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With help from corollaries that scientists were only beginning to explore in the early 20th century, this summary is designed to empower and entice readers with cool connections between introductory physics and mutual information measures applied widely today in: (i) tracking the divergence of evolving codes, (ii) quantum computation, (iii) dynamical study of complex systems, (iv) compressing data, and (v) forensic science including the detection of plagarism.

*Note:* The concepts used here cut across many technical
disciplines. As a result, no attempt is made in traditional
physics-exposition style to derive or even to teach the
relationships and concepts introduced. References will be
provided for
those interested in learning more. The objective of this summary,
rather, is to alert physics readers to relationships that
connect introductory thermal physics concepts to a subset
of related developments in other fields.

"Two objects allowed to equilibrate by
random sharing with a third object are likewise equilibrated with each
other." Random sharing in practice maximizes the collective
state-uncertainty S_{tot} = k *ln* Ω of these two objects,
where Ω is the number of accessible states and S is in
{nats,
bits, or J/K} respectively
if k is {1, 1/*ln*2, or 1.38×10^{-23}}.
Individual object uncertainty-slopes converge at this maximum.
Hence objects allowed to randomly share...

**thermal energy**seek a common*reciprocal temperature*dS/dQ = 1/T where T is absolute temperature,**volume**seek a common*free expansion coefficient*dS/dV = P/T where P is isostatic pressure, and**number of particles**seek a common*chemical affinity*dS/dN = -μ/T where μ is chemical potential.

If the *number of accessible states* Ω is
proportional to E^{νN/2}, this yields
*equipartition* E/N = (ν/2)kT
i.e. a prediction that the
average thermal energy per *degree-of-freedom* ν is kT/2 for
*quadratic systems*. If Ω is proportional to V^{N},
the foregoing yields the *ideal gas law*
PV=NkT useful for
low-density gases. For systems obeying the *law of mass action* where
Ω is proportional to ζ^{N}/N! and ζ is the
multiplicity of states accessible to a given molecule,
then dS/dN ~ ln[ζ/N].
Thus if a molecule has *stoichiometric coefficient* b in a reaction,
at equilibrium the b-weighted sum of slopes goes to zero and the
product of (ζ/N)^{b} for each reactant
goes to one. Along with equilibrium values for E, V and N,
such analyses also predict the size of fluctuations from one molecule to
the next.

"The change in an object's internal
energy equals the
disordered-energy (heat) transferred into that object minus the
ordered-energy (work) that the object
does on its surroundings." In equations, this is oft written
as ΔE = ΔQ - W_{out} where W_{out} = PΔV.

The ratio of total
energy E to kT is the
*multiplicity-exponent* d(S/k)/d(lnE) with
respect to energy, as well as the number of base-b units of
mutual information lost about the state of the system per b-fold
increase in thermal energy. For quadratic systems this is
the no-work specific heat C_{v}/k. More generally,
C_{v}/k is the multiplicity-exponent
with respect to absolute temperature, and thus the number of
base-b units of mutual information lost about the state of the
system per b-fold increase in absolute temperature. Thus for
each 2-fold increase in absolute temperature we lose:
~3/2 bits of mutual information per
atom in a monatomic gas (given kinetic energies in
each of ν=3 spatial directions), ~6/2 bits of mutual information
per atom in a metal, more than 18/2 bits of mutual information
per water molecule in the liquid (this is huge), and ν/2 bits
of mutal information in other systems that obey equipartition.
These "no-work" relationships
change under conditions of constant pressure, with
enthalpy H = E + PV replacing E above. In that case,
ideal-gas heat-capacity per molecule becomes
C_{p}/k = 1+ν/2 bits per 2-fold increase in T.

"The mutual information
that we share with an *isolated* system doesn't go up,
and hence its entropy doesn't decrease." Neglecting
creation or destruction of correlated subsystems, this means that
for non-isolated systems into which heat flows that
ΔS = ΔQ/T + ΔS_{irr} where the
irreversible entropy change ΔS_{irr} ≥ 0 over time.

If one
considers "steady-state engines" into and out of which
ordered and disordered energy flows while the engine itself
stays the same, we can write the first law for engines as
Q_{out} + W_{out} = Q_{in} + W_{in}.
The second law then takes the form
Q_{out}/T_{out} - Q_{in}/T_{in} -
ΔI_{c} = ΔS_{irr} ≥ 0 with
time. Along with zero increase in subsystem correlations ΔI_{c},
a *heat engine* has zero W_{in} while a *heat pump* has
zero W_{out}. These equations put powerful limits
on what cannot, and what *might*, be done. When they don't
rule something out, they often leave you with the job of
figuring out how to pull it off. For example, they suggest
that home heating in winter could be
factors of two
more efficient if we don't thermalize flame heat directly,
and that when cold enough outside the cooking of food might
be done at negligible energy cost if we get our act together.

