A mechanical approach to one-dimensional interacting gas

Accepted Manuscript

A Mechanical Approach to One-Dimensional Interacting Gas Chung-Yang Wang, Yih-Yuh Chen PII: DOI: Reference:

S0577-9073(18)30920-1 https://doi.org/10.1016/j.cjph.2018.09.016 CJPH 638

To appear in:

Chinese Journal of Physics

Received date: Revised date: Accepted date:

4 July 2018 15 September 2018 19 September 2018

Please cite this article as: Chung-Yang Wang, Yih-Yuh Chen, proach to One-Dimensional Interacting Gas, Chinese Journal of https://doi.org/10.1016/j.cjph.2018.09.016

A Mechanical Physics (2018),

Apdoi:

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Highlights • Traditional interpretation of the van der Waals equation is actually incorrect. • The concept of mean field is trickier than we thought.

AC

CE

PT

ED

M

AN US

CR IP T

• For a one-dimensional gas, the idea of particle trajectory can be introduced.

1

ACCEPTED MANUSCRIPT

A Mechanical Approach to One-Dimensional Interacting Gas

CR IP T

Chung-Yang Wang and Yih-Yuh Chen

Department of Physics, National Taiwan University, Taipei 106, Taiwan, Republic of China

September 20, 2018

AN US

Abstract

Derivations of the van der Waals equation typically use standard recipes involving ensemble averages of statistical mechanics. In this work, we study a box of weakly interacting gas particles in one-dimension by explicitly incorporating the mechanical point of view. This has the merit that it not only reproduces the van der Waals equation but also tells us some extra interesting physics not immediately clear from a pure statistical

M

mechanical approach. For example, we show that the traditional handwaving argument leading to van der Waals equation requires closer scrutiny if it is to get things right.

AC

CE

PT

ED

Keywords: statistical mechanics; thermodynamics; kinetic theory; equation of state; onedimensional interacting gas; van der Waals equation

2

ACCEPTED MANUSCRIPT

1

Introduction

AN US

CR IP T

Statistical mechanics and kinetic theory are two very powerful frameworks one adopts when studying a system containing many interacting particles in thermodynamic equilibrium. Researches in this field concern various systems, including hard sphere system [1–8], hard ellipse system [9], hard needle system [10, 11], van der Waals theory [12–15], granular particle system [16–18], etc. There are also researches concerning fundamental issues [19, 20]. However, because statistical mechanics and kinetic theory both are statistical in nature, which does not necessarily require the inclusion of the actual particle trajectories, a beginning student might find it difficult to build up good physics intuition and firmly grasp the idea, even after one has derived the desired results. It is the aim of this work to show that it might be helpful to also incorporate the concept of particle trajectories when one studies statistical physics. To be specific, we will use the one-dimensional van der Waals gas as an illustration. Along the way, we will also show that the conventional handwaving argument leading to the van der Waals equation really requires more scrutiny if one is to get the physics right. Before describing our mechanical approach, we first give a brief review of how one tackles a weakly interacting one-dimensional gas in statistical mechanics. Here, one first writes down the Hamiltonian of the system and computes its partition function, then following through standard recipes [21, 22] one easily arrives at the following equation of state Z

M

2

F = ρ(kB Treal ) − ρ (kB Treal )

∞



−k

dr e

0

U (r) B Treal

 − 1 + O(ρ3 ),

(1.1)

AC

CE

PT

ED

where F is the force exerted on each side of the confining box, ρ ≡ N/L is the linear number density (N and L are the particle number and the size of the one-dimensional box, respectively), kB is Boltzmann’s constant, Treal is the temperature of the system (It will be clear why we use the notation Treal instead of T .), and U (r) is the potential energy between two gas particles separated by a distance r. In the above, we have assumed a low density limit so that a Taylor expansion in ρ has been performed. We will specialize to the case when the potential representing particle-particle interaction is consisted of a hard core and an interaction tail, as shown in Fig. 1. Consider only the low density limit and keep up to the second order in the number density, and further assume the high temperature limit and weakly interacting limit. Approximate Eq.1.1 to first order of U (r)/kB Treal under these limits, one easily derives the following equation of state  Z F = ρ(kB Treal ) 1 + ρd1 − ρ

∞

d1

After rearrangement, the equation of state is given by

3



−U (r) dr kB Treal



.

(1.2)

ACCEPTED MANUSCRIPT

U(r) r=d1

CR IP T

r

AN US

Figure 1: Particle-particle interaction consisting of a hard core (at r = d1 ) and an interaction tail. U (r) is the potential describing the interaction between two particles.   Z N 2 ∞ dr [−U (r)] (L − d1 N ) = N kB Treal , F +( ) L d1

(1.3)

AC

CE

PT

ED

M

which is the van der Waals equation in one-dimension. This derivation provided by statistical mechanics is simple, and, in one scoop, easily relates the force with the temperature, the density, and the potential describing particleparticle interactions. On the other hand, one also loses track of exactly what has happened to the dynamical behavior of the constituent particles. For example, why does the interaction tail contributes a correction to the force to the wall that is quadratic in the particle density? Another related question is: If we “turn on” the interparticular interactions of an otherwise ideal gas, will that make the gas particles move faster or slower on the average? Statistical mechanics by itself does not directly provide such information. In this regard, the physics contained in the equation of state has been hidden under the carpet, all because the recipe is so deceptively simple. In this work, we will incorporate the mechanical point of view, considering trajectories and detailed mechanics of the gas particles. Besides helping us see more clearly why the equation of state assumes the form it does, it also offers something else that standard statistical mechanics doesn’t. For example, we will try to answer the two previous questions using our approach. The paper is organized as follows. In Section 2, we discuss the foundation of our mechanical point of view. In Section 3, we consider a special case of square well particle-particle interaction and show how the mechanical model is built. In Section 4, we generalize the previous results to generic interacting gas, with the help of physics insight we developed in Section 3. Finally, we summarize the result and give some possible implications in Section 5.

4

ACCEPTED MANUSCRIPT

2

Mechanical picture for one-dimensional interacting gas

X 2mvj0

F =

Tj0

j

CR IP T

Consider first a box of ideal gas in one-dimension. For visualization, we attach a velocity arrow to each particle. When two particles collide, they will exchange their velocity arrows. Thus, we may follow an arrow all the way as it propagates from left to right, totally ignoring the fact that it is actually the different particles that are carrying it at different times. To avoid possible misunderstanding, we want to emphasize that we are tracking a velocity arrow instead of a particle itself. The time-averaged force experienced by the wall is then given by ,

(2.1)

X j

AN US

where j is the index of each arrow hitting the wall, vj0 being the velocity associated with the j-th arrow, and Tj0 = 2L/vj0 is the total time it takes for a specific arrow to complete one circuit. The superscript 0 stands for free particle (the unperturbed state). Summation over j is given by 2N (...) = √ aπ

Z

∞

0

dvj0 e−

(vj0 ) a

2

(...) ,

(2.2)

