從力學觀點探討一維有交互作用之氣體：一個從力學觀點探討弱相互作用之多體系統的示範

國立臺灣大學理學院物理學研究所 碩士論文

Graduate Institute of Physics College of Science

National Taiwan University Master Thesis

從力學觀點探討一維有交互作用之氣體：一個從力學觀 點探討弱相互作用之多體系統的示範

A Mechanical Approach to One-dimensional Interacting Gas: A Demonstration of Investigating Weakly Interacting

N-body Systems from A Mechanical Viewpoint

王重陽

Chung-Yang Wang

指導教授：陳義裕 教授 Advisor: Prof. Yih-Yuh Chen

中華民國一百零六年三月

March 2017

致謝

首先，我要感謝我的指導教授，陳義裕老師。在過去三年的時間，和老師一 起做研究，一起討論，除了學到具體的物理知識，我也從老師身上學到一些看事 情的方式。像是，當我們在黑暗中摸索、還不知道答案的時候，要怎麼在沒有『事 後的先見之明』(老師的經典名言之一)的情況下，試著發展一些想法。

在研究的過程中，老師的風格是對底下的學生非常放任，讓我們有很大的自 由度(like degrees of freedom in a N-body system)嘗試自己的想法。從事後的先見 之明來看，有些嘗試雖然錯得離譜，但卻讓我從這樣一團糟的東西裡面想到一些 日後很有用的想法。在老師的團隊裡面，這樣自由放任的模式，讓我這三年來過 得很愉快。

我也要感謝我們團隊中的同學。在研究的過程中，我會和老師底下的其他同 學討論。即使他們沒有給我明確的意見(畢竟我們做的題目都是分散在不同領域)，

和大家一起討論，讓我在討論的過程中把一些東西想得更清楚。

最後，我要感謝我的父母。儘管他們不瞭解我在物理系到底在做什麼，並認 為這是一件既沒前途也沒錢途的事，他們依然讓我在大學畢業後在物理系繼續待 了三年。

雖然不知道自己之後到底會走向何方，在物理系這幾年，我真的過得很充實，

很愉快。

摘要

著名的凡得瓦方程式(van der Waals equation)是用來描述一箱有弱相互作用、

而且在特定條件下(高溫極限以及低密度極限)的氣體的狀態方程式。傳統上對凡 得瓦方程式的推導是採取統計力學中的標準做法，其中會牽涉到系綜的平均 (ensemble average)。在我們的研究中，我們從純粹力學的觀點切入，來探討在一 維空間中，一箱有弱相互作用之氣體的行為。因此，在我們的架構中，三個核心 的概念為：粒子的軌跡、粒子交互作用的次數、每一次交互作用產生的效果。這 樣的力學架構的優點是，除了推導出凡得瓦方程式，我們還可以得到一些在標準 的統計力學中無法告訴我們的有趣的物理。例如，目前對於凡得瓦方程式的詮釋 與圖像是採取了平均場(mean field approximation)的想法，在力學的觀點中，我們 發現這樣的標準圖樣其實是錯誤的。傳統上，對於一個古典的多體系統，物理學 家通常是採用統計力學或是分子動力論(kinetic theory)的框架。在這份研究中，

我們除了探討一維的有交互作用之氣體，也展示了如何從力學的觀點來探討有弱 相互作用的多體系統，並對於粒子之間的交互作用所產生的第一階的物理效應有 更深刻的理解。

Abstract

The famous van der Waals equation is the equation of state for a box of weakly interacting gas particles under certain limits (high temperature and low density).

Traditional 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 from a purely mechanical point of view. Thus, trajectories, number of particle-particle interactions, and effect of each particle-particle interaction are at the heart of the present approach. 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 mechanical approach.

For example, we find that the traditional handwaving interpretation of the van der Waals equation adopting mean field approximation is actually incorrect. In this investigation of one-dimensional interacting gas, we demonstrate the possibility taking a mechanical point of view and having deeper understanding for the physics of leading order effect of particle-particle interaction, for weakly interacting N-body systems that are usually studied in the framework of statistical mechanics or kinetic theory.

