Lecture 25 (Schroeder, chapter 2.3).
Interacting Systems.
Postulate of “Equal a Priori Probabilities”.
During previous two lectures we calculated the multiplicities of macrostates of two
physical systems: a two-state paramagnet and an Einstein solid. Let me remind you that if a twostate paramagnet is in the macrostate with the number N ↑ of elementary dipoles point up, then
the multiplicity of this macrostate is
 N 
N!
 =
Ω(N ↑ ) = 
,
 N↑  N↑!N↓!
(25.1)
where N = N ↑ + N ↓ is the total number of dipoles. The total energy of such a system is
determined by the total number of up and down dipoles, so specifying which macrostate this
system is in is the same as specifying its total energy. The general formula for the multiplicity of
an Einstein solid with N oscillators and q energy units is
 q + N − 1 (q + N − 1)!
 =
.
Ω ( N , q ) = 
q
q
!
N
−
1
!
(
)


(25.2)
Now let me discover the manner in which the multiplicity of the macrostate is related to
any of the leading thermodynamic quantities. To do this we consider the problem of “thermal
contact” between two physical systems.
Let the macrostate of a system A be represented by the parameters N A ,V A ,U A and the
macrostate of system B be represented by the parameters N B ,VB ,U B . The corresponding
multiplicities are Ω A ( N A ,V A ,U A ) and Ω B ( N B ,VB ,U B ) .
We now bring the two systems into thermal contact with each other, thus allowing the
possibility of an exchange of energy between the two. However, the respective volumes V A , VB
and the respective particle numbers N A , N B remain unchanged. For simplicity we assume that
the two systems are weakly coupled, so that the exchange of energy between them is much
slower than the exchange of energy among particles within each system. As a result, we can use
the concept of “macrostate of the combined system” as the state specified by the (temporary)
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constrained values of U A and U B . However, on longer time scales the values of U A and U B will
change with only the sum U = U A + U B being fixed. As a result, the multiplicity of a macrostate
of the combined system can be written as
Ω total (U A ,U B ) = Ω A (U A ) Ω B (U B ) = Ω A (U A ) Ω B (U − U A ) = Ω total (U ,U A ) .
(25.3)
Clearly, the number Ω total itself varies with U A . Then, the question arises at what value of the
variable U A will the combined system be in equilibrium? Or, in other words, at what stage of the
energy exchange will A and B be in mutual equilibrium?
As an example, let me consider two small Einstein solids. Each solid contains only three
harmonic oscillators, N A = N B = 3 ; there are six units of total energy, q total = q A + q B .
Now let me formulate a very strong statement about microstates of any isolated system in
thermal equilibrium – a so-called the postulate of “equal a priori probabilities”:
Over long time scales the energy gets passed around randomly in such a way that all
microstates of an isolated system in thermal equilibrium are equally probable.
Or put in another form: in an isolated system in thermal equilibrium, all accessible
microstates are equally probable.
For the above example of two small Einstein solids it means that all 462 microstates of the
combined system are equally probable. But, as a result, we can immediately conclude that some
macrostates are more probable than the others. In particular, the higher multiplicity of the
macrostate the more probable it is. The chance of finding the system in the 4th macrostate is
100/462~21.6%, the chance of finding the system in the 1st macrostate is 28/462~6.1%.
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