CE

PT

ED

M

where a ≡ 2kB T /m. The extra factor of two above comes from the fact that we consider the force felt by the right wall. Since an arrow can hit the right wall only if its velocity is positive, the limit of integration is taken as 0 ≤ vj0 < ∞. However, this takes into account only one half of the particles, the other half being carrying arrows pointing to the left at each instant, and we must multiply the result by an extra factor of 2 since the left going arrows eventually will be converted into right going arrows when they hit the left wall. We also √ define a dimensionless velocity u ≡ v/ a, which will be used in later calculations. Plugging Eq.2.2 into Eq.2.1, we have F = ρkB T, (2.3)

AC

which is the one-dimensional version for P V = N kB T . What happens when particle-particle interaction is turned on? Because particles in the bulk are interacting, their velocities fluctuate, which causes a modification in the flight time of a specific arrow we are visually tracking (physics in the bulk). And when each arrow propagates near the wall, particle-particle interaction may alter particle-wall collision, which then causes a modification to the momentum imparted to the wall (physics on the boundary). Both effects now work together to change the force F . Clearly, we have F =

X 2mvj j

Tj

=

X 2m(vj0 + ∆vj0 ) j

5

Tj0 + ∆Tj0

,

(2.4)

ACCEPTED MANUSCRIPT

where a ∆ preceding a variable means it is a small change from the ideal case described before, and vj = vj0 + ∆vj0 is called the collision velocity. In view of Eq.1.2, which shows that the modification to the equation of state due to interparticular interaction U is only kept to the first order, we may correspondingly expand Eq.2.4 to derive X 2mvj0 X 2mvj0 ∆Tj0 X 2m∆vj0 − + . (2.5) F ≈ 0 0 0 0 T T T T j j j j j j j

AC

CE

PT

ED

M

AN US

CR IP T

That is, we consider the physics perturbed around ideal gas (free particle) regime and retain the perturbation up to the first order correction. P In order to do the summation ( j ), we will make the explicit assumption that Maxwell’s velocity distribution is valid for vj0 . That is, in our mechanical model, the unperturbed part (free particle part) described by vj0 satisfies Maxwell’s velocity distribution of temperature T . But here one must proceed with extra caution, because T here refers to the temperature of the system before the interaction is turned on. Now, if we prepare an unperturbed (ideal) system at an original temperature T , and then turn on the interaction all of a sudden, the extra interparticular forces necessarily cause a change in the root mean square speed of the particles, which in turn means that the system is now at a slightly different temperature. This altered temperature (for an interacting gas) should be identified with the actual temperature Treal we want to put in van der Waals equation. Just how T and Treal are related will be studied in
P Section 3.3. Note that if we have T 6= Treal , then we have j 2mvj0 /Tj0 = ρkB T 6= ρkB Treal ,
P which means that it is too naive to identify j 2mvj0 /Tj0 (in our mechanical approach) with the ideal gas part in Eq.1.1 (in standard statistical mechanics). To avoid possible confusion, we should emphasize that we are talking about the real temperature of the system when taking standard statistical mechanics. This is why we use Treal in Eq.1.1. In the following, we will first focus on evaluating the three terms of Eq.2.5 with the idea of trajectory, using the square well potential of Fig. 2 as an illustration. The strategy is tracking a particle (actually, a velocity arrow) labeled by j (which means that its free particle velocity is vj0 ), called the main particle, and finding out the influence of the other particles, called the background particles, on the main particle. At the end of this section, we want to point out that interacting gas is different from ideal gas due to particle-particle interaction. Therefore, it is not surprising that “how many particle-particle interactions are there” and “what is the influence of a particle-particle interaction” are at the heart of our mechanical approach when studying interacting gas. We will see how to understand the behavior of a box of interacting gas by these two central ideas.

6

ACCEPTED MANUSCRIPT

U(r)


>0

r

CR IP T


r=d2

r=d1

Figure 2: Square well potential.

3

One-dimensional interacting gas with square well po-

AN US

tential

In this section, we consider one-dimensional interacting gas with square well potential, as shown in Fig. 2. By Eq.1.2, equation of state for square well potential is (3.1)

M

F = ρ (kB Treal ) + ρ2 (kB Treal ) d1 − ρ2 (d2 − d1 ) .

3.1

ED

Eq.3.1 is the standard result we wish to compare with using our mechanical approach.

Mechanics of interaction between two particles

AC

and

CE

PT

Consider generic particle-particle interaction with a finite interaction range r0 . Suppose the interaction takes time interval of ∆tinteraction . In the lab frame (of the confining box), the displacements of two interacting particles during ∆tinteraction are given by displacement of left particle = r0 + vCM ∆tinteraction

(3.2)

displacement of right particle = −r0 + vCM ∆tinteraction ,

(3.3)

where vCM is the center-of-mass velocity. For a potential shown in Fig. 1, ∆tinteraction (v1 , v2 ) is given by ∆tinteraction (v1 , v2 ) =

Z

r0

d1

q

dr
v1 −v2 2 2

−

U (r) m

.

(3.4)

For particle-particle repulsion, ∆tinteraction is more complicated in the sense that the lower 7

ACCEPTED MANUSCRIPT

bound of d1 of the integral in Eq.3.4 should be replaced by a function of v1 and v2 . Particleparticle repulsion will be discussed later in Section 4.2.

3.2

The total flight time (physics in the bulk)

AN US

CR IP T

Without loss of generality, we may follow a typical velocity arrow as it propagates from the left wall to the right wall. At each instant, the particle carrying the arrow will be termed the “main particle.” The total flight time due to particle-particle interaction can be computed by evaluating the number of collisions and how a collision affects the travel time. In Section 3.2.1 we argue that the number of collisions turns out not to have anything to do with the form of the particle-particle interaction, so that the details of the particle-particle interaction comes in only when we are dealing with the effect of each collision (Section 3.2.2). Whenever the main particle is moving outside the interaction range of the rest of the particles (the “background”), it regains its free space velocity vj0 . Thus, the total flight time Tj satisfies L − total displacement when it is not free X 1 Tj = + ∆tinteraction (vj0 , vk ), 0 2 vj k

(3.5)

CE

PT

ED

M

where the background particles are labeled by k, and vk is the velocity of the background particle when colliding with the main particle. Particle-particle interactions can be classified as forward collision and backward collision. By forward collision, we mean the background particle collides with the main particle from the right. By backward collision, we mean the background particle collides with the main particle from the left. The numbers of forward collisions and backward collisions of the main particle in its whole trip are denoted by #R and #L , respectively. Putting Eq.3.2 and Eq.3.3 into Eq.3.5, we have

AC

1 ∆T 0 = 2 j

  r0 vj0 − vk 0 − 0+ ∆tinteraction (vj , vk ) · #R vj 2vj0 avg   0 v − v r0 k j + + ∆tinteraction (vj0 , vk ) · #L 0 0 vj 2vj avg

(3.6)

(Recall that ∆Tj0 = Tj − Tj0 .), where the subscript “avg” refers to averaging over the background index k. We now proceed to find out #R and #L .