Contents

1 Introduction 1

2 Mechanical picture for one-dimensional interacting gas 5

3 One-dimensional interacting gas with square well potential 11

3.1 Mechanics of interaction between two particles . . . 12

3.2 Flying time period . . . 14

3.2.1 The idea of “mirror diagram” . . . 16

3.2.2 Counting the number of collisions . . . 16

3.2.3 Correction of the flying time period . . . 24

3.2.4 Correction of flying time period in equation of state . . . 29

3.3 Temperature modification . . . 30

3.4 Momentum transferred . . . 35

3.4.1 Toy bean model . . . 37

3.4.2 Probability of the last collision . . . 40

3.4.3 Situation around the wall . . . 46

3.4.4 Correction to the collision velocity . . . 57

3.4.5 Correction of momentum transferred in equation of state . . . 65

3.5 Equation of state . . . 71

4 One-dimensional interacting gas with generic particle-particle interaction 73

4.1 Particle-particle attraction . . . 73

4.2 Meaning and interpretation . . . 76

4.3 Particle-particle repulsion . . . 80

5 Conclusion 84

5.1 Summary . . . 84

5.2 Meaning and implication . . . 85

Appendix A . . . 88

Appendix B . . . 92

Appendix C . . . 94

Appendix D . . . 100

Appendix E . . . 103

Reference . . . 105

1 Introduction

Statistical mechanics and kinetic theory are two main frameworks one adopts when studying a system containing a huge number of degrees of freedom. Researches in this field study hard sphere systems [2, 12, 14, 15, 16, 17, 19, 20], hard ellipse systems [4], hard needle systems [6, 7], van der Waals theory [8], granular particle systems [1, 5, 11], etc. There are also researches concerning fundamental issues [9, 10].

Statistical mechanics and kinetic theory are powerful and systematic, but it often amounts to meaning that one has to pay the price of losing certain detailed dynam- ical information about the interparticular interactions. In order to study a weakly interacting N-body system, besides statistical mechanics, kinetic theory and numerical simulation, do we have other choice? At first glance, the idea of particle trajectory is useful only for systems containing small degrees of freedom. However, we can actually bring in the idea of trajectory for a N-body system if the effect of particle-particle interaction is weak. In this work, we study a weakly interacting one-dimensional gas to see how a direct approach investigating the detailed mechanical interaction between particles might shed some light on its merits as opposed to the traditional approach.

Before describing our mechanical approach, we first give a brief review of how one understands a weakly interacting one-dimensional gas in statistical mechanics. For a one-dimensional classical gas in the box, the equation of state at thermal equilibrium can be calculated by standard recipes in statistical mechanics. Specifically, one can start with a given Hamiltonian of the gas system and computes its partition function, then following through the standard recipes [3, 13] to arrive at the following equation of state

F = ρ(kBT ) − ρ²(kBT ) Z ∞

0

dr

 e⁻

U (r) kB T − 1



+ O(ρ³), (1)

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

r

U(r) r=d

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.

box, respectively), k_B is Boltzmann’s constant, T is the temperature of the system, and U (r) is the potential between two gas particles with r being their separation. In the above, one has assumed a low density limit so that a Taylor expansion in ρ is possible.

We will specialize to the case when the potential representing particle-particle in- teraction is consisted of a hard core and an interaction tail, as shown in Fig. 1.

By Eq.1, the equation of state for such a potential is given by

F = ρ(kBT ) + ρ²(kBT )d1− ρ²(kBT ) Z ∞

d1

dr

 e⁻

U (r) kB T − 1



+ O(ρ³). (2)

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.2 to first order of ^{U (r)}_k

BT under these limits, and the equation of state becomes

F = ρ(k_BT )+ρ²(k_BT )d₁−ρ² Z ∞

d1

dr [−U (r)] = ρ(k_BT )



1 + ρd₁− ρ Z ∞

d1

dr −U (r) kBT



(3)

After rearrangement, the equation of state is given by



F + (N L)²

Z ∞ d1

dr [−U (r)]



(L − d1N ) = N kBT, (4)

which is the van der Waals equation in one-dimension.

This is the statistical mechanics approach to the van der Waals equation, the equa- tion of state for interacting gas under the conditions of low density, high temperature and weak interaction. The derivation provided by statistical mechanics is simple, and in one scoop it easily relates the force with the temperature, the density and the po- tential describing particle-particle interaction. On the other hand, one also loses track of exactly what has happened to the dynamical behavior of the constituent particles.

For example, if we “turn on” the interparticular interactions of an otherwise ideal gas, will that make the gas particle move faster or slower? And, does the answer depend on the original velocity of a particle? Statistical mechanics by itself doesn’t provide such information. In this regard, the physics contained in an equation of state is quite limited.

In this work, we adopt an approach different from statistical mechanics. We take a mechanical point of view, considering the detailed mechanics of the gas particles.

