Two-state Random Walk Research Articles

Stochastic resetting can optimize the intermittent search strategy in a two-dimensional confined topography

Based on the intermittent two-state random walk model, by calculating and discussing the mean first capture time, the search efficiency of the intermittent search strategy in a two-dimensional confined topography in the absence and presence of stochastic resetting is studied. Results reveal that for the close target, the regular Brownian motion is the efficient search strategy, whereas, for the target far from the starting point, the typical intermittent search strategy with relocation times being power-law distributed turns out to be the advantageous one, and the search efficiency impressively enhances with the distance of the target from the starting point increasing. We demonstrate that though in the absence of stochastic resetting there does not exist a universal optimal search strategy in the confined topography, the introduction of stochastic resetting makes the Brownian motion surprisingly to be the universal optimal search process, and distinctly outperforms the ones without stochastic resetting.

Strong anomalous diffusive behaviors of the two-state random walk process.

The phenomenon of the two-state process is observed in various systems and is increasingly attracting attention, such that there is a need for a theoretical model of the process. In this paper, we present a prototypal two-state random walk (TSRW) model of a renewal process alternating between the continuous-time random walk (CTRW) state and Lévy walk (LW) state. The jump length distribution of the CTRW state is assumed to be Gaussian whereas the time distributions of the two states are both considered to follow a power law. The diffusive behavior is analyzed and discussed by calculating the mean squared displacement (MSD) analytically and numerically. The results reveal that it displays strong anomalous diffusive behaviors caused by random motions of both states, i.e., two anomalous diffusion terms coexist in the expression of the MSD, and the time distribution which has the heavier tail determines their forms. Moreover, because the two diffusion terms originate from different mechanisms, we find that the diffusion can be characterized by either the term with the largest diffusion exponent or the term with the largest diffusion coefficient at long timescales, which shows very different properties from the single-state process. In addition, the two-state nature of the process of the particle moving in a velocity field makes the TSRW model applicable to describe it. Results obtained from the two-state model reveal that the diffusion can even exhibit subdiffusive behavior, which is significantly different from known results obtained using the single-state model.

Coexistence of ergodicity and nonergodicity in the aging two-state random walks.

The two-state stochastic phenomenon is observed in various systems and is attracting more interest, and it can be described by the two-state random walk (TSRW) model. The TSRW model is a typical two-state renewal process alternating between the continuous-time random walk state and the Lévy walk state, in both of which the sojourn time distributions follow a power law. In this paper, by discussing the statistical properties and calculating the ensemble averaged and time averaged mean squared displacement, the ergodic property and the ultimate diffusive behavior of the aging TSRW is studied. Results reveal that because of the two-state intermittent feature, ergodicity and nonergodicity can coexist in the aging TSRW, which behave as the time scalings of the time averages and ensemble averages not being identically equal. In addition, we find that the unique state occupation mechanism caused by the diverging mean of the sojourn times of one state, determines the aging TSRW's ultimate diffusive behavior at extremely large timescales, i.e., instead of the term with the larger diffusion exponent, the diffusion is surprisingly characterized by the term with the smaller one, which is distinctly different from previous conclusions and known results. At last, we note that the Lévy walk with rests model which also displays aging and ergodicity breaking, can be generalized by the TSRW model.

Flagellar number governs bacterial spreading and transport efficiency.

Peritrichous bacteria synchronize and bundle their flagella to actively swim, while disruption of the bundle leads to a slow motility phase with a weak propulsion. It is still not known whether the number of flagella represents an evolutionary adaptation toward optimizing bacterial navigation. We study the swimming dynamics of differentially flagellated Bacillus subtilis strains in a quasi-two-dimensional system. We find that decreasing the number of flagella N f reduces the average turning angle between two successive run phases and enhances the run time and the directional persistence of the run phase. As a result, having fewer flagella is beneficial for long-distance transport and fast spreading, while having a lot of flagella is advantageous for the processes that require a slower spreading, such as biofilm formation. We develop a two-state random walk model that incorporates spontaneous switchings between the states and yields exact analytical expressions for transport properties, in remarkable agreement with experiments. The results of numerical simulations based on our two-state model suggest that the efficiency of searching and exploring the environment is optimized at intermediate values of N f. The optimal choice of N f, for which the search time is minimized, decreases with increasing the size of the environment in which the bacteria swim.

Modeling kinetic adsorption in porous media by a two-state random walk practical method

Modeling kinetic adsorption in porous media by a two-state random walk practical method

Classification of microtubule histories

A microtubule of a given length undergoes all possible scenarios of transitions between growing and shrinking phases, so-called microtubule dynamic instability. In this paper we utilize a minimal two-state model proposed by Hill [Proc. Natl. Acad. Sci. USA 81, 6728 (1984)] that is equivalent to a two-state random walk. Using a technique for classifying discrete random walk configurations by introducing a counting variable in evolution equations, we have derived expressions for probability densities (which contain information about all transition histories) of phase transitions before the complete disappearance of a microtubule. As a result, the mean lifetime of a microtubule turns out to be equal to the total lifetime of growing and shrinking phases times the average number of transitions. An attractive feature of this simple model is that elementary formulas relating statistical averages to rate parameters are obtained.