8

ACCEPTED MANUSCRIPT

forward collision : A, C backward collision : B right wall reflection : R1, R3 left wall reflection : R2

x=L

R1

R3 C B

A

R2

CR IP T

x

t

AN US

t=L/vj0

Figure 3: The x − t diagram for a typical motion of the main particle (solid line) and a background particle (dashed line). 3.2.1

Counting the number of collisions

AC

CE

PT

ED

M

Noticing that we are only interested in computing physical quantities correct up to the first order in the particle-particle interaction, and the (...)avg in Eq.3.6 are both already of first order, we know that #R and #L can be evaluated using (the unperturbed) ideal gases. When the effect of particle-particle interaction is kept to order one, in order to count the number of collisions, the free particle model (ideal gas) is sufficient. This observation greatly simplifies the following algebra. For an ideal gas, Fig. 3 shows the position versus time plot of the trajectory of a main particle and that of a background particle with specified initial velocity vk0 and initial position x0k . Clearly, the number of forward collisions and backward collisions provided by the background particle is just the number of line-crossings between the main particle and the background particle. With some elementary counting of the intersection points of straight lines (For example,
RL there are two forward collisions and one backward collision in Fig. 3.), L1 0 dx0k #L vj0 , vk0 , x0k is calculated and the result is shown in Fig. 4. (Because the probability of the k-th background particle’s being initially located inside the interval x0k and x0k + dx0k is dx0k /L, the

RL expectation value of #L vj0 , vk0 , x0k is just 0 dx0k #L vj0 , vk0 , x0k /L.) A moment’s reflection shows that 1 L

Z

0

L

dx0k #R

vj0 , vk0 , x0k




1 =1+ L 9

Z

0

L


dx0k #L vj0 , vk0 , x0k .

(3.7)

ACCEPTED MANUSCRIPT

4

4 3

3 2

CR IP T

2 1

1 0

-9 -8 -7 -6 -5 -4 -3 -2 -1

vk0/vj0

2 3 4 5 6 7 8 9 10

AN US


RL Figure 4: L1 0 dx0k #L vj0 , vk0 , x0k as a function of vk0 /vj0 . The line segments connecting two adjacent flat parts are straight lines with slopes being ±1.

AC

CE

PT

ED

M

The interpretation of Fig. 4 and Eq.3.7 is easy. For a backward collision to occur, the background particle must travel fast enough to catch up with the main particle, which can

RL be seen from L1 0 dx0k #L vj0 , vk0 , x0k = 0 for −1 < vk0 /vj0 < 2 in Fig. 4. Comparing the initial situation at t = 0 and the final situation at t = Tj0 /2, all background particles change from sitting at the right side of the main particle to its left side. Therefore, all background particles contribute at least one forward collision, and the number of forward collisions is larger than the number of backward collisions by one, as shown in Eq.3.7. Another way of viewing this is to consider the condition for a neighbor particle to collide with the main particle. The admissible velocities of neighbor particles causing forward colli

sions and backward collisions are −∞, vj0 and vj0 , ∞ , respectively, and hence the number of forward collisions is larger than that of backward collisions. For a given (x0k , vk0 ), the corresponding vk can be found, with the help of x − t diagram. With the assumption of uniform distribution in the initial position and the Maxwellian velocity distribution, plus the knowledge of #R , #L and vk , we may recast Eq.3.6 into

1 N N 1 ∆T 0 = −r0 0 + √
2 j vj 2 aπ vj0 2

3.2.2

Z

0

∞

dvk0

"

e

2

−

(vk0 −vj0 ) a

2

−

−e

(vk0 +vj0 ) a

#

vk0


2


∆tinteraction 0, vk0 .

(3.8)

Correction to the flight time

By virtue of Eq.3.4, we may compute ∆tinteraction (0, vk0 ) for square well potential and then substitute it into Eq.3.8. Expanding out in the small parameter /kB T to its first order, we 10

ACCEPTED MANUSCRIPT

1.2

velocity correction

1.0 0.8 0.6 0.4

0.0

0

2

6

4

dimensionless velocity

8

CR IP T

0.2 10

Figure 5: Correction to the average velocity due to particle-particle attraction for square well uj |average −u0 potential. The x-axis is u0j and the y-axis is ρ(d2 −d1 )  j given by Eq.3.10. kB T

AN US

have

√ 0 2 vj0 1 N d1 N (d2 − d1 ) aπ − (vj )  0 a . ∆Tj = − 0 − erfi( √ ) e
2 0 2 vj a kB T vj

(3.9)

CE

PT

ED

M

This is the correction to the flight time for square well potential. The various terms in Eq.3.9 can be interpreted as follows. The first term is the effect of the hard core, which reduces the distance a particle must travel (and thus shortens the flight time). The length of the box L is replaced by the effective L − N d1 , and the corresponding flight time is cut short by the amount N d1 /vj0 . The second term is the effect of the attractive potential. It is always negative, meaning that all particles are sped up a bit on the average. This intuitively obvious assertion can
be analyzed by looking at the average velocity vj |average = 2 (L − N d1 ) / Tj0 + ∆Tj0 . By Eq.3.9, we have √ uj |average −u0j 0 2 = πe−(uj ) erfi(u0j ).  ρ (d2 − d1 ) kB T

(3.10)

AC

√ (Recall that u ≡ v/ a.) The result is shown in Fig. 5. Fig. 5 shows that uj |average −u0j is always positive, and it increases when the velocity vj0 is small, but decreases when vj0 is large. The following paragraphs give a physical interpretation to these features. Due to the particle-particle attraction, when the main particle experiences a forward collision, its velocity is increased; but when the main particle experiences a backward collision, its velocity is decreased. Since forward collisions are more frequent than backward collisions (by Eq.3.7), a particle is sped up when particle-particle attraction is turned on. As to the increasing trend of Fig. 5 when vj0 is small, we may understand it this way. By 11

ACCEPTED MANUSCRIPT

M

AN US

CR IP T

Eq.3.7 and Fig. 4, the number of collisions contributed by a background particle can be written as 1 forward collision + n (forward collision, backward collision) pair. For a background particle, the effect of a (forward collision, backward collision) pair is decreasing the velocity of the main particle, since the interaction is most favorably felt when the relative velocity is small. The feature that nonvanishing n decreases as vj0 increases leads to the increasing trend in Fig. 5. If vj0 is very large, the background particles that can significantly affect the main particle necessarily have a large and comparable velocity. But Maxwellian distribution tells us that the population of these high speed particles is inherently small due to the Boltzmann factor. Thus, the effect of interaction actually becomes smaller as vj0 increases. This is the main reason for the decreasing trend for large vj0 in Fig. 5. In the traditional handwaving picture of an interacting gas, particle-particle interaction is thought to have been averaged out in the bulk (mean field approximation) [23]. The foregoing analysis shows that, in a well-defined sense, this is misleading. Though the background particles are uniform, the net effect is not zero, since the main particle flying towards the wall necessarily introduces asymmetry in the interaction. (The admissible velocities of neighbor

particles causing forward collisions and backward collisions are −∞, vj0 and vj0 , ∞ , respectively.) The effects of forward and backward collisions just don’t cancel out, and the idea of mean field is not valid. A particle is necessarily sped up when particle-particle attraction is turned on.