In order to deal with a system of an essentially infinite number of degrees of freedom, statistical mechanics hides the complexity in its fundamental principles and assumptions (such as the notion of partition function), so that we can focus on things like the equation of state that can be more readily measured. In our mechanical approach, in order to deal with the complex one-dimensional interacting gas system, we still need to put in by hand some basic assumptions. We are not pretending that we are going to derive the equation of state for a one-dimensional interacting gas from scratch and nothing else. However, we will do our best making assumptions that seem at least reasonable. Besides rebuilding the equation of state, we will have some physics that standard statistical mechanics doesn’t tell us. For example, we will try to answer the

two previous questions in our mechanical approach.

The thesis is organized as follows. In Chapter 2, we discuss the foundation of our mechanical point of view. In Chapter 3, we consider a special case of square well particle-particle interaction and show how the mechanical model is built. In Chapter 4, we generalize the previous results to generic interacting gas, with the help of physics insight we developed in Chapter 3. Finally, we summarize the result and give some possible implications in Chapter 5.

2 Mechanical picture for one-dimensional interact- ing gas

Consider a bunch of particles flying in a one-dimensional box, as shown in Fig. 2. As a particle hits the wall of the box, the wall experiences a force. The collision between the particle and the wall is assumed to be perfectly elastic. Because our description is a bit unorthodox, we first describe how we view the momentum transferred to the right-hand wall by the right-most particle in a one-dimensional box.

Physically we know the right-most particle is the only particle that is constantly bombarding the wall. But the momentum it transfers to the wall after each collision varies in time, because, prior to the collision, this particle gets its momentum from a collision with the particle immediate to its left. But where does that neighboring particle pick up that particular momentum? Clearly, It gets it through an even earlier collision with the immediate neighbor to its left. In this sense, we see that we may assign an arrow to the momentum carried by each particle, and think of each collision happening between two particles as a mechanism of exchanging the arrows of momentum. In other words, an arrow gets pushed to its right after each collision, all the way until it hits the right-hand wall.

In the picture described above, we see that the total force felt by the wall is the summation of the contributions of all particles in the box, namely

F = X

j

F_j =X

j

(momentum transferred)_j

(flying time period)_j =X

j

2mv_j

T_j (5)

where j is the index of particle, T_j is the time period it takes for an arrow to complete one round trip (between the two walls), and v_j is the velocity of a particle when it hits the wall, called collision velocity. By the lower index j, we actually label the velocity arrow, but not the particle itself, since a particle cannot pass through another particle in

L

Figure 2: Particles flying in a one-dimensional box of length L.

A

A

B

B time

v

interaction process

v

v

v

Figure 3: Interaction between particle A and particle B. Particle A is always on the left hand side of particle B, while the velocity arrow v₁ can penetrate.

one-dimensional space. When two particles collide, the velocity arrows will penetrate, but the left particle stays always on the left and the right particle stays always on the right, as shown in Fig. 3. A particle is just a mediator of a velocity arrow.

For a box of ideal gas, that is, the particles have no size and there is no particle- particle interaction, the particles are free, and the force is given by

F =X

j

2mv⁰_j

T_j⁰ =X

j

2mv_j⁰

2L v_j⁰

= 2 L

X

j

1

2m v_j⁰
2

(6)

Here the upper index of 0 stands for free particle. The notion of temperature is intro- duced via Maxwell’s velocity distribution

f (v) ≡ 1 q2kBT

m π

e⁻

1 2mv2

kB T = 1

√aπe^−v2a , Z ∞

−∞

dvf (v) = 2 Z ∞

0

dvf (v) = 1 (7)

where a ≡ ^2k_m^BT. Summation over j is given by

X

j

= N

 2

Z ∞ 0

dv_j⁰f (v_j⁰)



= 2N

√aπ Z ∞

0

dv⁰_je⁻(^v0j)²

a (8)

The factor of two comes from the fact that we consider the force felt by the right wall.

Since particle hits the right wall only if its velocity is positive, we consider the right half part (positive velocity) of Maxwell’s velocity distribution and then multiply a factor of two to recover the true particle number. Plugging Eq.8 into Eq.6, the force for an ideal gas is

F = 2 L

√2N aπ

Z ∞ 0

dve^−v2a 1

2mv² = N k_BT

L = ρk_BT (9)

which is the one-dimensional version for P V = N k_BT , the equation of state for an ideal gas. Note that Maxwell’s velocity distribution is put in by hand at the last stage just so that the population of particles with a specific velocity can be connected with temperature, provided that thermal equilibrium is reached.

