^{*}

A new alternative approach to the statistical behavior of particle-particle collisions is introduced. The alternative approach is derived rigorously from well known mechanical laws; and the results given by it, quantitatively and qualitatively different from what the standard kinetic theory yields, can be directly checked with computer-simulated or realistic experiments. More importantly, from the introduction of it, a number of new concepts and new methodologies emerge, which might turn out to be very significant to the future development of nonequilibrium statistical mechanics.

It is commonly believed that the statistical behavior of classical particle-particle collisions has been adequately codified into the equations of nonequilibrium gas dynamics (kinetic theory), and there is nothing truly new to today’s physicists. However, as has been pointed out by us [

What made us question the validity of the standard theory might be summarized as the following. In the context of gas dynamics, particle-particle collisions are supposed to be examined in the six-dimensional position- velocity phase space, but the space, though superficially simple, is inherently counterintuitive. When a six- dimensional statistical dynamics is of concern, it is often the case that our attention is invited to certain incomplete and misleading pictures. To get a taste of such trickiness, let’s briefly review a long-neglected fact associated with the Boltzmann equation [

In this paper, after advancing several typical examples, an alternative approach is proposed, which takes care of the position space and the velocity space in a relatively balanced way. The proposed formalism is strictly based on well-known mechanical laws, and the results given by it can be directly compared with computer simulations. More importantly, from the introduction of this alternative approach, a number of new concepts and new methodologies emerge, which might turn out to be very significant to the future development of nonequilibrium statistical mechanics.

The structure of this paper is the following. In Section 2, several examples are advanced in which particle- particle collisions cannot be well treated by the standard kinetic theory. In Section 3, an alternative approach is proposed. In Section 4, the newly proposed alternative approach is extended to more general situations. In Section 5, a brief summary is provided.

To investigate the collective behavior of particle-particle collisions, we shall in this paper concern ourselves with three different, but interconnected, situations shown in

To begin with, let’s first look at the simplest situation illustrated in

where

where

There are several reasons why we start our investigation with this particular example. Firstly, this is the case to which the standard collision theory is presumably applicable. Secondly, this is the case that we, as physicists, can easily realize and easily monitor in computer simulation. Thirdly, this is the case in which the complexity of the subject is largely reduced but the statistical characteristics of the subject remain intact (so the insight gained will be generally instructive).

It is further assumed that a virtual detector, marked as

Before entering the next section, where a new alternative approach will be introduced, let’s briefly go through what the Boltzmann equation has to say about the situation. Referring to

in which

In writing these two formulas, it is understood that the two initial gases are dilute enough so that the collision probability between

Surprisingly, although formally pertinent, expression (4) yields no meaningful result for the situation. Firstly, if we wish to compute the integral involved, we find no attainable way to get rid of the five

When treating the examples shown in

There has been a detailed investigation about the sources of the aforementioned problems [

In connection with the situation shown in

the aftermath of a collision is illustrated in the position space and in the velocity space respectively. The two figures are drawn quite generally and their full implications will manifest themselves in this and the next sections.

In

where

Concerning expression (5), there are essential things worth discussing. To ensure that the right side of the expression stands for the “exact” distribution function at the position point

To make expression (5) physically meaningful in rather general situations, we shall adopt the following two assumptions: 1)

Under these two assumptions, we redefine the distribution function as a mathematical hybrid:

where

At this point, one remark seems in order. In the standard theory the distribution function, as the primary concept of the theory, is often defined or interpreted in intentionally vague language. For instance, in one of the books on nonequilibrium statistical mechanics [

We take three steps to compute expression (6): 1) finding out the position region in which particle-particle collisions may possibly give contribution to

The first two steps can be accomplished rather easily. Since the solid angle

where

To accomplish the third step aforementioned, let’s first note that the information concerning how the scattered particles spread in the position space and the velocity space is stored in

Referring to

Since

To connect the center-of-mass frame and the laboratory frame, we have to deal with the energy-momentum

conservation law explicitly. Let all the particles in our consideration have the same mass (for simplicity); and let

where

The above expressions show that for a pair of

in which

With help of Equations (8), (9), (12) and (13), the limit-average-hybrid distribution function defined by Equation (6) is

in which the integration region of

Unlike expression (4), expression (14) is obviously computable. It yields a finite and definite result no matter whether

Now, we look at how to extend our proposed approach to the cases shown in

Let the gas on the left-hand side of

where

where

Again, we are interested in determining

position of the detector’s inlet in

By defining the up-zone,

in which

Now, the energy-momentum conservation law shows us that

and Equation (17) becomes

in which

is the Jacobian of the variable transformation. Notice that

where

in which the integration region of

namely, by

As for the general case shown in

where

in which

Notice that in Equation (25)

in which

velocity space,

sion can be determined by

which is interpreted similarly to the diagram given in expression (23).

Obviously, both expressions (22) and (27) are directly computable, and ready to be checked with computer-simulated or realistic experiments.

It is easy to see that if

In this paper, to formulate the statistical behavior of particle-particle collisions, a new alternative integral formalism has been introduced. The results given by the new formalism are quantitatively and qualitatively different from what the standard theory yields. If interested, readers may confirm or deny them by applying their own theoretical and/or numerical approaches.

More importantly, along with the introduction of the new approach, a set of new concepts and methodologies are proposed, which might turn out to be very significant to the future development of nonequilibrium statistical mechanics. Some of them are:

Due to the discontinuity concern, the distribution function in a nonequilibrium approach should be defined as an average in the position-velocity phase space, at least partially.

Instead of using differential-integral equations, approaches of completely integral type should be employed.

Instead of examining the events in a control volume element, what takes place in an upstream path zone should be investigated.

Collisional effects should be studied in both the position space and the velocity space. The energy- momentum conservation law should be fully incorporated.

The author is grateful to Drs. M. Berry, O. Penrose, R. Littlejohn, V. Travkin, W. Hoover, Hanying Guo, Tian- rong Zhang, Keying Guan, Xingren Ying for their direct or indirect encouragement. The discussions with them have been pleasant and helpful.