Question

Compute the mean departure of a one-dimensional random walker from its starting point. In particular, use the fact that the mean excursion can be written as \(\langle R\rangle=\left(\left\langle n_{\mathrm{r}}\right\rangle-\left\langle n_{\ell}\right\rangle\right) a\) and that the probability distribution for \(n_{\mathrm{r}}\) right steps out of a total of \(N\) steps is given by the binomial distribution.

Short answer

The mean excursion of a one-dimensional random walker can be computed using the given formula and the binomial distribution. The outcome relays the influence of the step size and the difference in step direction probabilities on the mean excursion.

Step-by-step solution

Calculate \(\langle n_{\mathrm{r}}\rangle\)

For a binomial distribution, the expected value of the number of successes, in this case, right steps, can be calculated as \(\langle n_{\mathrm{r}}\rangle = Np\), where N is the total number of steps and p is the probability of a right step.

Calculate \(\langle n_{\mathrm{\ell}}\rangle\)

Similar to step 1, we calculate the expected value of the number of left steps as \(\langle n_{\mathrm{\ell}}\rangle = N(1-p)\), where N is the total number of steps and (1-p) is the probability of a left step.

Calculate the mean excursion

Using the values of \(\langle n_{\mathrm{r}}\rangle\) and \(\langle n_{\mathrm{\ell}}\rangle\) obtained from steps 1 and 2 in equation \(\langle R\rangle = (\langle n_{\mathrm{r}}\rangle - \langle n_{\mathrm{\ell}}\rangle)a\), we find the mean excursion of the random walk.

Analyze the outcome

Studying the obtained result, during random walks, the influence of the walker's step size (a) and the dissimilarity in probabilities of step directions (right or left) on the mean excursion can be understood.

Key concepts

Binomial Distribution

In the context of a random walk, the binomial distribution plays a significant role. It is a probability distribution that describes the number of successes in a fixed number of independent trials. Each trial results in a success with probability \( p \) and a failure with probability \( 1-p \). For a random walker, a successful trial might be taking a step to the right, while a left step could be considered a failure. The total number of trials corresponds to the total number of steps \( N \) in the walk.
To compute the number of right steps, you apply the formula for the expected value of a binomial distribution: \( \langle n_{\mathrm{r}} \rangle = Np \). This formula represents the average number of right steps taken during repeated experiments, each consisting of \( N \) total steps.
The symmetry of the binomial distribution makes it particularly suited for modeling random walks, allowing for straightforward calculations of step expectancy in both directions. It highlights the random nature of each step the walker takes, demonstrating that even with equal probabilities for left and right steps, the outcome is spread over various possible walks.

Mean Excursion

The mean excursion of a random walker is a measure of the average distance the walker will deviate from the starting point after a series of random steps. It is a helpful concept in understanding how far a walker typically drifts due to randomness.
To calculate the mean excursion, utilize the formula \( \langle R \rangle = (\langle n_{\mathrm{r}} \rangle - \langle n_{\mathrm{\ell}} \rangle)a \). Here, \( \langle n_{\mathrm{r}} \rangle \) and \( \langle n_{\mathrm{\ell}} \rangle \) are the average counts of right and left steps respectively, and \( a \) designates the step size.
The formula breaks down to finding the difference between the expected right and left steps. This is then scaled by the step size to find the mean deviation from the starting point. If the probabilities for right and left are equal, the mean excursion naturally trends toward zero, representing a balanced random walk with no net movement direction over time.

Probability Theory

Probability theory forms the backbone of analyzing random walks by providing the tools to model and predict the behavior of different random events. At its core, it studies the likelihood of different outcomes in uncertain situations, such as those faced by a random walker.
In a random walk problem, probability theory helps in determining the likelihood of taking a right or left step. This can be directly related to their respective probabilities \( p \) and \( 1-p \).

• Probability of Right Step: \( p \)
• Probability of Left Step: \( 1-p \)

With this framework, we can easily compute the expected outcomes for a sequence of random events. For example, based on the calculated probabilities, one can find \( \langle n_{\mathrm{r}} \rangle = Np \) and \( \langle n_{\mathrm{\ell}} \rangle = N(1-p) \).
Probability theory, in this instance, offers a well-structured method to model the random variability and equip students with the tools to quantify and understand the uncertainty involved in real-life scenarios.