Section 6–3 The random walk

(Average distance / Root-mean-square distance / Empirical probability)

In this section, the three interesting concepts discussed are average distance, root-mean-square distance, and empirical probability.

1. Average distance:

“We might, therefore, ask what is his average distance traveled in absolute value, that is, what is the average of |D| (Feynman et al., 1963, section 6.3 The random walk).”

Dr. Sands explains that the problem of a random walk is related to the motion of atoms as well as the coin-tossing problem. He characterizes the random walk of a walker’s progress by the net distance D_N moved in N steps. We may expect the walker’s average progress in a one dimension walk to be zero because he is equally likely to move either forward or backward. One may intuitively feel that as the number of steps N increases, the walker is more likely to have strayed farther from the original position. This refers to the average distance moved in a random walk in absolute value or the average of |D|.

Importantly, the average distance in a random walk in one-dimension and two-dimensions is zero which means that it is possible that the walker returns to the original position. Conversely, in a three-dimensional random walk, the walker will unlikely return to the same initial position. Thus, Shizuo Kakutani, a mathematician, describes these two different consequences of the random walk as “[a] drunk man will find his way home, but a drunk bird may get lost forever (Durrett, 2010, p. 163).” In general, mathematicians may say that a random walk is recurrent if it moves to its original position infinitely often with probability one and the random walk is transient if it moves to its original position finitely often with the same probability.

Note: The term “random walk” was coined by Karl Pearson in 1905. In his own words, “A man starts from a point O and walks l yards in a straight line; he then turns through any angle whatever and walks another l yards in a second straight line. He repeats this process n times. I require the probability that after these n stretches he is at a distance between r and r+dr from this starting point, O (p. 294).” His question was answered by Lord Rayleigh (1905), who had solved a general form of this problem in 1880.

2. Root-mean-square distance:

“… to represent the ‘progress made away from the origin’ in a random walk, we can use the ‘root-mean-square distance’ D_rms: = √⟨D²⟩ = √N (Feynman et al., 1963, section 6.3 The random walk).”

Dr. Sands elaborates that it is more convenient to have a measure of “progress” or random wandering by using the square of the distance (D²) that is positive for both positive and negative motion. Mathematically, we can show that the expected value of D_N² is just the number (N) of steps taken. The “expected value” means the probable value that is based on our best guess on the expected average behavior in repeated sequences. We may represent this expected value by ⟨D_N²⟩ and refer to it as the “mean square distance.” In addition, we can use the “root-mean-square distance” Drms = √N to represent the “progress made away from the origin” in a random walk. These distances are measured in terms of a unit of one step (instead of meters or other units) for reasons of simplicity.

Alternatively, we can derive the root-mean-square distance of a one-dimensional random walker as follows:

If there are N steps, D² = (Dx₁ + Dx₂ + … + Dx_N)²

                                   = Dx₁² + Dx₂² + … + Dx_N² +2Dx₁Dx₂ + … + 2Dx₁Dx_N

On the average, the cross terms (2Dx₁Dx₂ + … + 2Dx₁Dx_N) would have the same amount of positive and negative quantities.

That is, we expect 2Dx₁Dx₂ + … + 2Dx₁Dx_N = 0

If we let every step to be the same unit distance, D² = Dx₁² + Dx₂² + … + Dx_N² = N

Therefore, the root-mean-square distance Ö(D²) = ÖN

Note: Feynman later explains the problem of random walk again as follows: “[i]t is like the famous drunken sailor problem: the sailor comes out of the bar and takes a sequence of steps, but each step is chosen at an arbitrary angle, at random… (Feynman et al., 1963, section 41–4 The random walk).”

3. Empirical probability:

“…An experimental physicist usually says that an ‘experimentally determined’ probability has an ‘error,’ and writes P(H) = N_H/N ± 1/2√N (Feynman et al., 1963, section 6.3 The random walk).”

According to Dr. Sands, an experimental physicist commonly says that an empirical probability has an “error,” and one may write this probability P(H) as N_H/N ± 1/2√N. In other words, this expression implies that there is a “correct” probability which could be computed if we have sufficient knowledge and that the observation may have an “error” due to a fluctuation. However, the empirical probability P(H) of an event may vary depending on the experimental conditions or the experimenter that performs the experiment. In essence, we should be cognizant of the subjectivity in the probability concept because it is always based on uncertain knowledge, and that its quantitative determination is subject to change as we obtain more information.

The concept of empirical probability is applicable in a casino, for example, a “loaded” die may be due to a loading with metals such that the unaltered side is more likely to land facing up. Furthermore, it is not always possible to perform experiments such that the same experimental conditions are maintained. Thus, Mayants (2012) explains that “experimental determination of probability distribution is in the general case a practically unsolvable problem (p. 70).” Nevertheless, physicists always need to use experiments to verify the correctness of a hypothetical probability distribution based on theoretical considerations. More importantly, a different physicist performing experiments under a slightly different condition may conclude that P(H) was different.

Questions for discussion:

1. What is the average distance of a walker in a one-dimensional random walk?

2. What is the root-mean-square distance of a walker in a one-dimensional random walk?

3. What is the empirical probability of a drunk walker located at the original position?

The moral of the lesson: the probability concept is in a sense subjective because it is always based on uncertain knowledge and an empirical probability has an experimental “error.”

References:

1. Durrett, R. (2010). Probability: theory and examples (4th ed.). Cambridge: Cambridge University Press.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Mayants, L. (2012). The enigma of probability and physics. Dordrecht: D. Reidel.

4. Pearson, K. (1905). The Problem of the Random Walk. Nature, 72(1865), 294.

5. Rayleigh, J. W. S. (1905). The Problem of the Random Walk. Nature, 72(1866), 318.