Entropy

[Roger D. Jones]

I usually just scan the FRIAM email traffic, but the recent discussion
on entropy is one that is dear to me and one that Roger Frye, Sven
Redsun and I have been working on recently.

 

The discussion is of
http://jchemed.chem.wisc.edu/Journal/Issues/1999/Oct/abs1385.html by
Frank Lambert. Lambert makes the following points:

1.      "simply changing the location of everyday macro objects from an
arrangement that we commonly judge as orderly (relatively singular) to
one that appears disorderly (relatively probable) is a "zero change" in
the thermodynamic entropy of the objects because the number of
accessible energetic microstates in any of them has not been changed."

2.      "Finally, although it may appear obvious, a collection of
ordinary macro things does not constitute a thermodynamic system as does
a group of microparticles. The crucial difference is that such things
are not ceaselessly colliding and exchanging energy under the thermal
dominance of their environment as are microparticles."

 

I would like to make the following points:

1. Lambert's definition of a thermodynamic system is unnecessarily
restrictive. A thermodynamic system need not be composed of
microparticles "ceaselessly colliding and exchanging energy." In
particular, a very common physical thermodynamic system would not be
included in Lambert's scheme. Lambert's definition precludes an ideal
gas in an insulated container from being a thermodynamic system.
2. The Maxwell Demon problem is a valuable tool for clarifying the
differences among entropy, information, and heat flow in physical
systems. Moreover, the Demon can provide a bridge for understanding the
thermodynamics of nonphysical systems.
3. There is a great deal to be gained by considering nonphysical
systems in a thermodynamic context. I would include here, not only
systems of macroscopic objects but also systems composed simply of
information such as trades in a stock market. To put it in more
technical terms, the concept of energy is not crucial in thermodynamic
treatments of nonphysical systems, but the concepts of entropy and
temperature are crucial.

 

Point 1: An ideal gas in an insulated container does not exchange energy
among the microparticles, yet it is describable in thermodynamic terms
such as energy, entropy, and temperature. The definition of an ideal gas
is that the particles travel freely without interacting. Moreover, if
the walls of the container are elastic, there is no energy transfer
between the particles and the walls of the container. Therefore, a
particle initially with energy E=mv^2/2 always has that energy. Yet, the
temperature is well defined as a parameter of the probability
distribution for the energies of particles in the box. And entropy is
well-defined in terms of the probability distribution for the energies
of particles in the box. Moreover, the probability distribution need not
be the common Maxwell-Boltzmann distribution. There is some
arbitrariness in the choice of probability distribution. The
thermodynamics of an ideal gas can be derived without reference to a
Maxwell-Boltzmann distribution. The key concept here is the idea of an
"ensemble" of possible states that the system can be in. In physics
typically three types of ensembles are used, the microcanonical, the
canonical, and the grand canonical. These three ensembles represent
three different choices for the probability distribution of particles.
All three typically yield the same thermodynamics for physical systems
and therefore which one is chosen is usually a matter of computational
convenience. In the microcanonical ensemble, for instance, the
temperature emerges as the change in total energy of the system with
respect to a change in entropy. The microcanonical ensemble does not
require that the microparticles exchange energy, only that the total
energy of the system is conserved. Therefore, an ideal gas in an
insulated container is an important thermodynamic system that does not
qualify as a thermodynamic system under Lambert's definition.

 

Point 2: The Maxwell Demon problem can be used to separate the concepts
of entropy, information, and heat flow and can help clarify the
relationships. Maxwell created his Demon in 1867 to help clarify the
issues associated with the Second Law of Thermodynamics. In particular,
Maxwell wished to address the question of the role of intelligence in
the flow of entropy. The Demon was an intelligent microscopic creature
that sat at a trapdoor separating a box into two sides. Particles
inhabited both sides of the box. The Demon observed the particles and
allowed fast particles to enter into one side of the box and slow
particles to enter into the other side of the box. The entropy of the
particles was thus decreased and a temperature gradient, capable of
producing useful work, was created. The intelligent Demon seemed to
violate the Second Law. Either the Second Law had to be abandoned or the
entropy of the Demon had to increase to compensate for the decrease in
entropy of the particles. A detailed discussion of the relationships
appears in  

 


physics/0311023 [abs <http://xxx.lanl.gov/abs/physics/0311023> , pdf
<http://xxx.lanl.gov/pdf/physics/0311023> ] :


Title: Entropy Generation by a Maxwell Demon in the Sequential Sorting
of the Particles in an Ideal Gas
Authors: Roger D.
<http://xxx.lanl.gov/find/physics/1/au:+Jones_R/0/1/0/all/0/1>  Jones,
Sven G. <http://xxx.lanl.gov/find/physics/1/au:+Redsun_S/0/1/0/all/0/1>
Redsun, Roger E.
<http://xxx.lanl.gov/find/physics/1/au:+Frye_R/0/1/0/all/0/1>  Frye
Comments: 14 pages, 2 figures
Subj-class: Classical Physics; General Physics
 

In this paper we regard the Demon as a physical computer and explicitly
calculate the entropy increase of the universe in both the system of
particles and in the physical computer. This approach is a purely
physical calculation of a physical problem. Zurek, however, has pointed
out (see reference in above paper) that an ensemble approach to
calculation of entropy is equivalent to a deterministic calculation of
the algorithmic complexity of the problem. This allows us to describe
thermodynamic problems in both physical terms, as in the above paper, or
in information terms - in the language of computation. This leads us to:

 

Point 3: There is a great deal to be gained by considering nonphysical
systems in a thermodynamic context. Zurek's Principle allows us to
generalize to nonphysical systems. It provides a recipe for moving
between physical systems in which ensembles of microstates are used to
calculate the entropy to deterministic computational systems in which
information is measured as the size of a computer program. A summary of
the program applied to the Maxwell Demon problem and to the stock market
is in


physics/0311074 [abs <http://xxx.lanl.gov/abs/physics/0311074> , pdf
<http://xxx.lanl.gov/pdf/physics/0311074> ] :


Title: The Maxwell Demon and Market Efficiency
Authors: Roger D.
<http://xxx.lanl.gov/find/physics/1/au:+Jones_R/0/1/0/all/0/1>  Jones,
Sven G. <http://xxx.lanl.gov/find/physics/1/au:+Redsun_S/0/1/0/all/0/1>
Redsun, Roger E.
<http://xxx.lanl.gov/find/physics/1/au:+Frye_R/0/1/0/all/0/1>  Frye,
Kelly <http://xxx.lanl.gov/find/physics/1/au:+Myers_K/0/1/0/all/0/1>  D.
Myers
Comments: 15 pages, 6 figures
Subj-class: Classical Physics; General Physics

 

The interesting thing to note is that the increase in entropy of the
universe is the same independently of whether the system is a box of
particles being sorted by a Maxwell Demon or whether the system is a
stock market being ordered by limit traders.
 

 

 

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