Nonergodicity of reset geometric Brownian motion

Impact of stochastic resetting on resource allocation: The case of reallocating geometric Brownian motion

We study the effects of stochastic resetting on the reallocating geometric Brownian motion (RGBM), an established model for resource redistribution relevant to systems such as population dynamics, evolutionary processes, economic activity, and even cosmology. The RGBM model is inherently nonstationary and non-ergodic, leading to complex resource redistribution dynamics. By introducing stochastic resetting, which periodically returns the system to a predetermined state, we examine how this mechanism modifies RGBM behavior. Our analysis uncovers distinct long-term regimes determined by the interplay between the resetting rate, the strength of resource redistribution, and standard geometric Brownian motion parameters: the drift and the noise amplitude. Notably, we identify a critical resetting rate beyond which the self-averaging time becomes effectively infinite. In this regime, the first two moments are stationary, indicating a stabilized distribution of an initially unstable, mean-repulsive process. We demonstrate that optimal resetting can effectively balance growth and redistribution, reducing inequality in the resource distribution. These findings help us understand better the management of resource dynamics in uncertain environments.

Controlling noise with self-organized resetting

Biological systems often consist of a small number of constituents and are therefore inherently noisy. To function effectively, these systems must employ mechanisms to constrain the accumulation of noise. Such mechanisms have been extensively studied and comprise the constraint by external forces, nonlinear interactions, or the resetting of the system to a predefined state. Here, we propose a fourth paradigm for noise constraint: self-organized resetting, where the resetting rate and position emerge from self-organization through time-discrete interactions. We study general properties of self-organized resetting systems using the paradigmatic example of cooperative resetting, where random pairs of Brownian particles are reset to their respective average. We demonstrate that such systems undergo a delocalization phase transition, separating regimes of constrained and unconstrained noise accumulation. Additionally, we show that systems with self-organized resetting can adapt to external forces and optimize search behavior for reaching target values. Self-organized resetting has various applications in nature and technology, which we demonstrate in the context of sexual interactions in fungi and spatial dispersion in shared mobility services. This work opens routes into the application of self-organized resetting across various systems in biology and technology.

Optimal conditions for first passage of jump processes with resetting

We investigate the first passage time beyond a barrier located at b≥0 of a random walk with independent and identically distributed jumps, starting from x0=0. The walk is subject to stochastic resetting, meaning that after each step the evolution is restarted with fixed probability r. We consider a resetting protocol that is an intermediate situation between a random walk (r=0) and an uncorrelated sequence of jumps all starting from the origin (r=1) and derive a general condition for determining when restarting the process with 0 Collapse

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Affiliation(s)

Mattia Radice Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
Giampaolo Cristadoro Dipartimento di Matematica e Applicazioni, Università degli Studi Milano-Bicocca, 20126 Milan, Italy
Samudrajit Thapa Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany

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Stochastic resetting in a nonequilibrium environment

This study examines the dynamics of a tracer particle diffusing in a nonequilibrium medium under stochastic resetting. The nonequilibrium state is induced by harmonic coupling between the tracer and bath particles, generating memory effects with an exponential decay in time. We explore the tracer's behavior under a Poissonian resetting protocol, where resetting does not disturb the bath environment, with a focus on key dynamical behavior and first-passage properties, both in the presence and absence of an external force. The interplay between coupling strength and diffusivity of bath particles significantly impacts both the tracer's relaxation dynamics and search time, with external forces further modulating these effects. Our analysis identifies distinct hot and cold bath particles based on their diffusivities, revealing that coupling to a hot particle facilitates the searching process, whereas coupling to a cold particle hinders it. Using a combination of numerical simulations and analytical methods, this study provides a comprehensive framework for understanding resetting mechanisms in non-Markovian systems, with potential applications to complex environments such as active and viscoelastic media, where memory-driven dynamics and nonequilibrium interactions are significant.

Granular gases under resetting

We investigate the granular temperatures in force-free granular gases under exponential resetting. When a resetting event occurs, the granular temperature attains its initial value, whereas it decreases because of the inelastic collisions between the resetting events. We develop a theory and perform computer simulations for granular gas cooling in the presence of Poissonian resetting events. We also investigate the probability density function to quantify the distribution of granular temperatures. Our theory may help us to understand the behavior of nonperiodically driven granular systems.

