138
3 Ensembles: Systems of Particles
The number of microstates tot will be maximal if the two subsystems A,
B brought into thermal contact have the same values of β, i.e., if β A = β B :
otherwise, the number of microstates for the composite system will not be at its
maximum value, and energy will be redistributed between the two subsystems
so as to maximize the product A B . If an amount q of energy is transferred
spontaneously from subsystem A into subsystem B, giving new values
A and
B for the numbers of microstates in the two subsystems, then the total number
of microstates for the composite system A⊕B will change from tot = A B to
tot =
A
B . Now, if we recall that we have seen in Eq. (3.1.5) that the amount,
q, of energy taken up by subsystem B (in our case) can be expressed in terms of the
natural logarithm of B as
ln B ≡ ln
B − ln B = β B q B = β B q ,
and similarly, this same amount of energy q given up by subsystem A can be
expressed in terms of ln A as
ln A = ln
A − ln A = β A q A = −β A q ,
with the negative sign indicating that energy (of amount q) has been transferred out
of subsystem A. If we now turn to the composite system A⊕B and ask about the
corresponding change for tot , we see that it is given by
ln tot = ln(( A B ) = (β B − β A )q .
(3.1.6)
This result establishes that if a spontaneous energy transfer out of subsystem A
and into subsystem B results in an increase in the total number of microstates
associated with the composite system A⊕B (so that ln tot > 0), then necessarily
β A < β B : in other words, energy is transferred spontaneously from the subsystem
having the smaller value of β to the subsystem having the larger value of β. As
has been pointed out by Schoepf, the expression (3.1.6) for ln tot provides an
important relationship between the values of β A and β B when thermal equilibrium
has been achieved for the composite system. Specifically, near equilibrium small
energy changes between two subsystems cause only minor changes in the value of
tot (and hence in the value of ln tot ) so that the right-hand side of Eq. (3.1.6) must
be vanishingly small, in which case β A = β B becomes an expression of thermal
equilibrium between two systems in (thermal) contact. We shall say more about this
in the next section.
When energy is transferred from one material object to another via thermal
contact, we say that the object that is giving up the energy, the donor, is ‘hotter’, and
that the object that is receiving the energy, the acceptor, is ‘colder’: by convention,
we assign a higher temperature to the energy donor, i.e., the ‘hotter’ object. Thus,
when two objects in thermal contact reach a (common) thermal equilibrium, they
must have the same temperature. If we now utilize this convention to assign
3 Ensembles: Systems of Particles
The number of microstates tot will be maximal if the two subsystems A,
B brought into thermal contact have the same values of β, i.e., if β A = β B :
otherwise, the number of microstates for the composite system will not be at its
maximum value, and energy will be redistributed between the two subsystems
so as to maximize the product A B . If an amount q of energy is transferred
spontaneously from subsystem A into subsystem B, giving new values
A and
B for the numbers of microstates in the two subsystems, then the total number
of microstates for the composite system A⊕B will change from tot = A B to
tot =
A
B . Now, if we recall that we have seen in Eq. (3.1.5) that the amount,
q, of energy taken up by subsystem B (in our case) can be expressed in terms of the
natural logarithm of B as
ln B ≡ ln
B − ln B = β B q B = β B q ,
and similarly, this same amount of energy q given up by subsystem A can be
expressed in terms of ln A as
ln A = ln
A − ln A = β A q A = −β A q ,
with the negative sign indicating that energy (of amount q) has been transferred out
of subsystem A. If we now turn to the composite system A⊕B and ask about the
corresponding change for tot , we see that it is given by
ln tot = ln(( A B ) = (β B − β A )q .
(3.1.6)
This result establishes that if a spontaneous energy transfer out of subsystem A
and into subsystem B results in an increase in the total number of microstates
associated with the composite system A⊕B (so that ln tot > 0), then necessarily
β A < β B : in other words, energy is transferred spontaneously from the subsystem
having the smaller value of β to the subsystem having the larger value of β. As
has been pointed out by Schoepf, the expression (3.1.6) for ln tot provides an
important relationship between the values of β A and β B when thermal equilibrium
has been achieved for the composite system. Specifically, near equilibrium small
energy changes between two subsystems cause only minor changes in the value of
tot (and hence in the value of ln tot ) so that the right-hand side of Eq. (3.1.6) must
be vanishingly small, in which case β A = β B becomes an expression of thermal
equilibrium between two systems in (thermal) contact. We shall say more about this
in the next section.
When energy is transferred from one material object to another via thermal
contact, we say that the object that is giving up the energy, the donor, is ‘hotter’, and
that the object that is receiving the energy, the acceptor, is ‘colder’: by convention,
we assign a higher temperature to the energy donor, i.e., the ‘hotter’ object. Thus,
when two objects in thermal contact reach a (common) thermal equilibrium, they
must have the same temperature. If we now utilize this convention to assign