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_The 2nd Law of Thermodynamics_

On the Second Law of Thermodynamics
* [1]Introduction
* [2]The Second Law of Thermodynamics (In Classical Thermodynamics)
* [3]A New Concept: Phase Space vs Coordinate Space
* [4]The Second Law of Thermodynamics (In Statistical Mechanics, I)
* [5]The Paradox of Irreversibility
* [6]The Second Law of Thermodynamics (In Statistical Mechanics, II)

_Introduction_

This page is the second part of a series that describes entropy and
the fabled second law of thermodynamics, the previous part being
"[7]The Definitions of Entropy". You should read that page before
this, if you are not already satisifed that you know what entropy is
(you might read it anyway, and find out that you really didn't know!).

The 2nd law of thermodynamics is the law that constrains how the
entropy of a thermodynamic system is allowed to change, given some
process or circumstances. The true meaning of the 2nd law is most
commonly lost in the confusion over the true meaning of entropy, or a
failure to appreciate the restrictions on "closed" and "open" systems.
The entropy problem is hopefully solved by reading the companion piece
already noted above. Here I will go on to show how the 2nd law was
derived, and how it works in nature.

_The Second Law of Thermodynamics (in Classical Thermodynamics)_

Here is how Maxwell explains the second law (_Theory of Heat_, p.
153).

Admitting heat to be a form of energy, the second law asserts that
it is impossible, by the unaided action of natural processes, to
transform any part of the heat of a body into mechanical work,
except by allowing heat to pass from that body into another at a
lower temperature. Clausius, who first stated the principle of
Carnot in a manner consistent with the true theory of heat,
expresses this law as follows: -

It is impossible for a self-acting machine, unaided by any external
agency, to convey heat from one body to another at a higher
temperature.

Thomson gives it a slightly different form: -

It is impossible, by means of inanimate material agency, to derive
mechanical effect from any portion of matter by cooling it below
the temperature of the coldest of the surrounding objects.

And again, later on in _The Theory of Heat_ (page 328), Maxwell has
this to say:

One of the best established facts in thermodynamics is that it is
impossible in a system enclosed in an envelope which permits
neither change of volume nor passage of heat, and in which both the
temperature and pressure are everywhere the same, to produce any
inequality of temperature or of pressure without the expenditure of
work.

A more modern statement of this classical second law may look more
complicated, but means the same thing:

_Processes in which the entropy of an isolated system would
decrease do not occur, or, in every process taking place in an
isolated system, the entropy of the system either increases or
remains constant_

That version of the 2nd law comes from the textbook _An Introduction
to Thermodynamics, the Kinetic Theory of Gases, and Statistical
Mechanics_ (2nd edition), by Francis Weston Sears, Addison-Wesley,
1950, 1953, page 111 (Chapter 7, "the Second Law of Thermodynamics").

The phrase _isolated system_ means that neither energy nor matter may
enter or leave the system; it is an embodiment of the word "_unaided_"
as used by Maxwell & Clausius. If the system is not isolated, then
energy can get in, and so can "aid". Hence, isolation is required to
uphold the restriction "_unaided_". The manner in which the
"transition" is accomplished is irrelevant; all possible transitions
are allowed.

In the earlier paper [8]On the Definition of Entropy, we already
encouterd the equation that defines change in classical entropy, _S =
Q/T_. The 2nd law contrains the change in entropy (_ S_) so as to give
us the fundamental equation for the 2nd law, in classical
thermodynamics.

_S 0_
Equation 1

In classical thermodynamics, the change in entropy indicates the
difference between the amount of energy that is available to perform
mechanical work, and the amount that is not. A larger entropy, means
less energy available for work. So this forced increase in entropy
means that less energy is available for work. Carried to the obvious
conclusion, one would presume that eventually there would be no energy
at all available for work. But if the system is not isolated, if it is
open, the "work energy" resevoir can be replenished, and new energy
made available for work. So the restriction that the system be
"isolated" is a necessary part of the second law; in classical
thermodynamics, it is always wrong to argue that the entropy of a
non-isolated (open) system must always increase.

