Model and Statistical Analysis of the Motion of a Tired Random Walker in Continuum

Muktish Acharyya^*

Department of Physics, Presidency University, Calcutta, India.

DOI: 10.4236/jmp.2015.614208                

Abstract

The model of a tired random walker, whose jump-length decays exponentially in time, is proposed and the motion of such a tired random walker is studied systematically in one-, two- and three-dimensional continuum. In all cases, the diffusive nature of walker breaks down due to tiring which is quite obvious. In one-dimension, the distribution of the displacement of a tired walker remains Gaussian (as observed in normal walker) with reduced width. In two and three dimensions, the probability distribution of displacement becomes nonmonotonic and unimodal. The most probable displacement and the deviation reduce as the tiring factor increases. The probability of return of a tired walker decreases as the tiring factor increases in one and two dimensions. However, in three dimensions, it is found that the probability of return is almost insensitive to the tiring factor. The probability distributions of first return time of a tired random walker do not show the scale invariance as observed for a normal walker in continuum. The exponents, of such power law distributions of first return time, in all three dimensions are estimated for normal walker. The exit probability and the probability distribution of first passage time are found in all three dimensions. A few results are compared with available analytical calculations for normal walker.

Keywords

Return Probability, Exit Probability, First Passage Time

1. Introduction

In statistical physics, process of polymerization [1] [2] , diffusion [3] in restricted geometry etc. are some classic phenomena, which have drawn much attention of the researcher in last few decades. The underlying mechanism of such physical phenomena is tried to explain by random walk [4] . Different types of random walk are studied on the lattice in different dimensions by the method of computer simulation. The absorbing phase transition in a conserved lattice gas with random neighbour particle hopping is studied [5] . Quenched averages for self avoid- ing walks on random lattices [6] , asymptotic shape of the region visited by an Eulerian walker [7] , linear and branched avalanches are studied in self avoiding random walks [8] ; effects of quenching are studied in quantum random walk recently [9] . The drift and trapping in biased diffusion on disordered lattices are also studied [10] .

Very recently, some more interesting results on random walk were reported. The average number of distinct sites visited by a random walker on the random graph [11] , statistics of first passage time of the Browian motion, conditioned by maximum value of area [12] are studied recently. It may be mentioned here that the first passage time in complex scale invariant media was studied [13] . The theory of mean first passage time for jump pro- cesses is developed [14] and verified by applying in Levy flights and fractional Brownian motion. The statistics of the gap and time interval between the highest positions of a Markovian one dimensional random walker [15] , the universal statistics of longest lasting records of random walks and Levy flights are also studied [16] .

The random walks in continuum are studied to model real life problems. The exact solution of a Brownian inchworm model and self-propulsion was also studied [17] ; theory of continuum random walks and application in chemotaxis was developed [18] . Random walks in continuum were also studied for diffusion and reaction in catalyst [19] . Very recently, the random walk in continuum is studied with uniformly distributed random size of flight [20] . The statistics of Pearson walk are studied [21] [22] in two dimensions for shrinking stepsize and found a transition of the endpoint distribution by varying the initial stepsize.

The living random walker in continuum gradually becomes tired as the time passes, in reality. This would reduce its energy, as a result the size of flight gets reduced gradually with time. The first-passage properties [23] of a walker are important in various aspects, namely, the fluorescence quenching in which a fluoresecent mole- cule stops while reacting with a quencher, firing neurons when the fluctuating voltage level first reaches a speci- fied value, in econophysics, the execution of buy/sale orders when a stock price first reaches a threshold. What will be the first passage properties if the stepsize of a Pearson walker decreases exponentially in time? In this paper, addressing this particular problem, a model of tired random walker is proposed in continuum and statistics of its motion are studied systematically in one-, two- and three-dimensional continuum. The first passage properties, return and exit probabilities are studied here. The numerical results of detailed statistical analysis of the motion of a tired random walker are also reported here. This paper is organised as follows: In the next section (Section 2) the model of tired random walk is proposed and the results obtained from numerical simulations are given. The paper ends with a summary given in Section 3.

