# Pathological limits in statistical mechanics

C. Y. Chen  Beihang University, PRC, Beijing 100191<EMAIL_ADDRESS>
###### Abstract

It is unveiled that some of the multi-variable limits used in statistical
mechanics have been ill-defined in the mathematical sense. Along the line, it
is suggested that significant progresses in non-equilibrium gas dynamics can
be made by redefining, reinterpreting, and reformulating those limits.

Keywords: Boltzmann’s equation, Phase space, Liouville’s theorem,
Discontinuity.

##  I Introduction

In mathematics, the term “pathological object”, or “monster”, refers to
something whose behavior is unexpectedly bad, counter-intuitive, and
inexplicable in terms of supposedly relevant theories. As a matter of fact,
there are popular bookscounter1 ; counter2 ; counter3  titled ”Counterexamples
in analysis”, ”Counterexamples in topology” and ”Counterexamples in
probability”, in which many of such counterexamples can be identified as
typical pathological objects.

One of the reasons behind those pathological objects is that basic concepts in
analysis, such as infinitely small, infinitely large, limit and continuity,
are simple but complex. If explained with help of examples, they appear to be
intuitive and rudimentary; however if explained by the textbook concepts,
which is intended to be theoretically rigorous and generally applicable, they
possess certain amount of abstraction and intricacy. After they are
incorporated with other concepts to build up more complex objects, the
abstraction and intricacy possessed by them may turn themselves into more
tricky ones. Professional mathematicians are fully aware of the risk. When
they propose a new theory in mathematics, they spend, quite often, a lot of
time and effort to recheck every building block of the theory.

Understandably, nonmathematicians pay less or no attention to things like
that. According to some prominent physicists, mathematics could be described
as a tool unreasonably effective in physical scienceswigner . This description
of mathematics has inspired and continue to inspire physicists to apply
mathematics more diligently and more creatively. But one question may still
remain in the mind of some physicists: In what sense and to what extent should
we, as physicists, also pursue the mathematical rigor? Considering that a
simple but ambiguous answer to this question would help no one, this paper is
devoted to an example exemplifying that pathological objects in mathematics
and half-baked ideas in physics may somehow find each other and result in
long-lasting mistakes (reminding us of Murphy’s law: anything that can go
wrong will go wrong).

The concrete topic of this paper is related to a number of multi-variable
limits used in statistical mechanics. Unlike the papers on multi-variable
limits in mathematics, the language, scope, perspective and objective of this
paper are mostly physics-oriented. Physicists and applied mathematician should
be able to find things fundamentally interesting and important.

##  II Partial limits of a multi-variable limit

In the conceptual sense the multi-variable limit is not much different from
the single-variable limit, but dealing with multi-variable limits is much more
complex. As far as mathematics is concerned, the main problem is that a
considerable number of multi-variable “limits”, though looking like and
seeming like a limit, are actually ill-defined and must be forbidden from
mathematical use. (This paper will show that the problem in physics becomes
how to define and use variants of such ill-defined “limits”.) In this section,
we look at this issue from the mathematical perspective.

To get some taste of it, let’s first examine the expression below:

$\lim\limits_{(x,y,z)\to(0,0,0)}\frac{x^{2}+y^{2}+\sin(3z^{2})}{x^{2}+z^{2}}.$
(1)

Its limiting behavior around the limit point $(x,y,z)=(0,0,0)$ can be
investigated through the following if-then (hypothesis-conclusion) analysis.
If we let $x$ and $y$ tend to zero much faster than $z$ does, then the
expression goes to $3$. If $x$ is kept relatively stationary while letting $y$
and $z$ go to the infinitesimal at a regular pace, independently or in any
combined way (e.g., $y^{2}=z^{2}\to 0$ or $y^{2}+z^{2}\leq\rho^{2}\to 0$),
then the expression approaches the value of $1$. If $x$ goes to zero at the
fastest pace, then the expression takes on a value from $3$ to $\infty$,
depending on in what way the two remaining variables, $y$ and $z$, go to their
own zeros. This analysis shows that a multi-variable “limit” can indeed have
multi behavior and the multi behavior can be revealed by performing several
if-then cycles (as long as they are designed properly).

In today’s mathematics, there is a special rule governing such multi-variable
“limits”courant ; thomas , and the rule may be described as follows: the limit
of a function can be considered as legitimately defined if and only if the
function approaches a unique and definite value no matter in what way the
function’s variables proceed with their limiting processes. Different people
may describe the rule in different manners. For instance, it can be
interpreted as saying that a well-defined limit must be path-independent, in
which the term path refers to a path in the variable space that leads to the
limit point. Or, it can be stated like this: a well-defined limit must be
hypothesis-free, by which it is meant that the unique and definite value of
the limit must not be associated with an explicit or implicit hypothesis
(presupposition).

As said already, revealing multi behavior of a multi-variable “limit” can be
accomplished by performing several well-designed if-then cycles. To make such
analysis be a symbolically concise approach, we introduce the following
concepts and notations.

Consider, for instance, a three-variable function $f(x,y,z)$ whose limiting
behavior at a limit point $(a,b,c)$ is of concern. The expression

$f([\hskip 1.0ptx\rangle_{a},[\hskip 1.0pty\rangle_{b},[\hskip
1.0ptz\rangle_{c})\equiv\lim\limits_{(x,y,z)\to(a,b,c)}f(x,y,z)$ (2)

will, hereafter, be called an intended limit (rather than a limit). By the
term “intended limit”, it is meant that there is a large chance that this
“limit” is ill-defined.

Each of the following expressions will be regarded as a partial limit produced
from $f([\hskip 1.0ptx\rangle_{a},[\hskip 1.0pty\rangle_{b},[\hskip
1.0ptz\rangle_{c})$:

$f([\hskip 1.0ptx\rangle_{a},y,z),\,f([\hskip 1.0ptx\rangle_{a},[\hskip
1.0pty\rangle_{b},z),\,f([\hskip 1.0ptx{\&}y\rangle_{(a,b)},z),\,\cdots,$ (3)

in which the variables enclosed in $[\hskip 1.0pt..\rangle$ are called the
active ones approaching their limits promptly, the variables not enclosed in
$[\hskip 1.0pt..\rangle$ are called the inactive ones whose limiting processes
are suspended until the active variables finish their limiting processes, and
the variables connected by “&” represent the ones behaving themselves in a
combined way. (When considering partial limits in physics, inactive variables
will be allowed to be inactive permanently.)

Take expression (1) as an example. It has quite a few partial limits, of which
the following three can be evaluated like single-variable limits:

$\displaystyle f(x^{2},[\hskip 1.0pty^{2}\rangle_{0},[\hskip
1.0ptz^{2}\rangle_{0})\to{x^{2}}/{x^{2}}\to 1,$ (4) $\displaystyle f([\hskip
1.0ptx^{2}\rangle_{0},y^{2}=z^{2})\to[z^{2}+\sin(3z^{2})]/z^{2}\to 4$ (5)

and

$f([\hskip 1.0ptx^{2}=z^{2}\rangle_{0},y^{2})\to{y^{2}}/{0}\to\infty.$ (6)

A simple observation of these expressions tells us that a partial limit’s
evaluation corresponds to an if-then cycle.

A conclusion is now evident: an intended limit can be identified as ill-
defined if an infinity or two different finite values emerge from the
evaluation of its partial limits. (In this paper, $\infty$ is not considered
as a legitimate value of a limit due to the two facts: i. $\infty$ is not a
definite value. ii. No measurable quantity in physics is $\infty$-valued.) As
an application of the conclusion, we can, by virtue of Eqs. (4) and (5) or by
virtue of Eq. (6) alone, deduce that expression (1) is ill-defined.

