Time-dependent Probability Distribution Research Articles

Diffusion of active Brownian particles under quenched disorder.

The motion of a single active particle in one dimension with quenched disorder under the external force is investigated. Within the tailored parameter range, anomalous diffusion that displays weak ergodicity breaking is observed, i.e., non-ergodic subdiffusion and non-ergodic superdiffusion. This non-ergodic anomalous diffusion is analyzed through the time-dependent probability distributions of the particle's velocities and positions. Its origin is attributed to the relative weights of the locked state (predominant in the subdiffusion state) and running state (predominant in the superdiffusion state). These results may contribute to understanding the dynamical behavior of self-propelled particles in nature and the extraordinary response of nonlinear dynamics to the externally biased force.

Time-dependent probability density functions, information geometry and entropy production in a stochastic prey–predator model of fusion plasmas

A stochastic, prey–predator model of the L–H transition in fusion plasma is investigated. The model concerns the regulation of turbulence by zonal and mean flow shear. Independent delta-correlated Gaussian stochastic noises are used to construct Langevin equations for the amplitudes of turbulence and zonal flow shear. We then find numerical solutions of the equivalent Fokker–Planck equation for the time-dependent joint probability distribution of these quantities. We extend the earlier studies [Kim and Hollerbach, Phys. Rev. Res. 2, 023077 (2020) and Hollerbach et al., Phys. Plasmas 27, 102301 (2020)] by applying different functional forms of the time-dependent external heating (input power), which is increased and then decreased in a symmetric fashion to study hysteresis. The hysteresis is examined through the probability distribution and statistical measures, which include information geometry and entropy. We find strongly non-Gaussian probability distributions with bi-modality being a persistent feature across the input powers; the information length to be a better indicator of distance to equilibrium than the total entropy. Both dithering transitions and direct L-–H transitions are (also) seen when the input power is stepped in time. By increasing the number of steps, we see less hysteresis (in the statistical measures) and a reduced probability of H-mode access; intermittent zonal flow shear is seen to have a role in the initial suppression of turbulence by zonal flow shear and stronger excitation of intermittent zonal flow shear for a faster changing input power.

Poisson distributions in stochastic dynamics of gene expression: What events do they count?

The Poisson distribution is the probability distribution of the number of independent events in a given period. Although the Poisson distribution appears ubiquitously in various stochastic dynamics of gene expression, both as time-dependent distributions and stationary distributions, underlying independent events that give rise to such distributions have not been clear, especially in the presence of the degradation of gene products, which is not a Poisson process. I show that the variable following the Poisson distribution is the number of independent events where biomolecules are created, destined to survive until the end of a given time duration. This new viewpoint enables me to rederive the Poisson distribution as a time-dependent probability distribution for molecule numbers in various monomolecular reaction models of stochastic gene dynamics. Additionally, it allows me to derive an analytic form of the time-dependent probability distribution for multispecies monomolecular reaction models with species whose lifetimes follow nonexponential distributions, which is the convolution of the Poisson distribution with the multinomial distribution. This distribution is then utilized for deriving a novel series expansion form of a time-dependent distribution for a model with a stochastic production rate.

Precision-dissipation trade-off for driven stochastic systems

Over the last few decades, stochastic thermodynamics has emerged as a framework to study the thermodynamics of small-scaled systems. The relation between entropy production and precision is one of the most prominent research topics in this field. In this paper, I answer the question how much dissipation is needed to follow a pre-determined trajectory. This will be done by deriving a trade-off relation between how precisely a mesoscopic system can follow a pre-defined trajectory and how much the system dissipates. In the high-precision limit, the minimal amount of dissipation is inversely proportional to the expected deviation from the pre-defined trajectory. Furthermore, I will derive the protocol that maximizes the precision for a given amount of dissipation. The optimal time-dependent force field is a conservative energy landscape which combines a shifted version of the initial energy landscape and a quadratic energy landscape. The associated time-dependent probability distribution conserves its shape throughout the optimal protocol. Potential applications are discussed in the context of bit erasure and electronic circuits.