2. Model and Results

Generally, the motion of a random walker is studied by considering the time (t) independent size (R) of flight in each move. In this study, the model of a tired random walker is proposed in such a way that the size of flight of a walker decreases exponentially as. A simple logic behind it may be stated as follows: if a living cell is moving continuously, its energy (basically kinetic energy) gradually decreases and hence the velocity, which in turn reduces its size of flight (i.e., jump-length per unit time). Here, is tiring factor. The statistical behaviour of such a tired random walker is studied in one-, two- and three-dimensional continuum. It may be noted here that such kind of behaviour of a tired random walker cannot be studied on the lattice.

In one dimension, the size of flight in each time step is. A walker starts its journey from the origin having the equal probability of choosing the left and right direction. The updating rule, in one dimem- sional tired walk, may be expressed as:

In two dimensions (planar continuum), the tired walker starts its journey from the origin and it has a uniform probability of choosing any random direction (θ) distributed between 0 and. Its motion can be represented mathematically as:

(1)

The displacement at time t is. In planar continuum, the area of the region visited by a tired walker is obviously shorter than that visted by a normal walker, in a specified course of time. A typical such comparison is shown in Figure 1 with α = 0.001. As a result, the mean square displacement does not show diffusive behaviour as shown by a normal walker. In long time, it gets saturated (motion stops practically).

A typical such comparison is shown in Figure 2 for α = 0.001 and α = 0.0005. The similar behaviours are also observed in one and three dimensions (not shown). The tired walk is not diffusive as observed in normal walk. It is also observed that the motion stops earlier if the tiring factor α increases.

Figure 1. A sample random walk in 2D continuum. The top one shows a walk for and the bottom one shows a tired walk for. Here, in both figures same sequence of random numbers are used. Note the scales of the axes of two figures. in both cases.

Now, let it be discussed systematically in one, two and three dimensions. In one dimensions, the probability distribution of the displacements of a walker are studied for (normal), 0.001(moderately tired) and 0.01(heavily tired). As usual, the distribution is normal (Gaussian) with zero mean in all the cases. However, as the tiring factor increases the distribution becomes sharper and sharper. These are depicted in Figure 3. Here, it may be mentioned that the values of α and the maximum time allowed are such that the walker gets frozen (due to exponential decrease of step-size after such long time). The distribution shown in Figure 3, is practically the density distribution of frozen walker. It would be interesting to study the density distribution of these frozen walker as a function of through the scaling.

What will be the probability of return in one dimension? First of all, in continuum one should be careful in defining the probability of return. In the lattice the probability of return is defined as the walker returns to its initial starting point. However, in continuum, it is quite unlikely that a tired walker returns to its initial starting point. Here, one may think that whether the tired walker returns within a linear zone centered around the origin. Now put a large number of walker at origin and allow them to walk (with different random sequence) upto a certain time and then check how many walkers return within the preassigned returning zone (of size). The calculated fraction is the probability of return (within time of observation) in this particular model. In the lattice model this probability is 1, which can also be derived from exact calculations [24] . In this model of tired walker, considering, for. This numerical estimate of returm probability agrees well with exact calculation of return probability (P_R = 1)

Figure 2. The mean square displacement versus time (t) in two dimensions for various values of (marked by different symbols). The tired walkers do not show the normal diffusive behaviour.

Figure 3. The distribution of the displacements (x) of a tired random walker in one dimension. (o) represents, (,) represents, and (*) represents. Note that tiring reduces the width of the distribution.