In mathematics (and in physics) there are quantities whose behavior is limit-
like from one perspective and function-like from another perspective.
Interestingly, whether they can be regarded as well-defined (or whether they
have multi behavior) can be investigated with help of the concept of partial
limit.

As an example, let’s suppose our task is to reveal the behavior of a two-
variable function defined as

$g(x,y)=\frac{1}{x^{2}y^{2}+1}$ (7)

around and along the $y$-axis.

It is obvious that

$g([\hskip 1.0ptx\rangle_{0},y)\equiv\lim_{x\to 0}g(x,y)=1\quad({\rm
for\;any\;definite}\;y).$ (8)

This result is impressive and may be of use for some purposes (such example in
physics will be given in the next section). But, can we thus claim that
$g(x,y)$ is an invariant around and along the $y$-axis? At first sight, this
claim can be proven with ease. Denoting $g([\hskip 1.0ptx\rangle_{0},y)$ as
$f(y)$, we obtain, from Eq. (8), $f(y_{0})=1$ and $f(y=y_{0}+\epsilon)=1$.
According to the usual conception, this can be rewritten as

$df/dy=0,$ (9)

which is synonymous to saying that $f(y)$ is an invariant along the $y$-axis.
From Eq. (9), one may further assert that $g(x,y)$ is an invariant around and
along the $y$-axis.

Figure 1: (a) $g=1/(x^{2}y^{2}+1)$ versus $y$ for different $x$; (b), (c) and
(d) $g=1/(x^{2}y^{2}+1)$ versus $x$ for different $y$.

Can this proof be taken for granted? In fact, we have two ways to refute it.
Firstly, let’s examine how the proof treats its hypothesis. Notice that
although $f(y)$ looks like a single-variable function of $y$, it is not. The
expression $f(y)$, or $g([\hskip 1.0ptx\rangle_{0},y)$, is and only is a
partial limit associated with the hypothesis that $[\hskip 1.0ptx\rangle_{0}$
prevails over $[\hskip 1.0pty\rangle_{\infty}$. When the above proof finally
concludes that $g$ is an invariant around and along the $y$-axis, this
hypothesis has been illogically disregarded and forgotten. Secondly, let’s
check out whether $g$ has multi behavior around and along the $y$-axis.
Obviously, the hypothesis that $[\hskip 1.0ptx\rangle_{0}$ prevails over
$[\hskip 1.0pty\rangle_{\infty}$ is not a unique hypothesis that we can
possibly have. If we let $x$ stay rather stationary and let $y$ go bigger and
bigger at a relatively fast pace, the value of $g$, denoted as $g(x,[\hskip
1.0pty\rangle_{\infty})$, will keep decreasing; namely, we have

$g(x,[\hskip 1.0pty\rangle_{\infty})\to 0\quad({\rm
for\;any\;definite\;}x\not=0).$ (10)

This behavior can be schematically seen from Fig. 1(a). The multi behavior
given by Eqs. (8) and (10) informs us that

$g([\hskip 1.0ptx\rangle_{0},[\hskip 1.0pty\rangle_{\infty})\to{\rm undef},$
(11)

in which undef means undefinable.

In summary, “the behavior of $g$ around and along the $y$-axis” involves two
competing processes: the first is related to “around” and the second to
“along”. If we choose either one of the two processes and let it prevail over
the other, the behavior of $g$ is well-defined. However, if we forget to make
the choice or allow the choice to be unclear, questing for “the behavior of
$g$ around and along the $y$-axis” is a self-misleading task.

It is also instructive to view the issue from the continuity and discontinuity
perspective. Figs. 1(b), 1(c) and 1(d) show that $g(x,y)$ behaves itself ever-
increasingly like a discontinuous function along the $y$-axis, which reminds
us of a plain fact that in a region where a certain type of discontinuity gets
involved defining a limit thereat is not really proper.

Several conclusive and/or speculative remarks are the following. Firstly, the
concept of partial limit is helpful in terms of identifying ill-defined multi-
variable limits. Secondly, if an “invariant” is established with help of
limiting processes, it is quite possible that the “invariant” is a partial
limit associated with an ad hoc hypothesis. Thirdly, considering a function’s
limiting behavior in a domain where a certain type of discontinuity gets
involved is in principle a risky business. Finally, if an intended limit in
physics proves to be ill-defined, we might be compelled to use some of its
partial limits as substitutes (with certain ad hoc hypotheses accepted).

##  III The distribution function for collision-free and force-free gas

When used to describe a gas in equilibrium, the concept of distribution
function is definitely a well-defined limit. But, can it be qualified as well-
defined when applied in gas dynamics? Since no paper has addressed the
question, we may reasonably believe there is no need to be concerned. However,
the issue is not that whether the distribution function is well-defined in the
academic sense but that if the distribution function involves multi behavior
in terms of describing gas dynamics we will have no choice but to deal with it
properly. We shall here investigate the issue with a collision-free and force-
free gas being used as the testbed.

One of the stadard ways of dealing with the dynamics of an ideal gas can be
called the Liouville-theorem approachliou ; nolte ; lerner , and the
Liouville-theorem approach has the following four-level structure:
i. Liouville’s theorem (path-constancy of phase volume).
ii. Path-invariance theorem of distribution function.
iii. The collisionless Boltzmann equation.
iv. The collisional Boltzmann equation.
(Note that level i listed above should be considered as the “ground level”,
level ii as the level immediately upon the ground level, and so on so forth…)
Historically, there were a few people who doubted the validity of the
collisional Boltzmann equation, but almost no serious scientists have openly
challenged this four-level structure.

Another way to derive the collisional Boltzmann equation is through the so-
called fluid-mechanics approachreif ; dorf , in which the involved phase space
is divided into many many small phase volume elements, and, then, how
particles get in, and out of, these volume elements is investigated and
formulated.

A common consensus in the community is that the Liouville-theorem approach and
the fluid-mechanics approach are essentially equivalent. Another common
consensus is that if a computational work of this theory needs to be
performed, the algorithm should be based on the fluid-mechanics approach.

Oddly, it is relatively easy to find out weak points of this conventional
wisdom. Suppose that our task is to examine the dynamic evolution of a
Boltzmann-type gas that suffers rather scarce particle-particle collisions.
According to the four-level structure listed above, two approaches can be
taken. The first will herein be called the path-approach and the second the
equation-approach; and the two approaches are respectively related to the
first half and second half of the four-level structure. In the path-approach,
the path-invariance theorem (level ii) plays the central role, and it works
like this: with help of two inputs, the path-information given by Newtonian
equations and the initial distribution function, the theorem yields the
distribution function $f^{(0)}(t)$. Since particle-particle collisions are
ignored, $f^{(0)}(t)$ may be regarded as the zeroth-order solution. (What
happens on the boundary in this approach is determined after $f^{(0)}(t)$ is
obtained.) However, in the equation-approach, the collisional Boltzmann
equation (level iv) plays the central role, and it works like the following:
with help of two inputs, the boundary condition and the initial condition
given in advance, the equation yields its solution $f(t)$. Then, interesting
questions arise: Are these two solutions, $f^{(0)}(t)$ and $f(t)$, basically
consistent? If not, can we reconcile them in a systematic way? Why the path-
information gets direct involvement only in the path-approach not in the
equation-approach? How come the two approaches treat the boundary condition so
differently?

Figure 2: Demonstration of how a phase volume element of a collisionless and
force-free gas moves and evolves.

