4.6. The big picture[[Student version, December 8, 2002]] 129
T 2 Section 4.6.5′on page 134 points out that an approximation used in Section 4.4.2 limits the
accuracy of our result in the far tail of the distribution.
The big picture
Returning to the Focus Question, this chapter has shown how large numbers of random, independent
actors can collectively behave in a predictable way. For example, we found that the purely random
Brownian motion of a single molecule gives rise to a rule of diffusive spreading forcollectionsof
molecules (Equation 4.5a) that is simple, deterministic, and repeatable. Remarkably, we also found
that precisely the same math gives useful results about the sizes of polymer coils, at first sight a
completely unrelated problem.
Wealready found a number of biological applications of diffusion and its other side, dissipation.
Later chapters will carry this theme even farther:
- Frictional effects dominate the mechanical world of bacteria and cilia, dictating the strategies
they have chosen to do their jobs (Chapter 5). - Our discussion in Section 4.6.4 about the conduction of electricity in solution will be needed
when we discuss nerve impulses (Chapter 12). - Variants of the random walk help explain the operation of some of walking motors mentioned
in Chapter 2 (see Chapter 10). - Variants of the diffusion equation also control the rates of enzyme-mediated reactions (Chap-
ter 10), and even the progress of nerve impulses (Chapter 12).
More bluntly, we cannot be satisfied with understanding thermal equilibrium (for example, the
Boltzmann distribution found in Chapter 3), becauseequilibrium is death. Chapter 1 emphasized
that life prospers on Earth only by virtue of an incoming stream of high-quality energy, which keeps
usfarfrom thermal equilibrium. The present chapter has provided a framework for understanding
the dissipation of order in such situations; later chapters will apply this framework.
Key formulas
- Binomial: The number of ways to choosekobjects out of a jar full ofndistinct objects is
n!/k!(n−k)! (Equation 4.1). - Stirling: The formula: lnN!≈NlnN−N+^12 ln(2πN)allows us to approximateN!for
large values of N (Equation 4.2). - Random walk: The average location after random-walkingN steps of lengthLin one
dimension is〈xN〉=0.The mean-square distance from the starting point is〈xN^2 〉=NL^2 ,or
2 Dt,whereD=L^2 /2∆tif we take a step every ∆t(Equation 4.5). Similarly, when taking
diagonal steps on a two-dimensional grid gives〈(xN)^2 〉=4Dt(Equation 4.6). Dis given
bythe same formula as before; this timeLis the edge of one square of the grid. (In three
dimensions the 4 becomes a 6.) - Einstein: An imposed forcefon a particle in suspension, if small enough, results in a slow
net drift with velocityvdrift=f/ζ(Equation 4.12). Drag and diffusion are related by the
Einstein relation,ζD=kBT(Equation 4.15). This relation is not limited to our simplified
model.