[24] in one dimensional normal random walk. It may be noted here that for α = 0, the walker returns at origin (the starting point also) and the probability of return can be compared to that obtained in random walk on one dimensional lattice. As the tiring factor increases, P_R decreases. For moderately tired walker, P_R = 0.955 and for heavily tired walker, P_R = 0.874. In Figure 4, the P_R is plotted against for various values of. Now, this probability of return (P_R) must depend on the size (r_z) of returning zone. To study the dependences of P_R on r_z, P_R is studied as a function of r_z for different values of and shown in Figure 5. It shows that the P_R grows as r_z increases in the case of tired walker, but P_R does not depend on r_z for normal walker. It is important to note here that even for heavily tired random walker, the size of the flight, after t = 10 is larger than 0.90. So, the range of values of r_z, chosen here, does not have any chance that the walker remains in the returning zone immediately after starting its journey. So, the choice r_z = 0.5 is quite safe to study the probability of return in this context.

How long a tired walker takes to return first time within returing zone? How does the distribution of this first returning time look like? The probability distribution of first returning time of a tired random

Figure 4. Probability of return (P_R) plotted against the maxi- mum time N_t in one dimension. Different symbols correspond to different values of. (o),(g) and (*). Here in all cases. The absolute dis- tance of returning zone is here.

Figure 5. Probability of return (P_R) plotted against the absolute distance r_z of returning zone in one dimension. Diffe- rent symbols correspond to different values of. (o),(g) and (*).

walker is shown in Figure 6. A normal walker shows a scale invariant distribution of first returning time. The exponent estimated is. This result agrees well with analytical result [23] , where it is found. However, this scale invariant nature of the distribution of first returning time, breaks down in the cases of tired walking (for) (see Figure 6). More detail investigation is required to propose any functional behaviour of for.

In one dimension, how long a tired walker takes to exit (first time) from a zone? The pro- bability distribution of first passage time (for a fixed value of), is studied for diffe- rent values of α and shown in Figure 7(a). As the α increases, the most probable first passage time decreases. It should be noted here that the probability of first passage is defined (in this study) as the probability to escape in a given time, from a bounded linear (in one dimension) region. If it would be defined as the pro-

bability to escape through a given point (say) the power law distribution in long time

Figure 6. Probability distribution of first returning time in one dimension. Different symbols correspond to different values of. (o),(g) and (*). The solid line is.

Figure 7. Probability distribution of first passage time in one dimension. (a) Different symbols correspond to different values of. (o),(g) and (*). Here, , and. In (a) the first passage is defined as the probability of exit first from a bounded linear region around the origin; (b) the first passage is defined to cross first a point (here). The solid line is supporting analytical prediction [23] .

limit is found which supports the analytical prediction [23] . This is shown in Figure 7(b).

What is the probability of exit of a tired walker one dimensional continuum? The exit probability (for a fixed time of observation) from a zone of absolute distance (measured from the origin) is also studied, in one dimension, as a function of and shown in Figure 8. Here, the exit probability, of a tired walker, was found to decreases as the absolute distance of zone increases. However, it remains fixed (nearly 1) for a normal walker [24] . It may also be noted that, the rate of fall of exit probability increases as the tiring factor (α) increases.

In two dimensions, the motion of a tired random walker is studied by using the rule given in Equation (1). Here, the mean square displacement is proportinal to the time t for, reveals the conventional diffusive behaviour. However, a moderately tired walker does not show long time diffusive behaviour. This is quite obvious and already shown in Figure 2.

The distribution of absolute displacement is nonmonotonic unimodal function. It is shown in Figure 9. It is observed that the maximum probability of finding the walker at a distance from the origin and the average distance () both decreases as the tiring factor (α) increases. In this case, , for, , for and, for. Here also this distri- bution is practically the density distribution of frozen walker.

What will be the probability of return of a tired walker in planar continuum? The probability of return within a circle of return having radius is studied as a function of maximum time of observation and shown in Figure 10. In planar continuum, a tired walker has a probability of return in a circle of radius as follows: for, , for, and for,.

For a fixed value of, the probability of return of a tired walker in planar contunuum, grows as the radius of returning zone increases. This is shown in Figure 11.

