The random walk (RW) is a very important model in science and engineering researches. It has been studied over hundreds years. However, there are still some overlooked problems on the RW. Here, we study the mean absolute distance of an N-step biased random walk (BRW) in a d-dimensional hyper-cubic lattice. In this short paper, we report the exact results for d = 1 and approximation formulae for d ≥ 2.
As a mathematical model, the random walk (RW) has been widely used in almost all branches of sciences and engineering [
In this short paper, we first give a brief description of the conventional results, and then report our study with some results on the BRW.
Let us consider the one dimensional BRW: a probability p of going forward and a probability (1 − p) of going backward with uniform step length L. Traditionally, the average distance gone in one step is expressed as:
The variance of a one step BRW can be calculated as:
After N such steps, the mean distance becomes
In the last expression,
variance of the N steps is
The standard deviation of the N steps is
In the case of the pure random walk (RW), i.e. when
RW is
This value, known widely in literature, is usually considered as the absolute distance of an N-step RW. This expression is independent of the dimensions of the lattice.
However, the mean absolute distance of the N-step RW in a d-dimensional hyper-cubic lattice cannot be expressed by (6), but is the following formula [
where
We compute the absolute distance for the N-step biased random walk (BRW). We find that (3) is a fairly good
approximation for a reasonably large N and p away from the neighborhood of
In Section 2, the exact results for d = 1 are presented. The approximation results for higher dimensions are shown in Section 3. A brief discussion is given afterward. A warning: it is possible that some of our results might have been already published in earlier literatures unknown to us.
For convenience, without loss of generality, we choose a step length of L = 1 in hereafter expressions.
For an N step biased random walker (BRW), if the walker moves forward n steps with probability p, and moves backwards (N ? n) steps with probability 1 − p, this is a binomial process with probability p. The absolute distance from the origin will be
After taking the weighted configuration average, the mean absolute distance of the one-dimensional BRW can be expressed as:
Using Mathematica [
where
Furthermore, we obtain the following relationship (via Mathematica):
The
For convenience, we have listed some exact results for small values of N as follows:
Further algebraic calculations yield the following recursion equations (for an even N, let N = 2 m in the following expressions):
i.e.
Additionally, because
If we let
When
In order to obtain these results, we use the following identity:
Therefore, Equation (15) can be expressed as a polynomial of x2:
In order to see the quantitative behavior of the averaged absolute distance as a function of
We have computed some typical values of the approximation error as follows (and partially shown in
In the range for which the linear approximation is invalid (the neighborhood of
where
The relationship between
by Equation (7). The second term coefficient
coefficient
We also compute the next three terms of Equation (18):
From
To verify the validity of the approximation (25), we plot
with exact results vs. the 3-term approximation Equation (25) for small values of x in
For a
along the i'th coordinate. We define
are the unit vectors. The absolute value of
nent is:
In the neighborhood of
Here,
Alternatively, we can consider the multidimensional BRW by transposing the coordinate system so that only one direction is biased. For instance, we can consider a diffusion process via an interface in which there is a pressure applied in the direction perpendicular to the interface. In this situation, all directions except the (biased) direction perpendicular to the interface follow a pure random walk. This model can be applied to many diffusion problems in physics and chemistry. Generally speaking, we can express the dimension, d as
distance of the g-dimensional
lattice, the (pure random walk) displacement is perpendicular to the (biased) direction of the BRW. In many problems, we are only concerned with the absolute distance in the (biased) direction, i.e., the projection portion of the total absolute distance. For this reason, we can model it as a modified 1d approach as follows: a probability q walking in the g-dimensional hyper-cubic lattice, a probability
We study two cases, one for reasonably large p and the other in the neighborhood of
Substituting this into (28) and using the results of the Appendix yields:
It is not surprising that the modification of higher dimensions on the 1d result requires only multiplication of the probability factor
For the second case, i.e., in the neighborhood of
where
Using the 1d results, (31) can be estimated to be a very simple formula:
where
The biased random walk has widely applications in various fields: for examples, a pressured diffusion process, an ionic injection with bombardment, a ballistic transport, financial market data, etc. For most natural phenomena and engineering processes, the particle number is about the order or a fraction of the Avogadro’s constant (~1023), the traditional treatment is good enough. However, the financial data and some high precision experimental data are far away from a large number, say 1010. For example in financial industry, the most active index futures, SP500, has only the order of 105 open interest contracts before rolling the date. The daily trading volume is one or two order smaller than the open interest. Therefore, when the particle number is not large enough, one has to consider the new behavior. The present results are just the better quantitative descriptions for those phenomena. In some high precision experiments in physical sciences, one may have to measure parameters with small amount of particles. To quantify the property, the present results can provide better mathematical expressions.
The authors would like to Angel Yang for her helpful discussion.
ZhongjinYang,CassidyYang, (2015) How Far Can a Biased Random Walker Go?. Journal of Applied Mathematics and Physics,03,1159-1167. doi: 10.4236/jamp.2015.39143
In this appendix, we present a very useful approximation formula: for a large enough M and
It is easy to see that
Numerically, we computed the values of
very good approximation.