Leaving these questions aside for time being, let’s start our long exploration
with the first half of the Liouville-theorem approach. Consider the phase
space shown by Fig. 2 (given only in the $x-v_{x}$ plane for technical
reasons), in which a material phase volume element, denoted as
$\delta\beta\equiv\delta x\delta v_{x}$, is moving and evolving. Suppose that
the initial distribution function in the concerned region is well defined.
[The term “well-defined” is not truly simple. For the discussion here, let’s
assume that the particles in and around $(\delta\beta)_{0}$ initially belong
to a gas in equilibrium.] Since the volume of $\delta\beta$ remains unchanged
in the process (due to Liouville’s theorem) and no particles move out of, or
move in, $\delta\beta$ (due to the fact that the gas is collision-free), the
particle density at the center of $(\delta\beta)_{t}$ will remain unchanged.
On the basis of this reasoning, almost all textbooks concludelandau ; stocker
; huang ; harris  that the distribution function in phase space behaves like
an incompressible fluid and the path-invariance theorem of distribution
function is a corollary of Liouville’s theorem.

Figure 3: An alternative view on how the distribution function evolves in the
$x$-$v_{x}$ phase space.

Interestingly, there is an alternative view from which we can get a different
conclusion. In Fig. 3, the initial distribution function $f_{0}(x,v_{x})$ has
been sliced into many thin slices according to their different $v_{x}$.
Obviously, as time proceeds each slice will move rightward with its own speed,
along its own path. Those slices, as a whole, will spread wider and wider in
the position dimension while each of them becomes thinner and thinner in the
velocity dimension. (We shall regard it as the velocity tearing effect
although the distribution function will not be truly torn apart.) If we are a
number of identical observers located one by one along a slice’s path, we will
definitely see that the number of the particles around a slice’s peak keeps
decreasing.

To comprehend these two figures, let’s do a bit of analytic work. When
thinking about the path-invariance theorem of distribution function, we must
concern ourselves with the following quantity:

$\hat{f}(\delta\beta,l)=\frac{\delta N(\delta\beta,l)}{|\delta\beta|},$ (12)

in which $l$ represents the length of a particle’s path and $\delta\beta$
stands for a small phase volume element whose “center” (defined by whatever
definition) is moving along the particle’s path, $|\delta\beta|$ is the volume
of $\delta\beta$ and $\delta N$ is the particle number in $\delta\beta$.

To most physicists who have been accustomed to the textbook treatment, the $l$
in Eq. (12) must be a definite (finite) one, and the $\delta\beta$ must be
infinitesimal (approaching zero rather freely). Namely, the distribution
function in their mind is nothing but the partial limit $\hat{f}([\hskip
1.0pt\delta\beta\rangle_{0},l)$. For convenience and clarity, we shall name
such distribution function the zero-volume distribution function; and denote
it as $f_{t}({\bf r},{\bf v})$ where $({\bf r},{\bf v})$ represents the
endpoint of the path $0\to l$. We then have, by virtue of Fig. 2,

$\hat{f}([\hskip 1.0pt\delta\beta\rangle_{0},l)=\hat{f}([\hskip
1.0pt\delta\beta\rangle_{0},0)$ (13)

or

$f_{t}({\bf r},{\bf v})=f_{0}({\bf r}_{0},{\bf v}_{0})$ (14)

in which $({\bf r}_{0},{\bf v}_{0})$ is the starting phase point of the path
$0\to l$. For historical reasons, we still consider Eq. (13) or (14) as the
path-invariance theorem of distribution function although the distribution
function here has been defined as the partial limit associated with the
specific hypothesis that $[\hskip 1.0pt\delta\beta\rangle_{0}$ prevails.

For the following four reasons, we realize that defining the zero-volume
distribution function is not sufficient. i. According to the last section, in
order to know the complete behavior of Eq. (12), we need to evaluate it under
different hypotheses. One of such different hypotheses is that $l$ becomes
larger and larger at a pace much faster than that of $\delta\beta\to 0$. ii.
In terms of measuring the distribution function along a path by physical
means, the involved $\delta\beta$ must be finite. iii. In any numerical
treatment, setting $\delta\beta$ to be infinitesimal is pointless. iv. It
turns out (in this paper and the other publications of the authorchen1 ) that
the gas dynamics of real gases possesses a number of mechanisms that can, and
will, produce particles distributing discontinuously so that defining the
zero-volume distribution function becomes very difficult or impossible. Thus,
we define

$f_{\delta\beta}(l)\equiv\hat{f}(\delta\beta,l),$ (15)

in which $\delta\beta$ stands for a phase volume element with a definite size
and a definite shape and $f_{\delta\beta}(l)$ represents a function of $l$
defined by Eq. (12). We shall regard such distribution function as a finite-
volume distribution function (or the average distribution function over the
volume element $\delta\beta$). As has been shown by Fig. 3, if such
$\delta\beta$ initially encloses a peak of the initial distribution function,
$f_{\delta\beta}(l)$ will decrease continuously along the path; and the end
behavior can be expressed by

$f_{\delta\beta}(l\to\infty)\equiv\hat{f}(\delta\beta,[\hskip
1.0ptl\rangle_{\infty})\to 0.$ (16)

(If one wishes, one can use the fluid-mechanics approach to confirm this path-
behavior.)

We now realize that neither the zero-volume distribution function nor the
finite-volume distribution function is sufficient in terms of describing the
path-behavior of a distribution function. While the zero-volume distribution
function can reveal more details of the local structure (if definable), the
finite-volume distribution function is more useful and more meaningful in many
other senses. [It turns out that in usual gases the velocity distribution will
vary more violently than the position distribution. Under this realization, we
may sometimes define $f([\hskip 1.0ptd^{3}{\bf r}\rangle_{0},d^{3}{\bf v})$
instead.]

As a matter of fact, we can use the zero-volume distribution function to
further uncover the necessity of using the finite-volume distribution
function. With reference to Fig. 3, let’s suppose that $f_{0}$ is well-defined
and that $\partial f_{0}/\partial v_{x0}$ is quite small
($\equiv\epsilon\approx 0$) in the region concerned. Then, we have

$\left.\frac{\partial f_{t}}{\partial
x}\right|_{x_{0}=x-v_{x}t,v_{x0}=v_{x}}=\left.\frac{\partial f_{0}}{\partial
x_{0}}\right|_{x_{0}}{\hskip 59.75095pt}$ (17)

and

$\left.\frac{\partial f_{t}}{\partial
v_{x}}\right|_{x_{0}=x-v_{x}t,v_{x0}=v_{x}}=-t\left.\frac{\partial
f_{0}}{\partial x_{0}}\right|_{x_{0}}+\epsilon.\hskip 30.0pt$ (18)

These two expressions are quite informative. In reference to Fig. 3 and Fig.
4, Eq. (17) shows that if the initial distribution function (deemed as the
source distribution function) has a definite profile in the position space,
the distribution function at a later time (deemed as the image distribution
function) will have infinitely many such profiles in the position space due to
the fact that each $\bf v$ can carry the source profile to a new place; and
Eq. (18) shows that when velocities carry the source distribution function to
a new place $\bf r$, a new profile in the velocity space will be produced, and
the new velocity profile is roughly the same as the source profile but the
size of it is expanded or compressed by the factor $t$.

Figure 4: Illustration of how the source distribution function creates many
new profiles (a) in the position space and (b) in the velocity space.