It is also important to remember that in classsical thermodynamics,
entropy is defined at all, only for systems in thermodynamic
equilibrium. So, naturally, the change in entropy that is restricted
by the second law, can only be determined when the system moves from
one state of thermodynamic equilibrium to another.

The second law in classical thermodynamics is derived from Carnot
cycles. The assumption of perfect quasistatic processes involved is an
idealism that can be approached, but not duplicated, in practical
reality. Hence, in practice, the change in entropy for an isolated
system can only be greater than zero, and equal to zero only in the
case of an ideally reversible Carnot cycle. The derivation is standard
fare in any thermodynamics textbook.

_A New Concept: Phase Space vs Coordinate Space_

Before we go on to explore how the second law works in statistical
mechanics, we need to understand a concept that will likely be a new
one for most readers, the idea of a _phase space_. The kind of "space"
we are used to working with, the kind that represents the volume of a
box, for instance, what we might call "physical space", is called a
_coordinate space_ in physics. A coordinate space is characterized by
positional coordinates. We could line up rulers along the side of the
box, and use the reading from the rulers to establish the (x,y,z)
location of a point anywhere inside the box. Here we are using the
word _space_ not in its colloquial sense, but in a restrictive sense
as mathematical jargon. A set of numbers has to satisfy certain
defining characteristics, in order to qualify in mathematics as a
_space_. I will not deal with those details here, because everything
that I will call "space" does in fact meet those qualifications. The
volume of the box qualifies as "space", but so do the momentum &
position of particles moving inside the box.

Consider that box to be full of gas molecules. Each molecule can be
located in the box by 3 position coordinates (x,y,x), and the momentum
of each particle also has 3 components (Lx, Ly, Lz; I will use "L" to
represent momentum). So the combined positions & momenta of the
particles constitutes a space with 6 dimensions, as opposed to our
normal concept of physical space, being only 3 dimensional. That 6
dimensional space is called the _phase space_ for the gas molecules.

Remember that the ability to do mechanical work is a key concept for
understanding entropy. Particles do work by virtue of their motion,
which is expressed by their momentum (I might have used kinetic
energy, but momentum is what you usually find in the text books). So
the entropy of the gas is determined by the behavior of the particles
in their 6 dimensional phase space, and not in the 3 dimensional
coordinate space ("physical space") of the box. Furthermore, while the
volume of coordinate space filled by the gas is just the physical
volume of the box, the volume of phase space occupied by the gas is
the 6 dimensional space occupied by one particle, times the total
number of particles. When you consider that one cubic centimenter of
sea level air holds about 1019 molecules, you can see that the phase
space occupied by the gas is conceptually quite large when compared to
the physical space of the box.

Now suppose that our box is really just one half of a larger box, with
a partition down the middle. When we remove the partition, the
physical space (coordinate space) of the box increases by a factor of
2. But what happens to the volume of phase space occupied by the gas?
It goes up by a factor of 2N, where N is the total number of particles
in the box (and the 2 would be a 3 if we tripled the physical volume
instead of doubling it). If that box was a one cubic centimeter box of
sea level air, while the physical volume doubles, the phase space
volume goes up by a factor of 21019 = 100.3x1019, quite a bit larger
than 2.

The enormous change in phase space is an important concept, which will
come back to rescue us (we hope), from the [9]paradox of
irreveribility, which I will discuss after we deal with the second law
in statistical mechanics.

_The Second Law of Thermodynamics (in statistical mechanics, I)_

The 2nd law is no different in statistical mechanics than it is in
classical thermodynmics. It remains a fact that _S 0_, as previously
derived by the founders of classical mechanics. But in classical
thermodynamics, heat and entropy are treated like fluids that flow
from one system to another. The asymmetrical 2nd law forces them to
flow always in one direction, but not the other. But gravity
asymmetrically forces water, for instance, to always flow down hill.
So the asymmetry of the 2nd law, and the need to pump heat or entropy
"uphill", as one would for water, don't seem like a fundamental
problem.

