66 Chapter 3. The molecular dance[[Student version, December 8, 2002]]
3.1 The probabilistic facts of life
Wewantto explore the idea that heat is nothing but random motion of molecules. First, though,
weneed a closer look at that slippery word, “random.” Selecting a person at random on the
street, you cannot predict that person’s IQ before measuring it. But on the other hand, youcanbe
virtually certain that her IQ is less than 300! In fact, whenever we say that a measured quantity
is “random,” we really implicitly havesomeprior knowledge of the limits its value may take, and
more specifically of the overall distribution that many measurements of that quantity will give, even
though we can say little about the result of any one measurement. This simple observation is the
starting point of statistical physics.
Scientists at the start of the twentieth century found it hard to swallow that sometimes physics
gives only the expected distribution of measurements, and cannot predict the actual measured
value of, say, a particle’s momentum. Actually, though, this is a blessing in disguise. Suppose we
idealize the air molecules in the room as tiny billiard balls. To specify the “state” of the system
at an instant of time, we would list the positions and velocity vectors of every one of these balls.
Eighteenth-century physicists believed that if they knew the initial state of a system perfectly, they
could in principle find its final state perfectly too. But it’s absurd—the initial state of the air in
this room consists of the positions and velocities of all 10^25 or so gas molecules. Nobody has that
muchinitial information, andnobody wantsthat much final information! Rather, we deal in average
quantities, such as “how much momentum, on average, do the molecules transfer to the floor in one
second?” That question relates to the pressure, which wecaneasily measure.
The beautiful discovery made by physicists in the late nineteenth century is that in situa-
tions where only probabilistic information is available and only probabilistic information is desired,
physics can sometimes make incredibly precise predictions. Physics won’t tell you what any one
molecule will do, nor will it tell you precisely when a molecule will hit the floor. But itcantell you
the precise probability distribution of gas molecule velocities in the room, as long as there are lots
of them. The following sections introduce some of the terminology we’ll need to discuss probability
distributions precisely.
3.1.1 Discrete distributions
Suppose some measurable variablexcan take only certain discrete valuesx 1 ,x 2 ,...(see Figure 3.1).
Suppose we have measuredxonNunrelated occasions, findingx=x 1 onN 1 occasions,x=x 2
onN 2 occasions, and so on. If we start all over with anotherNmeasurements we’ll get different
numbersNi′,but for large enoughNthey should be about the same; then we say theprobability
of observingxiisP(xi), where
Ni/N→P(xi) for largeN. (3.1)ThusP(xi)isalways a number between zero and one.
The probability that any given observation will yieldeitherx 5 orx 12 (say) is just (N 5 +N 12 )/N,
orP(x 5 )+P(x 12 ). Since the probability of observingsomevalue ofxis 100% (that is, 1), we must
have
∑
iP(xi)=(N 1 +N 2 +···)/N=N/N=1. normalization condition (3.2)Equation 3.2 is sometimes expressed in the words “Pis properly normalized.”