526 18. PROBABILITY AND STATISTICSat a higher temperature have molecules with higher kinetic energy—they are either
more massive or moving faster. When two bodies are in contact, their molecules
collide along the interface. Thermal energy then diffuses just as a gas diffuses when
the boundaries confining it are removed. James Clerk Maxwell (1831-1879) created
a theory of gases in which, he thought, the second law of thermodynamics could be
violated. In an 1867 letter to Peter Guthrie Tait (1831-1901), he imagined a per-
son or other agency, later dubbed "Maxwell's demon" by Willam Thomson (Lord
Kelvin, 1824-1907). The demon's job was to stand guard at a small interface be-
tween two objects at different temperatures and allow only those molecules to pass
through that would cause the temperature difference to increase.^16 Statistically,
it was possible that thermal energy might flow "uphill," so to speak. The question
was a quantitative one: How likely was that to happen?
The workings of this process can most easily be seen in the case of a sample
of an ideal monatomic gas, whose pressure (P), volume (V), and absolute Kelvin
temperature Ô satisfy the equation of state PV = nRT = kT, where ç is the
number of moles of gas present and R is a universal constant of proportionality.
The quantity S = kln(T3V/^2 ) is called the entropy of the sample.^17
The evolution of a thermally isolated system can be thought of as the effect
of bringing many small samples of gas at different temperatures into contact. If a
sample of nj moles of the ideal monatomic gas occupying volume V\ at temperature
Ôé is placed in thermally isolated contact with a sample of n 2 moles occupying
volume V 2 at temperature T 2 , the total internal energy will be |(fci + k 2 )T =
|fciTi + \k 2 T 2 , where Ô is the temperature after equilibrium is reached. Thus
Ô = (fci7i)/(fci + fc 2 ) + {k 2 T 2 )/{ki + k 2 ) = cTx + (1 - c)T 2. The ultimate entropy
of the combined system will then be(fc! + fc 2 ) In ((c7\ + (1 - c)T 2 )^1 (V, + V 2 )).This quantity is larger than the combined initial entropy of the two parts,fc) ln(T 1 3/V 1 )+fc 2 ln(r 2 3/2V 2 ),as one can see easily since c = (k\)l(k\ + fc 2 ).^18 Thus, entropy increases for this
system of two samples, and by extension in any thermally isolated system.
Maxwell began to urge a statistical view of thermodynamics in 1868, comparing
the velocities of gas molecules with the white and black balls in the urn models that
had been used for 150 years. In particular, he noted the tendency of these velocities
to assume the normal distribution, as a consequence of the central limit theorem.
When he became head of the Cavendish Laboratory at Cambridge in 1871, he said
in his inaugural lecture that the statistical method(^16) There is more to Maxwell's demon than is implied here. Consider, for example, what energy is
required for the demon to acquire the information about each molecule, decide whether to allow
it to pass, and enforce the decision.
(^17) Strictly speaking, it is not possible to take the logarithm of a quantity having a physical
dimension. The expression J"(l/V)dV = ln(V) should be interpreted in dimensionless terms.
That is, V is really V/Vu, where Vu is a unit volume, and likewise dV = l/VudV and ln(V) is
ln(V/Vu).
(^18) The function 31n(x)/2 is concave, so that a point on an arc of its graph lies above the
chord of that arc, and (assuming without loss of generality that V\ > V-J) In (l + (Vi)/(V2)) >
ln((Vi)/(V 2 )) >eIn((Vi)/(Va)).