Temperature modification

ED

3.3

AC

CE

PT

As pointed out in Section 3.2, particles do move faster on the average when particle-particle attraction is turned on (uj |average > u0j ). Therefore, the temperature Treal of the box of interacting gas is greater than it was (T ) before the interaction is turned on. What this means is that, one really should start with an ideal gas with a suitably chosen lower temperature to perform the perturbation calculation, so that in the end the combined effect of the added interaction will raise the temperature of the system to the desired temperature. The analysis of this idea can be done as follows. Statistical physics tells us that the real temperature Treal should be defined by the average kinetic energy hK.E.i through N hK.E.i = N kB Treal /2 in one-dimension. That is, the real temperature should be introduced by X1 j

2

m (vj |average )2 =

N kB Treal . 2

(3.11)



2 P Putting vj |average = 2 (L − N d1 ) / Tj0 + ∆Tj0 and j m vj0 /2 = N kB T /2 into Eq.3.11, 12

ACCEPTED MANUSCRIPT

we get

X
2 ∆Tj0 N N kB Treal = kB T (1 − 2ρd1 ) − . m vj0 2 2 Tj0 j

(3.12)

Eq.3.12 is the relation between Treal and T . Putting Eq.3.12 into Eq.2.5, the equation of state is modified to F = ρkB Treal + 2ρ (kB Treal ) d1 +

X 2mvj0 ∆Tj0 j

Tj0

Tj0

+

X 2m∆vj0 j

Tj0

.

CR IP T

2

(3.13)

Actually, the idea of temperature modification can also be seen by considering the total energy of the box of gas. In our current formulation, turning on the interaction does not change the total energy of the system. Thus, on the one hand, this total energy should be equal to the total kinetic energy of the unperturbed ideal gas, which implies X1

m vj0


2

=

N kB T. 2

AN US

total energy =

j

2

(3.14)

On the other hand, this total energy becomes the summation of the average kinetic energy and average potential energy of each particle, which reads N kB Treal + N hU i . 2 Comparing the above two expressions for the same total energy, we have

(3.15)

ED

M

total energy =

kB Treal = kB T − 2 hU i .

(3.16)

Momentum transferred to the wall (physics on the boundary)

AC

3.4

CE

PT

Therefore, Treal > T when particle-particle attraction is turned on, just as claimed before. Note that we can see that traditional interpretation of the van der Waals equation adopt
2 P P ing mean field approximation is incorrect simply by j m vj0 /2 = j m (vj |average )2 /2 + N hU i, without doing detailed calculations. When particle-particle attraction is turned on, potential energy is negative and hence particles move faster.

Collision velocity of the main particle is determined by its state hitting the right wall. The main particle experiences lots of collisions during its entire trip, and the last one determines its state hitting the right wall. It is conceivable that, while en route to the wall in its final approach, this main particle is still interacting with a neighbor particle (the last collision). When this happens, “normal,” uninterrupted collisions considered before are modified by the presence of the wall, and special care should be paid to this situation in order to get the momentum transfer right. 13

ACCEPTED MANUSCRIPT

CR IP T

Figure 6: The initial state of the last collision being a forward collision.

3.4.1

AN US

In order to deal with the nonvanishing ∆vj0 caused by this effect, we consider all the different scenarios that might lead to a nonvanishing ∆vj0 . The effect of ∆vj0 can then be decomposed into the product of two factors. The first factor is the probability for a certain scenario to occur during the last collision (Section 3.4.1). The second factor gives us the effect associated with a given scenario of the last collision (Section 3.4.2). Their composition yields the desired result, to be discussed in Section 3.4.3. Probability of the last collision

M

Let us assume that the last collision starts when the main particle is at x = L − d3 , and the background particle causing the last collision hits the main particle with vk0 . The probability of such scenario is studied in Appendix A, and is given by

Situation around the wall

PT

3.4.2

ED

" #! 2    0 √ 0 vj0 ) (

v v ∆d 1 a 3 j N 1 − k0 f vk0 ∆vk0 × exp − ρd3 1 + erf √ + √ 0 e− a . (3.17) vj L 2 a πvj

AC

CE

Let’s consider non-trivial situations (nonvanishing ∆vj0 ) of the last collision of the main particle around the wall. For square well potential, the relation between vin and vout (vin and vout are the speeds inside and outside the potential well, respectively, defined in the center-of-mass frame of the two interacting particles.) is r  vin = (vout )2 + . m

(3.18)

Particle-particle interaction is easy to analyze in the center-of-mass frame. But to investigate the role played by the wall, lab frame is a better choice. Therefore, we will switch between center-of-mass frame and lab frame. Note that collision velocity is defined in the lab frame. In the following analysis, cases having a forward and a backward collision as the last collision are separately treated.

14

ACCEPTED MANUSCRIPT

CR IP T

Figure 7: The initial state of the last collision being a backward collision.

AN US

When forward collision occurs The situation for the beginning of forward collision is shown in Fig. 6. Here, vk0 is not arbitrary. Firstly, the occurrence of forward collision implies vk0 < vj0 . Secondly, if vCM =
vj0 + vk0 /2 is not positive, the two particles don’t hit the wall until their interaction is over. Combining the two constraints, we obtain the condition for vk0 as −vj0 < vk0 < vj0 .

(3.19)

ED

M

For the main particle to hit the wall when it is still in the potential well of another particle, d3 should be relatively short enough. The range of d3 and the collision velocity is presented in Appendix B. We put the range of d3 and the collision velocity in appendix because it is the range of 0 vk that plays the truly significant role for the feature of physics on the boundary. The role of the range of vk0 will be clear in Section 3.4.3 and Section 4.1.

CE

PT

When backward collision occurs The situation for the beginning of backward collision is shown in Fig. 7. The occurrence of backward collision implies vk0 > vj0 .

(3.20)

AC

The range of d3 and the collision velocity is presented in Appendix C. 3.4.3

Correction to the collision velocity

Having considered all the factors, we are now in a position to actually calculate the combined result for ∆vj0 . The total effect is the product of “probability of a given situation of the last collision” (Section 3.4.1, Appendix A) and “effect associated with a given situation of the last collision” (Section 3.4.2, Appendix B and Appendix C). The derivation is in Appendix D, E, and F. Finally, we get

15

ACCEPTED MANUSCRIPT

in the bulk on the boundary forward collision −∞ ∼ vj0 −vj0 ∼ vj0 backward collision vj0 ∼ ∞ vj0 ∼ ∞ Table 1: Velocity domains of neighbor particles (surrounding the main particle) that cause forward and backward collisions.

Z

∞

−u0j

2

0 du0k e−(uk )

u0j + u0k + u0j − u0k

Z

∞

u0j

0 du0k e−(uk )

2

u0j − u0k u0j + u0k

!

CR IP T

∆u0j 1 = √ 0  ρ (d2 − d1 ) kB T 2 πuj

.