What happens when particle-particle interaction is turned on? In order to study the force felt by the wall, which is equivalent to the equation of state, there are three elements: summation over all particles (P

j), flying time period (Tj), and momentum transferred (2mv_j). For the summation over all the particles, we will make the explicit assumption that Maxwell’s velocity distribution is valid for our system. When particle- particle interaction is turned on, a particle interacts with other particles and hence its velocity is no longer a constant but influenced by particle-particle interactions. There- fore, we expect that the travel time period will be modified. For momentum transferred, due to particle-particle interaction, the velocity of a particle at the moment hitting the wall (collision velocity) may differ from its free particle velocity, and hence the mo- mentum collected by the wall should be modified in the presence of particle-particle interaction. The question we want to ask is, when there is particle-particle interaction, how do flying time period (physics in the bulk) and momentum transferred (physics on

the boundary) change, and hence lead to the modification of the equation of state? And if we stick with our mechanical approach (but with a set of extra assumptions), how far can we go? Note that standard statistical mechanics cannot tell us “what happens in the bulk” and “what happens on the boundary.” It just gives us the equation of state, the total result after combining “physics in the bulk” and “physics on the boundary.”

If the system has a high density or a long range interaction, many particles in the box are simultaneously coupled together. This makes things rather hard to track.

In order to have a working mechanical approach, we therefore consider the case of low density and particle-particle interaction being short-ranged and weak. Of course, particle-particle interaction by itself has a dimension, and we will see later that a more appropriate statement is that we consider ^{U (r)}_k

BT, a dimensionless quantity, to be small.

This requirement corresponds to assuming a weak particle-particle interaction and high temperature. Since ¹₂k_BT stands for average kinetic energy, O

U (r) kBT



< O (1) means that average kinetic energy is much larger than potential energy describing particle- particle interaction. So we consider one-dimensional interacting gas systems under the limits of low density (dilute gas), short-ranged and weak particle-particle interaction (weakly interacting) and high temperature. Note that these conditions are also required by the van der Waals equation. In this sense, then, we are not imposing more conditions than the standard statistical mechanics approaches require. Since the interaction is small in some sense, we treat the effect of particle-particle interaction as a perturbation, and a particle is a free particle like ideal gas most of the time.

Compared with ideal gas, namely F = ρk_BT , the correction due to particle-particle interaction in van der Waals equation of state is kept to order one, which can be seen in Eq.3. Therefore, in order to construct an equation of state from a mechanical model that can be compared with the van der Waals equation, we only need to retain the accuracy up to the first order in the particle-particle interaction. In other words, we consider the physics perturbed around the ideal gas (free particle) regime and need only

retain the perturbation up to the first order correction.

To be more concrete, our goal is to study the equation of state, which is equivalent to the force felt by the wall, and is given by

F = X

j

2mv_j Tj

=X

j

2m(v_j⁰+ ∆v_j⁰)

T_j⁰+ ∆T_j⁰ (10)

where v⁰_j is the free particle velocity, vj = v_j⁰+ ∆v⁰_j is the collision velocity, T_j⁰ = ^2L_v0 j

is the flying time period without interaction, and T_j = T_j⁰ + ∆T_j⁰ is the flying time period in the presence of interaction. Expanding things out to first order in the small corrections, we see that Eq.10 becomes

F =X

j

2mv⁰_j

T_j⁰ −X

j

2mv_j⁰ T_j⁰

∆T_j⁰

T_j⁰ +X

j

2m∆v⁰_j

T_j⁰ (11)

What we are going to do is trying to study the three terms in Eq.11 with the notion of particle trajectories. The first term corresponds to ideal gas, namely the unperturbed part. The second term arises from the modification of flying time period (physics in the bulk). The third term comes from the modification of collision velocity (physics on the boundary). The strategy is tracking a particle label by j (which means that its free particle velocity is v_j⁰), called the main particle, and find out the influence of the other particles, called the background particles, on the main particle. Apparently, the main particle is not special but just a notion for bookkeeping.

Here we want to mention that the idea of temperature is actually a subtle concept.

