40
THEORY OF THERMAL
EQUILIBRIUM
mechanically
connected into
one
single
system
that
has
the
same
temperature
function.
Let
two
mechanical
systems
E1
and
E2
be
merged
into
one
system,
but
in such
a
way
that
the
energy
terms
that
contain
state
variables
of
both
systems
be infinitesimally
small.
Let
E1
as
well
as
E2
be
connected with
an
infinitesimally
small
thermometer S.
The readings
H1
and
H2
of
the
latter
are
certainly
identical
up
to
the
infinitesimally
small because
they
refer
only to
different
locations
within
a
single
stationary
state. The
same
is of
course
true
of
the
quantities
h1
and
h2.
We now
imagine
that the
energy
terms
common
to
both
systems
decrease infinitely
slowly
toward
zero.
Thereby
the quantities
H and h
as
well
as
the distributions of
state of
the
two
systems change
infinitesimally because
they
are
determined
by
the
energy
alone. If then the
complete
mechanical separation
of
E1
and
E2
is
carried
out,
the relations
H1
=
H2,
h1
=
h2
continue
to
hold all the
same,
and the
distribution
of
states
changes
infin-
itesimally.
H1
and
h1,
however,
will
now
pertain
only
to
E1,
and
H2
and
[20]
h2
only to
E2.
Our
process
is
strictly
reversible,
as
it consists
of
a
sequence
of
stationary states.
We
thus obtain the
theorem:
Two
systems having
the
same
temperature
function
h can
be
merged
into
a
single
system
having
the
temperature
function
h
such
that their
distribution of
states
changes
infinitesimally.
Equality
of the quantities
h
is thus the
necessary
and
sufficient
condition for the
stationary
combination
(thermal equilibrium)
of
two systems.
From
this follows
immediately:
If the
systems
E1
and
E2,
as
well
as
E1
and
E3,
can
be combined in
a
stationary
fashion
mechanically
(in thermal
equilibrium),
then
so can
E2
and
E3.
I would
like
to note
here that until
now
we
have
made
use
of the
assump-
tion that
our
systems
are
mechanical
only
inasmuch
as we
applied
Liouville's
theorem and
the
energy
principle.
Probably
the basic
laws of
the
theory
of
heat
can
be
developed
for
systems
that
are
defined
in
a
much
more
general
way.
We
will
not
attempt
to
do
this
here,
but will
rely
on
the
equations
of