(3.21)

3.4.4

ED

M

AN US

The result is shown in Fig. 8. The profile of ∆u0j in Fig. 8 can be understood if we decompose the profile into two main trends, an increasing trend and a decreasing trend. From Table 1, for physics on the boundary, as u0j increases, number of relevant forward collisions increases and number of relevant backward collisions decreases. So ∆u0j increases as u0j increases. As for the
decreasing trend, the physics is the same as the decreasing trend for uj |average −u0j . The
analysis here is like what we do for the profile of uj |average −u0j in Fig. 5. The underlying physics are both collision number and exponential decay suppression. This is why ∆u0j and
uj |average −u0j have similar profiles in Fig. 8. We can see that average velocity is always larger than collision velocity. We will come back to the physical meaning of this feature later when studying generic particle-particle interaction. Correction to the momentum transferred in equation of state

CE

PT

We now walk the last mile to a full description of the correction of the momentum transferred
P in equation of state. Plugging Eq.3.21 into j 2m∆vj0 /Tj0 , we get P

X 2m∆vj0 j


0

Tj0

= 0.

(3.22)

AC

In order to obtain j 2m∆vj0 /Tj , we went through the calculations in all the appendices, and yet the final result is just zero! This suggests that there should be a much simpler underlying principle to the whole problem. What is it then?
P Note that what we have is j 2m∆vj0 /Tj0 = 0 but not 2m∆vj0 /Tj0 = 0. A zero after summing over all the contributions from its constituents often signals a certain conservation law. This is the first hint for the physics behind Eq.3.22. In our construction, vj0 + ∆vj0 is the collision velocity that a particle hits the wall. Though not explicitly spelt out, this also means that the wall is supplying some momentum to the box of gas every time a particle hits the wall. That is, the wall is a momentum provider. So let us 16

ACCEPTED MANUSCRIPT

2

0 -1 -2 -3 correction of average velocity correction of collision velocity

-4 -5

0

1

3

2

4

dimensionless velocity

5

CR IP T

velocity correction

1

6

7

Figure 8: Correction to the average velocity and correction to the collision velocity for square well potential. The x-axis is u0j . The upper curve is the correction to the average velocity uj |average −u0j ρ(d2 −d1 ) k  T

in Eq.3.10. The lower curve is the correction to the collision velocity

AN US

given by

B

given by

(u0j +∆u0j )−u0j ρ(d2 −d1 ) k  T

in Eq.3.21.

B

AC

CE

PT

ED

M

check the total positive momentum carried by some of the particles in the system. In order to have an idea of total positive momentum that is separate from total negative momentum, we take a snapshot for the box of gas and pull apart all particle pairs, and then define total positive momentum and total negative momentum by these N decoupled particles. If we now allow the system to evolve for a time ∆t, what is the total positive momentum
now? For a particle with vj0 > 0 and having the initial position in L − vj0 ∆t, L , its velocity will be negative at t = ∆t. When counting the total positive momentum at t = ∆t, such particle has no contribution since it no longer belongs in the class of particles having positive momentum. From the basic assumption that position distribution is homogeneous, such

particles have the population of vj0 ∆t /L = ∆t/ 12 Tj0 . When counting the total positive momentum at t = ∆t, momentum loss due to the effect that a particle can change its direction is given by momentum loss =

X ∆t
0 1 0 −mvj . T j 2 j

(3.23)

On the other hand, we also have momentum gain due to the reflection provided by the
left wall. For each collision, the left wall provides a momentum of m vj0 + ∆vj0 to the total positive momentum of the system. Momentum gain provided by the left wall during ∆t is momentum gain =

X ∆t

0 0 . 1 0 m vj + ∆vj T j 2 j 17

(3.24)

ACCEPTED MANUSCRIPT

When the box of gas is at thermal equilibrium, it is a steady state. For steady state, we expect the total positive momentum to be time-independent. By Eq.3.23 and Eq.3.24, we have X 2m∆vj0 j

Tj0

(3.25)

= 0.

3.5

CR IP T


P By requiring the system to be a steady state, we can get j 2m∆vj0 /Tj0 = 0. However, it cannot tell us the vj0 − dependence of ∆vj0 , since we have summed over all vj0 and thus lost such information. Such information can be figured out only if we study individual ∆vj0 . This is one merit of the mechanical model for calculating ∆vj0 (Eq.3.21 and Fig. 8).

Equation of state

AN US

Now we are ready to write down the equation of state for a one-dimensional gas with square well potential. Putting Eq.3.9 and Eq.3.25 into Eq.3.13 and keeping the physics to the lowest order of particle-particle interaction, we have F = ρkB Treal + ρ2 (kB Treal ) d1 − ρ2 (d2 − d1 ) .

(3.26)

ED

M

This is the equation of state for square well potential in our mechanical approach, and it gives the same answer as statistical mechanics (Eq.3.1). The physical meaning behind Eq.3.26 is a little subtle, though. Putting Eq.3.9 and Eq.3.25 into Eq.2.5, we have the equation of state of form (3.27)

PT

F = ρkB T + ρ2 (kB T ) d1 + ρ2 (d2 − d1 ) .

CE

As a limiting case, let us neglect the hard core and consider only the effect of the particleparticle attraction. By Eq.3.26 and Eq.3.27, we have ρkB T < F < ρkB Treal .

(3.28)

AC

Thus, to the naive question of “When particle-particle attraction is turned on, is the force increased or decreased?” the answer is that the force is actually increased. But one should follow by the immediate qualifier that, “the temperature of the system is also increased.” However, if we are comparing the force exerted by an ideal gas and that by a van der Waals gas at the same temperature, but with the particle size effect ignored, then the van der Waals gas necessarily exerts a smaller force.

18

ACCEPTED MANUSCRIPT

4

One-dimensional interacting gas with generic particleparticle interaction

4.1

CR IP T

With the lessons learnt from one-dimensional interacting gas with square well potential, we can generalize the mechanical model to generic one-dimensional interacting gas. By generic particle-particle interaction, we mean the potential is consisted of a hard core and an interaction tail with finite range r0 . In the following, we study attraction tail and then repulsion tail.

Particle-particle attraction

Obtain ∆tinteraction (0, vk0 ) by Eq.3.4 and then put it into Eq.3.8. Expand with U (r)/kB T to O (U (r)/kB T ). We have

AN US

Z r0   √ 0 2 vj0 N d1 N aπ − (vj ) U (r) 1 0 ∆T = − 0 − dr − .
2 e a erfi( √ ) 2 j vj kB T a d1 vj0

(4.1)

M

R  r Eq.4.1 is just Eq.3.9, with (d2 − d1 ) /kB T replaced by d10 dr [−U (r)/kB T ] . Therefore, the physics is all the same. Putting Eq.3.25 and Eq.4.1 into Eq.3.13, we obtain 2

F = ρkB Treal + ρ (kB Treal ) d1 − ρ

2

Z

r0

d1

 dr [−U (r)] .

