First-passage time: a conception leading to superstatistics

The study of the First Passage Time (FPT) problem (also known as first passage problem, FPP) started more than a century ago, but its diverse applications in science and engineering mostly emerged in the last two to three decades. Assuming that X(t) is a one-dimensional stochastic process, the First Passage Time is defined as the time (T) when X(t) first crosses a threshold. Engineering reliability is obviously a suitable application domain, and indeed applications such as optimal dam design in hydrology and analysis of structural failure in civil and mechanical engineering are typical examples. Although we envision that the FPT problem has great potential in network and software reliability, it should be more useful for network security and survivability because the approaches developed for the FPT problem are mostly analytical. The assumption for this inference is that in reliability analysis, experimental or historical data are often more readily available, which makes statistical approaches such as survival analysis more convenient and likely more realistic. In contrast, data is generally more difficult to obtain in security and survivability analyses, and analytical approaches can be leveraged to play more important roles. Furthermore, security and survivability often have to deal with malicious actions that may be driven by sophisticated cognition and behavioral processes, which are highly variable over time and very difficult to detect with short term data. If the behavior of an intruder can be characterized with some stochastic process such as Brownian motion, then the FPT approach may be applied to find the closed-form solution of the probability density function (PDF) of the first passage time, which can be the time when the system breaks down or when the hacker is successful in compromising a network. In addition, the solutions to FPT depend on boundary and initial conditions of the corresponding partial differential equations, and they also describe the evolution of PDF over time. This may suggest that it is possible to model the behavior changes of an intruder over time and circumstances. Another advantage of FPT analysis is that it may help solve some non-Markov stochastic process problems in reliability analysis and survival analysis. In this article, we first briefly introduce the FPT problem with Brownian motion as an example, and then suggest its potential applications in software reliability and network security.

Waiting for Returns: Using Space-Time Duality to Calibrate Financial Diffusions

A molecular motor is made of either a single macromolecule or a macromolecular complex.Just like their macroscopic counterparts, molecular motors "transduce" input energy into mechanical work.Their diffusion is biased by the energy pumped in by the "burning fuel".In this talk I'll begin by giving an elementary introduction to the most essential and relevant features of molecular motors [1, 2] and then present three sets of examples to establish the ubiquity of first-passage times in intracellular processes.Some motors walk along filamentous proteins carrying molecular cargo.The stochastic pause-and-translocation of such a porter is characterized by the distribution of the times of its dwell at the successive positions on its track.The dwell times are essentially first-passage times and their fluctuation provides a lower bound on the number of kinetic states of the motor.For a motor that can step both forward and backward the kinetics of translocation is characterized by the distributions of four distinct conditional dwell times.DNA polymerase is a motor that can not only step forward and backward mechanically, but also step backward by catalyzing a different chemical reaction; its stepping kinetics is characterized by nine distinct conditional dwell times.As the first example of first passage times, I'll define these conditional dwell times and summarize our analytical results as well as their physical implications [1, 2].In the nucleus DNA is stored in a hierarchical structure called chromatin.The basic unit of chromatin consists of 146 base pairs of a double stranded DNA (dsDNA) wrapped around a spool formed by proteins.The helical path of the dsDNA on a spool is also called its "footprint".We have investigated the mechanism of footprint traversal by a chromatin-remodeling enzyme (CRE), a motor that translocates along the dsDNA thereby unwrapping it from the spool.A CRE-induced biased diffusion of a small DNA loop along the footprint can lead to the sliding of the spool along the DNA.The average time needed for the traversal of the footprint, for the first time, by sliding is essentially the corresponding mean first passage time.As an example of second class of first-passage problems, I'll present our model and fuel-dependence of the mean foot-print traversal time [3].The third type of example I'll present occurs in chromosome segregation before cell division.Each of the sister chromatids, that result from chromosome replication, is bound to a structure, called kinetochore (kt) that, in turn, is coupled to the plus ends of stiff protein filaments called microtubules (MT).The lifetime of a kt-MT attachment is also a first-passage time.I'll present our recent results that explain the experimentally observed, apparently counter-intuitive, dependence of the mean lifetime on the externally applied tension that tends to detach MT from the kinetochore [4,5].

The distribution of the time elapsed before a random variable reaches a threshold value for the first time, called the first passage time (FPT) distribution, is a fundamental characteristic of stochastic processes. Here, by solving the standard macroscopic diffusion equation, we derive analytical expressions for the FPT distribution of a diffusing particle hitting a spherical object in two dimensions (2D) and three dimensions (3D) in the course of unrestricted diffusion in open space. In addition, we calculate, analytically, the angular dependence of the FPT, known as the hit distribution. The analytical results are also compared to simulations of the motions of a random walker on a discrete lattice. This topic could be of wide pedagogical interest because the FPT is important not only in physics but also in chemistry, biology, medicine, agriculture, engineering, and finance. Additionally, the central equations often appear in physics and engineering with only trivial variations, making the solution techniques widely applicable.

