6.6. Microscopic systems[[Student version, January 17, 2003]] 193
B
aFigure 6.7:(Schematic.) A small subsystem “a” is in thermal contact with a large system “B.” Subsystem “a”
may be microscopic, but “B” is macroscopic. The total system a+B is thermally isolated from the rest of the world.
6.6.1 The Boltzmann distribution follows from the Statistical Postulate
The key insight needed to get a simple result is that any single molecule of interest (for example
amolecular motor in a cell) isin contact with a macroscopic thermal reservoir(the rest of your
body). Thus we want to study the generic situation shown in Figure 6.7. The figure shows a tiny
subsystem in contact with a large reservoir at temperatureT.Although the statistical fluctuations
in the energies of “a” and “B” are equal and opposite, they’re negligible for “B” but significant for
“a.” We would like to find the probability distribution for the various allowed states of “a.”
The number of states available to “B” depends on its energy via ΩB(EB)=eSB(EB)/kB
(Equation 6.5). The energyEB,inturn, depends on the state of “a” by energy conservation:
EB=Etot−Ea.Thusthe number of joint microstates where “a” is in a particular state and “B”
is inanyallowed state depends onEa:Itequals ΩB(Etot−Ea).
The Statistical Postulate says that all allowed microstates of the joint system have the same
probability; call itP 0 .The addition rule for probabilities then implies that the probability for “a”
to be in a particular state, regardless of what “B” is doing, equals ΩB(Etot−Ea)P 0 ,orinother
words is a constant times e−SB(Etot−Ea)/kBT.Wecan simplify this result by noting thatEais much
smaller thanEtot(because “a” is small), and expanding:
SB(EB)=SB(Etot−Ea)=SB(Etot)−Ea
dSB
dEB+···. (6.22)
The dots refer to higher powers of the tiny quantityEa,which we may neglect. Using the fun-
demental definition of temperature (Equation 6.9) now gives the probability to observe “a” in a
particular state as eSB(Etot)/kBe−(Ea/T)/kBP 0 ,or:
The probability for the small system to be in a state with energyEais a normal-
ization constant timese−Ea/kBT,whereTis the temperature of the surrounding
big system andkBis the Boltzmann constant.(6.23)
Wehave just found the Boltzmann distribution, establishing the proposal made in Chapter 3 (see
Section 3.2.3 on page 76).
What makes Idea 6.23 so powerful is that it hardly depends at all on the character of the
surrounding big system: The properties of system “B” enter via onlyone number, namely its
temperatureT.Wecan think ofTas the “availability of energy from “B”: When it’s big, system
“a” is more likely to be in one of its higher-energy states, since then e−Ea/kBTdecreases slowly as
Eaincreases.