Asymptotic results of a multiple-entry reinforcement process

We introduce a class of stochastic processes with reinforcement consisting of a sequence of random partitions {Pt}t≥1, where Pt is a partition of {1,2,…,Rt}. At each time t, R numbers are added to the set being partitioned; of these, a random subset (chosen according to a time-dependent probability distribution) joins existing blocks, and the others each start new blocks on their own. Those joining existing blocks each choose a block with probability proportional to that block’s cardinality, independently. We prove results concerning the asymptotic cardinality of a given block and central limit theorems for associated fluctuations about this asymptotic cardinality: these are proved both for a fixed block and for the maximum among all blocks. We also prove that with probability one, a single block eventually takes and maintains the leadership in cardinality. Depending on the way one sees this partition process, one can translate our results to Balls and Bins processes, Generalized Chinese Restaurant Processes, Generalized Urn models and Preferential attachment random graphs.

Improving estimation of entropy production rate for run-and-tumble particle systems by high-order thermodynamic uncertainty relation.

Entropy production plays an important role in the regulation and stability of active matter systems, and its rate quantifies the nonequilibrium nature of these systems. However, entropy production is hard to experimentally estimate even in some simple active systems like molecular motors or bacteria, which may be modeled by the run-and-tumble particle (RTP), a representative model in the study of active matters. Here we resolve this problem for an asymmetric RTP in one dimension, first constructing a finite-time thermodynamic uncertainty relation (TUR) for a RTP, which works well in the short observation time regime for entropy production estimation. Nevertheless, when the activity dominates, i.e., the RTP is far from equilibrium, the lower bound for entropy production from TUR turns out to be trivial. We address this issue by introducing a recently proposed high-order thermodynamic uncertainty relation (HTUR), in which the cumulant generating function of current serves as a key ingredient. To exploit the HTUR, we adopt a method to analytically obtain the cumulant generating function of the current we study, with no need to explicitly know the time-dependent probability distribution. The HTUR is demonstrated to be able to estimate the steady state energy dissipation rate accurately because the cumulant generating function covers higher-order statistics of the current, including rare and large fluctuations besides its variance. Compared to the conventional TUR, the HTUR could give significantly improved estimation of energy dissipation, which can work well even in the far from equilibrium regime. We also provide a strategy based on the improved bound to estimate the entropy production from a moderate amount of trajectory data for experimental feasibility.

Estimating the recurrence of earthquakes with statistical methods in the city of Bingöl, Eastern Turkey: a district-based approach

This study discusses the temporal distribution of earthquake magnitudes in the city of Bingöl, near Karlıova Triple Junction. We determine the probability distributions and return periods of earthquakes for all districts of Bingöl. Bingöl has eight districts; namely Adaklı, Central, Genç, Karlıova, Kiğı, Solhan, Yayladere, and Yedisu. In six of them, active faults were mapped previously (Adaklı, Central, Genç, Karlıova, Solhan, and Yedisu). We consider 5 time-dependent probability distributions for analysis. Using the annual maximum earthquake magnitudes, the best fit arises from the Gumbel distribution for Central, Karlıova, and Adaklı Districts. For the Genç District, where the least maximum earthquake magnitude is reported, the Weibull distribution gives the best fit. The return period and maximum annual earthquake magnitude relations suggest the following results. For the Central and Karlıova Districts along which maximum earthquake magnitudes are reported, every 250 years a 6.7 M, and 7.2 M occurs respectively. These results are compatible with the results of paleo-seismological data reported along the NAFZ and the EAFZ. For a 10-year return period, earthquake magnitudes reach 3.9 and 5.1 in all districts. It is important to note that in the Yedisu District, the maximum earthquake magnitudes seem as 5.1 M for the 1000-year return period, incompatible with previously published findings probably because low quality seismic data in this region.

Quantum Equilibrium in Stochastic de Broglie–Bohm–Bell Quantum Mechanics

This paper investigates dynamical relaxation to quantum equilibrium in the stochastic de Broglie–Bohm–Bell formulation of quantum mechanics. The time-dependent probability distributions are computed as in a Markov process with slowly varying transition matrices. Numerical simulations, supported by exact results for the large-time behavior of sequences of (slowly varying) transition matrices, confirm previous findings that indicate that de Broglie–Bohm–Bell dynamics allows an arbitrary initial probability distribution to relax to quantum equilibrium; i.e., there is no need to make the ad-hoc assumption that the initial distribution of particle locations has to be identical to the initial probability distribution prescribed by the system’s initial wave function. The results presented in this paper moreover suggest that the intrinsically stochastic nature of Bell’s formulation, which is arguable most naturally formulated on an underlying discrete space-time, is sufficient to ensure dynamical relaxation to quantum equilibrium for a large class of quantum systems without the need to introduce coarse-graining or any other modification in the formulation.