Here, like in one dimensional tired walker, the probability distribution of first returning time shows a scale invariance for with. However, the analytic result [23] suggests

. The possible reason of disagreement may be stated as follows: in the analytic calcu-

lation of, it was defined as the probability of return exactly at the origin from where the walker has started its journey. However, in the numerical simulation, is defined as the probability of return (first time) within a circular zone of radius. As the tiring factor increases, the scale invariance nature of the distribution on first returning time breaks down. This is demonstrated in Figure 12.

In two dimensions, the distribution of first passage time (for a fixed distance r_e = 25.0), is studied for different values of α and shown in Figure 13. As the α increases, the most probable first passage time and mean first

Figure 8. Exit probability P_e plotted against r_e in one dimension. Different symbols correspond to different values of α. α = 0.0 (o),(g) and (*). Here, and.

Figure 9. The distribution of displacements (r) of a tired random walker in two dimensions. (o) represents, , , (,) represents, , and (*) represents, ,.

Figure 10. Probability of return plotted against N_t in two dimen- sions. Different symbols correspond to different values of. (o),(,) and (*). Here in all cases. The radius of circular returning zone is.

passage time decreases.

The exit probability (for a fixed time of observation) from a circular zone of radius (measured from the origin) is also studied, in two dimensions, as a function of and shown in Figure 14. Here, the exit pro- bability was found to decreases as the radius of circular zone increases. Here also, the rate of fall of exit probability increases as the tiring factor (α) increases. However, the exit probability of a normal walker remains unchanged (nearly 1) as increases.

The tired walk in three dimensional continuum can also be generalized. The updating of coordinates obey the following rule:

(2)

where, θ is uniformly distributed random angle between 0 and and is uniformly distributed

Figure 11. Probability of return plotted against the radius of return- ing zone in two dimensions. Different symbols correspond to diffe- rent values of. (o),(,) and (*).

Figure 12. Probability distribution of first returning time of two dimensions. Different symbols correspond to different values of. (o),(,) and (*). Solid line represents. The dotted line is [23] .

random angle between 0 and. The displacement at time t is.

In 3D continuum, the motion of a tired random walker is studied by using the rule given in Equation (2). Here, the mean square displacement is proportinal to time t for, reveals the diffusive behaviour (not

shown). However, a tired walker does not show long time diffusive behaviour (not shown).

The probability distribution of absolute displacement (or the density distribution of frozen walker in reality) in 3D continuum is observed to be a nonmonotonic unimodal function. It is shown in Figure 15. It is observed that the maximum probability of finding the walker at a distance from the origin and the mean displacement both decreases as the tiring factor (α) increases. In this case, , for, , for and, for.

What will be the probability of return in 3D continuum? The probability of return within a sphere of return having radius is studied as a function of maximum time of observation and shown in Figure 16. In 3D continuum, unlike the cases in 1D and 2D continuum, a tired walker has a probability of return in a sphere of radius is almost insensitive of the tiring factor α.

Figure 13. Probability distribution of first passage time in two dimensions. Different symbols correspond to different values of α. α = 0.0 (o), α = 0.001 (g) and α = 0.02 (*). Here, , and.

Figure 14. Exit probability P_e plotted against r_e in two dimensions. Diffe- rent symbols correspond to different values of α. α = 0.0 (o), α = 0.001 (g) and α = 0.01 (*). Here, and.

For a fixed value of, the probability of return of a tired walker in 3D contunuum, grows as the radius of returning zone increases keeping the independence on tiring factor. This is shown in Figure 17.

In 3D continuum, the probability distribution of first returning time shows a scale invariance

for. The exponent estmated. Accidentally, this is close the analytical pre-

diction [23] . As the tiring factor increases, the scale invariance of the distribution on first returning time, breaks down. This is demonstrated in Figure 18.

In three dimensions, the distribution of first passage time (for a fixed distance), is studied for different values of α and shown in Figure 19. As the increases, the most probable first passage time and the mean first passage time decreases.

