  To illustrate Boltzmann's construction of an entropy function that is defined
for a microstate of a macroscopic system, we present here the simple example of
the free expansion of a one dimensional gas of non-interacting point particles.
The construction requires one to define macrostates, corresponding to
macroscopic variables. We define a macrostate $M$ by specifying the fraction of
particles in rectangular boxes $\Delta x \Delta v$ of the single particle
position-velocity space $\{x,v\}$. We verify that when the number of particles
is large the Boltzmann entropy, $S_B(t)$, of a typical microstate of a
nonequilibrium ensemble coincides with the Gibbs entropy of the coarse-grained
time-evolved one-particle distribution associated with this ensemble. $S_B(t)$
approaches its maximum possible value for the dynamical evolution of the given
initial state. The rate of approach depends on the size of $\Delta v$ in the
definition of the macrostate, going to zero at any fixed time $t$ when $\Delta
v \to 0$. Surprisingly the different curves $S_B(t)$ collapse when time is
scaled with $\Delta v$ as: $t \sim \tau/\Delta v$. We find an explicit
expression for $S_B(\tau)$ in the limit $\Delta v \to 0$. We also consider a
different, more hydrodynamical, definition of macrostates for which $S_B(t)$ is
monotone increasing, unlike the previous one which has small decaying
oscillations near its maximum value. Our system is non-ergodic, non-chaotic and
non-interacting; our results thus illustrate that these concepts are not as
relevant as sometimes claimed, for observing macroscopic irreversibility and
entropy increase. Rather, the notions of initial conditions, typicality, large
numbers and coarse-graining are the important factors. We demonstrate these
ideas through extensive simulations as well as analytic results.
