## Saturday, October 1, 2011

### The Second Law, Microscopic Reversibility, and Small Systems

This post is part of a series, Nonsense and the Second Law of Thermodynamics. The previous post is entitled The Second Law, Radiative Transfer, and Global Warming.

On a small scale, individual physical events are reversible; yet on a macroscopic scale, it is not so.  I used to find that confusing.  I'd like to try to cut through some of the confusion.  In so doing, the underlying mechanism of the second law may become clearer.

Figure Source (Monopoly by Hasbro).

Microscopic Reversibility

The principle of microscopic reversibility is simple.  It  states that at a microscopic level a reverse reaction takes place by the same mechanism as the forward reaction, only it is reversed.

It is perhaps not so simple to understand without an example.  Consider a billiard table, one ball comes from the left and hits another and imparts all its momentum to that ball.  The first ball takes the place of the second ball, and the second ball continues to roll.

If we assume a friction-less, drag-less ideal billiard table, we can play the film backwards.  The second ball comes from the right, imparts its momentum to the first ball and takes its place.  The first ball keeps moving.

Energy is conserved.  All of the momentum can be transferred forward or backward. Of course with a macroscopic billiard table, our idealization matters.  In real life, friction occurs and we can tell the difference between the forward process and the reverse process even if we play the film backwards.

On a microscopic level, however, processes are, in general, reversible.

Why does taking a process that is reversible microscopically and scaling it up to the  macroscopic scale make it suddenly "care" about the forward and backward direction?

If you understand the answer to that question, you understand the basis of the second law of thermodynamics itself.

A Modified Monopoly Game

To understand the this question better, I am going to modify the rules to the game of Monopoly.  Ordinarily, in Monopoly, one has to roll doubles to get out of jail.  Rolling two dice gives 36 possible combinations, only six of which are doubles.  So the probability of rolling doubles on any turn is 1/6.

In  ordinary Monopoly, the path by which one ends up in jail is different than the way one gets out of jail. I am going to change the rules to make it symmetrical.

To enter jail, one lands on the "Go to Jail," square.  Then one goes to jail, if and only if one rolls doubles.  To leave jail, one rolls doubles, and then takes a turn starting from the "Go to Jail" square.

To make it completely symmetrical, one can move around the board either clockwise or counter-clockwise. if one enters jail clockwise, one must leave counter-clockwise and vice-versa.

Community Chest cards and Chance cards that send a player to jail are to be taken out of the deck.

There are 40 squares on the Monopoly board.  The probability of landing on a given square, of course, depends on what square a player starts on, and in this modified game, the direction the player is headed.  I am going to make a simplifying assumption that we have no information about what square a player was on previously.

For a small number of players (and let N = the number of players), this assumption seems silly, but as N gets large the probabilities converge.

So the chances of landing on any one square in a turn, by assumption, are 1/40.

Note that the path into jail is the reverse of the path out of jail:

Into jail: 1) Land on "Go to Jail." 2) Roll doubles.

Out of jail: 1) Roll doubles. 2) Move from "Go to Jail."

Let's make one more modification: all players start the game together in jail instead of "Go."  There is only one possible arrangement of the players on the board with all of them in jail.  It can be considered the state of zero entropy.

Consider a 1-player game.  Eventually, the player will roll doubles and get out of jail.  It is certainly possible that the player will end up back in jail.  So, one could consider the system reversible.

Consider N=2.  The state where both players are in jail has only one possible arrangement.  The state with only one player in jail has 80 possible arrangements (if we don't know which player is in jail).  There are 1600 possible states with both players out of jail ("Just Visiting" is a square like any other).

Eventually, both players get out of jail.  It is certainly likely that eventually one or the other player will end up in jail.  It is not too improbable that from time-to-time both players may wind up in jail at the same time.  In fact, if the game goes on long enough, as Monopoly games seem to do, it will probably happen that both players are in jail at the same time again.