To investigate effects of preoperative walking on bowel function recovery for patients after gynecological malignancy laparoscopy. The 156 patients with gynecoligical cancers after laparoscopy in Jiangsu from June 2020 to September 2021 were selected as research subjects, who were randomized into an experimental group (n = 78) and a control group (n = 78). Both of the groups received routine nursing care during the study. In addition, the experimental group underwent low‐moderate intensity walking exercise 1 week before surgery. The bowel function (including the time of first defecation, the time of first passage of flatus/”gas‐out time” and the recovery time of bowel sound), adverse events (nausea, vomiting abdominal distension and abdominal pain), as well as postoperative complications (ileus symptoms, deep venous thrombosis, infections and etc.), were measured daily. The time of first defecation, the time of first passage of flatus and the recovery time of bowel sound in experimental group were less than the control group after treatment (p < .05). Repeated measures analysis of variance showed that the adverse reactions (nausea, vomiting, abdominal distension, abdominal pain, and ileus symptoms) of the experimental group were weaker than those of the control group at different time points after the intervention (p < .05). Walking before surgery can effectively promote the recovery of bowel function and reduce the adverse reactions, as well as the risk of ileus related to gynecological malignancy laparoscopy.

Recent developments in the measurement of radioactive gases in passive diffusion motivate the analysis of Brownian motion of decaying particles, a subject that has received little previous attention. This paper reports the derivation and solution of equations comparable to the Fokker-Planck and Langevin equations for one-dimensional diffusion and decay of unstable particles. In marked contrast to the case of stable particles, the two equations are not equivalent, but provide different information regarding the same stochastic process. The differences arise because Brownian motion with particle decay is not a continuous process. The discontinuity is readily apparent in the computer-simulated trajectories of the Langevin equation that incorporate both a Wiener process for displacement fluctuations and a Bernoulli process for random decay. This paper also reports the derivation of the mean time of first passage of the decaying particle to absorbing boundaries. Here, too, particle decay can lead to an outcome markedly different from that for stable particles. In particular, the first-passage time of the decaying particle is always finite, whereas the time for a stable particle to reach a single absorbing boundary is theoretically infinite due to the heavy tail of the inverse Gaussian density. The methodology developed in this paper should prove useful in the investigation of radioactive gases, aerosols of radioactive atoms, dust particles to which adhere radioactive ions, as well as diffusing gases and liquids of unstable molecules.

Randomness accelerates the dynamic clearing process of the COVID-19 outbreaks in China

The spread of influence in a network is a fundamental issue in many complex systems. For instance, the word of-mouth effect in a social network is crucial to the emerging viral market. The process was previously analyzed with a voter model where an individual weighs the opinions of his acquaintances equally. However, one would expect that an individual's opinion is often more swayed by her/his better connected peers. As such, the influence of a node x on another node should be proportional to x's degree. This paper studies the spread of influence under the latter model - called Preferential Voter Model (PVM) for easier reference. We first present the exact form of the random walk stationary distribution for PVM, based on which, we show that the First Passage Time (FPT) of PVM follows an exponential decay. Furthermore, compared against the conventional (nonpreferential) voter model, we show that nodes with larger degrees exhibit faster decay rate and lower mean First Passage Time in PVM. These new results have applications to network-based complex systems. For instance, in the context of viral marketing in a social network, picking nodes of larger degrees as the starting nodes of a sales campaign not only maximizes the expected number of influenced nodes in the network, but also reduces the expected time to spread the influence. Thus, our result confirms analytically that well-connected individuals indeed exert faster and more effective influences in social networks under the preferential model.

The classical theory of chemical reactions can be understood in terms of\ndiffusive barrier crossing, where the rate of a reaction is determined by the\ninverse of the mean first passage time (FPT) to cross a free energy barrier.\nWhenever a few reaction events suffice to trigger a response or the energy\nbarriers are not high, the mean first passage time alone does not suffice to\ncharacterize the kinetics, i.e., the kinetics do not occur on a single\ntime-scale. Instead, the full statistics of the FPT are required. We present a\nspectral representation of the FPT statistics that allows us to understand and\naccurately determine FPT distributions over several orders of magnitudes in\ntime. A canonical narrowing of the first passage density is shown to emerge\nwhenever several molecules are searching for the same target, which was termed\nthe 'few-encounter limit'. The few-encounter limit is essential in all\nsituations, in which already the first encounter triggers a response, such as\nmisfolding-triggered aggregation of proteins or protein transcription\nregulation.\n

