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

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 contin- uum. In all cases, the diffusive nature of walker, breaks down due to tiring which is quite obvious. In one dimension, the distribution of the displace- ment 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 reduces 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 almost insensitive to the tiring factor. The prob- ability 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.


I. 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 are 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 avoiding 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 is studied in quantum random walk recently [9]. The drift and trapping in biased diffusion on disordered lattices is also studied [10].
Very recntly, 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 thoery of mean first passage time for jump processes are 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 was 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 is 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 [24] of a walker is important in various aspects, namely, the fluorescence quenching in which a fluoresecent molecule stops while reacting with a quencher, firing neurons when the fluctuating voltage level first reaches a specified 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 is 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 is also reported here. This paper is organised as follows: In the next section (section-II) 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-III.

II. 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 R(t) = e −αt . 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, α 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 x l (t) = e −αt . A walker starts its journey from the origin having the equal probability of choosing the left and right direction. The updating rule, in one dimemsional 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 2π. Its motion can be represented mathematically as: The displacement at time t is r(t) = x 2 (t) + y 2 (t). 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 Fig-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 Fig-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 (< r 2 >∼ t) as observed in normal (α = 0) walk. It is also observed that the motion stops earlier if the tiring factor α increases. 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 α = 0.0 (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 Fig-3. Here, it may be mentioned that the values of α and the maximum time allowed (N t ) are such that the walker gets frozen (due to exponential decrease of stepsize after such long time). The distribution shown in Fig-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 α through the scaling.
What will be the probability of return(P R ) 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 ([−r z , r z ]) centered around the origin. Now put a large number (N s ) of walker at origin and allow them to walk (with different random sequence) upto a certain time (N t ) and then check how many walkers return within the preassigned returning zone (of size r z ). The calculated fraction is the probability of return (within time of observation N t ) in this particular model. In the lattice model this probability is 1, which can also be derived from exact calculations [23]. In this model of tired walker, considering r z = 0.5, P R = 0.992 for α = 0.0. This numerical estimate of returm probability agrees well with exact calculation of return probability (P R = 1) [23] 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 (α = 0.001) walker, P R = 0.955 and for heavily tired walker (α = 0.01), P R = 0.874. In Fig-4, the P R is plotted against N t for various values of α. 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 α and shown in Fig-5. It shows that the P R grows as r z increases in the case of tired walker (α = 0.001, 0.01), but P R does not depend on r z for normal walker. It is important to note here that even for heavily tired (α = 0.01) 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 (t r ) look like ? The probability distribution of first returning time (t r ) of a tired random walker is shown in Fig-6. A normal walker (α = 0.0) shows a scale invariant (P (t r ) ∼ t −β r ) distribution of first returning time (t r ). The exponent estimated is β ≃ 1.49. This result agrees well with analytical result [24], where it is found P (t r ) ∼ t −3/2 r . However, this scale invariant nature of the distribution of first returning time, breaks down in the cases of tired walking (for α = 0.001, 0.01) (see Fig-6). More detail investigation is required to propose any functional behaviour of P (t r ) for α = 0.
In one dimension, how long (t e ) a tired walker takes to exit (first time) from a zone ([−r e , r e ]) ? The probability distribution (P (t e )) of first passage time (t e ) (for a fixed value of r e = 25.0), is studied for different values of α and shown in Fig-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 ([−r e , r e ]) linear (in one dimension) region. If it would be defined as the probability to escape through a given point (say x = r e ) the power law (P (t e ) ∼ t −1.49 e ) distribution in long time limit (t e → ∞) is found which supports the analytical prediction (P (t e ) ∼ t −3/2 e ) [24]. This is shown in Fig-7 What is the probability of exit (P e ) of a tired walker one dimensional continuum ? The exit probability (for a fixed time of observation N t ) from a zone of absolute distance r e (measured from the origin) is also studied, in one dimension, as a function of r e and shown in Fig-8. Here, the exit probability, of a tired walker, was found to decreases as the absolute distance of zone (r e ) increases. However, it remains fixed (nearly 1) for a normal walker [23]. 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 < r 2 > is proportinal to the time t for α = 0.0, reveals the conventional diffusive (< r 2 >∼ t) behaviour. However, a moderately tired (α = 0.001&0.0005) walker does not show long time diffusive behaviour. This is quite obvious and already shown in Fig-2.