Armed with the notion that $\partial f/\partial({\bf r},{\bf v})\sim\infty$
can be used to characterize the discontinuity of distribution function, we
should be able to view the above gas dynamics from a new perspective. If
$\partial f_{0}/\partial{\bf r}_{0}$ is fairly large somewhere, the
distribution functions along paths will be increasingly like discontinuous
ones in the velocity space (due to the fact that $t|\partial
f_{0}/\partial{\bf r}_{0}|\to\infty$). We shall call this phenomenon the
dynamic quasi-discontinuity. Furthermore, if $f_{0}$ is already discontinuous
somewhere in the position space (caused by the initial condition or by the
boundary condition), the whole phase space will be full of discontinuity,
which can also be shown by Fig. 4.

From Fig. 4(b), it should be noted that the velocity distribution function at
any position $\bf r$ is a compressed image of the entire position space
(different position points have different affecting times though); and thus a
large or overwhelming fraction of a gas’s particles must involve quasi-
discontinuity and ordinary discontinuity. In this sense, the gas dynamics of
real gas must be based on the finite-volume distribution function which can
treat continuous and discontinuous particles in a unifying way.

Figure 5: Schematic of how to determine the finite-volume distribution
functions in the phase elements $(\delta\beta)_{t}^{i}$,
$(\delta\beta)_{t}^{ii}$, $(\delta\beta)_{t}^{iii}$ and
$(\delta\beta)_{t}^{iv}$ from the given initial distribution function.

Now, we give an example of how to evaluate the finite-volume distribution
function under the condition that the initial distribution function is of
piecewise continuity. In reference to Fig. 5, $(\delta x)_{1}$ and $(\delta
x)_{2}$ are two regions in each of which there exists a gas in its own
equilibrium, and the other regions are in the vaccum. Suppose that after $t=0$
all the walls confining the gases disappear suddenly and completely. Our task
is to determine the finite-volume distribution functions at a later time $t>0$
in the phase elements $(\delta\beta)_{t}^{i}$, $(\delta\beta)_{t}^{ii}$,
$(\delta\beta)_{t}^{iii}$ and $(\delta\beta)_{t}^{iv}$. The discussion given
in this section suggests that we should first locate the source regions of
those phase elements. By performing the time reversal, we find that the source
regions are $(\delta\beta)_{0}^{i}$, $(\delta\beta)_{0}^{ii}$,
$(\delta\beta)_{0}^{iii}$ and $(\delta\beta)_{0}^{iv}$ respectively. Since the
initial distribution function is well-defined in the piecewise sense, the
finite-volume distribution functions in $(\delta\beta)_{t}^{i}$,
$(\delta\beta)_{t}^{ii}$, $(\delta\beta)_{t}^{iii}$ and
$(\delta\beta)_{t}^{iv}$ can be easily evaluated with help of

$\displaystyle f_{\delta\beta}(t)$
$\displaystyle=\int_{(\delta\beta)_{t}}f_{t}({\bf r},{\bf v})d^{3}{\bf
r}d^{3}{\bf v}\Big{/}|(\delta\beta)_{t}|$
$\displaystyle=\int_{(\delta\beta)_{0}}f_{0}({\bf r}_{0},{\bf v}_{0})d^{3}{\bf
r}_{0}d^{3}{\bf v}_{0}\Big{/}|(\delta\beta)_{0}|.$ (19)

If the initial distribution function is not well-defined (for instance if the
particles come from the boundary-particle interaction), we can determine
$f_{\delta\beta}(t)$ by regarding each patch of the boundary surface as a
point-wise source of particleschen1 .

In some sense, the investigation done in this section has clearly shown the
advantages of the path-approach. i. Instead of giving an unsolved equation, It
gives directly treatable formulas. ii. The involved cause-effect relationship
is very clear and very straightforward and no unnecessary mistakes can be
made. iii. Dealing with continuity and discontinuity with almost the same
accuracy and efficiency.

In next several sections, we shall step by step take particle-particle
collisions into account.

##  IV Scattering cross sections in different reference frames

To deal with particle-particle collisions between particle beams, several
types of scattering cross sections are employed in textbooks of statistical
mechanicsreif ; kubo . It turns out that one of the cross sections therin is
also ill-defined.

For simplicity, all particles considered in this paper are distinguishable and
subject only to short-range elastic interaction (in terms of classical
mechanics), but have the same mass, size and shape. Furthermore, throughout
this paper, the assumption of molecular chaos is taken for granted whenever a
gas system is in the consideration.

Although the usual examination in textbooks starts with a beam of particles
falling upon a resting target particle, we shall, for consistence of this
paper, directly discuss what will happen when two particle beams meet in
space.

Imagine the setup illustrated in Fig. 6a where two narrow particle beams move
towards each other. The particles in the first beam, with the velocity ${\bf
v}$, will be referred to as incident particles; and the particles in the
second beam, with the velocity $\bf w$, as target particles. Inside the two
beams, the positions of these particles are completely randomized. If a
collision between an incident particle and a target particle occurs, their
incoming velocities $\bf v$ and ${\bf w}$ will be changed to the outgoing
velocities ${{\bf v}^{\prime}}$ and ${\bf w}^{\prime}$ respectively. The small
region where those collisions may take place is denoted as $dU$.

Figure 6: (a) Particles of two beams colliding in a small region $dU$; and (b)
a pair of the outgoing particles shown in the velocity space.

We then define the velocity of the center-of-mass and the velocity of the
incident particles relative to the center-of-mass as

${\bf c}\equiv({\bf v}+{\bf w})/2\quad{\rm and}\quad{\bf u}\equiv({\bf v}-{\bf
w})/2$ (20)

respectively; after a collision, these two velocities become

${\bf c}^{\prime}\equiv({{\bf v}^{\prime}}+{\bf w}^{\prime})/2\quad{\rm
and}\quad{\bf u}^{\prime}\equiv({{\bf v}^{\prime}}-{\bf w}^{\prime})/2$ (21)

respectively. Since elastic collisions are our concern, the energy-momentum
conservation law reads

${\bf c}^{\prime}={\bf c}\quad{\rm and}\quad|{\bf u}^{\prime}|=|{\bf u}|\equiv
u.$ (22)

It should be mentioned that for a velocity pair $({\bf v},{\bf w})$ there
exist infinitely many spreading velocity pairs $({{\bf v}^{\prime}},{\bf
w}^{\prime})$, and the paired velocities ${{\bf v}^{\prime}}$ and ${\bf
w}^{\prime}$ will symmetrically fall on two sides of the spherical shell
labeled as the energy-momentum shell (EMS) shown in Fig. 6b.

Armed with these notions, we have the differential scattering cross section in
the center-of-mass reference frame taking the form

$\sigma({\bf u},{\bf
u}^{\prime})=\sigma(\Omega)=\frac{dN}{d\Omega}=u^{2}\frac{dN}{dS},$ (23)

in which the solid angle $\Omega$ is defined by the direction of ${\bf
u}^{\prime}$ with respect to ${\bf u}$, $d\Omega$ is an infinitesimal solid-
angle element about $\Omega$, $dS$ is the infinitesimal surface element on the
EMS subtending $d\Omega$ and $dN$ represents the average number of the
incident particles emerging after collisions within $d\Omega$, or on $dS$, per
unit incident flux (in terms of the relative speed $2u$), unit target
($n_{{\bf w}}|dU|=1$) and unit time.

Eq. (23) shows that $\sigma$ is a two-variable $0/0$ limit. It is easy to see
that the incident particles will, after collisions, distribute on the EMS
continuously (in the statistical sense); and thus the magnitude of $\sigma$
will be independent of how $d\Omega$, or $dS$, shrinks to an infinitesimal
one. Based on this realization, we admit that $\sigma$ is well-defined.

