The Mean Square Distance Traveled by a Random-Walking Particle

APPENDIX 4.1. THE MEAN SQUARE DISTANCE TRAVELED BY A RANDOM-WALKING PARTICLE 

In the one-dimensional random-walk problem, the expression for j is found by mathematical induction as follows. Consider that after N - 1 steps, the sailor has progressed a distance If he takes one more step, the distance from the origin will be either 

This is the result for when the distance traveled after A - 1 steps is exactly acj, . hi general, however, one can only expect, for the value of the square of the distance at the (A - l)th step, an averaged value xl , in which case one must write 

At the start of the random walk, i.e., after zero steps, progress is given by 

This equation has been derived for a one-dimensional random walk, but it can be shown to be valid for three-dimensional random flights, too. 

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