When there is particle-particle interaction, the velocity of a particle is no longer a constant, but fluctuates as the particle interacts with other particles. How do we introduce the idea of Maxwell’s velocity distribution, and hence the temperature of the system, in this situation? The idea is, the velocity distribution of v_j⁰ (the unperturbed part) is described by Maxwell’s velocity distribution with temperature T . That is, we identify the first term of the right hand side of Eq.11 with ρk_BT , since the picture in

mind is that we are doing perturbation and things should go back to ideal gas when there is no particle-particle interaction. But is this T equal to “the real temperature of the box of interacting gas?” We don’t know. At this stage, the relation between “T ” and “the real temperature of the box of interacting gas” is unclear in our mechanical model. Later (Section 3.3) we will find out what is the real temperature of the box of interacting gas. At this stage, T is just a parameter appears inP

j (Eq.8), and we don’t know how to deal with the real temperature of the system. However, we want to point out that temperature is a thermodynamic concept instead of a mechanical concept, and hence calculations of the effect of particle-particle interaction can be done in our mechanical approach, without dealing with the thermodynamic issue of “What is the real temperature of the system?”

To avoid possible confusion, we should emphasize that we are talking about the real temperature of the system when taking standard statistical mechanics. Therefore, although we don’t know how to introduce the idea of temperature in our mechanical approach (the real temperature of the system may be larger or smaller than T in our mechanical approach), the symbol T in Eq.1 to Eq.4 is the real temperature of the system. Later we will use T_realfor the real temperature of the system in our mechanical approach. Note that if we have T 6= T_real, then we have P

j 2mv_j⁰

T_j⁰ 6= ρk_BT_real, which means that it is too naive to identifyP

j 2mv⁰_j

T_j⁰ (our mechanical approach) with the ideal gas part in Eq.1 to Eq.4 (standard statistical mechanics). Of course, T is close to T_real even if T 6= T_real, since the effect of particle-particle interaction is weak.

At the end of this chapter, 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.

r

U(r)

Ɛ

Ɛ>0

r=d

r=d

Figure 4: Square well potential.

3 One-dimensional interacting gas with square well potential

In this chapter, we consider one-dimensional non-ideal gas with square well interpar- ticular potential, as shown in Fig. 4. We work out the details in the mechanical model for this specific particle-particle interaction. The advantage of square well potential is that when two particles come close enough and interact, the motion is easy to analyze, since the two particles are actually two free particles inside the potential well (namely, they are free particles for d₁ < r < d₂). Interaction is nonvanishing only at r = d₂ and r = d₁. This is why we choose square well potential. In the next chapter, a more general potential is considered.

By the recipe in standard statistical mechanics (Eq.3), equation of state for square well potential is

F = ρ (k_BT ) + ρ²(k_BT ) d₁− ρ²(d₂− d₁) . (12) Eq.12 is what we are going to compare with when obtaining an equation of state by our mechanical approach.

3.1 Mechanics of interaction between two particles

Consider the interaction between two particles described by a generic potential with the only condition that the interaction has a finite range r₀. Since there is no in- teraction for r > r₀, we consider the physics with the initial state being the start of the interaction (r = r₀)(incoming) and the final state being the end of the interaction (r = r₀)(outgoing), as shown in Fig. 5. Suppose the interaction takes time interval of ∆tinteraction, which is a function of the two input velocities and the form of the po- tential. Now we want to ask, what are the displacements of the two particles during

∆tinteraction? In center-of-mass frame, the left particle takes a displacement of r0 and the right particle takes a displacement of −r₀. We want to emphasize again that by particle we actually mean the velocity arrow, but not the particle itself. In the lab frame (of the confining box), the displacements are given by

displacement of left particle = r0+ vCM∆tinteraction (13)

displacement of right particle = −r₀+ v_CM∆tinteraction (14) where v_CM is the center-of-mass velocity.

What is this ∆tinteraction? Consider a potential having a hard core shown in Fig. 1.

By conservation of energy, the speed of the two particles in their center-of-mass frame is given by

v(r) = s

 v₁− v₂ 2

2

+U (r₀) − U (r)

m (15)

where m is the mass of the particle, r is the separation of the two particles. So

∆tinteraction(v₁, v₂) is given by

∆tinteraction(v₁, v₂) = Z r0

d1

dr q

v1−v₂ 2

2

− ^{U (r)}_m

(16)

In lab frame :

In center-of-mass frame : initial state

initial state final state

final state

v

v

(v

-v

)/2 -(v

-v

)/2

r

v

v

r

r

r

(v

-v

)/2 -(v

-v

)/2

Figure 5: A particle with velocity v₁ interacts with a particle with velocity v₂. The interaction range is denoted as r₀. The upper half part shows the situation in the lab frame. The lower half part shows the situation in center-of-mass frame.

where we have used the condition of U (r₀) = 0.

