Biological Physics: Energy, Information, Life

72 Chapter 3. The molecular dance[[Student version, December 8, 2002]]

σ 2 σ 3 σ 4 σ

0.5/σ

r

P(

r)

Figure 3.3:(Mathematical function.) The probability distribution for the distancerfrom the origin, when bothx
andyare independent distributions with varianceσ^2.

vvz

vx

u vy

u+d

u

Figure 3.4:(Sketch.) The set of all vectorsvof lengthuis a sphere. The set of all vectors with length betweenu
andu+duis a spherical shell.

Your Turn 3f

a. Repeat the above example for athree-component vectorv,each of whose components is an

independent, random variable distributed as a Gaussian of varianceσ^2 .That is, letudenote the

length ofvand findP(u)du.[Hint: Examine Figure 3.4.]

b. Graph your answer to (a) with a computer math package. Again try various values ofσ.

3.2 Decoding the ideal gas law

Let us try to interpret the ideal gas law (Equation 1.11 on page 23), and its universal constantkB,
in the light of the working hypothesis that heat is random motion. Once we make this hypothesis
precise, and confirm it, we’ll be in a position to understand many physical aspects of the nanoworld.