Then, let’s look at another scattering cross section in textbooksreif ; kubo ,
which is in this paper referred to as the cross section in the laboratory
frame. The cross section, denoted as $\sigma^{l}$ instead of $\sigma$, is
defined in the form

$\sigma^{l}({\bf v},{\bf w}\to{{\bf v}^{\prime}},{\bf
w}^{\prime})=\frac{dN}{d^{3}{{\bf v}^{\prime}}d^{3}{\bf w}^{\prime}}$ (24)

where $d^{3}{{\bf v}^{\prime}}\equiv
dv_{x}^{\prime}dv_{y}^{\prime}dv_{z}^{\prime}$ is a small velocity element
enclosing ${{\bf v}^{\prime}}$, $d^{3}{\bf w}^{\prime}\equiv
dw_{x}^{\prime}dw_{y}^{\prime}dw_{z}^{\prime}$ is a small velocity element
enclosing ${\bf w}^{\prime}$, and $dN$ is the number of the incident particles
scattered into $d^{3}{{\bf v}^{\prime}}$ per unit incident flux, unit target
and unit time (while the involved target particles fall into $d^{3}{\bf
w}^{\prime}$ presumably). The fact that $\sigma^{l}$ has been defined and
applied without any specification about $d^{3}{\bf v^{\prime}}$ and $d^{3}{\bf
w^{\prime}}$ tacitly implies that this $\sigma^{l}$ is supposed to be a
legitimate $0/0$ limit.

Now, we inspect whether or not $\sigma^{l}$ is indeed well-defined.

A simple comparison betwee these two defining equations gives us rather useful
information. Eq. (23) says that the collision-produced particles have a
2-dimensional distribution in the velocity space (which is related to the fact
that there are 6 unknown variables concerning ${\bf v^{\prime}}$ and $\bf
w^{\prime}$ and there are only 4 energy-momentum equations to constrain them).
But, Eq. (24) seemingly suggests that ${\bf v}^{\prime}$ and ${\bf
w}^{\prime}$ have six free space to move in. For convenience, we shall call it
the problem of “dimension mismatch”.

With help of the concepts presented in the last section, the dimension
mismatch can be analyzed easily. It is obvious that $\sigma^{l}$ defined by
Eq. (24) is just an intended limit. One of its partial limits takes the form

$\sigma^{l}([\hskip 1.0ptd^{3}{{\bf v}^{\prime}}\rangle_{0},d^{3}{\bf
w}^{\prime})=\lim\limits_{d^{3}{\bf v^{\prime}\to 0}}\frac{dN}{d^{3}{\bf
v^{\prime}}d^{3}{\bf w^{\prime}}}.$ (25)

If we let $d^{3}{\bf w^{\prime}}$ be a spherical ball with a definite radius
and let $[\hskip 1.0ptd^{3}{{\bf v}^{\prime}}\rangle_{0}$ be an ever-shrinking
spherical ball in the velocity space with the radius $\rho\to 0$. Assuming the
centers of the two balls to lie on the EMS symmetrically as shown in Fig. 6b,
Eq. (25) becomes

$\sigma^{l}([\hskip 1.0ptd^{3}{{\bf v}^{\prime}}\rangle_{0},d^{3}{\bf
w}^{\prime})=\frac{\sigma\pi\rho^{2}/u^{2}}{(4\pi\rho^{3}/3)d^{3}{\bf
w^{\prime}}}=\frac{3\sigma}{4u^{2}d^{3}{\bf w^{\prime}}}\cdot\frac{1}{\rho},$
(26)

in which we have used $dN=\sigma\cdot dS/u^{2}=\sigma\pi\rho^{2}/u^{2}$ given
by Eq. (23).

Alternatively, if we let $[\hskip 1.0ptd^{3}{{\bf v}^{\prime}}\rangle_{0}$
stand for an ever-shrinking cube whose side length is $b\to 0$, whose center
lies on the EMS, and whose top and bottom are parallel with the EMS locally,
we get, in, otherwise, the same context,

$\sigma^{l}([\hskip 1.0ptd^{3}{{\bf v}^{\prime}}\rangle_{0},d^{3}{\bf
w}^{\prime})=\frac{\sigma b^{2}/u^{2}}{b^{3}\cdot d^{3}{\bf
w}^{\prime}}=\frac{\sigma}{u^{2}d^{3}{\bf w^{\prime}}}\cdot\frac{1}{b}.$ (27)

These results show that $\sigma^{l}$ is multi-valued; furthermore, if $\rho\to
0$ or $b\to 0$, $\sigma^{l}$ is $\infty$-valued. In fact, by changing the
shape of $d^{3}{\bf v^{\prime}}$, the value of $\sigma^{l}$ can be any value
from $0$ to $\infty$chen2 . Therefore, $\sigma^{l}$ cannot, and should not, be
used in any mathematical and physical formalism.

In the usual textbook treatment,

$\sigma^{l}({\bf v},{\bf w}\to{\bf v^{\prime}},{\bf
w^{\prime}})=\sigma^{l}({\bf v^{\prime}},{\bf w^{\prime}}\to{\bf v},{\bf w})$
(28)

is given as the expression for the time-reversibility of particle-particle
collision. The discussion given above has shown that this expression is
misleading. To express the time-reversibility in a correct way, we need at
least to avoid the problem of dimension mismatch. An inspection of Eq. (23)
tells us that the equation can be rewritten as

$\sigma(\Omega_{\bf uu^{\prime}})=u^{2}\frac{dN}{dS_{\bf
v^{\prime}}}\equiv\sigma({\bf v},{\bf w}\to dS_{\bf v^{\prime}},dS_{\bf
w^{\prime}}),$ (29)

in which $dS_{\bf v^{\prime}}$ and $dS_{\bf w^{\prime}}$ stand for two
infinitesimal symmetric patches located symmetrically on the two side of the
EMS (allowing $\bf v^{\prime}$ and $\bf w^{\prime}$ to fall on them
respectively). This new definition of the cross section gives a good picture
about what happens with the velocities $\bf v$, $\bf w$, $\bf v^{\prime}$ and
$\bf w^{\prime}$ in the laboratory reference frame. With help of this new
definition of $\sigma$, we have

$\sigma({\bf v},{\bf w}\to dS_{\bf v^{\prime}},dS_{\bf
w^{\prime}})=\sigma({\bf v^{\prime}},{\bf w^{\prime}}\to dS_{\bf v},dS_{\bf
w}),$ (30)

whose validity can be understood from $\sigma(\Omega_{\bf
uu^{\prime}})=\sigma(\Omega_{\bf u^{\prime}u})$.

We have seen the usefulness and powerfulness of $\sigma$, we shall in the next
two sections use $\sigma$ alone to solve various problems involving particle-
particle collisions.

##  V the scattering-out rate from a particle beam

The most fundamental collision problem has been considered in the last section
with reference to Fig. 6. Apart from the ill-defined cross section in the
laboratory fram, there seems nothing really new to physicists. Strangely, when
we include some of the errorproof formulas into the standard kinetic theory,
we shall still meet with unexpected difficultieschen2 . In this section, we
reinvestigate the issue and give a better solution for it.

According to textbooks of statistical mechanicslerner ; reif , the time
evolution of the distribution function of an ideal dilute gas should be
described by the Boltzmann equation

$R_{t}=R^{\bf v}+R^{\bf F}+R^{\rm s.in}-R^{\rm s.out},$ (31)

in which $R_{t}\equiv(\partial f/\partial t)_{{\bf r},{\bf v}}$ is the change
rate of the local distribution function at $({\bf r},{\bf
v})\equiv(x,y,z,v_{x},v_{y},v_{z})$, $R^{\bf v}=-{\bf v}\cdot(\partial
f/\partial{\bf r})$ represents the particles driven by the velocity $\bf v$,
$R^{\bf F}=-{\bf F}/m\cdot(\partial f/\partial{\bf v})$ stands for the
influence of the macroscopic force ${\bf F}\equiv(F_{x},F_{y},F_{z})$, and
$R^{\rm s.in}-R^{\rm s.out}$ is called the Boltzmann scattering operator.