But the transition to statistical mechanics, and the idea that a gas
is made up of moving particles creates a large, fundamental problem.
The motion of the particles is controlled by Newton's equations, and
they are all time symmetrical, they work just as well "backwards" as
they do "forwards". Indeed, there is no intrinsic difference between
the two at all. That means that a gas should be able to run
"backwards" as well as "forwards", and entropy, as a state variable
for a gas system should be able to do likewise. But the 2nd law
forbids it, setting up a fundamental conflict. Why should an
essentially Newtonian system not be time symmetric? The problem has
earned a name: _The Paradox of Irreversibility_.

_The Paradox of Irreversibility_

Consider a tank of gas. It is full of gas molecules, the motion of
each one describing a trajectory in both physical space, and in phase
space. Each particle moves in accordance with the laws of Newtonian
mechanics, which are fully reversible in time; a "time-backward"
trajectory cannot be distinguished from a "time-forward" trajectory,
in the absence of some outside context used to define "backward" and
"forward". However, when those particles are all gathered together in
the ensemble of a gas tank, their combined motion is not time
reversible! We know this because the second law of thermodynamics
enforces it.

Here is how [10]William Thomson (Lord Kelvin) describes the problem
(Proceedings of the Royal Society of Edinburgh, 8, 325 (1874):

The essence of Joule's discovery is the subjection of physical
phenomena to dynamical law. If, then, the motion of every particle
of matter in the universe were precisely reversed at any instant,
the course of nature would be simply reversed for ever after. The
bursting bubble of foam at the foot of the waterfall would reunite
and fall into the water ... Physical processes, on the other hand,
are irreversible: for example, the friction of solids, conduction
of heat, and diffusion. Nevertheless, the principle of dissipation
of energy [read irreversible behavior] is compatible with a
molecular theory in which each particle is subject to the laws of
abstract dynamics.

This apparent paradox caused much controversy in the early days of
statistical mechanics, and in fact remains today a topic of open
interest & controversy. Maxwell, Thomson, but mostly Boltzmann,
together offered an explanation that satisfied them, but not
necessarily everyone else.

As shown above, in the section on [11]phase space & coordinate space,
a mere doubling of the physical volume of one cubic centimeter of air,
will enlarge the phase space of the gas by a factor of 103x1018. So
the random odds that the gas in the tank might return to one of the
previous phase space volumes (one that corresponds to all of the gas
molecules suddenly gathering in one half of the box) are about 1 in
103x1018. Those are pretty slim odds in any practicle sense, and they
are even slimmer for more likely and larger volumes. An ensemble of
closer to a mole, say 1023 molecules will have its phase space
enlarged by a factor of 103x1022, and so on. This argument will
explain why violations of the second law are not encountered, they are
just too unlikely. But, while the argument works (maybe), it is not
very satisfying to many that the argument is built around "unlikely",
rather than some more fundamental concept.

But maybe there is a better answer, "chaos". Otherwise known as
"nonlinear dynamics", but popularly (and misleadingly) called "chaos",
this field of mathematical physics comes out of the pioneering work of
the French mathematician [12]Henri Poincaré. Poincaré published the
first description of chaotic motion in 1890, but it was not until the
latter half of the 20th century that the scope of application of chaos
theory to Newtonian mechanics was fully realized. But chaos theory
will explain the second law of thermodynamics, by virtue of the
infinite sensitivity to initial conditions that characterize such
systems. The gas molecules cannot return to the previous small phase
space volume, because they are a chaotic system, which cannot recover
the initial conditions.

Jean Bricmont, mathematician & physicst from [13]l'Université
catholique de Louvain, in Belgium, holds to this point of view (
"[14]Science of Chaos or Chaos in Science?", J. Bricmont, Annals of
the New York Academy of Sciences 775: 131-175, 1996). Likewise
[15]Joel Lebowitz (Department of [16]Physics and astronomy at
[17]Rutgers University (_Microscopic Origins of Irreversible
Macroscopic Behavior_, J.L Lebowitz, Physica A 263(1-4): 516-527,
February 1 1999, or "[18]Microscopic Reversibility and Macroscopic
Behavior: Physical Explanations and Mathematical Derivations", from a
lecture on nonequilibrium statistcal mechanics in 1994).