The exit probability (for a fixed time of observation N_t) from a spherical zone of radius r_e (measured from the origin) is also studied, in two dimensions, as a function of and shown in Figure 20. Here, the exit pro- bability, of a tired walker, was found to decreases as the radius of circular zone increases. However, like the earlier cases, it reamins same (nearly 1) for all. Here also the rate, of fall of the exit probability of a tired walker, increases as the tiring factor α increases.

Figure 15. The distribution of the displacements of a tired random walker in three dimensions. (o) represents α = 0.0, , , (,) re- presents α = 0.001, , and (*) represents α = 0.01, ,.

Figure 16. Probability of return (P_R) versus N_t in three dimensions. Diffe- rent symbols correspond to different values of α. α = 0.0 (o), α = 0.001 (,) and α = 0.01 (*). Here in all cases N_s = 10⁵. The radius of spherical re- turning zone is r_z = 0.5.

Figure 17. Probability of return (P_R) versus r_z in three dimensions. Diffe- rent symbols correspond to different values of α. α = 0.0 (o), α = 0.001 (,) and α = 0.01 (*).

Figure 18. Probability distribution of first returning time in three dimensions. Different symbols correspond to different values of α. α = 0.0 (o), α = 0.001 (,) and α = 0.01 (*). Solid line represents.

Figure 19. Probability distribution of first passage time in three dimensions. Different symbols correspond to different values of α. α = 0.0 (o), α = 0.001 (g) and α = 0.002 (*). Here, , and.

Figure 20. Exit probability P_e plotted against r_e in three dimensions. Diffe- rent symbols correspond to different values of α. α = 0.0 (o), α = 0.001 (g) and α = 0.01 (*). Here, and.

3. Summary

In this article, a model of tired random walker in continuum is proposed. Generally, a random walker moves with constant size of flight. However, as the time passes, if the walker gets tired, one should think of a time dependent size of flight. Here, this size of flight decays exponentially with time. The motion of such a tired walker is studied in one-, two- and three-dimensional continuum. In this statistical investigation, the distribution of the absolute displacement, mean displacement, probability of return (within a specified zone), distribution of time of first return are studied systematically. In one- and two-dimensional continuum, the probability of return decreases as the tiring factor increases. However, in three-dimensional continuum, this probability of return seems to be independent of the tiring factor. The distribution of first returning time in all dimensions (for normal walker with tiring factor α = 0), shows power law behaviours. This scale invariance of the distribution of first returning time breaks down for in all dimensions. In the study of first returning probability, a very important point should be mentioned. For α = 0, the probability of return could be compared with that calculated analytically [23] in one dimension only, where the walker can return to the initial point. In higher dimensions, it returns within a circular (spherical) zone in two (three) dimensions.

The exit probability and the distribution of first passage time are studied. In all dimensions, the exit pro- bability is found to decrease as the size of the zone (from where the tired walker exits out) increases. The rate of decrease of the exit probability was found to increase as the tiring factor α increases. Here also the probability of first passage (for α = 0) can only be compared with analytical calculations [23] in one dimension, if it is defined as the probability of escape through a particular point.

The first passage time is defined (in this simulational study) as the time required by a walker to exit from a specified zone. This time has a distribution and this distribution is studied for various values of α. It is observed that, in all dimensions, the most probable first passage time decreases as α increases. A rigorous analysis and possible scaling behaviour (if any) may be investigated.

Some more interesting studies can be done in this field. In this paper, only the numerical results are reported. A rigorous mathematical formulation of first passage properties for tired walk has to be developed following the same already developed [23] for normal walk.

The possibilities of scaling of distribution of return time, distribution of first passage time, distribution of distances and exit probabilities with respect to the tiring factor (α) have also to be explored.

Acknowledgements

Author would like to express sincere gratitudes to D. Dhar, S. S. Manna and P. Sen for important discussions. The library facilities of Calcutta University is gratefully acknowledged.

Conflicts of Interest

The authors declare no conflicts of interest.

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