What's the highest possible ambient
temperature T_{r}
from which an unpowered device may convert boiling water
(at T_{h}=373K) into ice water (at T_{c}=273K).
We already know that it happens automatically if the
ambient temperature is below freezing. But what if it's
warm outside, and the last thing you need to cool you off
is a cup of boiling water?

Given that water is relatively uncompressible at standard
temperature and pressure, in doing
this calculation we might as
well neglect expansion-related work and further assume that the
heat capacity of water is a constant between freezing and
boiling. Perhaps also stick with cooling the
water from *arbitrarily close to boiling* down
to *arbitrarily
close to freezing* so that life isn't complicated by phase changes.

Secondly
calculate heat, work and entropy flows into and out
of the device as it cools the water from room
temperature down to near freezing (T_{c}). These are
illustrated in
the bottom half of the same figure.

The five small black circles in the figure
represent five equations: (i)
two first-law (heat and work) equations
including e.g. Q_{in}=Q_{out}+W_{out},
(ii) two 2nd-law (entropy) equations, set as equalities
so as to get values for the limiting case of reversible
processes, and (iii) a work equality
W_{out}=W'_{in} to guarantee that the
invention neither requires (nor acts as a supply of) net
available work.

Solving these five equations for the unknowns
Q_{out}, W_{out}, Q'_{out},
W'_{in}, and T_{r} yields the
desired value: the requested upper limit or maximum
room temperature T_{r} at which one can
pull this off without a separate power source. There is
of course no lower limit on T_{r}. If it's below
freezing in the room, it's trivial to turn boiling water
into ice water: Just sit it out and wait until it's ice.

Books on physical chemistry
might prompt their readers to solve the problem with
intermediate results from a 1st and 2nd law analysis of
specific ideal engines. For example, begin with a *Carnot
heat engine* to recover work W_{out} while cooling the
coffee to room temperature. Its well-known efficiency is
dW_{out}/dQ_{in} = (1 - T_{r}/T),
where T is the varying temperature of the hot reservoir.
Total W_{out} is then calculated by multiplying by C dT,
and integrating over T between T_{r} and T_{h}.

This approach returns to
basic principles, taking advantage of *net surprisal*
as a dimensionless* free-energy
analog designed to track finite departures from expected. It is proving
useful in both thermal and purely information-based
investigations. For example a special case of it, *mutual information*,
is extensively used in otherwise quite unrelated
studies of stuff like quantum computing, dynamical attractors hidden
in non-linear systems, and the phylogeny of evolved
codes like a mitochondrial DNA sequence, or
the
text of a chain letter.

We begin with
two generally useful
observations about a pair of connected sub-systems (like the cup's
contents and
environment in this problem), each of which
may be treated as internally
equilibrated:
(i) Reversible processes will hold net surprisal constant
even though its form might change, and (ii) net surprisal for
simple constant heat-capacity systems with respect to an ambient
heat bath at temperature T_{o} is
I_{net}=C*Ξ[T/T_{o}]
where Ξ[*x*]≡*x*-1-ln[*x*] and C is
the system's heat capacity.
For larger context on these facts (e.g. to find out why you
might want to know more, or for derivations and/or
caveats) see the "thermal roots" note
here.

Given this context, the answer can be obtained by solving one equation
for one unknown. To be specific, find the value of
T_{r} by equating surprisals before and after, i.e. from
C*Ξ[T_{h}/T_{r}]=C*Ξ[T_{c}/T_{r}].
It's easy to see here that the specific heat C of the cup's contents,
as well as any intermediate forms of net surprisal
involved**, are
irrelevant to the final answer.

The approaches above all yield the same
result. We'll leave it for you
to figure out:
*How cool does the room have to be
in order for an unpowered device to convert boiling water
reversibly into icy water?* We've
given you the equations for several approaches -- which
will you try to apply first?

The rationale underlying the surprisal-based solution
is also perhaps simpler and more intuitive.
For example, one might say that both
boiling water and ice water have *some* net surprisal with respect
to typical room temperature ambients.
If one converts hot water to cold water having
comparable or less net surprisal, there is obviously
no thermodynamic requirement for added surprisal
(or batteries that provide net surprisal in the form
of available work). Of course figuring out
how to do it in practice, for a reasonable manufacturing
cost, remains a challenge (as far as I know) that has
not yet been met...

* Like entropy or average surprisal, cross-entropy or net surprisal may be expressed in natural information units like [bits] and [nats], or in historical thermodynamic units like [J/K]. Boltzmann's constant converts between [nats] and [J/K].

** For example, the other approaches described here assume intermediate storage of net surprisal specifically as available work (i.e. ``ordered energy''). Although there may in principle be other ways, use of available work is easiest since the number of bits of surprisal stored is huge e.g. when converting a cup of boiling water to ice water.

- More notes on the connection between statistical inference and thermal physics.
- Notes on the related subject of layer-multiplicity analysis and useful information
- Papers for
*Complexity*on evolving correlations and niche networks. - Notes and writeup for the NECSI meeting in Oct/Nov 2007.
- Fun uses for bits and bytes, with links on other uses for surprisal-based information measures.
- Notes on a related course for engineering students at MIT.
- Talk notes on code-expression and multiscale thinking.
- A java-based quiz for assessing your own attention focus toward community correlations.
- Notes about a cross-disciplinary course on complex-system informatics.
- Web-notes on the relevance of layered niche-network models to mediating community health.
- Earlier paper and recent web-note on information-units in thermal physics.

This page is put together and maintained by P. Fraundorf. Let us know if you can suggest other approaches as well. How for example would solutions in a geology class, a mechanical engineering class, and a physical chemistry class, compare? There have been many contributors to this dialog. Thanks in particular to Professor Keith Stine of