Anomalous diffusion of active Brownian particles in responsive elastic gels: Nonergodicity, non-Gaussianity, and distributions of trapping times

Understanding actual transport mechanisms of self-propelled particles (SPPs) in complex elastic gels-such as in the cell cytoplasm, in in vitro networks of chromatin or of F-actin fibers, or in mucus gels-has far-reaching consequences. Implications beyond biology/biophysics are in engineering and medicine, with a particular focus on microrheology and on targeted drug delivery. Here, we examine via extensive computer simulations the dynamics of SPPs in deformable gellike structures responsive to thermal fluctuations. We treat tracer particles comparable to and larger than the mesh size of the gel. We observe distinct trapping events of active tracers at relatively short times, leading to subdiffusion; it is followed by an escape from meshwork-induced traps due to the flexibility of the network, resulting in superdiffusion. We thus find crossovers between different transport regimes. We also find pronounced nonergodicity in the dynamics of SPPs and non-Gaussianity at intermediate times. The distributions of trapping times of the tracers escaping from "cages" in our quasiperiodic gel often reveal the existence of two distinct timescales in the dynamics. At high activity of the tracers these timescales become comparable. Furthermore, we find that the mean waiting time exhibits a power-law dependence on the activity of SPPs (in terms of their Péclet number). Our results additionally showcase both exponential and nonexponential trapping events at high activities. Extensions of this setup are possible, with the factors such as anisotropy of the particles, different topologies of the gel network, and various interactions between the particles (also of a nonlocal nature) to be considered.

A Deep Learning-Enhanced Compartmental Model and Its Application in Modeling Omicron in China

The mainstream compartmental models require stochastic parameterization to estimate the transmission parameters between compartments, whose calculation depend upon detailed statistics on epidemiological characteristics, which are expensive, economically and resource-wise, to collect. In addition, infectious diseases spread in three dimensions: temporal, spatial, and mobile, i.e., they affect a population through not only the time progression of infection, but also the geographic distribution and physical mobility of the population. However, the parameterization process for the mainstream compartmental models does not effectively capture the spatial and mobile dimensions. As an alternative, deep learning techniques are utilized in estimating these stochastic parameters with greatly reduced dependency on data particularity and with a built-in temporal-spatial-mobile process that models the geographic distribution and physical mobility of the population. In particular, we apply DNN (Deep Neural Network) and LSTM (Long-Short Term Memory) techniques to estimate the transmission parameters in a customized compartmental model, then feed the estimated transmission parameters to the compartmental model to predict the development of the Omicron epidemic in China over the 28 days for the period between 4 June and 1 July 2022. The average levels of predication accuracy of the model are 98% and 92% for the number of infections and deaths, respectively. We establish that deep learning techniques provide an alternative to the prevalent compartmental modes and demonstrate the efficacy and potential of applying deep learning methodologies in predicting the dynamics of infectious diseases.

Two coupled population growth models driven by Gaussian white noises

Exact solution for the probability density function is considered for two coupled population growth models driven by Gaussian white noises. Moreover, n-moments of interactions of the Gompertz and Verhulst logistic models are obtained and analyzed. It is shown that interactions can modify the behaviors of the population growth models, i.e, the species may collaborate and/or compete between them.

Quest for optimal quantum resetting: Protocols for a particle on a chain

In the classical context, it is well known that, sometimes, if a search does not find its target, it is better to start the process anew. This is known as resetting. The quantum counterpart of resetting also indicates speeding up the detection process by eliminating the dark states, i.e., situations in which the particle avoids detection. In this work, we introduce the most probable position resetting (MPR) protocol, in which, at a given resetting step, resets are done with certain probabilities to the set of possible peak positions (where the probability of finding the particle is maximum) that could occur because of the previous resets and followed by uninterrupted unitary evolution, irrespective of which path was taken by the particle in previous steps. In a tight-binding lattice model, there exists a twofold degeneracy (left and right) of the positions of maximum probability. The survival probability with optimal restart rate approaches 0 (detection probability approaches 1) when the particle is reset with equal probability on both sides path independently. This protocol significantly reduces the optimal mean first-detected-passage time (FDT), and it performs better even if the detector is far apart compared to the usual resetting protocols in which the particle is brought back to the initial position. We propose a modified protocol, an adaptive two-stage MPR, by making the associated probabilities of going to the right and left a function of steps. In this protocol, we see a further reduction of the optimal mean FDT and improvement in the search process when the detector is far apart.