The timescales of many physical, chemical, and biological processes are determined by first passage times (FPTs) of diffusion. The overwhelming majority of FPT research studies the time it takes a single diffusive searcher to find a target. However, the more relevant quantity in many systems is the time it takes the fastest searcher to find a target from a large group of searchers. This fastest FPT depends on extremely rare events and has a drastically faster timescale than the FPT of a given single searcher. In this work, we prove a simple explicit formula for every moment of the fastest FPT. The formula is remarkably universal, as it holds for d-dimensional diffusion processes (i) with general space-dependent diffusivities and force fields, (ii) on Riemannian manifolds, (iii) in the presence of reflecting obstacles, and (iv) with partially absorbing targets. Our results rigorously confirm, generalize, correct, and unify various conjectures and heuristics about the fastest FPT.

This work is a contribution towards the understanding of certain features of mathematical models of single neurons. Emphasis is set on neuronal firing, for which the first passage time (FPT) problem bears a fundamental relevance. We focus the attention on modeling the change of the neuron membrane potential between two consecutive spikes by Gaussian stochastic processes, both of Markov and of non-Markov types. Methods to solve the FPT problems, both of a theoretical and of a computational nature, are sketched, including the case of random initial values. Significant similarities or diversities between computational and theoretical results are pointed out, disclosing the role played by the correlation time that has been used to characterize the neuronal activity. It is highlighted that any conclusion on this matter is strongly model-dependent. In conclusion, an outline of the asymptotic behavior of FPT densities is provided, which is particularly useful to discuss neuronal firing under certain slow activity conditions.

The first-passage-time (FPT) problem is studied for superstatistical models assuming that the mesoscopic system dynamics is described by a Fokker-Planck equation. We show that all moments of the random intensive parameter associated to the superstatistical approach can be put in one-to-one correspondence with the moments of the FPT. For systems subjected to an additional uncorrelated external force, the same statistical information is obtained from the dependence of the FPT moments on the external force. These results provide an alternative technique for checking the validity of superstatistical models. As an example, we characterize the mean FPT for a forced Brownian particle.

This paper introduces two techniques for computing the distribution of the absorption or first passage time of the drifted Wiener diffusion subject to Poisson resetting times to an upper hard wall barrier and to a lower absorbing barrier. The first method, which we call the Padé-partial fraction approximation, starts with the Padé approximation to the Laplace transform of the first passage time distribution, which is then exactly inverted by means of the partial fraction decomposition. The second method, which we call the multiresolution algorithm, is a Monte Carlo technique that exploits the properties of the Wiener process to generate Brownian bridges at increasing levels of resolution. Our numerical study reveals that the multiresolution algorithm has higher efficiency than standard Monte Carlo, whereas the faster Padé-partial fraction method is accurate in various circumstances and provides an analytical formula. Also, a closed-form exact expression for the expected first passage time is derived.

An exact and efficient first passage time algorithm for reaction–diffusion processes on a 2D-lattice

Protein folding has been a challenging puzzle for decades but it is still not fully understood. One important way to gain insights of the mechanism is to study how kinetics in the unfolded state affects protein folding. The answer to this fundamental issue hinges on the time scale to equilibrate the unfolded state and the energy landscape of the unfolded state. We construct Markov state models (MSMs) of several mini-proteins to study the kinetics of their unfolded state ensemble and find that the folding kinetics are two-state even though there are multiple folding pathways with nonuniform barriers, which are the direct consequences of rapid mixing within the unfolded state. Also, we introduce a time integral of a proper correlation function, namely relaxation time, to characterize the time scale of equilibration within the unfolded state. However, the mean first passage times (MFPTs) between different regions of the unfolded state are observed to be orders of magnitude longer than the folding time. This seeming paradox is solved by the derivation of a simple relation that shows the mean first passage time to any state is equal to the relaxation time of that state divided by its equilibrium population. This simple relation explains why MFPTs among unfolded states can be very long but the energy landscape can still be smooth (minimally frustrated). As a matter of fact, when the folding kinetics is two-state, all of the unfolded state relaxation times are faster than the folding time. This result supports the well-established funnel-like energy landscape picture and resolves an apparent contradiction between this model and the recently proposed kinetic hub model of protein folding. Markov state model is a powerful tool but we seek for alternative ways of studying kinetics when MSM does not work very well. For example, diffusion maps of dimensionality reduction and discrete transition-based reweighting analysis method, are very useful in determining a geometrical measure that preserves intrinsic dynamics and in fully utilizing enhanced sampling simulation data.