Poincaré Recurrence Explained

This problem is sometimes called (among statistical mechanics) Boltzmann's paradox, in part because Ludwig Boltzmann [that's he at right], one of the architects of modern statistical mechanics, and a giant among scientific minds, struggled with this issue at the close of the last century. He endured a storm of criticism among the scientific and philosophical community for his efforts to solve the paradox in favor of the simple, mechanical model of matter that is modern atomic theory, and died (by his own hand) an extremely bitter man.

Boltzmann's solution to the paradox of irreversibility is in many ways as drastic and astonishing as Einstein's solution to the paradox of privileged frames of reference, which led to relativity. Indeed, Boltzmann asserts that the paradox is resolved not (as would be the most natural guess) by some delicate irreversibility of microscopic atomic motions, that only becomes significant and observable when zillions of atoms conspire to produce macroscopic phenomena; rather Boltzmann asserts that the paradox is resolved by the fact that macroscopic phenomenon are not in fact irreversible! They merely appear that way. To put this bluntly: any macroscopic cause and effect not only can but must operate equally well backwards as forwards: that the forest fire spontaneously produces smoke and ashes from trees necessarily implies that under the proper initial circumstances smoke and ashes will spontaneously form living trees.

How can this rather peculiar assertion be reconciled with reality? By the observation that the number of different ways in which molecules can "reverse cause and effect" is enormously smaller (and hence fantastically less likely) than the number of ways in which they can trace out the normal ordering. For example, there are enormously many ways in which molecules might spontaneously rearrange after lightning strikes a tree, all of which would be interpreted macroscopically as "the tree begins to burn." By contrast there are hardly any ways -- there need not be more than one to satisfy Boltzman's assertion -- in which molecules of a burning tree might spontaneously rearrange in a way that would be interpreted macroscopically as "smoke and ash recombine to form wood." And let us stress that when we say "enormously many" we are not speaking lightly --- the numbers involved are so large they are impossible even to write down. The number of atoms in the entire Universe or seconds since the Big Bang is essentially zero by comparison.

Because of the huge number of ways in which "normal" events can occur, compared to those in which "reversed" events can occur, if we observe ever so many times lightning striking strees, we will see uncounted variations on the theme of wood turns to smoke and flame, but many many lifetimes of the Universe are likely to go by before we see the one, peculiar result of a tree reconstituting itself from smoke and flame. Being limited to human lifetimes of experiments, we have naturally concluded from our collective experience that the burning of trees is "irreversible" rather than, as in fact it is, merely awfully unlikely to undo itself.

But if you watch the simulation for a while, you will discover that Boltzmann was right. All the atoms will eventually come precisely back together again on a square grid in the middle of the box, spontaneously retracing exactly their initial steps and completely reversing the original "irreversible" expansion. (You may also notice the formation of some unnatural-looking partially ordered states before the final collapse.)

Why does this happen? Since any closed system has a certain finite number of states in which it can exist, if you wait long enough the system will have to return to any state you care to name. The maximum time this takes is called the Poincaré recurrence time, and of course for most systems it is unbelievably large. But for this peculiar system, it turns out to be quite short. So short, in fact, that the unusual initial state (all the atoms at a square in the center) necessarily recurs on a time scale that is observable. The recurrence of the initial state is not "programmed" into the simulation by a clever choice of the initial conditions -- each time you start the simulation the initial velocities are randomly chosen. The recurrence is observed simply by the fact that this system exhausts rather rapidly its entire repertoire of possible states and must then begin repeat visiting them.

Incidentally, if you are wondering why this system has such as small number of possible states, it's because the initial velocity components vx and vy of the particles are all integers ± 0,1,2,3.... It takes 12800 time steps (ticks) to finish one cycle. Here's the explanation: The moving frame of the simulation has width and height of 12800 steps, each (the frame shown on screen has 400 pixels on each side, it needs 32 steps to move one pixel). The smallest velocity component (apart from zero) is one step per tick. It takes 12800 ticks for a molecule moving at 1 step/tick to cross the width and/or height of the box and come back exactly to its starting location. Faster molecules, with vx = n steps/tick and vy = m steps/tick cross the width of the box n times and the height m times, when the slowest moving makes one pass, i.e. 12800 ticks is the duration for all particles to come back at least once to their starting point. This assumes toroidal boundary conditions (escape at right, come in at left, and similar for the other walls). If the molecules reflect at the walls, the time doubles. Since collisions among the molecules are not simulated, the velocity components are constant and equal to the random numbers given at the start. After every pause of the Java applet another array of random velocity components is computed.- Hence, the recurrence time is 12800 ticks or about 12 seconds on a modern PC with 3.0 GHz. The cycle time is ~inversely proportional to this frequency. The program makes a one second pause every 12800 (toroidal) or 25600 (reflecting) ticks to let you see the initial arrangement of the molecules being revisited. If you switch to another mode by clicking on one of the buttons, this takes over at the next pause. The cycles can go on indefinitely because integer math on a computer is exact. However, the CPU is driven at 100% power during this simulation! Don't heat the universe excessively!

The area of research of statistical mechanics is occupied by people from many backgrounds and disciplines, but physical chemists play one of the leading roles, and professional training in physical chemistry is an excellent way to prepare to take on an active part in the remarkable science of understanding complexity.