Although different approaches deal with the Boltzmann equation differently,
the Boltzmann collision operator $R^{\rm s.in}-R^{\rm s.out}$ is formulated
unanimously by examining what happens in a small phase volume element
$d\beta\equiv d^{3}{\bf r}d^{3}{\bf v}$ fixed in phase space. This type of
approach may be referred to as the Eulerian approach; and every term in Eq.
(31) should be understood likewise.

On the understanding, the change rate of the local distribution function
$R_{t}$ is defined by

$R_{t}=\lim\limits_{dt\to 0,d\beta\to
0}\frac{(dN)_{\delta\beta,0-dt}}{dtd\beta}=\left.\frac{\partial f}{\partial
t}\right|_{{\bf r},{\bf v}},$ (32)

in which $(dN)_{\delta\beta,0-dt}$ is the net increment of the particle number
in $d\beta$ from time $0$ to time $dt$, which shows us that $R_{t}$ is
supposed to be a multi-variable limit with $dt\to 0$ and $\delta\beta\to 0$.
Actually, every term in Eq. (31) can be expressed similarly.

To be less burdensome in this section, we take on the following
simplifications: i. ${\bf F}=0$. ii. The direction of $\bf v$ is consistent
with the direction of the $x$-axis. And, iii. $R^{\rm s.in}$ needs not to be
taken into account (which means in the consideration of this section there
will be no particle-particle collisions that can produce particles with $\bf
v$). Thus, Eq. (31) becomes $R_{t}=R^{v_{x}}-R^{\rm s.out}$, or

$R_{t}=\frac{[{(dN)^{v_{x}\rm.in}-(dN)^{v_{x}\rm.out}}]}{dtd\beta}-\frac{(dN)^{\rm
s.out}}{dtd\beta},$ (33)

in which the first term $R^{v_{x}}$ caused by $v_{x}$ will be called the fluid
term and the second term $R^{\rm s.out}$ caused by particle-particle
collisions will be called the kinetic term. In what follows, our task is to
inspect whether each of the two terms is well-defined and inspect whether the
events related to the two terms are mutually exclusive (in order to obey the
addition rule for probabilities).

Figure 7: Dynamics of particle number: (a) Particles moving in and out of
$d\beta$ due to $v_{x}dt$. (b) Particles scattered out of $d\beta$ by $f_{\bf
w}$. (c) The combination of these two effects.

The fluid term $R^{v_{x}}$ can indeed be analyzed by fluid mechanics. From
Fig. 7a, we obtain:

$\displaystyle(dN)^{v_{x}\rm.in}-(dN)^{v_{x}\rm.out}$
$\displaystyle=v_{x}dtdydzd^{3}{\bf v}(f^{l}-f^{r})$
$\displaystyle=-v_{x}\frac{\partial f}{\partial x}dtd\beta,$ (34)

where $f^{l}$ and $f^{r}$ are the local distribution functions of $f$ on the
left and right ends of $dx$ respectively. Hence, we have

$R^{v_{x}}=\frac{(dN)^{v_{x}\rm.in}-(dN)^{v_{x}\rm.out}}{dtd\beta}=-v_{x}({\partial
f}/{\partial x}).$ (35)

It is noted that this result is independent of how $dt$ and $d\beta$ shrink to
their zeros (by choice or by chance), and thus $R^{v_{x}}$ is indeed well-
defined in the mathematical sense.

The kinetic term $R^{\rm s.out}$ can be analyzed with help of Fig. 7b, which
is essentially the same as Fig. 6a except that the possible collisions are
marked with letters x in Fig. 7b. In reference to Fig. 6, we find, from Eq.
(23), that the number of the particles initially belonging to $n_{\bf v}$ and
eventually scattered into the patch $dS$ on the EMS is

${(dN)^{dS}}=\sigma{(d\Omega)\cdot(2un_{\bf v})\cdot(n_{{\bf w}}d^{3}{\bf
r})\cdot(dt)},$ (36)

where $2un_{\bf v}$, $n_{\bf w}d^{3}{\bf r}$ and $dt$ are no longer unit ones,
and $d\Omega=dS/u^{2}$. The total number of the scattering-out particles is

$(dN)^{\rm s.out}=\int 2u\sigma n_{\bf v}n_{{\bf w}}d\Omega d^{3}{\bf r}dt.$
(37)

By replacing $n_{\bf v}$ with $f_{\bf v}d^{3}{\bf v}$ and replacing $n_{{\bf
w}}$ with $\int d^{3}{\bf w}f_{{\bf w}}$ (assuming there are more beams
entering $\delta\beta$ and involving the collisions), we arrive at

$R^{\rm s.out}=\lim\frac{(dN)^{\rm s.out}}{dtd^{3}{\bf r}d^{3}{\bf v}}=\int
2u\sigma f_{\bf v}f_{{\bf w}}d\Omega d^{3}{\bf w}.$ (38)

To get it, no hypothesis about how $d^{3}{\bf r}$, $d^{3}{\bf v}$ and $dt$
shrink to their zeros has been introduced, which shows that the $R^{\rm
s.out}$ here is a legitimate $0/0$ limit.

So far the discussion given in this section is completely consistent with the
standard textbook treatment.

Let’s now check out whether or not the above two terms are “mutually
exclusive”. In Fig. 7c, the particles related to the fluid term and the
kinetic term are shown in a combined way. The figure shows that if we pay
attention to the particles in the shaded region we find that there is double
counting concerning $(dN)^{v_{x}\rm.out}$ and $(dN)^{\rm s.out}$.

The direct consequence of this double counting is that $R_{t}$, cannot be
recognized as a well-defined limit, although the standard theory, represented
by expression (32), seems to suggest otherwise.

To reveal the problem more explicitly, let’s do a thought experiment. Consider
that the experiment shown in Fig. 7c is done in a computer simulation. By
letting the incoming particle beam represented by $f_{\bf v}$ be a stationary
one during the experiment, and letting the incoming colliding particle beam
represented by $f_{\bf w}$ be applied to $\delta\beta$ after $t=0$, a direct
evaluation of Eq. (32) gives us that

$R_{t}([\hskip 1.0ptvdt<dx\rangle_{0},dydzd^{3}{\bf v})=-R^{\rm s.out}+{\kappa
R^{\rm s.out}}/2,$ (39)

where $\kappa\equiv v_{x}dt/(dx)$ is a real number between 0 and 1. In
physics, this expression means that the value of $R_{t}$ strongly depends on
the experiment parameters $dt$ and $dx$. In mathematics, it means that $R_{t}$
is an ill-defined limit. All these are not expected from the viewpoint of the
standard theory.

As a matter of fact, there is a simpler way to find out what the problem is.
According to the Eulerian approach, equation (31) is constructed on the basic
assumption that the particles in $\delta\beta$ can be partitioned into several
groups, such as the particles driven into $\delta\beta$ by $v_{x}$, the
particles driven out of $\delta\beta$ by $v_{x}$, the particles driven out of
$\delta\beta$ by collisions, the particles driven into $\delta\beta$ by
collisions and so on so forth. However, when $\delta\beta\to 0$ and $dt$ is
slightly long, dividing the particles therein into such groups will become a
mission impossible (those particles are no longer “mutually exclusive”).

