Howard said: Joel, I have always been bothered by the transition from one or two particle deterministic models to N-particle statistical models. Bohr gave the story about the boy who goes into a candy store and asks for a penny’s worth of mixed sweets. The store owner says I don’t have any mixed sweets, but here are three sweets, you can mix them yourself. Whether reality is deterministic, as Einstein and Ira believe, or whether it is stochastic, as most quantum physicists believe, is a metaphysical difference that does not make any practical or empirically testable difference in our models.
Joel responds: After thinking it through, I think you and many others have good reason to be bothered. I felt bothered too and now believe I have come to a conclusion concerning the paradox. I believe that the sign reversibility of the equations of motion has been misinterpreted as thermodynamic or mechanical reversibility. Please look at this thought experiment involving just two perfectly elastic billiard balls on a odd-shaped pool table with perfectly elastic cushions. We recognize that if one ball (A) is stationary and the other (B) is aimed at it on centers, then B will transfer its momentum to A and become stationary. If a cushion is arranged at right angle to the path of A, then A will be reflected and head back toward B. In the collision that follows, A will return its momentum to B and B will head back toward the shooter. Let another cushion be interposed at right angle to the path of B so that B is again reflected toward A. The net result of all this is that we have a reversible or naturally reversing process. We see similar approximations to this description in pendulums and celestial orbiting bodies. The reversibility of the billiard ball process is a consequence of the sign reversibility of the velocity in the momentum equation PLUS the purposeful adjustment of the cushion to reverse the sign of the velocity at some point.
Let's look at a very slightly more complicated experiment. Let everything be the same except that the first cushion is oriented to cause ball A to go off in a direction which will not cause a return collision with B. Since we have a "closed system," A will eventually strike a cushion, i.e., part of the container wall. Let this cushion be oriented so that it is at right angles to the path of A. The result is that A will be reflected and follow its same path in reverse, and striking the first wall and head back toward B. In other words, it will all be played out in reverse as the momentum (and energy equation) demand, but only because the second wall was PURPOSELY oriented to cause the reflection in the proper direction. One could extend this logic to any number of acute reflections preceding the 180 degree reversing collision.
It might seem as though only the Nth (final before reversal) wall segment has to be fixed. However, its orientation cannot be calculated without a knowledge of all the other reflections that preceded it, since the direction of ball A must be known. Hence, N wall segments are fixed by the motion of just a single collision pair. Like Monte Hall, with his knowledge of what is behind each of the doors, we must have knowledge of the entire particle path in order to design the reflecting container to insure reversal after the Nth wall collision. As long as these conditions are satisfied, mechanical reversibility and entropy are not a problem. The entropy neither increases nor decreases.
I think it would be instructive to design a wall for which an entire class of interactions will be reversible. For instance, consider a container made of two parabolas with a common axis and common foci. A collision which causes ball or particle A to travel through the focus and then continue to the wall will be reflected parallel to the axis of the parabolas, strike the opposite parabola and then back through the common focus. My gut tells me that by continuing in this vein, one would find that elastic balls in an elastic container with several degrees of freedom will spread their initial energy irreversibly with asymptotically increasing entropy. I'm going to work on such a proof. Thanks again Howard for opening Pandora's Elastic Box for me.
3 comments:
Joel - Sorry for posting my "Einstein" topic on the same day as your new topic, but I promised the Philo Club here in The Villages that my Powerpoint charts for tomorrow's talk would be up in advance.
On your points about reversibility, I look forward to Howard's reply. However, I was thinking similar thoughts about perfectly elastic and non-friction balls and walls and came to the same conclusions as Joel.
Say you have a perfectly elastic ball on a friction-free table with perfectly elastic walls.
Take a ball and place it anywhere you want and hit it with any force and at any angle you want. Call these the Initial (I) conditions and record them exactly.
The ball will bounce from wall to wall to wall to wall forever unless you stop it.
So, after a few bounces, measure the exact velocity (speed and angle) and rotation of the ball when it is at any point of its path. Call these the End (E) conditions.
OK, stop the ball and place it at that exact location E. Now strike it such that it has the exact opposite velocity and rotation (-E) and, if all is friction-free and elastic, it will retrace its exact path and bounces and, eventually return to location I with the exactly opposite (-I) velocity and rotation. QED!
Anticipating Howard's reply, and recalling our extensive and long ago discussions of chaos theory, I believe he will say that (since his world is continuous, unlike mine that is discrete :^) no matter how hard you try, you cannot get the EXACT Initial conditions nor the EXACT End conditions. Even if you could, you cannot strike the ball with the EXACT -E conditions, nor, when it returns to the I location, measure its EXACT -I conditions.
Indeed, Howard may say, if you let the ball bounce more than X times (ten or twenty or some number related to your capability of exactitude), chaos theory takes over and its path is almost totally random. In other words, if you try to reproduce I or -E conditions, you will get a very different velocity and location after X bounces.
How about that Howard, am I (still) a good student?
Ira Glickstein
It's not just chaos. Heisenberg’s Uncertainty Principle is at the foundation of quantum theory. There is no possibility of escaping it without undermining the entire structure of modern physics. Einstein never questioned it. It applies to all objects observable by their position and momentum. Billiard balls are obviously included.
Max Born in 1959 showed that after only a dozen classical (deterministic) collisions this inexorable uncertainty of the position and momentum of just the first ball has grown so large that all future trajectories are unpredictable. That is, you can’t be sure it will even hit another single ball. All you can do is give some statistics given a distribution of balls.
Howard
Sorry guys, but I'm uncomfortable with your approach. I believe quantum theory and its effectiveness. I just think it's used in this case like a sledge hammer to kill a fly. If all it's going to accomplish is to add some randomness to the collision process, I'd just as well put some imperfections on the containing wall. Random errors don't change the basic philosophical problem of decreasing entropy and the boundary between impossibility and improbability. I think that the answer lies in the "principle of equipartition of energy." The conflict between the idea of perfectly reflecting insulating walls and the tendency of energy to spread from macroscopic levels to microscopic levels seems to me to be at the root of the reversibility problem.
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