Section 6–4 A probability distribution

(Distribution of distances / Normal distribution / Distribution in velocity)

In this section, the three interesting points discussed are a distribution of distances, normal distribution, and distribution in velocity.

1. Distribution of distances:

“What would we expect now for the distribution of distances D? What is, for example, the probability that D = 0 after 30 steps? The answer is zero! (Feynman et al., 1963, section 6.4 A probability distribution).”

Dr. Sands modifies the previous random walk by varying the length of each step such that the average step length is one unit. It can be mathematically represented as root-mean-square distance S_rms = 1 in which the length of a step S may have any value or possibly close to one unit. In essence, this modification helps to model the thermal motion of a gas molecule. In this case, physicists define P(x, Δx) as the probability that distances D will lie in an interval Δx located at x (say from x to x+Δx). One may write P(x, Δx) = p(x)Δx in which the function p(x) is the probability density for ending up at the distance D from the original position.

According to Dr. Sands, the probability for any particular value of D is zero because there is no chance at all that the sum of the backward steps (of varying lengths) would exactly equal the sum of forward steps. However, I would emphasize that the concept of probability is now defined for continuous random variables instead of discrete random variables. Thus, the probability can be calculated by using an integral such that it is always zero for any single value. In other words, the probability of any possible step is exactly zero because the integral over a single point (or the area under a point) is zero.

Note: You may prefer Feynman’s insightful explanation of random walk in chapter 41: “We have already answered this question, because once we were discussing the superposition of light from a whole lot of different sources at different phases, and that meant adding a lot of arrows at different angles (Chapter 32). There we discovered that the mean square of the distance from one end to the other of the chain of random steps, which was the intensity of the light, is the sum of the intensities of the separate pieces… (Feynman et al., 1963, section 41–4 The random walk).”

2. Normal distribution:

“…The probability density function we have been describing is one that is encountered most commonly. It is known as the normal or Gaussian probability density (Feynman et al., 1963, section 6.4 A probability distribution).”

Dr. Sands briefly describes the normal or Gaussian probability density in which the total probability for all possible events between x = −∞ and x = +∞ is surely 1. The probability density function can be represented as p(x) = (1/σ√[2π]) (exp[−x²/2σ²]), where σ is the standard deviation. There are five characteristics of the normal distribution: (1) The bell curve is symmetric about the mean, m. (2) The mode occurs at x = m. (3) The curve approaches the horizontal axis asymptotically. (4) The curve has its points of inflection at x = m ± σ. (5) The total area under the curve is equal to 1 (Walpole & Myers, 1985). However, the term normal distribution is a misnomer because it is actually a family of distributions and has a connotation that other distributions are abnormal.

In Theory of the motion of the heavenly bodies moving about the sun in conic sections, Gauss (1809) uses the method of least squares to deduce the orbits of celestial bodies. Historically speaking, his work supersedes Laplace’s method of estimation by using the method of least squares with principles of probability and the normal distribution that minimizes the error of estimation. That is, the least squares estimates of orbital paths are the same as the maximum likelihood estimates if the errors due to observations follow a normal distribution.

Note: In his seminal paper titled On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat, Einstein (1905) derives the probability distribution of a molecule’s resulting displacement x in a given time t as “f(x, t) = (n/Ö[4πD])(exp[-x²/4Dt])/(t^½).”

3. Distribution in velocity:

“…We call Np(v) the ‘distribution in velocity.’ The area under the curve between two velocities v₁ and v₂ … represents the expected number of molecules with velocities between v₁ and v₂ (Feynman et al., 1963, section 6.4 A probability distribution).”

Physicists may want to know how fast some molecules are moving from organic compounds in a bottle as a result of collisions with other molecules. Dr. Sands clarifies that the spread of molecules in still air may be detected from its color or odor (e.g. colored smoke grenades). In general, these molecules have different velocities and they continue to change their velocities after collisions. Thus, we describe the probability that any particular molecule will have velocities between v and v+Δv is p(v)Δv, where p(v), a probability density, is a function of speed. Importantly, they are described by Maxwellian velocity distribution instead of normal distribution.

Dr. Sands mentions that we shall see later how Maxwell, using common sense and the ideas of probability, to find a mathematical expression for p(v). However, in footnote 1 of Chapter 39, it is stated that “[t]his argument, which was the one used by Maxwell, involves some subtleties. Although the conclusion is correct, the result does not follow purely from the considerations of symmetry that we used before, since, by going to a reference frame moving through the gas, we may find a distorted velocity distribution. We have not found a simple proof of this result (Feynman et al., 1963).” Interested readers may want to read Peliti’s (2007) refinement of an argument due to Maxwell for the equipartition of kinetic energy in a mixture of ideal gases with different masses.

Questions for discussion:

1. Why should we ask what is the probability of obtaining distances D near 0, 1, or 2 instead of 0, 1, or 2?

2. Should we define Gaussian distribution such that it is not exactly the same as a normal distribution?

3. Could we speak of the speed of a molecule instead of using a probability description?

The moral of the lesson: the distribution in velocities of gas molecules is not described by a normal distribution.

References:

1. Einstein, A. (1905). On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Annalen der physik, 17, 549-560.

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. Gauss, K. F. (1809). Theory of Motion of the Heavenly Bodies Moving About the Sun in Conic Sections: A Translation of Theoria Motus. New York: Dover Phoenix Editions.

4. Peliti, L. (2007). On the equipartition of kinetic energy in an ideal gas mixture. European journal of physics, 28(2), 249-254.

5. Walpole, R. E., & Myers, R. H. (1985). Probability and Statistics for Engineers and Scientists (3rd ed.). New York: Macmillan.