Interestingly, although the problem indicated above is a fundamental one, it
is still surmountable. If we look at $R^{\rm s.out}$ from the perspective of
the path-approach (or look at it in the Lagrangian reference frame), the
difficulties associated with $R^{\rm s.out}$ disappear almost entirely. In
fact, there are two pieces of good news. The first is that, for the particles
in a moving beam, the only leaving mechanism is the one that has been
formulated by $R^{\rm s.out}$. The second is that Eq. (38) can be integrated
meaningfully and easily. To see it, let’s rewrite Eq. (38) as

$\left.\frac{df_{\bf v}}{dt}\right|_{l}=v\frac{df_{\bf v}}{dl}=-f_{\bf v}\int
d^{3}{\bf w}d\Omega 2u\sigma f_{{\bf w}},$ (40)

where $l$ represents the particles’ path (a curve in the position space,
generally speaking) and $v\equiv|{\bf v}|$ the particles’ speed. This formula
is essentially the same as another one in the textbook treatmentreif :

$\frac{dp}{pdt}=v\frac{dp}{pdl}=-\tau^{-1}=-\int d^{3}{\bf w}d\Omega 2u\sigma
f_{\bf w},$ (41)

in which $p$ is the survival probability of a test particle and $\tau$ is the
average collision time (or relaxation time).

These two formulas inform us that for a test particle or for a small group of
similar particles represented by $f$ the path-dynamics is exactly the same.

The integration of Eq. (41) over a finite path, rather than over a period of
time at a fixed phase point, gives us the following path-survival probability

$p(\Delta l)=\exp\left(-\int_{\Delta l}\frac{dl}{|\bf v|}\int d^{3}{\bf
w}d\Omega 2u\sigma f_{\bf w}\right).$ (42)

This path-survival probability formula sheds a lot of light on the gas
dynamics that we wish to explore. It greatly highlights that all the
inferences associated with Fig. 4a and 4b in Sect. III make sense even in
presence of particle-particle collisions as long as certain path-survival
probabilities are taken into account. In particular, it tells us that if a
local distribution involves discontinuity (for whatever reasons), the
discontinuity will affect almost everywhere of the gas system.

##  VI The particles scattered into a phase volume element

In Sect. IV, we have shown that the standard expression for time-reversibility
of particle-particle collision is a misunderstanding. In this section, we
shall try to formulate the particles scattered into a phase volume element
without taking any help from the time-reversibility.

Let’s first follow the textbook treatment and try to define the scattering-in
rate in an explicit way. In reference to Fig. 6, we accept the notation in
which the two colliding particle beams are represented by $n_{\bf v^{\prime}}$
and $n_{\bf w^{\prime}}$ and the phase volume element used to collect the
scattered particles is $d^{3}{\bf v}$ (which means that the collected
particles initially belong $n_{\bf v^{\prime}}$). Under the notation, the
scattering-in rate is supposed to be defined by

$R^{\rm s.in}=\lim\frac{(dN)^{\rm s.in}}{dtd^{3}{\bf r}d^{3}{{\bf v}}},$ (43)

where $(dN)^{\rm s.in}$ is the number of the particles collected by $d^{3}{\bf
v}$ during $dt$ and $d^{3}{\bf r}$ is a position volume element in the
collision region.

Evidently, like $\sigma^{l}$, this definition suffers the problem of dimension
mismatch: while $d^{3}{\bf v}$ is a three-dimensional volume element, the
scattered particles will distribute only on a two-dimensional surface in the
velocity space. Without any difficulty, we find that if $d^{3}{\bf v}$ is a
spherical ball with radius $\rho$ whose center lies on the EMS determined by
$\bf v^{\prime}$ and $\bf w^{\prime}$, then we obtain, with help of Eq. (36),

$R^{\rm s.in}([\hskip 1.0ptdt\rangle_{0},[\hskip 1.0ptd^{3}{\bf
r}\rangle_{0},d^{3}{\bf v})=\frac{3\sigma n_{\bf v^{\prime}}n_{\bf
w^{\prime}}}{2u}\frac{1}{\rho}.$ (44)

By letting $\rho\to 0$, we know that this scattering-in rate is ill-defined,
if the defining equation (43) is insisted upon.

Figure 8: (a) Two gases meeting in a collision region. (b) The collision-
produced distribution function on the half-line OA.

To find a way out of the difficulty, let’s consider the situation shown by
Fig. 8, where $f_{{\bf v}^{\prime}}$ and $f_{{\bf w}^{\prime}}$ represent the
distribution functions of two “colliding” gases. Our goal is to determine the
distribution function $f_{\bf v}$ which is employed to stand for the
collision-produced particles initially belong to $f_{\bf v^{\prime}}$.

For the following two reasons, we shall first use the collisional Boltzmann
equation to deal with the situation: i. In view of that many of us still
belive that the collisional Boltzmann equation can give a qualitatively and
quantitatively correct answer to the problem, we should take it as the first
step. ii. The solution given by the standard theory must give us a number of
helpful hints.

According to the Boltzmann equation in the textbook treatmentreif , $f_{\bf
v}$ on the half-line OA satisfies (with the half-line OA chosen to be the
positive $x$-axis)

$v_{x}\frac{\partial f_{\bf v}(x)}{\partial x}=\int 2u\sigma f_{\bf
v^{\prime}}f_{\bf w^{\prime}}d^{3}{\bf w}d\Omega,$ (45)

in which we have set $v_{y}=v_{z}=0$, $(\partial f_{\bf v}/\partial t)=0$ and
${\bf F}=0$, and we have disregarded all higher-order collisions. This leads
to

$f_{\bf v}(x)=\frac{1}{v_{x}}\int{dx}\int 2u\sigma f_{{\bf v}^{\prime}}f_{{\bf
w}^{\prime}}d^{3}{\bf w}d\Omega.$ (46)

This expression can be numerically integrated, and the final result is
schematically shown by the dotted curve in Fig. 8b.

But, our physical sense would tell us a different story. If we, as an
observer, move further and further away along the OA-axis, what shall we
observe? Considering that the collision region will gradually behave like a
point-like particle source, the inverse square law must eventually take
control. In other words, the solid curve shown in Fig. 8b is supposed to
describe the practical behavior of $f_{\bf v}(x)$.

Although Eq. (46) does not give us a correct result, it still hints us that
the particle-particle collisions along the segment OA can serve as a particle
source giving contribution to the distribution function somewhere else (named
as the contributed distribution function hereafter). A careful observation of
Eq. (46) unveils two shortcomings of it: i. The source region has been
confined on a line segment, which has zero volume and cannot be considered as
a reasonable source region. And ii. The velocity volume element used to
collect the collision-produced particles has been implicitly assumed to be an
infinitesimal three-dimensional one, which is not proper according to the
discussion that has been just given.

Figure 9: (a) A detector with an inlet area $d\alpha$ is placed outside the
collision region shown in Fig. 8a. (b) Only the particles whose velocities are
within $v^{2}dvd\Omega_{0}$, where $d\Omega_{0}=d\alpha/|{\bf r}-{\bf
r}_{0}|^{2}$, can be detected by the detector.