"As an object's reciprocal-temperature
approaches ±Infinity, the number of states accessible to it
become few so that its entropy approaches zero."
Thus as 1/T approaches +∞ (that is T decreases to 0K),
*and* as 1/T approaches -∞ for
population
inversions in spin systems and lasers (where
T>∞),
S approaches 0 or at least
something small.

Surprisals add where probabilities
multiply. The
surprisal
for an event of probability p is
defined as s=k*ln*[1/p], so that there are N bits of surprisal
for landing all "heads" on a toss of N coins.
State uncertainties are maximized by finding the largest
*average-surprisal* for a given set of control
parameters (say three of the set E, V, N, T, P, μ).
This optimization minimizes Gibbs *availability* in
entropy units A=-k*ln*Z where Z is a constrained
multiplicity or *partition function*. When temperature T
is fixed, *free-energy* defined as T times A is also
minimized. Thus if T, V and N
are constant, the Helmholtz free energy F=E-TS is
minimized as a system "equilibrates". If T, P and N are held
constant (say during processes in your body), the Gibbs
free energy G=E+PV-TS is minimized instead. The change in
free energy under these conditions is a measure of
*available work* that might be done in the process.
More generally, the work available relative
to some *ambient* state follows from
multiplying ambient temperature T_{o} by
*net-surprisal*
ΔI_{n}≥0,
defined as the average value
of k*ln*[p/p_{o}] where p_{o} is the probability
of a given state under ambient conditions. From the engine corollary,
the work available in thermalizing an object of fixed heat-capacity C
from temperature T to ambient T_{o} is thus
W=T_{o}ΔI_{n}.
Here net-surprisal
ΔI_{n}=CΘ[T/T_{o}]
where Θ[x]=x-1-*ln*x≥0. The
contours
in the figure at right
for example, put limits on the conversion of hot to cold
as in flame-powered A/C and the ice-water invention depicted above.
The economic market value of "specials" is also linked to
perceptions about what is "ordinary" (ambient).
Mutual information is the average
of joint/marginal probabilities
k*ln*[p_{ij}/p_{i}p_{j}] or the
net-surprisal that results from learning *only* that two subsystems
are correlated e.g. that DNA molecule strings from two different
animals are similar, or that entries in an
atlas correspond to stars visible in the night sky.
As in the latter example, mutual information can thus help
quantify the truth of assertions about the world around.

Mutual information ΔI_{c} on subsystem correlations
(in the same units as S) extracts a price in availability,
or from the engine corollary
ΔI_{c} = W_{in}/T_{o}. Thus
steady-state excitations, by reversibly thermalizing
some ordered-energy to heat at ambient temperature,
can support the emergence of a layered series
of correlations with respect to subsystem boundaries.
Such physical boundaries include gradients (as in the formation of
galaxies, stars, and planets), surfaces (as in the context of molecules
and multi-cell tissues), membranes
(as in the development of biological cells and metazoan skins),
and molecular or memetic code-pool boundaries (as in the
definition of genetic families and idea-based cultures).
Thus, for example, a significant flow of ordered-energy is needed
by metazoans who choose to support subsystem-correlations
focused inward and outward from the boundaries of
self, family, and culture. Molecule and idea codes capable
of replication, both discrete and analog, assist with information
storage in this process. Thus as non-sustainable *free-energy
per capita* drops, you might expect idea sets to thrive
which give short-shrift to some of the boundaries that distinguish
observation, belief, consensus, family, friendship, and self.

- Notes on a related course for engineering students at MIT.
- Notes on how different disciplines might analyze an ice-water invention.
- Tales of heat
capacity in bits from
*Amer. J. Phys.***71**(2003) 1142-1151. - Maxent reveals that thermal physics is mostly statistical inference, plus...
- Possibility problems for putting the corollary on engines to use.
- Notes on the importance of multiscale thinking in the study of complex systems.
- Notes on our network-layer multiplicity project.
- Niche-network spiders for visualizing your connections to the world around.
- Comments
on a
*Discover Magazine*article about units for "coldness".

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