(4.2)

AC

CE

PT

ED

This is exactly the equation of state derived using standard statistical mechanics, as we show in Eq.1.2. The physics behind Eq.4.2 is the same as what we present in Section 3. Now we want to analyze the physics from another viewpoint, which is based on the idea of average velocity. Instead of separating ∆Tj0 from Tj = Tj0 + ∆Tj0 (as we did from Eq.2.4 to Eq.2.5), we can regard Tj as an entire entity of its own and then work with the idea of average velocity (vj |average = 2 (L − N d1 ) /Tj ) directly. In such an approach, the property of the bulk is automatically embedded in the idea of average velocity. By our very construction, the only effect to be considered is what happens when a particle hits the wall. Rewriting the equation of state with average velocity and collision velocity, and starting from Eq.2.4 and keeping the physics to the first order in the particle-particle interaction, this time we get

F = ρkB Treal + ρ2 (kB Treal ) d1 +

mX (vj |average ) L j



vj0 + ∆vj0 − vj |average .

(4.3)

Compared with Eq.4.2 for the case when particle-particle interaction is attractive, the third term in Eq.4.3 should be negative. The reason hides in Table 1. From Table 1, we can see 19

ACCEPTED MANUSCRIPT

that, on the average, a particle experiences more forward collisions from its neighbors when it is in the bulk than it is on the boundary. Therefore, we have
vj0 + ∆vj0 − vj |average < 0.

(4.4)

4.2

PT

ED

M

AN US

CR IP T


The feature that vj0 + ∆vj0 − vj |average < 0 can be seen in Fig. 8. So now we understand that it is caused by the fact that the two kinds of velocities have different numbers of forward collisions. In a naive description trying to explain the form of the van der Waals equation [23], one is tempted to claim that a particle hitting the wall is slowed down due to the attraction of other gas particles from the back, and hence the force felt by the wall is decreased. Put differently, this seemingly plausible theory claims that the velocity hitting the wall (collision velocity) is smaller than the velocity in the bulk (average velocity), which also holds true in our detailed mechanical approach. However, we must quickly point out that, as particle-particle attraction is turned on, velocity in the bulk is different from the free particle velocity and particles in the box do fly faster. The reason that we don’t need to consider average velocity correction in the conventional description of the van der Waals equation is not because forward collisions and backward collisions cancel out exactly (they don’t), but lies in the fact that they are all lumped into the idea of vj |average . Now it is clear that traditional picture of the van der Waals equation actually uses vj |average but not vj0 in the argument. For physics in the bulk, it just takes the net result (vj |average ) and then bypasses all the issues related to “what happens when a particle flies in the middle of the box;” and it also avoids the necessity of including the temperature modification. It is in this sense that the traditional picture makes sense.

Particle-particle repulsion

AC

CE

For particle-particle repulsion, we just reverse the physics of attraction tail. However, one feature does stand out that is characteristic to particle-particle repulsion alone. When the relative velocity of two interacting particles is small, they don’t have enough kinetic energy (in center-of-mass frame) to overcome the potential barrier and reach r = d1 . Nevertheless, the population of such particle pairs is quite small since particle-particle interaction is weak by our construction, and hence such effect is negligible in equation of state. Another issue related to particle-particle repulsion is soft core. An example of soft core is Lennard-Jones potential, which has a r−12 repulsive term. Can we attack soft core in our mechanical approach? The answer is a sadly no. If a soft core is to stop two colliding particles, the perturbation incurred must necessarily be large, and a nonperturbative approach must be resorted to.

20

ACCEPTED MANUSCRIPT

5

Conclusion

AC

CE

PT

ED

M

AN US

CR IP T

In this work, we have taken a mechanical approach to study a one-dimensional interacting gas with particle-particle interaction made up of a hard core and an interaction tail. Perturbations from an ideal gas are considered, and the physics is computed correct up to O (U/kB T ). Equation of state derived using this mechanical approach matches the result given by statistical mechanics. Besides having been able to reproduce the equation of state, this approach has provided us with further physics insight which traditional statistical mechanics doesn’t readily tell us. Almost all the physics in the mechanical model is hidden in Fig. 8 and Table 1. Three of the more salient features for the case when the interaction tail is attractive are: (1) Due to the fact that the number of forward collisions is larger than the number of backward collisions, on the average, all particles move faster in the presence of particleparticle attraction. (2) Everything else being equal, the increased particle speed means the temperature of the system increases. (3) When one is interested in the change in the force, as compared with the ideal gas value of ρkB T , the force increases when the attraction is turned on. But if one insists that the temperature be kept fixed, then the force decreases. But the point of this study is not just to rederive the well-known results. Instead, we chose the more tedious route of tracking the trajectories of the interacting particles so that we can more easily visualize which factor is causing what. And this added bonus is traditionally hidden behind the more elegant approach of standard statistical mechanics. In this regard, it suggests to us that supplementing the traditional approach with a more detailed look of how the individual molecules move and behave might also teach us more interesting physics.

21

ACCEPTED MANUSCRIPT

A

Probability of the last collision

In this appendix, we calculate the probability of the last collision. To find the probability for such a scenario, we put in the background particles one by one. The probability of the background particle that causes the last collision is given by   0
0
0 1 − vk ∆d3 ∆vk . f v k vj0 L

CR IP T

(A.1)

Z

L−d3

0

dxel L

Z

AN US

For the remaining N − 2 background particles, they should be put in the box in the manner that they cannot touch the main particle after the main particle has its last collision with the first background particle. The position of one of the remaining N − 2 background particles with free particle velocity vl0 at t = (L − d3 ) /vj0 is denoted as xel . The requirement for xel is 0 < xel < L−d3 and the admissible velocity is − (L + xel ) vj0 /d3 < vl0 < (L − xel ) vj0 /d3 . The probability of a single background particle in the admissible zone is thus given by L−f xl 0 vj d3

−

L+f xl 0 vj d3

1 dvl0 √

aπ

2

−

e

(vl0 ) a

.

(A.2)

When thermodynamic limit is considered, we have

ED

M

" #!  0 √ 0 2 vj 1 a − (vj ) . probability of (N −2) background particles = exp − ρd3 1 + erf √ + √ 0e a 2 a πvj (A.3) The complete probability of the last collision is the product of Eq.A.1 and Eq.A.3 and the permutation factor N , and is given by

AC

CE

PT

" #! 2    0 √ 0 vj0 ) (

v v a ∆d 1 3 j N 1 − k0 + √ 0 e− a f vk0 ∆vk0 × exp − ρd3 1 + erf √ . (A.4) vj L 2 a πvj

22

CR IP T

ACCEPTED MANUSCRIPT

B

AN US

Figure 9: Nonvanishing ∆vj0 if forward collision occurs. Nonvanishing ∆vj0 appears in region0. Detailed information is given by Eq.B.3 and Eq.B.4.

Physics on the boundary for forward collision

M

In this appendix, we study the range of d3 and the collision velocity for forward collision. By Eq.3.2 and Eq.3.18, the condition on d3 is

ED

d3 < d2 + (d2 − d1 ) r

vj0 +vk0 2

(B.1)

.