Based on these understandings, we arrive at several new guidelines concerning
the would-be treatment of Fig. 8a: i. Each spot of the collision region will
be deemed as a point-like “particle source”. And, ii. The contributed
distribution function will be defined as the finite-volume distribution
function in the velocity space to avoid the infinity difficulty caused by
dimension mismatch. In relation to Fig. 9a, to make the source-contribution
relationship as clear as possible, it is also assumed that the position point
$\bf r$, at which the contributed distribution function is of interest, is
located outside the collision region (for clarity of thinking), and that an
“idealized” detector, as a substitute for a human observer, is placed at $\bf
r$ to collect the collision-produced particles. With respect to the detector,
we investigate the following “distribution function”:

$\bar{f}([\hskip 1.0ptd^{3}{\bf r}\rangle_{0},\delta
v,\delta\Omega)\equiv\frac{1}{{\bar{v}}^{2}\delta
v\delta\Omega}\lim\limits_{dtd\alpha\to 0}\int_{\delta
v\delta\Omega}\frac{dN}{vdtd\alpha},$ (47)

where $[\hskip 1.0ptd^{3}{\bf r}\rangle_{0}$ represents an infinitesimal
position volume element just at the detector’s inlet, $\delta v$ is a small
but finite speed increment, $\delta\Omega$ is a small but finite solid-angle
element in the velocity space,
$\bar{v}^{2}\equiv(v_{l}^{2}+v_{l}v_{h}+v_{h}^{2})/3$ with $v_{l}$ and $v_{h}$
being the lowest and highest bounds of $\delta v$ respectively, $dN$ is the
number of the particles registered during $dt$ by the detector whose
velocities are confined in $\bar{v}^{2}\delta
v\delta\Omega=(v_{h}^{3}-v_{l}^{3})\cdot\delta\Omega/3$ and $d\alpha$ stands
for the infinitely small opening area of the detector’s inlet.

Explanation is needed about this definition. As implied by the form of
$\bar{f}$, $\bar{f}$ here is nothing but the partial limit associated with the
assumption that $d^{3}{\bf r}$ is infinitesimal and $\bar{v}^{2}\delta
v\delta\Omega$ is finite. In other words, it is an ordinary distribution
function in the position space, but an average distribution function in the
velocity space. While its differentiation in the velocity space makes no
sense, its integration can be used to obtain macroscopic quantities as if it
is a regular distribution function.

By examining the upstream paths of the particles collected by the detector in
Fig. 9a, it is found that only the collisions taking place in the spatial cone
$\\{-\delta\Omega\\}_{\bf r}$ are relevant to $\bar{f}$; and thus the region
included in $\\{-\delta\Omega\\}_{\bf r}$ will be called the effective zone.
Supposing ${\bf r}_{0}$ is a spatial point in the effective zone, how do the
collisions taking place around ${\bf r}_{0}$ contribute to $\bar{f}$? Due to
the infinite smallness of $d\Omega_{0}$ which is defined as

$d\Omega_{0}=d\alpha/|{\bf r}-{\bf r}_{0}|^{2},$ (48)

and the finiteness of $\delta\Omega$, any collision-produced particle that can
enter the detector at a speed within $\delta v$ is qualified as a one
belonging to $dN$ in Eq. (47).

Thus, from Eq. (36), the contribution to $dN$ due to the collisions in
$d^{3}{\bf r}_{0}$ can be written as

$dt\int_{\delta v}\int_{v^{2}dvd\Omega_{0}}2uf_{{\bf v}^{\prime}}d^{3}{\bf
v}^{\prime}d^{3}{\bf r}_{0}f_{{\bf w}^{\prime}}d^{3}{\bf
w}^{\prime}\sigma\cdot(dS/u^{2}),$ (49)

where $dS$ is the surface element lying on the EMS defined by ${\bf
v}^{\prime}$ and ${\bf w}^{\prime}$ and enclosed by $v^{2}dvd\Omega_{0}$ as
shown in Fig. 9b.

To truly integrate Eq. (49), we invoke the variable transformation $d^{3}{\bf
v}^{\prime}d^{3}{\bf w}^{\prime}\to d^{3}{\bf c}^{\prime}d^{3}{\bf
u}^{\prime}$, where ${\bf c}^{\prime}=({\bf v^{\prime}}+{\bf w^{\prime}})/2$
and ${\bf u}^{\prime}=({\bf v^{\prime}}-{\bf w^{\prime}})/2$, and obtain

$dtd^{3}{\bf r}_{0}\int_{\delta v}\int_{v^{2}dvd\Omega_{0}}d^{3}{\bf
c}^{\prime}d^{3}{\bf u}^{\prime}2uJf_{{\bf v}^{\prime}}f_{{\bf
w}^{\prime}}\sigma\cdot(dS/u^{2}),$ (50)

in which $J$ is the Jacobian

$J=\left\|\frac{\partial({\bf v}^{\prime},{\bf w}^{\prime})}{\partial({\bf
c}^{\prime},{\bf u}^{\prime})}\right\|=8.$ (51)

Noticing that

$\int d^{3}{\bf u}^{\prime}=\int u^{\prime 2}du^{\prime}d\Omega^{\prime}=\int
d\Omega^{\prime}\int u^{2}du,$ (52)

where $\Omega^{\prime}$ is the direction of ${\bf u}^{\prime}$, and that

$\int_{v^{2}dvd\Omega_{0}}dudS=v^{2}dvd\Omega_{0}=\frac{v^{2}dvd\alpha}{|{\bf
r}-{\bf r}_{0}|^{2}},$ (53)

we arrive at

$\bar{f}=\frac{1}{{\bar{v}}^{2}\delta v\delta\Omega}\int d^{3}{\bf
r}_{0}dvd^{3}{\bf c}^{\prime}d\Omega^{\prime}\frac{16uv\sigma}{|{\bf r}-{\bf
r}_{0}|^{2}}f_{\bf v^{\prime}}f_{{\bf w}^{\prime}},$ (54)

where $\int d^{3}{\bf r}_{0}$ is over the effective zone given by
$\\{-\delta\Omega\\}_{\bf r}$, $\int dv$ is over $\delta v$, $\int d^{3}{\bf
c}^{\prime}$ is over the entire velocity space, and $\int d\Omega^{\prime}$ is
over $0-4\pi$. It is noted that several quantities in this formula, such as
$\sigma$, $f_{{\bf v}^{\prime}}$ and $f_{{\bf w}^{\prime}}$, are defined with
help of ${\bf u}$ and ${\bf u}^{\prime}$, in which ${\bf u}={\bf v}-{\bf
c}={\bf v}-{\bf c}^{\prime}$ with ${\bf v}=v({\bf r}-{\bf r}_{0})/|{\bf
r}-{\bf r}_{0}|$, and ${\bf u}^{\prime}$ is defined by the magnitude of ${\bf
u}$ and the direction of $\Omega^{\prime}$.

If the collision region is relatively small, the result given by Eq. (54)
evidently satisfies the inverse square law. If interested, readers may use
computational means to verify (or deny) this approach. With slight
modifications, the formulation can be applied to many other situationschen3 .

##  VII Summary

In conclusion, we have shown that

  * •

A number of multi-variable limits used in the existing kinetic theory are
actually ill-defined.

  * •

By redefining, reinterpreting and reformulating these ill-defined limits, we
can make significant progresses in gas dynamics.

  * •

To formulate the quantities introduced in this pepr, we need to give up the
equation-type approach and accept the path-integral approach.

  * •

In real dynamic gases there are a considerable number of particles that are
discontinuously distributed. The newly introduced path-approach can treat them
with almost the same accuracy and effectiveness.

The concepts, methodologies and conclusions given by this paper are
fundamentally different from those in the conventional theory. Possible impact
on fluid mechanics, turbulence studies, plasma physics or quantum statistical
mechanics remains to be seen.

The author is very grateful to Drs. M. Berry, O. Penrose, Robert G.
Littlejohn, Guo Han-ying, Zhang Tian-rong, Guan Ke-ying, Ying Xing-ren, Wen
Duanzhi, V. Travkin, W. Hoover and many others for their direct or indirect
encouragement. Discussions with them have been pleasant and stimulating.

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