Thermal interaction between macrosystems

The combined system
is assumed to be
isolated (*i.e.*, it neither does work on
nor exchanges heat with its surroundings). It follows from the first law of
thermodynamics that the total energy is constant.
When
speaking of thermal contact between two *distinct*
systems, we usually assume that the mutual interaction is
sufficiently weak for the energies to be additive. Thus,

According to Eq. (157), if the energy of lies in the range to
then the energy of must lie between
and .
Thus, the number of microstates accessible to each system is
given by
and
, respectively.
Since every possible state of can be
combined with every possible state of to form a distinct microstate,
the total number of distinct states
accessible to when the energy of lies in the range to
is

Consider an ensemble of pairs of thermally interacting systems,
and , which are left undisturbed
for many relaxation times so that they can attain thermal equilibrium.
The principle of equal *a priori* probabilities is applicable to
this situation (see Sect. 3).
According to this principle, the probability of occurrence of a given macrostate
is proportional to the number of accessible microstates, since all microstates are
equally likely. Thus, the probability that the system has an energy lying in
the range to can be written

We know, from Sect. 3.8, that the typical variation of the number of accessible
states with energy is of the form

Let us
Taylor expand the logarithm of in the vicinity of its maximum value, which
is assumed to occur at . We expand the relatively slowly varying
logarithm, rather than the function itself, because the latter varies so rapidly
with the energy that the radius of convergence of its Taylor expansion
is too small for this expansion to be of any practical use.
The expansion of
yields

(162) | |||

(163) | |||

(164) |

Now, since , we have

(165) |

where and are defined in an analogous manner to the parameters and . Equations (161) and (166) can be combined to give

(167) |

(168) |

(169) |

where

(171) |

(172) |

According to Eq. (170), the probability distribution function is
a Gaussian. This is hardly surprising, since the central limit theorem ensures that
the probability distribution for any macroscopic variable, such as , is Gaussian
in
nature (see Sect. 2.10). It follows that the mean value of corresponds to
the situation of maximum probability (*i.e.*, the peak of the Gaussian curve), so
that

(173) |

(174) |

(175) |