Re: [Phys-l] Negative kelvin temperatures

["David Bowman" <David_Bowman@georgetowncollege.edu>]

Regarding Daniel's question about negative absolute temperatures:

> On another list, the claim was made that in a system with a finite
> number of energy levels, population inversion implies a negative
> kelvin temperature.

Although that's essentially correct, the statement could be kibbutzed
a bit.  Instead of just finite state systems, the real criterion that
distinguishes whether or not a negative absolute temperature can be
sustained or not is whether or not the system in question has an
upper bound to the energy spectrum of its Hamiltonian or not.  Of
course, every finite state system has such an upper bound to its
energy spectrum, so finite-state systems *can* acquire a negative
absolute temperature.  But it is conceivable that a system whose
energy spectrum is bounded from above still has an infinite number
of orthogonal microstates (such as the bound-state subspace of
internal states of a hydrogen-like 1-electron atom).  Such a system
with an infinite number of allowed microstates all with an
upper-bounded energy could *still*, typically, be allowed to acquire
a negative absolute temperature.  (Of course, that is not the case
for a real hydrogen-like atom because its Hamiltonian *does* have
the continuum ionized subspace as well as the bound-state subspace,
and that certainly does not have an upper bound to the energy.)

Also, having a population inversion is a prerequisite for a negative
absolute temperature, but it is not necessarily a sufficient
condition for it.  Not only must there be a population inversion but
the levels in that inversion need to be distributed with a
Boltzmanesque distribution where the exponents of the exponentials are
essentially positive (rather than the usual negative exponent for a
usual positive temperature).  If such a distribution of levels is
not the case then ther system doesn't have any unique temperature at
all in the first place whether or not the occupancy probability
increases or decreases with increasing energy of the levels.

> The following Wikipedia article was cited as evidence:
>
> http://en.wikipedia.org/wiki/Negative_temperature
>
> I had never heard this claim of negative kelvin temperature before.
> My first reaction is that a system with population inversion is not
> in thermal equilibrium, so it doesn't make sense to determine the
> temperature of a system based on population inversion.

A system with an inverted Boltzmann distribution of levels *can* be,
in principle, (and is) in equilibrium with itself *if* the system is
strictly *isolated* from its environment.  But such a system can
*not* be in equilibrium with any other system whose Hamiltonian has
no upper bound to *its* energy spectrum.  In particular, a system of
magnetic spins could be in equilibrium with itself, but it can't be
in equilibrium with the lattice vibration degrees of freedom for the
atoms that those magnetic spins reside on.  But if there is a
sufficiently weak interactive coupling between such systems then
the spin degrees of freedom could be considered as in a metastable
quasi-equilibrium with themselves, although *eventually* those
neglected interactions would cause that system to re-equilibrate at
a finite positive absolute temperature with its surroundings (since
heat bath reservoirs necessarily have a finite positive absolute
temperature because their infinite heat capacity requires an
unavailable infinite amount of thermal energy be supplied to them to
heat them up to or beyond T = [infinity] as is necessary to get them
to the regime of negative absolute temperatures).

> I'm not even sure that the system has a well defined temperature.

Such mostly-isolated, quasi-equilibrium, metastable systems can be
well-approximated as being effectively in equilibrium for a fairly
long time before their inevitable interactions with their
surroundings cause their temperatures to return to the realm of
positive values.

It should be emphasized that a system with a negative absolute
temperature is most definitely *not* colder than T = 0.  Rather,
such a system is actually *hotter* than T = [infinity].  All systems
capable of achieving a negative absolute temperature have the
property that as their (positive) temperature is increased their heat
capacity decreases toward zero as their temperature approaches
T = [infinity], so they only require a *finite* amount of additional
thermal energy to cause them to reach an infinite temperature.  If
they continue to absorb more thermal energy than this then their
absolute temperature goes negative.  This is because the state of
positive infinite temperature and the state of negative infinite
temperature is actually the *same* (macro)state.

When a negative temperature system interacts with a normal
positive temperature environmental system what happens is that the
negative temperature system (being hotter than the positive
temperature one) cools off by giving up heat to the positive
temperature system (thus raising the positive temperature of that
other environmental system a bit).  As the negative temperature
system cools down its temperature plummets toward negative infinity
(as its heat capacity is vanishing).  At the point its temperature
reaches negative infinity that temperature is also the same as
*positive* infinity, and the system continues to cool off down from
positive infinity, eventually reaching a common positive temperature
with the normal system it is interacting with.

> Can someone familiar with this concept please discuss its merits?
>
> Thanks,
>
> Daniel Crowe
> Loudoun Academy of Science
> dan.crowe@loudoun.k12.va.us

David Bowman