“Random Walk--2-D”的概念、定义、习题解题方法及翻译-数学辞典

单词

Random Walk--2-D

释义

Random Walk--2-D

In a Plane, consider a sum of 2-D Vectors with random orientations. Use Phasornotation, and let the phase of each Vector be Random. Assume unit steps are takenin an arbitrary direction (i.e., with the angle uniformly distributed in and not on aLattice), as illustrated above. The position in the Complex Plane after steps is then given by

(1)

which has Absolute Square


			(2)

Therefore,

(3)

Each step is likely to be in any direction, so both

and

are Random Variables with identical Means of zero, and their difference is also a random variable. Averaging over thisdistribution, which has equally likely Positive and Negative values yields an expectation value of 0, so

(4)

The root-mean-square distance after

unit steps is therefore

(5)

so with a step size of

, this becomes

(6)

In order to travel a distance

(7)

steps are therefore required.

Amazingly, it has been proven that on a 2-D Lattice, a random walk has unity probability of reaching any point(including the starting point) as the number of steps approaches Infinity.

See also Pólya's Random Walk Constants, Random Walk--1-D, Random Walk--3-D

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