6.2. Entropy[[Student version, January 17, 2003]] 177
The constraints just mentioned include the facts that the total energy is fixed and the system is
confined to a box of fixed size.
Tosay it a third time, equilibrium corresponds to the probability distribution expressing great-
estignoranceof the microstate, given the constraints. Even if initially we had some additional
knowledge that the system was in a special class of states (for example, that all molecules were in
the left half of a box of gas), eventually the complex molecular motions wipe out this knowledge
(the gas expands to fill the box). We then know nothing about the system except what is enforced
bythe physical constraints.
In some very special systems it’s possible to prove Idea 6.4 mathematically instead of taking it
as a postulate. We won’t attempt this. Indeed, it’s not even always true. For example, the Moon
in its orbit around the Earth is constantly changing its velocity, but in a predictable way. There’s
no need for any probability distribution, and no disorder. Nevertheless our postulate does seem
the most reasonable proposal for a very large, complex system. It’s also easy to use; as always,
wewill look for experimentally testable consequences, finding that it applies to a wide range of
phenomena relevant for life processes. The key to its success is that even when we want to study
asingle molecule (a small system with relatively few moving parts), in a cell that molecule will
inevitably be surrounded by a thermalenvironment,consisting of a huge number of other molecules
in thermal motion.
The Pyramids at Giza are not in thermal equilibrium: Their gravitational potential energy could
bereduced considerably, with a corresponding increase in kinetic energy and hence disorder, if they
were to disintegrate into low piles of gravel. This hasn’t happened yet. So the phrase “long enough”
in our Postulate must be treated respectfully. There may even be intermediate time scales where
some variables are in thermal equilibrium (for example, the temperature throughout a pyramid is
uniform), while others are not (it hasn’t yet flowed into a low pile of sand).
Actually the Pyramids aren’t even at uniform temperature: Every day the surface heats up,
but the core remains at constant temperature. Still, every cubic millimeterisof quite uniform
temperature. So the question of whether we may apply equilibrium arguments to a system depends
both on time and on sizescales.Tosee how long a given length scale takes to equilibrate, we use
the appropriate diffusion equation—in this case the law of thermal conduction.
T 2 Section 6.2.1′on page 205 discusses the foundations of the Statistical Postulate in slightly more
detail.
6.2.2 Entropy is a constant times the maximal value of disorder
Let us continue to study an isolated statistical system. (Later we’ll get a more general formulation
that can be applied to everyday systems, which are not isolated.) We’ll denote thenumberof
allowed states ofNmolecules with energyEby Ω(E, N,...), where the dots represent any other
fixed constraints, such as the system’s volume. According to the Statistical Postulate, in equilibrium
asequence of observations of the system’s microstate will show that each one is equally probable;
thus Equation 6.1 gives the system’s disorder in equilibrium asI(E, N,.. .)=Kln Ω(E, N,...)bits.
As usual,K=1/ln 2.
Now certainly Ω is very big for a mole of gas at room temperature. It’s huge because molecules
are so numerous. We can work with less mind-boggling quantities if we multiply the disorder per
observation by a tiny constant, like the thermal energy of a single molecule. More precisely, the
traditional choice for the constant iskB/K,which yields a measure of disorder called theentropy,