Auxiliary Materials
Here we present the analytic expressions for the position and momentum of an ideal
gas particle suffering adiabatic expansion into vacuum and those of an ideal gas particle
suffering adiabatic expansion against a partition in which the position of the partition
increases in a constant speed. The results are used to depict the phase space extensions,
a (tS ) and b (tS ) , of a gas particle shown in Fig. 1.
(a) Adiabatic expansion into vacuum
Let x 0 and v0 denote the initial position and the initial speed of the gas particle in
the system in state A in Fig. 1. We assume the gas particle has a unit mass. For the
adiabatic expansion into vacuum, it is easy to integrate the equation of motion of the
one-dimensional ideal gas particle. The result is given by
v(t )  (1) n v0
x(t )  v(t )(t  t n ) 
1  (1) n
L1 ,
2
for time interval t n  t  t n1 (n  1, 2, ) . Here, t n with even (odd) n is the time at
which the particle makes a collision with the left (right) wall, and is given by
nL1  (1) k x0
tn 
where k  1 if the initial velocity is positive; otherwise, k  0 .
v0
When t is less than t1 , x(t )  x0  v0 t if the initial velocity is positive; otherwise,
x(t )  x0  v0 t (t  t 0 ) and x(t )  v0 t  x0 (t 0  t  t1 ) with t 0  x0 / v0 .
(b) Adiabatic expansion against the partition in which the position of the partition
increases with a constant speed v P in time interval [0, t S ] .
If the initial speed v0 of a gas particle is greater than v P , the particle makes at


least one collision with the moving partition at time t n  nL0  (1) k x0 (v0  nv P ) ,
(n  1, 2, ) , where k  1 if the initial velocity is positive; otherwise, k  0 . An even
n accounts for a collision with the static left wall and an odd n for a collision with the
moving partition. When t is less than t1 , x(t )  x0  v0 t if the initial velocity is
positive; otherwise, x(t )  x0  v0 t
(t  t 0 ) and x(t )  v0 t  x0
(t 0  t  t1 ) with
t 0  x0 / v0 . The position x(t ) of the particle between the collisions with the moving
partition is given by x(t )  (1) n vl t  ul  for (t n  t  t n1 ) .  l and  l are given by
vl  v0  2lv P and ul  2lL0  (1) k x0 , respectively. Here, l is the number of
collisions with the moving wall, and  vl is the velocity of the particle after the l-th
collision to the moving wall. The number l of collisions to the moving partition in

 1  v0 t  (1) k x0

time interval is given by l   
 1 , where [ x ] denotes the largest
 2  L0  v P t

integer that is not greater than x .
Since the speed of gas particle reduces by 2v p upon each collision to the moving
partition, there exists the maximum number M of collisions to the moving wall
allowed to a gas particle with initial speed v0 if the partition moves forever; M is the
smallest integer satisfying vM  v P or the integer closest from v0 / v P (  R ). If
( R  1) / 2  M  R / 2 , x(t )  vM t  xM with a positive v M for t  t 2 M . On the other
hand, if R / 2  M  ( R  1) / 2 , x(t )  vM t  xM with a negative v M for t  t 2 M 1 .