changye chen (Dept. of Physics, Beijing University of Aeronautics & Astronautics)

Published in physic.philica.com

Abstract
It is a new attempt to make the paradox raised by Ref. 1 more explicable. If you are serious about statistical mechanics, high-dimensional physics and other micro-or-macro dynamic events, you should know the essence of this important, but relatively simple, issue. (If you have difficulty to see the figures, use Firefox please. )

Article body

One of the fundamental tasks in the dynamical theory of classical statistical physics is to investigate how particles enter, or leave, a small volume element in the 6-dimensional position-velocity space. To make the 6-dimensional volume element less abstract, one of the inside particles is sketched in Fig. 1, where the position of the particle is in and the velocity is within . The aim of this paper is, however, to show that there exist intrinsic obstacles to formulating the changing rate of such particles.

As you may notice, the aim aforementioned is directly against the conventional wisdom. In terms of classical mechanics, we should be able to determine the position and velocity of any particle at any moment with any desired accuracy. How can there be any problem?

Before entering our detailed discussion, it is helpful to take a look at the following expression:

which, rather obviously, approaches +1 when y tends to zero much faster and approaches -1 when x tends to zero much faster. Infact, it can approach any specific value if the path of the limiting process is manipulated purposely. This paradoxical example reminds us of a well-known mathematical fact that a function whose value is path-dependent at a point should be considered to have no limit at that point (see any textbook concerning limits of multi-variable function).

Now, let be defined as

in which is the number of the particles that can possibly be found in . Then, the question simply becomes: whether or not is well defined? (The discontinuity problem associated with has been discussed elsewhere[2].)

The current theory, Boltzmann's equation for instance, asserts that can be formulated even in the presence of collisions[3]. Referring to Fig. 2, the philosophy of the theory concerning the particles entering the volume element due to collisions is explained in what follows. Consider what happens in during . If a colliding particle acquires a velocity within , the particle is identified as a one entering during . (Like anyone else, when I was a graduate student of physics, I accepted the logic in doing my homework without any question asked.)

Ironically enough, Fig. 2 can also be employed to reveal the hidden loophole of the standard theory. Since each of the identified particle has a new velocity within , represented by a single velocity in the figure, they will keep moving and tend to get out of right after the collisions. If is truly small, so small that its length scale is much shorter than , then almost all the "entering particles" will complete the escaping process from before the end of (except those involving collisions at the very final stage of ). That is to say, in order for the standard concept to work,  the length scale of has to be kept relatively large,  much larger than . According to the mathematical fact mentioned after (1), this is simply unacceptable.

As a stimulating exercise, interested readers could themselves construct a similar paradox concerning the particles leaving the volume element due to collisions[1].

The author has proposed some of experimentally verifiable counterexamples against the standard theory[4] and will soon give a detailed survey in this e-journal.

Special thanks to Prof. Keying Guan for discussion on several mathematical aspects of this subject.

References:

[1] C. Y. Chen, Mathematical investigation of the Boltzmann collisional operator, I1 Nuovo Cimento B, V117B, 177-181(2002) and Article 88 in Philica.com.

[2] C. Y. Chen, Article 111 in Philica.com.

[3] See, for instance, F. Reif, Fundamentals of Statistical and Thermal Physics, (McGraw-Hill Book Company, 1965, 1987, 1988 in English and German).

[4] C. Y. Chen, A new uncertainty principle, Arxiv: 0812.4343.

P.S. If you disagree on the point above, leave your criticism right after this article (exploiting the advantage of this e-journal). If you are not certain whether or not this article makes sense, why not invite more colleagues to join the discussion and to help clarify the issue. In addition, I will be very happy to hear from you directly (310-658-6459 in the US or chen4607@gmail.com).

Information about this Article
This Article has not yet been peer-reviewed
This Article was published on 25th November, 2009 at 02:40:48 and has been viewed 690 times.

This work is licensed under a Creative Commons Attribution 2.5 License.

The full citation for this Article is: chen, c. (2009). A graphic illustration of the misconcept in statistical mechanics. PHILICA.COM Article number 175.