Note that as N increases, the number of states with increasing players outside of jail, also increase dramatically.  Try calculating for N=3, the number of states with all players in jail, the number with one of the three players out of jail, the number with 2 players out of jail, and the number with 3 players out of jail.

Now imagine that N = 6 x 1023.  Suddenly the game seems different.

The games starts with all players in jail. On the first turn, 1/6 of the players will get out of jail.  The fluctuations from that prediction will be so small as to be negligible. See my post on fluctuations to understand why in more detail.

As the game goes on, the players will distribute themselves in a more statistically likely distribution.  There will always be players in jail, and there will always be players on every square.  The most likely configuration will eventually prevail, and the probability of going to a less likely configuration will be so minuscule as to be negligible.

In other words, the macroscopic "reaction" of players leaving jail is irreversible, even though it is reversible on a microscopic level.  This fact is really the basis of the second law of thermodynamics.

Small Systems

The astute reader may be asking about small systems.  What if N=2, both players have been freed from jail, and spontaneously end up back in jail.  Did they not violate the law of increasing entropy?

The more important point to understand is that such a movement of pieces does not violate the statistical basis of increasing entropy.  The probability of decreasing entropy in this small system is large enough that it can happen on occasion.

In real microscopic  physical systems, it is possible for the system to spontaneously move from a higher density of states to a lower density of states, as long as such a move is not too statistically improbable.

Thermodynamics is really the study of macroscopic systems.  The second law works because macroscopic systems involve such large numbers.

Time

So, on a microscopic scale, a reaction can be reversible. We can run the film backwards on our friction-less billiard table and not be sure whether time is moving forward or backward.  When we scale things up to the macroscopic, on the other hand, it is obvious which direction time goes.  The broken glass never leaves the floor to become a fully formed glass on top of the table.  Entropy seems to have a preferred direction forward in time.

The next post in this series is entitled the Arrow of Time.

Sources
• Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
• McQuarrie, Donal d A., Statistical Thermodynamics,  University Science Books, Mill Valley, CA, 1973
• Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
• Child, M.S., Molecular Collision Theory, Dover, Mineola, NY, 1974, 1984 Reprint.
• Tolman, R.C., The Principles of Statistical Mechanics, Oxford University Press, Oxford 1938
• Monopoly by Hasbro

Eddie Devere said...

I enjoy reading your blog and look forward to the future pots; however, I wanted to point out that there was nothing "reversible" about your monopoly analogy.

Rolling dice is probabilistic and is not reversible.
The question is: what known force would act like "throwing dice"?

It appears that the only force that could act like "throwing dice" would be the weak nuclear force.
The weak nuclear force appears to be the reason why entropy of the universe increases rather than stays constant.

Entropy generation is negligible in systems in which the weak nuclear force is negligible (such as non-tidal-planetary motion or the movement of bosons such as photons, superfluids, and superconductors.)

If your next post will be on the arrow of time, I'd be interested to see whether you think that there can be an "arrow of time" without the "time asymmetry" of the weak nuclear force.

Rich said...

Thanks for your comments. Monopoly is not a perfect analogy, but consider that classical and quasi-classical systems also follow the second law.

The Monopoly game is an analogy for the type of systems that I am used to thinking about involving molecular dynamics. A single trajectory on a chemical reaction coordinate is surely reversible (look up the M.S. Child reference for a little more depth on this). When you look at all the trajectories, however, and consider the effect of molecular collisions randomizing the energy distributions in an ensemble of molecules, all of a sudden reactions are really governed by statistics. Again, this result holds true in classical mechanics treating molecules like hard spheres and anharmonic oscillators. This randomization is not a force, and I don't think it matters if we are dealing with Boltzon, Bosons, or Fermions (once you take into account indistinguishability, and wavefunction symmetry.

I think the examples are a bit like a friction-less, penduluum. If you assume away friction, entropy doesn't increase, but the world has friction.

Of course all of that assumes that time actually does have a preffered direction. You have somewhat anticipated some of the discussion in my next post w.r.t. CPT-symmetry violation. I probably will not be as definitive as you would like on that issue, but I will raise some interesting questions.

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