Optimizing leapover lengths of Lévy flights with resetting

We consider a one-dimensional search process under stochastic resetting conditions. A target is located at b≥0 and a searcher, starting from the origin, performs a discrete-time random walk with independent jumps drawn from a heavy-tailed distribution. Before each jump, there is a given probability r of restarting the walk from the initial position. The efficiency of a "myopic search"-in which the search stops upon crossing the target for the first time-is usually characterized in terms of the first-passage time τ. On the other hand, great relevance is encapsulated by the leapover length l=x_{τ}-b, which measures how far from the target the search ends. For symmetric heavy-tailed jump distributions, in the absence of resetting the average leapover is always infinite. Here we show instead that resetting induces a finite average leapover ℓ_{b}(r) if the mean jump length is finite. We compute exactly ℓ_{b}(r) and determine the condition under which resetting allows for nontrivial optimization, i.e., for the existence of r^{*} such that ℓ_{b}(r^{*}) is minimal and smaller than the average leapover of the single jump.

Fractional heterogeneous telegraph processes: Interplay between heterogeneity, memory, and stochastic resetting

Fractional heterogeneous telegraph processes are considered in the framework of telegrapher's equations accompanied by memory effects. The integral decomposition method is developed for the rigorous treating of the problem. Exact solutions for the probability density functions and the mean squared displacements are obtained. A relation between the fractional heterogeneous telegrapher's equation and the corresponding Langevin equation has been established in the framework of the developed subordination approach. The telegraph process in the presence of stochastic resetting has been studied, as well. An exact expression for both the nonequilibrium stationary distributions/states and the mean squared displacements are obtained.

Short-time expansion of one-dimensional Fokker-Planck equations with heterogeneous diffusion

We formulate a short-time expansion for one-dimensional Fokker-Planck equations with spatially dependent diffusion coefficients, derived from stochastic processes with Gaussian white noise, for general values of the discretization parameter 0≤α≤1 of the stochastic integral. The kernel of the Fokker-Planck equation (the propagator) can be expressed as a product of a singular and a regular term. While the singular term can be given in closed form, the regular term can be computed from a Taylor expansion whose coefficients obey simple ordinary differential equations. We illustrate the application of our approach with examples taken from statistical physics and biophysics. Furthermore, we show how our formalism allows us to define a class of stochastic equations which can be treated exactly. The convergence of the expansion cannot be guaranteed independently from the discretization parameter α.

Multifractal spectral features enhance classification of anomalous diffusion

Anomalous diffusion processes, characterized by their nonstandard scaling of the mean-squared displacement, pose a unique challenge in classification and characterization. In a previous study [Mangalam et al., Phys. Rev. Res. 5, 023144 (2023)2643-156410.1103/PhysRevResearch.5.023144], we established a comprehensive framework for understanding anomalous diffusion using multifractal formalism. The present study delves into the potential of multifractal spectral features for effectively distinguishing anomalous diffusion trajectories from five widely used models: fractional Brownian motion, scaled Brownian motion, continuous-time random walk, annealed transient time motion, and Lévy walk. We generate extensive datasets comprising 10^{6} trajectories from these five anomalous diffusion models and extract multiple multifractal spectra from each trajectory to accomplish this. Our investigation entails a thorough analysis of neural network performance, encompassing features derived from varying numbers of spectra. We also explore the integration of multifractal spectra into traditional feature datasets, enabling us to assess their impact comprehensively. To ensure a statistically meaningful comparison, we categorize features into concept groups and train neural networks using features from each designated group. Notably, several feature groups demonstrate similar levels of accuracy, with the highest performance observed in groups utilizing moving-window characteristics and p varation features. Multifractal spectral features, particularly those derived from three spectra involving different timescales and cutoffs, closely follow, highlighting their robust discriminatory potential. Remarkably, a neural network exclusively trained on features from a single multifractal spectrum exhibits commendable performance, surpassing other feature groups. In summary, our findings underscore the diverse and potent efficacy of multifractal spectral features in enhancing the predictive capacity of machine learning to classify anomalous diffusion processes.

