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Consider two systems, and , which can interact by exchanging heat energy
and
doing work on one another. Let the system have energy and adjustable external
parameters
. Likewise, let the
system have energy and adjustable
external parameters
. The combined system
is
assumed to be isolated. It follows from the first law of thermodynamics that

(198) 
Thus, the energy of system is determined once the energy of
system is given, and vice versa. In fact, could be regarded as
a function of . Furthermore, if the two systems can interact mechanically then,
in general, the parameters are some function of the parameters . As
a simple example, if the two systems are separated by a movable partition in
an enclosure of fixed volume , then

(199) 
where and are the volumes of systems and , respectively.
The total number of microstates accessible to is clearly a function of
and the parameters (where runs from 1 to ), so
.
We have already demonstrated (in
Sect. 5.2) that
exhibits a very pronounced maximum at one particular
value of the energy
when is varied but the external parameters are held constant.
This behaviour comes about because of the very strong,

(200) 
increase in the number of accessible microstates of (or )
with energy. However, according to Sect. 3.8, the number of
accessible microstates exhibits a similar strong increase with
the volume, which is a typical external parameter, so that

(201) 
It follows that the variation of
with a typical parameter ,
when all the other parameters and the energy are held constant, also exhibits
a very
sharp maximum at some particular
value
. The equilibrium situation
corresponds to the configuration of maximum probability, in which virtually all
systems in the ensemble have values of and very close
to and
. The mean values of these quantities are
thus given by
and
.
Consider a quasistatic process in which the system is brought from an equilibrium
state described by and
to an infinitesimally different
equilibrium state described by
and
. Let us calculate the resultant change in the
number of microstates accessible to . Since
, the change in
follows from standard mathematics:

(202) 
However, we have previously demonstrated that

(203) 
[from Eqs. (186) and (197)],
so Eq. (202) can be written

(204) 
Note that the temperature parameter and the mean conjugate forces
are only welldefined for equilibrium states. This is
why we are only considering quasistatic changes
in which the two systems are always
arbitrarily close to equilibrium.
Let us rewrite Eq. (204)
in terms of the thermodynamic temperature ,
using the relation
. We obtain

(205) 
where

(206) 
Equation (205) is a differential relation which enables us to calculate
the quantity
as a function of the mean energy and the mean external parameters
, assuming that we can calculate the temperature and mean
conjugate forces
for each equilibrium state. The function
is termed the entropy of system . The word
entropy is derived from the Greek en+trepien, which means ``in change.''
The reason for this etymology
will become apparent presently. It can be seen from Eq. (206)
that the entropy is merely a parameterization
of the number of accessible microstates.
Hence, according to statistical mechanics,
is essentially
a measure of the relative probability
of a state characterized by values of the mean energy and mean external parameters
and
, respectively.
According to Eq. (129), the net amount of work performed during a quasistatic
change is given by

(207) 
It follows from Eq. (205) that

(208) 
Thus, the thermodynamic temperature is the integrating factor for the
first law of thermodynamics,

(209) 
which converts the inexact differential
into the exact
differential (see Sect. 4.5).
It follows that the entropy difference between any two macrostates
and can be written

(210) 
where the integral is evaluated for any process through which the system is brought
quasistatically via a sequence of nearequilibrium configurations
from its initial to its final macrostate. The process has to be quasistatic
because the temperature , which appears in the integrand, is only welldefined
for an equilibrium state. Since the lefthand side of the above equation only depends
on the initial and final states, it follows that the integral on the righthand side
is independent of the particular sequence of quasistatic changes used to get
from to . Thus,
is independent of the
process (provided that it is quasistatic).
All of the concepts which we have encountered up to now in this course, such
as temperature, heat, energy, volume, pressure, etc., have been fairly
familiar to us
from other branches of Physics.
However, entropy, which turns out to be of crucial importance
in thermodynamics, is something quite new. Let us consider the following
questions. What does the entropy of a system actually signify? What use is
the concept of entropy?
Next: Entropy
Up: Statistical thermodynamics
Previous: Mechanical interaction between macrosystems
Richard Fitzpatrick
20060202