2

+

vj0 − vk0 2

2

vj0 −vk0 2

 m

CE

PT

If the main particle hits the wall when it is in the potential well, by Eq.3.18, the collision velocity is vj0 + vk0 0 0 + vj + ∆vj = 2

s 

 . m

+

(B.2)

AC

By Eq.3.19, Eq.B.1 and Eq.B.2, the nonvanishing ∆vj0 for forward collision to occur is summarized in Fig. 9, with

γ1 : d3 = d2 , γ2 : d3 = d2 + (d2 − d1 ) r −vj0 + vk0 0 ∆vj |region−0 = + 2 23

s 

vj0 +vk0 2 vj0 −vk0 2

2

vj0 − vk0 2

+

2

,

(B.3)

 . m

(B.4)

 m

+

ACCEPTED MANUSCRIPT

C

Physics on the boundary for backward collision

In this appendix, we study the range of d3 and the collision velocity for backward collision. By Eq.3.2 and Eq.3.18, the condition for d3 is

vj0 −vk0 2

2

(C.1)

. +

 m

CR IP T

(d2 + d3 ) < d2 + (d2 − d1 ) r

vj0 +vk0 2

AN US

Eq.3.20 and Eq.C.1 are necessary conditions to have nonvanishing ∆vj0 . Under the two necessary conditions, the situations can be further classified as three classes (Class 1, Class 2 and Class 3), each with a different form for the collision velocity. The critical situation separating the three classes is shown in Fig. 10, with d3 |critical given by 

 d3 |critical = (d2 − d1 )   r



vj0 +vk0 2

vj0 −vk0 2

2

+

 m

 − 1 .

(C.2)

M

Class 1: In Class 1, d3 < d3 |critical . In this class, the main particle hits the wall when it is in the potential well. The collision velocity is

ED

vj0 + vk0 0 0 vj + ∆vj = − 2

s 

vj0 − vk0 2

2

+

 . m

(C.3)

AC

CE

PT

Class 2: In Class 2, d3 > d3 |critical . In this class, the main particle hits the wall after it decouples with the first background particle and becomes a free particle again. However, the velocity of the main particle is no longer vj0 , since the wall comes into play and interrupts the particleparticle interaction via particle-wall collision of the first background particle at the end of step-1. With the help of Eq.3.18, the new free particle velocity of the main particle, namely the collision velocity, is given by vj0

+

∆vj0

=

s 

vj0 + vk0 2

2

 − − m

s 

vj0 − vk0 2

2

+

 . m

(C.4)

Class 3: In Fig. 10, we actually already assume that the main particle always has a positive velocity. But this is not always guaranteed. If the main particle has a negative velocity, it cannot hit the wall, and the collision velocity in such case is zero. There are two situations when no 24

AC

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Figure 10: Critical situation that separates the three classes (Class 1, Class 2 and Class 3). The figure shows the motion of the main particle and the first background particle in the lab frame. Step-1 starts with the beginning of backward collision and ends with the event that the first background particle hits the wall. Step-2 starts with the end of step-1 and ends with the moment that the two particles are going to decouple. The value of d3 in this critical situation is denoted as d3 |critical .

25

CR IP T

ACCEPTED MANUSCRIPT

AN US

Figure 11: Nonvanishing ∆vj0 if backward collision occurs. Region-1,2, and 3 corresponds to Class 1,2, and 3, respectively. In addition, region for vk0 < vj0 is forbidden. Detailed information is given by Eq.C.5 and Eq.C.6.

AC

CE

PT

ED

M

collision happens between the main particle and the wall. The first situation is that the velocity of the main particle is negative at the moment right after the collision starts. That is, vCM − vin < 0. In this case, vk0 is given by vk0 < (/m) /vj0 . The second situation is that the velocity of the main particle is negative after they decouple. That is, the new free particle velocity in Eq.C.4 is negative. In this case, vk0 is given by (/m) /vj0 < vk0 < (2/m) /vj0 . In Class 1, the main particle hits the wall during its interaction with the first background particle. In Class 2, the main particle hits the wall when it becomes a free particle after it decouples with the first background particle. In Class 3, the main particle simply cannot hit the wall. The nonvanishing ∆vj0 when backward collision occurs is summarized in Fig. 11, with 

 γ3 : d3 = (d2 − d1 )   r γ4 : d3 = (d2 − d1 ) r

26



vj0 +vk0 2 vj0 −vk0 2

vj0 +vk0 2

vj0 −vk0 2

2

2

+

 m

, +

 m

 − 1 , (C.5)

ACCEPTED MANUSCRIPT

s 2  0 0 0 vj − vk0 + v −v  j k 0 − + , Class 1 : ∆vj |region−1 = 2 2 m s s  0 2  0 2 vj + vk0 vj − vk0   0 0 Class 2 : ∆vj |region−2 = −vj + − − + , 2 m 2 m

(C.6)

CR IP T

Class 3 : ∆vj0 |region−3 = −vj0 .

D

AN US

Of course, we also have the condition of Eq.3.20, and hence the region that vk0 < vj0 is forbidden. Therefore, Class 3 in backward collision part occurs only for small u0j satisfying p u0j < /kB T .

Total effect of the correction to the collision velocity

Z

ED

" #! 2  0 √ 0 vj0 ) (
v v 1 a j = ρ d(d3 ) dvk0 f vk0 1 − k0 exp − ρd3 1 + erf √ + √ 0 e− a vj 2 a πvj
× ∆vj0 |region−0 +∆vj0 |region−1 +∆vj0 |region−2 +∆vj0 |region−3 . (D.1) Z

PT

∆vj0

M

In this appendix, we finally calculate the total effect of the correction of collision velocity. The task we are going to do is putting Eq.3.17 (probability), Eq.B.4 (∆vj0 for forward collision) and Eq.C.6 (∆vj0 for backward collision) together and then integrate over the four regions shown in Fig. 9 and Fig. 11. That is, we are going to deal with the integral

AC

CE

Let’s stop and think whether Eq.D.1 can be simplified further. The value of d3 in region0,1,2,3 is quite small. We might make the guess that maybe it is all right to replace the exponential decay part in the probability distribution by unity. As it turns out, this is indeed the case, due to the feature that the probability decays slowly. The argument is in p Appendix E. Furthermore, since ∆vj0 |region−3 exists only for small velocity (u0j < /kB T ), the contribution is negligible for equation of state. The argument is in Appendix F. After the two simplifications, Eq.D.1 is now simplified to ∆vj0

=ρ

Z

d(d3 )

Z

dvk0 f

vk0

0

v k 1 − ∆vj0 |region−0 +∆vj0 |region−1 +∆vj0 |region−2 . (D.2) vj0

Calculate ∆vj0 via Eq.D.2 and expand the result with /kB T to O (/kB T ). We get

27

ACCEPTED MANUSCRIPT

∆u0j

Z

∞

−u0j

2

0 du0k e−(uk )

u0j + u0k + u0j − u0k

Z

∞

u0j

2

0 du0k e−(uk )

u0j − u0k u0j + u0k

!

 kB T

. (D.3)