Random Walks on Comb-like Structures under Stochastic Resetting

We study the long-time dynamics of the mean squared displacement of a random walker moving on a comb structure under the effect of stochastic resetting. We consider that the walker's motion along the backbone is diffusive and it performs short jumps separated by random resting periods along fingers. We take into account two different types of resetting acting separately: global resetting from any point in the comb to the initial position and resetting from a finger to the corresponding backbone. We analyze the interplay between the waiting process and Markovian and non-Markovian resetting processes on the overall mean squared displacement. The Markovian resetting from the fingers is found to induce normal diffusion, thereby minimizing the trapping effect of fingers. In contrast, for non-Markovian local resetting, an interesting crossover with three different regimes emerges, with two of them subdiffusive and one of them diffusive. Thus, an interesting interplay between the exponents characterizing the waiting time distributions of the subdiffusive random walk and resetting takes place. As for global resetting, its effect is even more drastic as it precludes normal diffusion. Specifically, such a resetting can induce a constant asymptotic mean squared displacement in the Markovian case or two distinct regimes of subdiffusive motion in the non-Markovian case.

Multifractal foundations of biomarker discovery for heart disease and stroke

Any reliable biomarker has to be specific, generalizable, and reproducible across individuals and contexts. The exact values of such a biomarker must represent similar health states in different individuals and at different times within the same individual to result in the minimum possible false-positive and false-negative rates. The application of standard cut-off points and risk scores across populations hinges upon the assumption of such generalizability. Such generalizability, in turn, hinges upon this condition that the phenomenon investigated by current statistical methods is ergodic, i.e., its statistical measures converge over individuals and time within the finite limit of observations. However, emerging evidence indicates that biological processes abound with nonergodicity, threatening this generalizability. Here, we present a solution for how to make generalizable inferences by deriving ergodic descriptions of nonergodic phenomena. For this aim, we proposed capturing the origin of ergodicity-breaking in many biological processes: cascade dynamics. To assess our hypotheses, we embraced the challenge of identifying reliable biomarkers for heart disease and stroke, which, despite being the leading cause of death worldwide and decades of research, lacks reliable biomarkers and risk stratification tools. We showed that raw R-R interval data and its common descriptors based on mean and variance are nonergodic and non-specific. On the other hand, the cascade-dynamical descriptors, the Hurst exponent encoding linear temporal correlations, and multifractal nonlinearity encoding nonlinear interactions across scales described the nonergodic heart rate variability more ergodically and were specific. This study inaugurates applying the critical concept of ergodicity in discovering and applying digital biomarkers of health and disease.

Anomalous diffusion, nonergodicity, non-Gaussianity, and aging of fractional Brownian motion with nonlinear clocks

How do nonlinear clocks in time and/or space affect the fundamental properties of a stochastic process? Specifically, how precisely may ergodic processes such as fractional Brownian motion (FBM) acquire predictable nonergodic and aging features being subjected to such conditions? We address these questions in the current study. To describe different types of non-Brownian motion of particles-including power-law anomalous, ultraslow or logarithmic, as well as superfast or exponential diffusion-we here develop and analyze a generalized stochastic process of scaled-fractional Brownian motion (SFBM). The time- and space-SFBM processes are, respectively, constructed based on FBM running with nonlinear time and space clocks. The fundamental statistical characteristics such as non-Gaussianity of particle displacements, nonergodicity, as well as aging are quantified for time- and space-SFBM by selecting different clocks. The latter parametrize power-law anomalous, ultraslow, and superfast diffusion. The results of our computer simulations are fully consistent with the analytical predictions for several functional forms of clocks. We thoroughly examine the behaviors of the probability-density function, the mean-squared displacement, the time-averaged mean-squared displacement, as well as the aging factor. Our results are applicable for rationalizing the impact of nonlinear time and space properties superimposed onto the FBM-type dynamics. SFBM offers a general framework for a universal and more precise model-based description of anomalous, nonergodic, non-Gaussian, and aging diffusion in single-molecule-tracking observations.

