Callisto. The signal-to-noise ratio (S/N) of an out-of-planetary-transit folding technique is reduced compared to a full photodynamical model to about 38% (Io), 45% (Europa), 65% (Ganymede), and 85% (Callisto), respectively. For the Earth’s Moon, we find an uncontaminated data fraction of typically just 18% and a resulting S/N reduction to 42%. These values are astonishingly small and suggest that the gain in speed for any exomoon transit search algorithm that ignores the planetary in-transit data comes at the heavy price of losing a substantial fraction of what is supposedly a tiny signal in the first place. We conclude that photodynamical modeling of the entire light curve has substantial, and possibly essential, advantages over folding techniques of exomoon transits outside the planetary transits, in particular for small exomoons comparable to those of the Solar System.

Colossal Brownian yet non-Gaussian diffusion in a periodic potential: Impact of nonequilibrium noise amplitude statistics.

Last year, Białas et al. [Phys. Rev. E 102, 042121 (2020)] studied an overdamped dynamics of nonequilibrium noise driven Brownian particle dwelling in a spatially periodic potential and discovered a novel class of Brownian, yet non-Gaussian diffusion. The mean square displacement of the particle grows linearly with time and the probability density for the particle position is Gaussian; however, the corresponding distribution for the increments is non-Gaussian. The latter property induces the colossal enhancement of diffusion, significantly exceeding the well known effect of giant diffusion. Here, we considerably extend the above predictions by investigating the influence of nonequilibrium noise amplitude statistics on the colossal Brownian, yet non-Gaussian diffusion. The tail of amplitude distribution crucially impacts both the magnitude of diffusion amplification and the Gaussianity of the position and increments statistics. Our results carry profound consequences for diffusive behavior in nonequilibrium settings such as living cells in which diffusion is a central transport mechanism.