For particle-particle interaction that is repulsive, ∆tinteraction is more complicated in the sense that the lower bound of d₁ of the integral in Eq.16 should be replaced by a function of v1 and v2. For example, when the two particles are not energetic enough in the center-of-mass frame (that is, v₁ − v₂ is small), they don’t have enough kinetic energy to overcome the potential barrier and reach r = d₁. Particle-particle repulsion will be studied later when we study generic particle-particle interaction (Section 4.3).

3.2 Flying time period

Without loss of generality, we may track a main “particle” (in fact, a velocity arrow) which flies from the left wall to the right wall. When particle-particle interactions are taken into account, the main particle changes its velocity along the trip from the left wall to the right wall, and hence its flying time period is changed. Since low density limit (dilute gas) is considered, it is a free particle moving with its free particle velocity v_j⁰ for most of the time.

How is the flying time period modified by particle-particle interaction? Well, we may separate the effect into two parts, that is, how many collisions there are in total, and what is the effect of each collision. So there are two steps. First, we find out how many collisions there are for the main particle to fly from the left wall to the right wall during the trip (Section 3.2.1 to Section 3.2.2). This number turns out not to have anything to do with the form of the particle-particle interaction, as we will show in the following. The details of the particle-particle interaction comes in only when we are dealing with the effect of each collision (Section 3.2.3).

We separate the flying time period into free particle part and interaction part. The

flying time period is the sum of the two parts, which is given by

1

2Tj = L − total displacement of v_j⁰ in interaction

v_j⁰ +X

k

∆tinteraction(v_j⁰, vk)

= L

v_j⁰ +X

k

∆tinteraction(v⁰_j, v_k) − total displacement of v⁰_j in interaction

v_j⁰ (17)

where the background particles are labeled by k, and v_kis 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 number of forward collisions and backward collisions are denoted as #_R(R stands for right) and #_L(L stands for left), respectively.

For forward collision, by Eq.13, we have

∆tinteraction(v⁰_j, vk) − displacement of v⁰_j in interaction v_j⁰

= ∆tinteraction(v_j⁰, v_k) − r₀+^v

0 j+vk

2 ∆tinteraction(v_j⁰, v_k) v⁰_j

= −r₀

v_j⁰ + v_j⁰− v_k

2v⁰_j ∆tinteraction(v⁰_j, v_k) (18)

Similarly, for backward collision, by Eq.14, we have

∆tinteraction(v_j⁰, v_k) −displacement of v_j⁰ in interaction v_j⁰

= ∆tinteraction(v_j⁰, v_k) − −r₀+^v

0 j+vk

2 ∆tinteraction(v_j⁰, v_k) v⁰_j

= r₀

v_j⁰ +v⁰_j − v_k

2v_j⁰ ∆tinteraction(v⁰_j, v_k) (19)

Plug Eq.18 and Eq.19 into Eq.17, and so ∆T_j⁰ is given by

1

2∆T_j⁰ = 1

2 Tj− T_j⁰
= 1

2Tj − L v_j⁰

= #_R



−r0

v_j⁰ +v⁰_j − v_k

2v_j⁰ ∆tinteraction(v_j⁰, v_k)



+ #_L r₀

v_j⁰ +v⁰_j − v_k

2v_j⁰ ∆tinteraction(v_j⁰, v_k)



(20)

So we need to find out #_R, #_L, v_k and ∆tinteraction(v⁰_j, v_k). Now we are going to build a mechanical model to find out #_R and #_L.

3.2.1 The idea of “mirror diagram”

In order to proceed, we now introduce the idea of “mirror diagram” as a useful graphical tool we will use frequently in later analysis. The idea is quite simple. Mirror diagram is a diagram helping us visualize the trajectory of a particle in the one-dimensional box.

Consider the case of an ideal gas, namely a box of free particles. For a particle flying in the box, it changes its direction when it hits the walls (the left wall and the right wall) due to the reflection provided by the walls. For a particle having a positive velocity at t = 0, as time goes on, the velocity will change from positive to negative, and negative to positive, and then positive to negative, so on and so forth. To visualize the motion of a particle, a common way is drawing its x − t diagram. Mirror diagram is an idea built on ordinary x − t diagram. An example of mirror diagram is presented in Fig. 6.

It turns out that such diagram is quite helpful in dealing with one-dimensional particle system.

