An illustrative disproof of the collisionless Boltzmann equation

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

Published in physic.philica.com

Abstract
Continuing my previous two papers on Boltzmann’s equation in this open-access journal (Ref. 1 and 2), it is illustrated that the force term in the collisionless Boltzmann equation actually involves dissipation and randomness, which in turn implies that the equation itself needs to be reconsidered.

$(1)\hspace{20pt} \displaystyle \left[\frac{\partial }{\partial t}+ {\bf v}\cdot\frac{\partial }{\partial
{\bf r}}+ \frac{{\bf F}}m\cdot \frac{\partial } {\partial {\bf v}}\right]f({\bf r},{\bf v},t)=0,$

in which $\bf F$ represents the force acting on a particle that is located at the position $\bf r$ and possesses the velocity $\bf v$. This equation, called also the Vlasov equation in the fusion and astrophysics researches, is extensively employed to investigate the statistical behavior of charged particles (whenever the right side, the local collision term, can be ignored). With help of Column's potential, the force term is expressed as[3]

$(2)\hspace{20pt} {\bf F}=q{\bf E}({\bf r})=q\left[{\bf E}_{\rm ext}({\bf r}) -\nabla\int\frac{q f({\bf r}',{\bf v}',t)}{|{\bf r}-{\bf r}'|}d{\bf r}'d{\bf v}'\right] ,$

where the magnetic force has been neglected for simplicity. The form of Eq. 2 tells us two things. The first is that the force term has been considered to be of conservative type. The second is that the force has been regarded as an average quantity in view of that the distribution function in expression 2 is in itself an average one.

Our recent studies, however, inform us that, due to the fact that the equation is constructed along a segment of a particle's path, the force there should be regarded as the one averaged over the path segment, which in turn means that dissipation and randomness must come into play and the foundation of the equation is unsound.

Before proceeding any further, let's recall why the usual friction force, originated from the conservative electric force, is a dissipative one. Referring to Figure 1, we focus on a charged particle that enters a group of other charged particles. If the instant force acting upon it is in consideration, Column's law or Column's potential is genuinely applicable. However, if we care about what happens along a segment of a particle's path, things change considerably. The average force experienced by the particle will then depend on the particle's speed $v$, taking the form

$(3)\hspace{20pt} F=-\alpha v - \beta v^2 - \gamma v^3 -
\cdots ,$

where $\alpha, \beta, \gamma$ are certain constants. Furthermore, if the group's particles have a certain amount of thermal energy (they do in reality), the incident particle will experience stochastic forces as well.

To find out of which type the force term in Eq. 1 is, we take a close look at the textbook derivations. In one of such derivations[4], the behavior of the distribution function along a small segment of a particle's path is investigated. Based on an ad hoc assumption that the force experienced by such particles is a function of ${\bf r}$ only, the textbook derivation arrives at the path-invariance theorem

$(4)\hspace{20pt} f(t,{\bf r},{\bf v})= f(t',{\bf r}',{\bf v}'),$

where $(t,{\bf r},{\bf v})$ and $(t',{\bf r}',{\bf v}')$ stand for two points, nearby or not, along a particle's path. Eq. 4 is considered to be equivalent to the collisionless Boltzmann equation. The derivation summarized above shows that nothing but the force averaged over the path segment should be used in Eq. 1 and therefore the ad hoc assumption of the derivation is an illusive one.

If turning attention to more realistic situations, the problem can be seen more directly. Referring to Figure 2, we consider a gas system that consists of two parts: one is a moving beam of particles, expressed by $f_1$, and the other a relatively stationary group of particles, expressed by $f_2$. As has been indicated, the particles of $f_1$ must experience dissipative and stochastic forces given by $f_2$. The particles of $f_1$ lose energy due to the dissipative forces and thus $f_1$ increases. The particles of $f_1$ diverge due to the stochastic forces and thus $f_1$ decreases. The two effects cannot be balanced away in general.

Anyone who is good at computer simulation (or real experiments) will easily find that in almost all situations the path-invariance expressed by Eq. 4 cannot be observed as long as the energy exchange are taken into account in treating Column's interaction between particles.

Many negative arguments concerning Boltzmann-type equations have been presented in our papers. Ref. 1 shows that introducing whatever collision terms to the collisionless Boltzmann equation does not help. Ref. 2 points out that there are many realistic situations in which the distribution functions are highly discontinuous and operations of differential type can hardly be performed. This paper illustrates that the force term of the equation involves in general dissipation and randomness, which also ruins the equation's foundation. But, some of us may still wonder why there exists the `rigorous derivation' in which Liouville's theorem of classical mechanics is used to obtain Boltzmann-type formalisms (so-called the BBGKY approach). In what follows, a brief mention will be made about this concern.

It is well-known that Liouville's theorem is based on the Hamiltonian formalism in classical mechanics; for it to hold water, the system has to be a closed one and the boundaries have to be treated as a bunch of conservative potentials. A non-equilibrium system in statistical mechanics, however, is fundamentally different due to the facts that i) dissipation and randomness get involved when any of the system's particles collide with real boundaries; ii) boundaries may move or change with time; and iii) it is possible there exist sources of particle and/or energy. In view of such sharp context differences, the adequateness of using Liousville's theorem cannot be justified (let alone the discontinuity problems analyzed briefly in Ref. 2).

Further questions may come into your mind. What about other textbook derivations of the collisionless Boltzmann equation? How to formulate the statistical behavior of charged particles in a correct way? How to extend this discussion to the quantum regime and so on? As for such questions, the author (cychen@buaa.edu.cn) is looking forward to hearing from you.

The author is grateful to Oliver Penrose, Ke Wu and Keying Guan for stimulating and helpful discussions.

References:

1. C. Y. Chen, "An interesting disproof of Boltzmann's equation". PHILICA.COM Article number 88(2007).

2. C. Y. Chen, "A simpler disproof of Boltzmann's equation". PHILICA.COM Article number 111(2007).

3. See for instance S. Ichimaru, "Statistical Plasma Physics", p39, (Addison-Wesley, 1992).

4. See for instance F. Reif, "Fundamentals of Statistical and Thermal Physics", p498, (McGraw-Hill Book Company, 1965).