New applications of the two-state random model

The two-state random walk model is used for the investigation of the kinetics of tracers in capillaries, dynamics of calcium ions in blood plasma and Brownian motion in fluctuating media.

Low-density expansion for a two-state random walk in a random environment

A nearest neighbor random walk on Z2 is considered where points of the lattice are labeled ‘‘good’’ or ‘‘bad.’’ A particle takes a vertical step with probability aG or aB and a horizontal step with probability 1−aG or 1−aB, depending on whether its present site is good or bad. Steps of size +1 are as probable as steps of size −1. If the good and bad sites are placed randomly, with density ρ of bad sites, it is known that there exists an ā such that for almost every placement of sites, the random walk in the long run behaves like a homogeneous walk with vertical probability ā and horizontal probability 1−ā. Here the problem of estimating ā as a function of ρ is considered; in particular when aG = 1/2 , aB fixed, the first two terms of the expansion of ā(ρ) at ρ=0 are rigorously derived.

Two-state random walk model of diffusion. 2. Oscillatory diffusion

Continuing our study of interrupted diffusion, we consider the problem of a particle executing a random walk interspersed with localized oscillations during its halts (e.g., at lattice sites). Earlier approaches proceedvia approximation schemes for the solution of the Fokker-Planck equation for diffusion in a periodic potential. In contrast, we visualize a two-state random walk in velocity space with the particle alternating between a state of flight and one of localized oscillation. Using simple, physically plausible inputs for the primary quantities characterising the random walk, we employ the powerful continuous-time random walk formalism to derive convenient and tractable closed-form expressions for all the objects of interest: the velocity autocorrelation, generalized diffusion constant, dynamic mobility, mean square displacement, dynamic structure factor (in the Gaussian approximation), etc. The interplay of the three characteristic times in the problem (the mean residence and flight times, and the period of the ‘local mode’) is elucidated. The emergence of a number of striking features of oscillatory diffusion (e.g., the local mode peak in the dynamic mobility and structure factor, and the transition between the oscillatory and diffusive regimes) is demonstrated.

Two-state random walk model of lattice diffusion. 1. Self-correlation function

Diffusion with interruptions (arising from localized oscillations, or traps, or mixing between jump diffusion and fluid-like diffusion, etc.) is a very general phenomenon. Its manifestations range from superionic conductance to the behaviour of hydrogen in metals. Based on a continuous-time random walk approach, we present a comprehensive two-state random walk model for the diffusion of a particle on a lattice, incorporating arbitrary holding-time distributions for both localized residence at the sites and inter-site flights, and also the correct first-waiting-time distributions. A synthesis is thus achieved of the two extremes of jump diffusion (zero flight time) and fluid-like diffusion (zero residence time). Various earlier models emerge as special cases of our theory. Among the noteworthy results obtained are: closed-form solutions (ind dimensions, and with arbitrary directional bias) for temporally uncorrelated jump diffusion and for the ‘fluid diffusion’ counterpart; a compact, general formula for the mean square displacement; the effects of a continuous spectrum of time scales in the holding-time distributions, etc. The dynamic mobility and the structure factor for ‘oscillatory diffusion’ are taken up in part 2.

Theory of the Gorsky effect for low interstitial concentrations

The formalism of the preceding paper is applied to work out the theory of the Gorsky effect, or anelastic relaxation due to the long-range diffusion of interstitials in a host lattice, for non-interacting (low-concentration) interstitials (e.g., H in Nb). It is shown how linear response theory (LRT) provides a number of advantages that simplify the solution of the problem and permit the handling of complications due to specimen geometry and stress inhomogeneity. The multiple-relaxation time creep function of Alefeldet al is first re-derived. Next, the dynamic responseand the short-time behaviour of the creep function are deduced exactly, and theω−1/2 fall-off of the internal friction at high frequencies is exhibited. Finally, it is pointed out that the true asymptotic behaviour of the dynamic response must be found by going beyond the diffusion equation model. A two-state random walk analysis is used to predict a cross-over to a trueω−1 asymptotic behaviour, and the physical reasons for this phenomenon are elucidated.

The two-state random walk

We develop asymptotic results for the two-state random walk, which can be regarded as a generalization of the continuous-time random walk. The two-state random walk is one in which a particle can be in one of two states for random periods of time, each of the states having different spatial transition probabilities. When the sojourn times in each of the states and the second moments of transition probabilities are finite, the state probabilities have an asymptotic Gaussian form. Several known asymptotic results are reproduced, such as the Gaussian form for the probability density of position in continuous-time random walks, the time spent in one of these states, and the diffusion constant of a two-state diffusing particle.