3.2.2 Counting the number of collisions

Back to the issue of computing #R and #L. Consider the case of an ideal gas. How many collisions does the main particle experience in the whole trip flying from the left

x=0

x=L

t

t x

x x=L

x=L

x=0

Figure 6: An example of mirror diagram. The lower diagram is an ordinary x − t diagram. The upper diagram is the mirror diagram of the lower diagram. To construct a mirror diagram, we just reflect line segments in ordinary x − t diagram with x = 0 and x = L, as they are two mirrors.

particle several times. On the contrary, for a background particle with low speed, it may hit the main particle only once. For a background particle with specified velocity and initial position, the numbers of forward collisions and backward collisions provided by the background particle are completely determined, as schematically shown in Fig.

7.

So much for the case of an ideal gas. But what about interacting gas particles?

Because we are only computing corrections correct to the first order of particle-particle interaction, it turns out that the counting of the number of particle collisions can be essentially done by assuming that the particles behave just like ideal gas particles! To be more precise, let’s look at Fig. 8.

Fig. 8 is the mirror diagram of the main particle and a background particle, in time interval t = ¹₂T_j⁰. The trajectory of the main particle in the bottom block is the real trajectory, and the other above trajectories are its mirror images. We also draw two kinds of trajectory for the background particle: one being the trajectory of a free particle (ideal gas trajectory) and the other being the trajectory when particle-particle

A B C x=L

t=L/v

forward collision : A, C backward collision : B

R1

R2

R3

right wall reflection : R1, R3 left wall reflection : R2

t x

Figure 7: The x − t diagram for a typical motion of the main particle (solid line) and a background particle (dash line) in the time interval that the main particle flying from the left wall to the right wall. There are two forward collisions and one backward collision in this figure.

interaction is turned on (real trajectory). Due to particle-particle interaction, the real background particle trajectory is different from its ideal gas trajectory. With the help of Fig. 8, we can see two things.

The first thing is: The number of collisions for the case when the background particle trajectory is that for an ideal gas is essentially the same as that for the case when the weak and short-ranged interaction is turned on. Their difference is expected to be of first order of particle-particle interaction. Referring back to Eq.(11), we see that this difference does not need to be explicitly included while performing the various summations, which themselves are already of first order in accuracy. For instance, to compute ∆T_j⁰, which in itself is already a first order correction, we can use ideal gas particles as the background for the main “particle.” Reprise: One can safely use the free particle model in counting the number of forward collisions and the number of backward collisions in our theory.

t=L/v

forward

forward backward collision

collision

collision

x=x

x=L

t x

x=L

x=L x=0

x=0

x=0

main particle and its mirror images background particle as free particle

background particle as interacting particle

Figure 8: An example of mirror diagram containing the main particle and a background particle. For the background particle, we consider two kinds of trajectory: one is the trajectory being a free particle (ideal gas trajectory) and the other is the trajectory when particle-particle interaction is turned on (real trajectory). The initial position of the background particle is denoted as x⁰_k.

Mathematically speaking, consider a function A(δ) = B(δ)C(δ), where δ is small.

Expand A(δ) with δ

A(δ) = B0+ δB1+ O(δ²)

C0 + δC1+ O(δ²)

(21) If B0 = 0 and A(δ) is kept to O (δ), we have

A(δ) = (δB₁) C₀ (22)

Since B(δ) contributes at least one order of δ, it is sufficient to preserve C(δ) to the zeroth order of δ.

Total effect of particle-particle interaction, effect of each collision, number of colli- sions and strength of particle-particle interaction play the role of A(δ), B(δ), C(δ) and δ in Eq.21 and Eq.22, respectively. 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.

The second thing is: Based on the above, we see that, in order to count the number of collisions, we can just resort to the mirror diagram and count the numbers of crossing between the straight line of Fig. 8 (corresponding to the background particle) and the mirror-reflected arrows of the main particle of Fig. 8. To do the actual counting, we proceed as follows.

The number of collisions provided by a background particle depends on its ve- locity and initial position, that is, #_R = #_R v_j⁰, v_k⁰, x⁰_k

= #_R_v0 k

v_j⁰, x⁰_k

and #_L =

#_L v_j⁰, v_k⁰, x⁰_k

= #_L_v0 k

v⁰_j, x⁰_k

. With the help of mirror diagrams and some elemen- tary counting of the intersection points of some straight lines (for example, there are two forward collisions and one backward collision in Fig. 8), _L¹ RL