Exploring Factors Influencing Changes in Incidence and Severity of Multisystem Inflammatory Syndrome in Children

Multisystem inflammatory syndrome (MIS-C) is a rare condition associated with COVID-19 affecting children, characterized by severe and aberrant systemic inflammation leading to nonspecific symptoms, such as gastrointestinal, cardiac, respiratory, hematological, and neurological disorders. In the last year, we have experienced a progressive reduction in the incidence and severity of MIS-C, reflecting the worldwide trend. Thus, starting from the overall trend in the disease in different continents, we reviewed the literature, hypothesizing the potential influencing factors contributing to the reduction in cases and the severity of MIS-C, particularly the vaccination campaign, the spread of different SARS-CoV-2 variants (VOCs), and the changes in human immunological response. The decrease in the severity of MIS-C and its incidence seem to be related to a combination of different factors rather than a single cause. Maturation of an immunological memory to SARS-CoV-2 over time, the implication of mutations of key amino acids of S protein in VOCs, and the overall immune response elicited by vaccination over the loss of neutralization of vaccines to VOCs seem to play an important role in this change.

An Epidemiological Analysis for Assessing and Evaluating COVID-19 Based on Data Analytics in Latin American Countries

This research provides a detailed analysis of the COVID-19 spread across 14 Latin American countries. Using time-series analysis and epidemic models, we identify diverse outbreak patterns, which seem not to be influenced by geographical location or country size, suggesting the influence of other determining factors. Our study uncovers significant discrepancies between the number recorded COVID-19 cases and the real epidemiological situation, emphasizing the crucial need for accurate data handling and continuous surveillance in managing epidemics. The absence of a clear correlation between the country size and the confirmed cases, as well as with the fatalities, further underscores the multifaceted influences on COVID-19 impact beyond population size. Despite the decreased real-time reproduction number indicating quarantine effectiveness in most countries, we note a resurgence in infection rates upon resumption of daily activities. These insights spotlight the challenge of balancing public health measures with economic and social activities. Our core findings provide novel insights, applicable to guiding epidemic control strategies and informing decision-making processes in combatting the pandemic.

Ornstein-Uhlenbeck process and generalizations: Particle dynamics under comb constraints and stochastic resetting

The Ornstein-Uhlenbeck process is interpreted as Brownian motion in a harmonic potential. This Gaussian Markov process has a bounded variance and admits a stationary probability distribution, in contrast to the standard Brownian motion. It also tends to a drift towards its mean function, and such a process is called mean reverting. Two examples of the generalized Ornstein-Uhlenbeck process are considered. In the first one, we study the Ornstein-Uhlenbeck process on a comb model, as an example of the harmonically bounded random motion in the topologically constrained geometry. The main dynamical characteristics (as the first and the second moments) and the probability density function are studied in the framework of both the Langevin stochastic equation and the Fokker-Planck equation. The second example is devoted to the study of the effects of stochastic resetting on the Ornstein-Uhlenbeck process, including stochastic resetting in the comb geometry. Here the nonequilibrium stationary state is the main question in task, where the two divergent forces, namely, the resetting and the drift towards the mean, lead to compelling results in the cases of both the Ornstein-Uhlenbeck process with resetting and its generalization on the two-dimensional comb structure.

Infinite ergodicity in generalized geometric Brownian motions with nonlinear drift

Geometric Brownian motion is an exemplary stochastic processes obeying multiplicative noise, with widespread applications in several fields, e.g., in finance, in physics, and biology. The definition of the process depends crucially on the interpretation of the stochastic integrals which involves the discretization parameter α with 0≤α≤1, giving rise to the well-known special cases α=0 (Itô), α=1/2 (Fisk-Stratonovich), and α=1 (Hänggi-Klimontovich or anti-Itô). In this paper we study the asymptotic limits of the probability distribution functions of geometric Brownian motion and some related generalizations. We establish the conditions for the existence of normalizable asymptotic distributions depending on the discretization parameter α. Using the infinite ergodicity approach, recently applied to stochastic processes with multiplicative noise by E. Barkai and collaborators, we show how meaningful asymptotic results can be formulated in a transparent way.

Spectral Method in Epidemic Time Series: Application to COVID-19 Pandemic

BACKGROUND

The age of infection plays an important role in assessing an individual's daily level of contagiousness, quantified by the daily reproduction number. Then, we derive an autoregressive moving average model from a daily discrete-time epidemic model based on a difference equation involving the age of infection. Novelty: The article's main idea is to use a part of the spectrum associated with this difference equation to describe the data and the model.