Quasi-Entropy Closure: a fast and reliable approach to close the moment equations of the Chemical Master Equation.

The Chemical Master Equation is a stochastic approach to describe the evolution of a (bio)chemical reaction system. Its solution is a time-dependent probability distribution on all possible configurations of the system. As this number is typically large, the Master Equation is often practically unsolvable. The Method of Moments reduces the system to the evolution of a few moments, which are described by ordinary differential equations. Those equations are not closed, since lower order moments generally depend on higher order moments. Various closure schemes have been suggested to solve this problem. Two major problems with these approaches are first that they are open loop systems, which can diverge from the true solution, and second, some of them are computationally expensive. Here we introduce Quasi-Entropy Closure, a moment-closure scheme for the Method of Moments. It estimates higher order moments by reconstructing the distribution that minimizes the distance to a uniform distribution subject to lower order moment constraints. Quasi-Entropy Closure can be regarded as an advancement of Zero-Information Closure, which similarly maximizes the information entropy. Results show that both approaches outperform truncation schemes. Quasi-Entropy Closure is computationally much faster than Zero-Information Closure, although both methods consider solutions on the space of configurations and hence do not completely overcome the curse of dimensionality. In addition, our scheme includes a plausibility check for the existence of a distribution satisfying a given set of moments on the feasible set of configurations. All results are evaluated on different benchmark problems. Supplementary data are available at Bioinformatics online.

Giant diamagnetism of a quantum charged particle after inversion of the magnetic field

We consider a quantum charged particle moving in the $xy$ plane under the action of a uniform perpendicular time-dependent magnetic field, in the presence of a parabolic binding potential. The time-dependent probability distribution of the magnetic moment is calculated analytically and numerically in the case of initial thermodynamic equilibrium state. In the high-temperature regime, the initial distribution is almost symmetric, resulting in a tiny mean diamagnetism. However, if the magnetic field eventually changes its sign, the fragile balance between the diamagnetic and paramagnetic ``wings'' of the probability distribution becomes broken, resulting in a giant mean diamagnetic moment, exceeding the initial one by several orders of magnitude. The final mean value of the magnetic moment is inversely proportional to the strength of the binding potential, and it does not depend on the Planck constant in the high-temperature regime. Strong fluctuations of the magnetic moment (described in terms of the variance) exist in all temperature regimes.

"Computational approach to Transient solution for single unreliable server Retrial Queue under non-preemptive priority"

"Single server Retrial Queueing model with system repair and breakdown under non-preemptive priority is considered in this paper. The study of the model so far is mostly explored only for steady state solutions with few exceptions to Transient solutions. Though some researchers have attempted transient solution to the model analytically, the solution thus obtained is more complicated that it cannot be solved. In this context, we have found the transient numerical solution to the model using eigenvalues and eigenvectors. Time dependent system performance measures and probability distributions are evaluated and validated with steady state solutions. Single server Retrial Queueing model with system repair and breakdown under non-preemptive priority is considered in this paper. The study of the model so far is mostly explored only for steady state solutions with few exceptions to Transient solutions. Though some researchers have attempted transient solution to the model analytically, the solution thus obtained is more complicated that it cannot be solved. In this context, we have found the transient numerical solution to the model using eigenvalues and eigenvectors. Time dependent system performance measures and probability distributions are evaluated and validated with steady state solutions."

Efficient approach to time-dependent super-diffusive Lévy random walks on finite 2D-tori using circulant analogues

This work resumes the investigation on discrete-time super-diffusive in Lévy random walks defined on networks by using a inverse problem approach, with a focus on 2D-tori. Imposing that the mean square displacement of the walker should be proportional to tγ, we use a Markov Chain formalism to evaluate a fine tuned time-dependent probability distribution of long-distance jumps the walker should use to meet this dependency. Despite its wide applicability, calculations are time-intensive, with a computing time proportional to the number of nodes in the graph to a power >3.4. Here it is shown that, by using the circulant property satisfied by the adjacency matrices of a class of tori, it is possible to significantly speed up the calculations. For the purpose of comparison, the inverse super-diffusion problem is solved for two tori based on finite patches of the two-dimensional square lattice, namely the usual (non-circulant) and the helical (circulant) ones. The results of the latter, based on derived new expressions to compute the mean square displacement valid for circulant tori, are in complete agreement with those derived using general expressions, even if the computing time increases with respect to the number of nodes with a significantly smaller exponent ≳2.1. Numerical simulations in both tori types also reproduce super-diffusion when using the time-dependent probability distributions obtained for the helical one. The results suggest that this time efficient approach can be extended to model super-diffusion on cubic and hyper-cubic lattices.

