Random flights. A particle follows in space a path that...

Random flights. A particle follows in space a path that consists of $N$ equal steps, each of length $s .$ The direction in space of each step is entirely random, with no relation or correlation between any two steps. The total displacement is $$ \mathbf{S}=\sum_{i=1}^{N} \mathbf{s}_{i} $$ Show that the mean square displacement between initial and final positions is $\left\langle\mathrm{S}^{2}\right\rangle=\mathrm{Ns}^{2}$, where \langle\rangle denotes mean value. [Hint: The assumption that the direction of every step is independent of the direction of every other step means that $\left\langle s_{i} \cdot \mathrm{s}_{j}\right\rangle=0$ for all $i$ and $i$, except $\left.i=j .\right]$

Random flights. A particle follows in space a path that consists of $N$ equal steps, each of length $s .$ The direction in space of each step is entirely random, with no relation or correlation between any two steps. The total displacement is
$$
\mathbf{S}=\sum_{i=1}^{N} \mathbf{s}_{i}
$$
Show that the mean square displacement between initial and final positions is $\left\langle\mathrm{S}^{2}\right\rangle=\mathrm{Ns}^{2}$, where \langle\rangle denotes mean value. [Hint: The assumption that the direction of every step is independent of the direction of every other step means that $\left\langle s_{i} \cdot \mathrm{s}_{j}\right\rangle=0$ for all $i$ and $i$, except $\left.i=j .\right]$

Key Concepts

Random Walk

A random walk is a process where a particle or entity takes successive steps, with each step determined by random, uncorrelated directions. This concept models various physical and financial systems and is fundamental in studying diffusive processes, as it establishes that the displacement of the particle accumulates in a way that can often be described statistically.

Mean Square Displacement

Mean square displacement (MSD) is a statistical measure used to quantify the average of the squared distances of a particle from its initial position over time. In the context of random walks, it provides a clear indication of the spread of the particle's position, showing that MSD increases linearly with the number of steps, which is critical in characterizing diffusive behavior.

Independence of Steps

The concept of step independence asserts that the direction of each step is completely independent of the others. This lack of correlation between steps means that when calculating the overall displacement, cross-terms involving different steps average out to zero, simplifying the analysis and leading to the conclusion that the mean square displacement is the sum of the squares of the individual steps.

Vector Addition in Displacement Analysis

Representing each step as a vector enables the application of vector algebra to determine the particle’s total displacement. The properties of vector addition and the dot product are used to calculate the mean square displacement by expanding the square of the sum of vectors, where the independence of the directions ensures that only the self dot products contribute to the final result.

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