RESULTS

We present some results of the parameters' identification of the model when all the eigenvalues are known. This method was applied to Japan's third epidemic wave of COVID-19 fails to preserve the positivity of daily reproduction. This problem forced us to develop an original truncated spectral method applied to Japanese data. We start by considering ten days and extend our analysis to one month.

CONCLUSION

We can identify the shape for a daily reproduction numbers curve throughout the contagion period using only a few eigenvalues to fit the data.

Diffusion with partial resetting

Inspired by many examples in nature, stochastic resetting of random processes has been studied extensively in the past decade. In particular, various models of stochastic particle motion were considered where, upon resetting, the particle is returned to its initial position. Here we generalize the model of diffusion with resetting to account for situations where a particle is returned only a fraction of its distance to the origin, e.g., half way. We show that this model always attains a steady-state distribution which can be written as an infinite sum of independent, but not identical, Laplace random variables. As a result, we find that the steady-state transitions from the known Laplace form which is obtained in the limit of full resetting to a Gaussian form, which is obtained close to the limit of no resetting. A similar transition is shown to be displayed by drift diffusion whose steady state can also be expressed as an infinite sum of independent random variables. Finally, we extend our analysis to capture the temporal evolution of drift diffusion with partial resetting, providing a bottom-up probabilistic construction that yields a closed-form solution for the time-dependent distribution of this process in Fourier-Laplace space. Possible extensions and applications of diffusion with partial resetting are discussed.

Time-averaging and nonergodicity of reset geometric Brownian motion with drift

How do near-bankruptcy events in the past affect the dynamics of stock-market prices in the future? Specifically, what are the long-time properties of a time-local exponential growth of stock-market prices under the influence of stochastically occurring economic crashes? Here, we derive the ensemble- and time-averaged properties of the respective "economic" or geometric Brownian motion (GBM) with a nonzero drift exposed to a Poissonian constant-rate price-restarting process of "resetting." We examine-based both on thorough analytical calculations and on findings from systematic stochastic computer simulations-the general situation of reset GBM with a nonzero [positive] drift and for all special cases emerging for varying parameters of drift, volatility, and reset rate in the model. We derive and summarize all short- and long-time dependencies for the mean-squared displacement (MSD), the variance, and the mean time-averaged MSD (TAMSD) of the process of Poisson-reset GBM under the conditions of both rare and frequent resetting. We consider three main regions of model parameters and categorize the crossovers between different functional behaviors of the statistical quantifiers of this process. The analytical relations are fully supported by the results of computer simulations. In particular, we obtain that Poisson-reset GBM is a nonergodic stochastic process, with generally MSD(Δ)≠TAMSD(Δ) and Variance(Δ)≠TAMSD(Δ) at short lag times Δ and for long trajectory lengths T. We investigate the behavior of the ergodicity-breaking parameter in each of the three regions of parameters and examine its dependence on the rate of reset at Δ/T≪1. Applications of these theoretical results to the analysis of prices of reset-containing options are pertinent.

Anomalous transport tuned through stochastic resetting in the rugged energy landscape of a chaotic system with roughness

Stochastic resetting causes kinetic phase transitions, whereas its underlying physical mechanism remains to be elucidated. We here investigate the anomalous transport of a particle moving in a chaotic system with a stochastic resetting and a rough potential and focus on how the stochastic resetting, roughness, and nonequilibrium noise affect the transports of the particle. We uncover the physical mechanism for stochastic resetting resulting in the anomalous transport in a nonlinear chaotic system: The particle is reset to a new basin of attraction which may be different from the initial basin of attraction from the view of dynamics. From the view of the energy landscape, the particle is reset to a new energy state of the energy landscape which may be different from the initial energy state. This resetting can lead to a kinetic phase transition between no transport and a finite net transport or between negative mobility and positive mobility. The roughness and noise also lead to the transition. Based on the mechanism, the transport of the particle can be tuned by these parameters. For example, the combination of the stochastic resetting, roughness, and noise can enhance the transport and tune negative mobility, the enhanced stability of the system, and the resonant-like activity. We analyze these results through variances (e.g., mean-squared velocity, etc.) and correlation functions (i.e., velocity autocorrelation function, position-velocity correlation function, etc.). Our results can be extensively applied in the biology, physics, and chemistry, even social system.