The Fundamental Theorem of Natural Selection.

Suppose we have n different types of self-replicating entity, with the population of the ith type changing at a rate equal to times the fitness of that type. Suppose the fitness is any continuous function of all the populations . Let be the fraction of replicators that are of the ith type. Then is a time-dependent probability distribution, and we prove that its speed as measured by the Fisher information metric equals the variance in fitness. In rough terms, this says that the speed at which information is updated through natural selection equals the variance in fitness. This result can be seen as a modified version of Fisher’s fundamental theorem of natural selection. We compare it to Fisher’s original result as interpreted by Price, Ewens and Edwards.

Solving the chemical master equation for monomolecular reaction systems and beyond: a Doi-Peliti path integral view.

The chemical master equation (CME) is a fundamental description of interacting molecules commonly used to model chemical kinetics and noisy gene regulatory networks. Exact time-dependent solutions of the CME-which typically consists of infinitely many coupled differential equations-are rare, and are valuable for numerical benchmarking and getting intuition for the behavior of more complicated systems. Jahnke and Huisinga's landmark calculation of the exact time-dependent solution of the CME for monomolecular reaction systems is one of the most general analytic results known; however, it is hard to generalize, because it relies crucially on special properties of monomolecular reactions. In this paper, we rederive Jahnke and Huisinga's result on the time-dependent probability distribution and moments of monomolecular reaction systems using the Doi-Peliti path integral approach, which reduces solving the CME to evaluating many integrals. While the Doi-Peliti approach is less intuitive, it is also more mechanical, and hence easier to generalize. To illustrate how the Doi-Peliti approach can go beyond the method of Jahnke and Huisinga, we also find an explicit and exact time-dependent solution to a problem involving an autocatalytic reaction that Jahnke and Huisinga identified as not solvable using their method. Most interestingly, we are able to find a formal exact time-dependent solution for any CME whose list of reactions involves only zero and first order reactions, which may be the most general result currently known. This formal solution also yields a useful algorithm for efficiently computing numerical solutions to CMEs of this type.

A tool for assessing demand side management and operating strategies for isolated microgrids

Globally, power system operators are exploring ways to leverage the capabilities of smart meters to implement fine-grained demand side management (DSM). This paper presents a new simulation tool to evaluate operating strategies for smart-meter-enabled (SME) islanded microgrids being advanced for sustainable rural electrification in emerging economies. Based on MATLAB/Simulink, the tool's component-level models of small microgrids can be easily configured. Alternative rule-based operating strategies are implemented with controllable supply and storage components as well as other components driven by external factors (e.g., weather and time of day). To account for the wide range of possible operating conditions in real-world applications, statistical metrics can be evaluated using Monte Carlo (MC) methods based on user-defined, time-dependent probability distributions for demands, supplies, and environmental variables. Features of the tool are demonstrated with case studies for a representative microgrid in rural Rwanda.

SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems

The Chemical Master Equation is a standard approach to model biochemical reaction networks. It consists of a system of linear differential equations, in which each state corresponds to a possible configuration of the reaction system, and the solution describes a time-dependent probability distribution over all configurations. The Stochastic Simulation Algorithm (SSA) is a method to simulate sample paths from this stochastic process. Both approaches are only applicable for small systems, characterized by few reactions and small numbers of molecules. For larger systems, the CME is computationally intractable due to a large number of possible configurations, and the SSA suffers from large reaction propensities. In our study, we focus on catalytic reaction systems, in which substrates are converted by catalytic molecules. We present an alternative description of these systems, called SiCaSMA, in which the full system is subdivided into smaller subsystems with one catalyst molecule each. These single catalyst subsystems can be analyzed individually, and their solutions are concatenated to give the solution of the full system. We show the validity of our approach by applying it to two test-bed reaction systems, a reversible switch of a molecule and methyltransferase-mediated DNA methylation.

Time-dependent probability distribution for number of infection in a stochastic SIS model: case study COVID-19.

