Random Walk Model for Diffusing Molecules

Q5. A simple model of a diffusing molecule is a “random walk.” Suppose that a molecule is confined to one dimension and takes random steps along a number line. It starts at x = 0, and after one step it is either at x = +1 with probability 1/2, or x = -1 with probability 1/2. Complete the table to show the probability of finding the molecule at each position (x = -4 to x = +4) after 1, 2, 3, and 4 steps.

Background

Topic: Statistical Physics / Random Walks

This question explores the concept of a random walk, which is a fundamental model for diffusion and Brownian motion in physics. The random walk describes how a particle moves step by step, with each step being independent and equally likely to go in either direction. This is a key idea in understanding how molecules spread out over time due to random motion.

Key Terms and Formulas

• Random Walk: A process where an object moves in steps, each of which is determined randomly.

• Probability: The likelihood of the molecule being at a certain position after a given number of steps.

• Binomial Distribution: The probability of being at position $x$ after $n$ steps is given by: $P(x, n) = \binom{n}{k} \left(\frac{1}{2}\right)^n$ where $k = \frac{n + x}{2}$ (number of steps to the right), and $n - k$ is the number of steps to the left. This formula is valid only if $n + x$ is even and $|x| \leq n$.

Step-by-Step Guidance

1. Start by considering the possible positions after each step. After 1 step, the molecule can only be at $x = +1$ or $x = -1$.

2. For each subsequent step, the molecule can move one unit to the left or right from its current position. For example, after 2 steps, possible positions are $x = +2, 0, -2$.

3. Use the binomial distribution formula to calculate the probability of being at each position $x$ after $n$ steps. Remember, $k = \frac{n + x}{2}$ must be an integer between 0 and $n$.

4. Fill in the table for each value of $x$ and $n$ (steps), using the formula above. For positions where $n + x$ is odd or $|x| > n$, the probability is zero.

5. Continue this process for 1, 2, 3, and 4 steps, but stop before calculating the final probabilities for each position. Set up the expressions you would use for each cell in the table.

Try solving on your own before revealing the answer!

Final Answer: Probabilities for Each Position After 1–4 Steps

Here is the completed table with the probabilities for each position after 1, 2, 3, and 4 steps:

Position	1 step	2 steps	3 steps	4 steps
x = +4	0	0	0	1/16
x = +3	0	0	1/8	0
x = +2	0	1/4	0	1/4
x = +1	1/2	0	3/8	0
x = 0	0	1/2	0	3/8
x = -1	1/2	0	3/8	0
x = -2	0	1/4	0	1/4
x = -3	0	0	1/8	0
x = -4	0	0	0	1/16

These probabilities are calculated using the binomial distribution formula for a symmetric random walk. For example, after 4 steps, the probability of being at $x = 0$ is $P(0,4) = \binom{4}{2} (1/2)^4 = 6/16 = 3/8$.