The distribution of absolute displacement is nonmonotonic unimodal function. It is shown in Fig-9. It is observed that the maximum probability of finding the walker at a distance (r m ) from the origin and the average distance (r) both decreases as the tiring factor (α) increases. In this case, r m = 64.0,r = 89.13 for α = 0.0, r m = 15.0,r = 20.04 for α = 0.001 and r m = 5.0,r = 6.42 for α = 0.01. Here also this distribution is practically the density distribution of frozen walker.
What will be the probability of return (P R ) of a tired walker in planar continuum ? The probability of return within a circle of return having radius r z = 0.5 is studied as a function of maximum time of observation N t and shown in Fig-10. In planar continuum, a tired walker has a probability of return in a circle of radius r z = 0.5 as follows: for α = 0.0, P R = 0.737, for α = 0.001, P R = 0.620 and for α = 0.01, P R = 0.524.
For a fixed value of N t , the probability of return of a tired walker in planar contunuum, grows as the radius of returning zone increases. This is shown in Fig-11.
Here, like in one dimensional tired walker, the probability distribution of first returning time (t r ) shows a scale invariance (P (t r ) ∼ t −β r ) for α = 0.0 with β ≃ 1.10. However, the analytic result [24] suggests P (t r ) ∼ 1/(t r (ln(t r ) 2 ). The possible reason of disagreement may be stated as follows: in the analytic calculation of P (t r ), 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, P (t r ) is defined as the probability of return (first time) within a circular zone of radius r z . As the tiring factor (α == 0) increases, the scale invariance nature of the distribution on first returning time breaks down. This is demonstrated in Fig-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 Fig-13. As the α increases, the most probable first passage time and mean first passage time decreases.
The exit probability (for a fixed time of observation N t ) from a circular zone of radius r e (measured from the origin) is also studied, in two dimensions, as a function of r e and shown in Fig-14. Here, the exit probability was found to decreases as the radius of circular zone (r e ) 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 (α = 0) remains unchanged (nearly 1) as r e increases.
The tired walk in three dimensional continuum can also be generalized. The updating of coordinates obey the following rule: where R(t) = e −αt , θ is uniformly distributed random angle between 0 and π and φ is uniformly distributed random angle between 0 and 2π. The displacement at time t is r(t) = x 2 (t) + y 2 (t) + z 2 (t).
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 < r 2 > is proportinal to time t for α = 0.0, reveals the diffusive behaviour (not shown). However, a tired (α = 0) 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 Fig-15. It is observed that the maximum probability of finding the walker at a distance (r m ) from the origin and the mean displacement (r) both decreases as the tiring factor (α) increases. In this case, r m = 72.0,r = 91.59 for α = 0.0, r m = 16.0, r = 20.65 for α = 0.001 and r m = 3.0,r = 6.96 for α = 0.01.
What will be the probability of return in 3D continuum ? The probability of return within a sphere of return having radius r z = 0.5 is studied as a function of maximum time of observation N t and shown in Fig-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 r z = 0.5 is almost insensitive (P R = 0.226) of the tiring factor α.
For a fixed value of N t , 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 α. This is shown in Fig-17.
In 3D continuum, the probability distribution of first returning time (t r ) shows a scale invariance (P (t r ) ∼ t −β r ) for α = 0.0. The exponent estmated β ≃ 1.48. Accidentally, this is close the analytical prediction (P (t r ) ∼ t −3/2 r ) [24]. As the tiring factor (α = 0) increases, the scale invariance of the distribution on first returning time, breaks down. This is demonstrated in Fig-18.
In three 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 Fig-19. As the α 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 r e and shown in Fig-20. Here, the exit probability, of a tired walker, was found to decreases as the radius of circular zone (r e ) increases. However, like the earlier cases, it reamins same (nearly 1) for all r e . Here also the rate, of fall of the exit probability of a tired walker, increases as the tiring factor α increases.

III. 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 α = 0 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 [24] 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 probability was 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 [24] 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 was observed, 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 formulations of first passage properties for tired walk has to be developed following the same already developed [24] for normal walk (α = 0).