We derive the time-dependent probability distribution for the number of infected individuals at a given time in a stochastic Susceptible-Infected-Susceptible (SIS) epidemic model. The mean, variance, skewness, and kurtosis of the distribution are obtained as a function of time. We study the effect of noise intensity on the distribution and later derive and analyze the effect of changes in the transmission and recovery rates of the disease. Our analysis reveals that the time-dependent probability density function exists if the basic reproduction number is greater than one. It converges to the Dirac delta function on the long run (entirely concentrated on zero) as the basic reproduction number tends to one from above. The result is applied using published COVID-19 parameters and also applied to analyze the probability distribution of the aggregate number of COVID-19 cases in the United States for the period: January 22, 2020-March 23, 2021. Findings show that the distribution shifts concentration to the right until it concentrates entirely on the carrying infection capacity as the infection growth rate increases or the recovery rate reduces. The disease eradication and disease persistence thresholds are calculated.

Postponing production exponentially enhances the molecular memory of a stochastic switch

Delayed production can substantially alter the qualitative behaviour of feedback systems. Motivated by stochastic mechanisms in gene expression, we consider a protein molecule which is produced in randomly timed bursts, requires an exponentially distributed time to activate and then partakes in positive regulation of its burst frequency. Asymptotically analysing the underlying master equation in the large-delay regime, we provide tractable approximations to time-dependent probability distributions of molecular copy numbers. Importantly, the presented analysis demonstrates that positive feedback systems with large production delays can constitute a stable toggle switch even if they operate with low copy numbers of active molecules.

Stochastic time-dependent enzyme kinetics: Closed-form solution and transient bimodality.

We derive an approximate closed-form solution to the chemical master equation describing the Michaelis-Menten reaction mechanism of enzyme action. In particular, assuming that the probability of a complex dissociating into an enzyme and substrate is significantly larger than the probability of a product formation event, we obtain expressions for the time-dependent marginal probability distributions of the number of substrate and enzyme molecules. For delta function initial conditions, we show that the substrate distribution is either unimodal at all times or else becomes bimodal at intermediate times. This transient bimodality, which has no deterministic counterpart, manifests when the initial number of substrate molecules is much larger than the total number of enzyme molecules and if the frequency of enzyme-substrate binding events is large enough. Furthermore, we show that our closed-form solution is different from the solution of the chemical master equation reduced by means of the widely used discrete stochastic Michaelis-Menten approximation, where the propensity for substrate decay has a hyperbolic dependence on the number of substrate molecules. The differences arise because the latter does not take into account enzyme number fluctuations, while our approach includes them. We confirm by means of a stochastic simulation of all the elementary reaction steps in the Michaelis-Menten mechanism that our closed-form solution is accurate over a larger region of parameter space than that obtained using the discrete stochastic Michaelis-Menten approximation.

Run-and-tumble particle in inhomogeneous media in one dimension

We investigate the run-and-tumble particle (RTP), also known as persistent Brownian motion, in one dimension. A telegraphic noise σ(t) drives the particle which changes between ±1 values at certain rates. Denoting the rate of flip from 1 to −1 as R1 and the converse rate as R2, we consider the position- and direction-dependent rates of the form and with α ⩾ 0. For α = 0 and 1, we solve the master equations exactly for arbitrary γ1 and γ2 at large t. From our analytical expression for the time-dependent probability distribution P(x, t) we find that for γ1 > γ2 the distribution relaxes to a steady state exponentially, whereas for γ1 ⩽ γ2 the distribution does not reach a steady state and can be described by a non-trivial scaling form. We interestingly find that these features of the probability distribution P(x, t) in the two regimes γ1 > γ2 and γ1 ⩽ γ2 also remain valid for general α > 0. In particular, for general α, we argue and numerically demonstrate that the approach to the steady state in γ1 > γ2 case is exponential. On the other hand, for γ1 ⩽ γ2, the distribution P(x, t) remains time dependent and possesses certain scaling behavior. For γ1 = γ2 we derive the scaling behavior as well as the scaling function rigorously, whereas for γ1 < γ2 we provide heuristic arguments to obtain the scaling behavior and the corresponding scaling functions. We also study the dynamics on a semi-infinite line with an absorbing barrier at the origin. For α = 0 and 1, we analytically compute the survival probabilities and the corresponding first-passage time distributions. For general α ⩾ 0, we provide approximate calculations to compute the behavior of the survival probability for t → ∞ in which limit it approaches a finite value for γ1 < γ2 but goes to zero for γ ⩾ γ2. We also study the approach to the large t value in both cases. Finally, we consider RTP in a finite interval [0, M] and compute the associated exit probabilities from that interval for all α. All our analytic results are verified with a numerical simulation.