The "loyal opposition" is non other than [19]Ilya Prigogine, the
Godfather of nonequilibrium thermodynamics himself. He does not like
the statistical argument, and claims that the second law can be
derived from the mathematics of functions directly (for instance,
_Laws of Nature, Probability and Time Symmetry Breaking_, Physica A
263(1-4): 528-539, February 1 1999).

Not all questions in science necessarily have definitive answers
(yet), and this may well be one of them. Perhaps the paradox is a real
paradox, but it does not look that way. Though Bricmont & Prigogine
disagree on the solution, they both offer compelling arguments for
their point of view. It may well be a paradox with too many solutions,
as opposed to a paradox with no solutions at all. But it is important
to keep in mind that science does not consist of nothing but questions
with canned answers. The paradox of irreversibility remains an issue
in statistical mechanics, and a field of active exploration.

_The Second Law of Thermodynamics (in statistical mechanics, II)_

Now lets go back to the definition of entropy that we learned in the
earlier chapter [20]on the defintion of entropy,

_S = -k· [Pilog(Pi)]_
Equation 2

Those probabilities _Pi_ are calculated in _phase space_, not in
_coordinate space_. That's why I tried my hand at organizing the
[21]section on phase space above, so the reader might gain some
understanding that can be applied to the paradox of irreversibility.
Under physically normal conditions, one can easily encounter changes
in phase space volume that are on the order of 1060, which certainly
emphasizes the meaning of "unlikely". Once chance in 1060 are slim
odds in anyone's book.

So the real essence of making the transition from classical mechanics
is that the 2nd law makes a transition, from something thought to be
comfortably absolute, to something that is uncomfortably
probabilistic, even if the odds of a violation are that small.

Go on to [22]Entropy and the Second Law in Open Systems
Go back to [23]The Definitions of Entropy
Go back to [24]The Collected Writings of Tim Thompson
Go all the way to [25]Tim Thompson's Home Page
_Page dated 19 February 2002_

setstats 1

References

1. file://localhost/www/sat/files/tim_thompson/entropy2.html#howintro
2. file://localhost/www/sat/files/tim_thompson/entropy2.html#2lotc
3. file://localhost/www/sat/files/tim_thompson/entropy2.html#phase
4. file://localhost/www/sat/files/tim_thompson/entropy2.html#2lots1
5. file://localhost/www/sat/files/tim_thompson/entropy2.html#paradox
6. file://localhost/www/sat/files/tim_thompson/entropy2.html#2lots2
7. file://localhost/www/sat/files/tim_thompson/entropy1.html
8. file://localhost/www/sat/files/tim_thompson/entropy1.html
9. file://localhost/www/sat/files/tim_thompson/entropy2.html#paradox
10. http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Thomson.html
11. file://localhost/www/sat/files/tim_thompson/entropy2.html#phase
12. http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Poincare.html
13. http://www.ucl.ac.be/home.html
14. http://xxx.lanl.gov/abs/chao-dyn/9603009
15. http://www.physics.rutgers.edu/people/hpgs/Lebowitz.html
16. http://www.physics.rutgers.edu/
17. http://www.rutgers.edu/
18. http://xxx.lanl.gov/abs/cond-mat/9605183
19. http://order.ph.utexas.edu/people/Prigogine.htm
20. file://localhost/www/sat/files/tim_thompson/entropy1.html
21. file://localhost/www/sat/files/tim_thompson/entropy2.html#phase
22. file://localhost/www/sat/files/tim_thompson/entropy3.html
23. file://localhost/www/sat/files/tim_thompson/entropy1.html
24. file://localhost/www/sat/files/tim_thompson/faqs.html
25. file://localhost/www/sat/files/tim_thompson/index.html