50
FOUNDATIONS OF THERMODYNAMICS
physical
system,
i.e.,
of
a
system
that
assumes
a
stationary
state,
then for
each
region
T
the
quantity r/T has
a
definite
limiting
value for
T
= od.
For
any
infinitesimally
small
region
this
limiting
value is
infinitesimally
small.
The following
consideration
can
be based
on
this postulate.
Let
there
be
very
many
(N)
independent
physical
systems,
all
of which
are
represented
by
the
same
system
of equations
(1).
We
select
an
arbitrary
instant
t and
inquire
after the distribution
of
the
possible states
among
these
N
systems,
assuming
that the
energy E
of
all
systems
lies
between
E*
and
the
infinitesimally
close value
E*
+
6E*. From
the postulate introduced
above,
it follows
immediately
that the
probability
that the
state
variables
of
a
system randomly
selected
from
among
N systems
will lie within the
region
T
at
time
t
has
the value
lim
T
=
const.
T
=
00
The number
of
systems whose state
variables lie within
the
region
T
at
time
t
is thus
N
•
lim
T/T,
T
=
oo
1
i.e.,
a
quantity
independent
of
time.
If
g
denotes
a
region
of the
coordi-
nates
p1...pn
that is
infinitesimally
small in all variables,
then the
number
of
systems
whose
state
variables fill
up an
arbitrarily
chosen
infinitesimally small
region
g
at
an
arbitrary
time will
be
[5]
(2)
dN
=
e(p1...pn)
fg
dp1...dpn
.
The
function
e
is obtained
by
expressing
in
symbols
the
condition that
the distribution
of
states expressed
by
equation
(2)
is
a
stationary
one.
Specifically, the
region
g
shall
be chosen such
that
p1
shall lie
between
the definite values
p1
and
p1
+
dp1,
p2
between
p2
and
p2
+
dp2,...Pn
between
pn
and
pn
+ dpn;
then
we
have at
the time
t
dNt
=
e(p1...pn).dpl.dp2...dpn,