32
THEORY OF
THERMAL
EQUILIBRIUM
[7]
system an
additional condition of the kind
u(P1,...,qn)
=
const.
that
cannot be
reduced
to
the
form
p(E) =
const.,
then it
would obviously be
possible to choose
initial conditions
such
that
each of
the
N systems
could
have
an
arbitrarily
prescribed
value for
p.
However,
since
these values
do
not
vary
with
time, it
follows,
e.g.,
that
for
a
given
value
of
E
any
arbitrary value
might
be
assigned to
S(p,
extended
over
all
systems,
through
appropriate
selection of initial conditions.
On
the other
hand,
Yjp
is
uniquely
calculable
by
the distribution of
states,
so
that other distributions
of
states correspond to
other
values
of
Yip.
It is thus clear that the exis-
tence
of
a
second such integral
(p
would
necessarily
have
the
consequence
that
the
state
distribution
would
not be
determined
by
E
alone but
would
necessarily have
to
depend
on
the initial
state of
the
systems.
If
g
denotes
an
infinitesimally
small
region
of
all
state
variables
P1,...Pn,
q1,...qn,
which
is
chosen such
that
E(p1...qn)
lies
between
E
and
E
+
SE
when
the
state
variables
belong
to
the
region
g,
then the
distribution
of
states
is characterized
by an
equation
of the
form
[8]
dN
=
iKpj,...,?
)
dpv..dq
,
J9
where
dN
denotes the
number
of
systems whose state
variables
belong
to
the
region
g
at
a
given
time.
The
equation
expresses
the condition that the
distribution is
stationary.
We now
choose such
an
infinitesimal
region
G.
The
number
of
systems
whose
state
variables
belong
to
the
region
G
at
a
given
time
t
=
0
is then
dN
=
*{Pv...q)
dP*
•• •
dQ
G
1 n
where the
capital
letters
indicate that the
dependent
variables
pertain
to
time
t
=
0.
We now
let
elapse
some
arbitrary time
t.
If the
system
possessed
the
specific state
variables
P1,...Qn at
time
t
=
0,
then it will
possess
the
specific state
variables
P1,...,qn
at
time
t
=
t.
Systems
whose state