Simplification of the exponential decay probability dis-

CR IP T

E

∆vj0 ρ (d2 − d1 ) √ = √ = a 2 πu0j

tribution

In this appendix, we argue that the exponential decay probability distribution in Eq.D.1 can be replaced by unity. The basic idea is trying to show that

AN US

" #!  0 √ 0 2 vj a − (vj ) 1 + √ 0e a exp − ρd3 1 + erf √ 2 a πvj

decays with d3 in a slow manner. To deal with the idea of “decays in a slow manner,” we define the “decay depth” as (E.1)

M

decay depth ≡

#!−1 "  0 √ 0 2 vj 1 a − (vj ) ρ 1 + erf √ . + √ 0e a 2 a πvj

ED

On the other hand, from Fig. 9 and Fig. 11, the maximum of relevant d3 to be considered is

maximum of relevant d3 decay depth

AC

O



CE

So we have

PT

maximum of relevant d3 = d2 + (d2 − d1 )



u0j +u0k 2 r 2 0 0 u −u j

k

2









(E.2)

. +

 2kB T

 1 −(u0j )2  d2 + (d2 − d1 ) = O ρ 1 + e   u0j

u0j +u0k 2 r 2 0 0 u −u j

k

2

 +

 2kB T

  . 

(E.3) We want the left hand side (LHS) to be < O (1). If LHS is not smaller than unity, we will
encounter difficulty. With some analysis for Eq.E.3, we can find that there are some u0j , u0k that don’t respect LHS < O (1). However, the violations appear only for small u0j or the case that u0k is very close to u0j . But since the contributions given by these “trouble makers” are negligible in the order being considered, due to the fact that their populations are small, we have saved the day. 28

ACCEPTED MANUSCRIPT

F

Contribution of ∆vj0 |region−3

"

X 2m∆vj0 j

Tj0

4 = − ρ2 kB T π

#

u0j <

"Z √ 0

√

 kB T

 kB T

CR IP T

In this appendix, we argue that the contribution of ∆vj0 |region−3 is negligible when we sum p over all velocities to get the correction in momentum transferred. For u0j < /kB T , we have ∆vj0 |region−3 = −vj0 . So we have

,Class 3 2 u0j

du0j e−( )


2 u0j

Z

 1 kB T u0 j

u0j

2 u0k

du0k e−( )

By Eq.C.5 and Fig. 11, the integral of d3 is bounded by

And so we have

u0 1 − k0 uj

#

Z

d(d3 ) .(F.1)

u0j + u0k d(d3 ) < (d2 − d1 ) q .
2 u0j − u0k + 2 kB T

AN US

Z



M

" # 0 π X 2m∆vj 0 4 Tj √  0 j uj < k T ,Class 3 "Z √ B Z 2 kB T
2 0 −(u0j ) 0 2 < ρ kB T (d2 − d1 ) duj e uj

u0j

ED

0

PT

+ρ2 kB T (d2 − d1 )

"Z √ 0

 kB T

0 du0j e−(uj )

2

u0j

Z

1  kB T u0 j

1  kB T u0 j

u0j

2 u0k

du0k e−( ) 2

0 du0k e−(uk )

(F.2)

#

#

u0k .

(F.3)

AC

CE

The  term in the right hand side of Eq.F.3 are both bounded by  first term andpthe second O [ρ2 (d2 − d1 ) ] /kB T and thus are negligible.

29

ACCEPTED MANUSCRIPT

References [1] J. W. Dufty, A. Santos, J. J. Brey, Practical kinetic model for hard sphere dynamics, Phys. Rev. Lett. 77 (7) (1996) 1270. [2] R. D. Rohrmann, A. Santos, Structure of hard-hypersphere fluids in odd dimensions, Phys. Rev. E 76 (5) (2007) 051202.

CR IP T

[3] L. Tonks, The complete equation of state of one, two and three-dimensional gases of hard elastic spheres, Phys. Rev. 50 (10) (1936) 955. [4] M. Uranagase, Effects of conservation of total angular momentum on two-hard-particle systems, Phys. Rev. E 76 (6) (2007) 061111.

AN US

[5] M. Uranagase, T. Munakata, Statistical mechanics of two hard spheres in a box, Phys. Rev. E 74 (6) (2006) 066101. [6] I. Urrutia, Two hard spheres in a spherical pore: Exact analytic results in two and three dimensions, J. Stat. Phys. 131 (4) (2008) 597–611.

M

[7] P. Visco, F. van Wijland, E. Trizac, Collisional statistics of the hard-sphere gas, Phys. Rev. E 77 (4) (2008) 041117.

ED

[8] X. Z. Wang, van der waals–tonks-type equations of state for hard-disk and hard-sphere fluids, Phys. Rev. E 66 (3) (2002) 031203.

PT

[9] M. E. Foulaadvand, M. Yarifard, Two-dimensional system of hard ellipses: A molecular dynamics study, Phys. Rev. E 88 (5) (2013) 052504.

CE

[10] P. Gurin, S. Varga, Towards understanding the ordering behavior of hard needles: Analytical solutions in one dimension, Phys. Rev. E 83 (6) (2011) 061710.

AC

[11] Y. Kantor, M. Kardar, One-dimensional gas of hard needles, Phys. Rev. E 79 (4) (2009) 041109. [12] J. Largo, J. Solana, Generalized van der waals theory for the thermodynamic properties of square-well fluids, Phys. Rev. E 67 (6) (2003) 066112. [13] A. Barra, A. Moro, Exact solution of the van der waals model in the critical region, Annals of Physics 359 (2015) 290–299. [14] F. Giglio, G. Landolfi, A. Moro, Integrable extended van der waals model, Physica D: Nonlinear Phenomena 333 (2016) 293–300.

30

ACCEPTED MANUSCRIPT

[15] W. Zhong, C. Xiao, Y. Zhu, Modified van der waals equation and law of corresponding states, Physica A: Statistical Mechanics and its Applications 471 (2017) 295–300. [16] J. J. Brey, J. W. Dufty, Hydrodynamic modes for a granular gas from kinetic theory, Phys. Rev. E 72 (1) (2005) 011303.

CR IP T

[17] I. Goldhirsch, S. Noskowicz, O. Bar-Lev, Nearly smooth granular gases, Phys. Rev. Lett. 95 (6) (2005) 068002. [18] D. Risso, P. Cordero, Dynamics of rarefied granular gases, Phys. Rev. E 65 (2) (2002) 021304. [19] S. Liu, C. Zhong, Investigation of the kinetic model equations, Phys. Rev. E 89 (3) (2014) 033306.

AN US

[20] G. F. Mazenko, Fundamental theory of statistical particle dynamics, Phys. Rev. E 81 (6) (2010) 061102. [21] R. P. Feynman, Statistical Mechanics: A Set of Lectures (Advanced Book Classics), Section 4.2, Westview Press Incorporated, 1998.

M

[22] S. Salinas, Introduction to statistical physics, Section 6.4, Springer Science & Business Media, 2013.

AC

CE

PT

ED

[23] J. D. van der Waals, J. S. Rowlinson, On the continuity of the gaseous and liquid states, Courier Corporation, 2004.

31