62 STATISTICAL PHYSICSSuppose that H decreases in the course of time. Then for a second system, which
differs from the first one only in that the initial conditions are time-reversed, H
must increase in the course of time. Thus, the law of increase of entropy cannot
be an absolute law. Boltzmann immediately recognized the importance of this
observation [B5] and in a major paper, published in 1877 [B6], finally arrived at
the modern view: in the approach to equilibrium the increase in entropy is not the
actual but the most probable course of events. Just as Loschmidt's remark guided
Boltzmann, so, twenty years later, did Boltzmann play a similar role for Planck,
who at that time was trying to derive the equilibrium distribution for blackbody
radiation under the assumption that the increase in entropy is an absolute law. In
the course of a polemic between these two men, Boltzmann became the first to
prove the property of time-reversal in electromagnetic theory: the Maxwell equa-
tions are invariant under the joint inversion of the directions of time and of the
magnetic field, the electric field being left unaltered [B7]. More generally, we owe
to Boltzmann the first precise statement that for a time-reversal invariant dynam-
ics, macroscopic irreversibility is due to the fact that in the overwhelming majority
of cases a physical system evolves from an initial state to a final state which is-
almost never less probable.* Boltzmann was also the first to state explicitly that
this interpretation might need reconsideration in the presence of time-asymmetric
dynamic forces.*
I turn next to Boltzmann's definition of the concept of thermodynamic proba-
bility. Actually, one finds two such definitions in his writings. The first one dates
from 1868 [B9]: Consider a system of N structureless particles with fixed total
energy. The evolution in time of this system can be represented as an orbit on a
surface of constant energy in the 6A^-dimensional phase space (later called the F
space [E3]). To a state S,{i = 1,2. ... ) of the system corresponds a point on the
orbit. The state S, shall be specified up to a small latitude, and thus the corre-
sponding point is specified up to a small neighborhood. Observe the system for a
long time r during which it is in S, for a period r,. Then T,/T (in the limit T -
oo) is defined to be the probability of the system being in the state S,. This we
shall call Boltzmann's first definition of probability.
I alluded earlier to Einstein's critical attitude toward some of Boltzmann's ideas.
That has nothing to do with the first definition of probability. In fact, that very
definition was Einstein's own favorite one. He independently reintroduced it him-
*See [P4] for a quantum mechanical version of the H theorem.
**See [B8]. The most important initial condition in our physical world is the selection of the Fried-
mann universe—in which, it seems, we live—as the one realized solution of the time-reversal invar-
iant gravitational equations. It has been speculated that this particular choice of actualized universe
is one indication of the incompleteness of our present physical laws, that the actual physical laws are
not all time-symmetric, that the time-reversal violation observed in the neutral K-particle system is
only a first manifestation of this asymmetry, and that the conventional view on the statistical arrow
of time may indeed need revision. For a discussion of all these topics, see the review by Penrose [P5].