0 dx⁰_k#_L v_j⁰, v_k⁰, x⁰_k
is

2 2 1 1

2 3 4 5 6 -1

-2 -3 -4 -5

0 3 3

4 4

7 8 9 10

-9 -8 -7 -6 v _k ⁰ /v _j ⁰

Figure 9: _L¹ RL

0 dx⁰_k#_L v_j⁰, v_k⁰, x⁰_k
as a function of ^v_v^k00

j. The line segments connecting two adjacent flat parts are straight lines with slopes being ±1.

calculated and the result is shown in Fig. 9. For #_R, we have

1 L

Z L 0

dx⁰_k#_R v_j⁰, v_k⁰, x⁰_k
= 1 + 1 L

Z L 0

dx⁰_k#_L v⁰_j, v_k⁰, x⁰_k

(23)

Let’s see what Fig. 9 and Eq.23 mean. Without loss of generality, the main particle is set to start from x = 0 flying to x = L in our construction. At t = 0, all background particles are between the right wall (x = L) and the initial position of the main particle (x = 0). Therefore, in the trip of the main particle flying from the left wall to the right wall, all background particles collide with the main particle at least one time, being forward collision of the main particle. To have a backward collision with the main particle, a background particle needs to fly fast enough to catch up with the main particle again, which can be seen from _L¹ RL

0 dx⁰_k#_L v⁰_j, v⁰_k, x⁰_k
= 0 for −1 < ^v_v^0k0 j

< 2 in Fig. 9. Furthermore, a backward collision implies that the background particle hits the main particle from the left side and then passes through the main particle to its right side. Since now the background particle is between the main particle and the right wall, there will be a forward collision before the main particle arrives at the right wall. In

1

forward collision : backward collision : forward collision :

backward collision :

v

=v

v

=v

v

=-v

v

=-v

x _k ⁰ =L

v _k ⁰ /v _j ⁰ x _k ⁰

Figure 10: Velocity of a background particle when colliding with the main particle. It is a function of x⁰_k and ^v_v^k00

j

.

other words, a backward collision is always followed by a forward collision, due to the construction that the end of the trip of the main particle is hitting the right wall, which implies that no one can exist between the main particle and the right wall at the end of the trip. Comparing the initial situation at t = 0 and the final situation at t = ¹₂T_j⁰, all the background particles change from sitting at the right side of the main particle to its left side, and hence the number of forward collisions is larger than the number of backward collisions by one.

Now we have #_R and #_L. In view of Eq.20, the next thing to study is v_k, the velocity of a background particle when colliding with the main particle.

A background particle is specified by its initial position and free particle velocity, namely labeled by (x⁰_k, v⁰_k). For a given (x⁰_k, v⁰_k), what is the velocity of the background particle when it collides with the main particle? For example, consider a background particle with v_k⁰ = ¹₂v_j⁰. If x⁰_k ∼ 0, forward collision occurs when the velocity of the background particle is v_k⁰. On the other hand, if x⁰_k ∼ L, that is, the background particle takes an initial position near the right wall, forward collision occurs when the velocity of the background particle is −v_k⁰. The minus sign comes from the reflection provided

by the right wall. The velocity of the background particle when it collides with the main particle depends on both x⁰_k and v⁰_k. With the help of mirror diagram, we can easily find out the result shown in Fig. 10.

With Fig. 10, we define

H v_k⁰
≡











−v⁰_k for ^v_v^k00 j

+^x_L^0k > 1 v_k⁰ for ^v_v^k00

j

+^x_L^0k < 1

(24)

For forward collision, v_k = H (v⁰_k). For backward collision, v_k= −H (v_k⁰).

Substituting the position distribution, the velocity distribution, and H (v⁰_k) into Eq.20, we have

1

2∆T_j⁰ = 1 L

Z L 0

dx⁰_k Z ∞

−∞

dv_k⁰ N f (v_k⁰)



#_R



−r₀

v_j⁰ + v_j⁰− H (v_k⁰)

2v_j⁰ ∆tinteraction v⁰_j, H v_k⁰



+#_L r₀

v_j⁰ +v⁰_j + H (v_k⁰)

2v_j⁰ ∆tinteraction v⁰_j, −H v_k⁰



(25)

Since the dependence of x⁰_k appears only in #_R and #_L, we may rearrange the integral as

1

2∆T_j⁰ = N v_j⁰

Z ∞

−∞

dv_k⁰f (v_k⁰)1 L

Z L 0

dx⁰_k



(−r₀) (#_R− #_L)

+#_Rv⁰_j − H (v⁰_k)

2 ∆tinteraction v_j⁰, H v_k⁰

+#L

v_j⁰+ H (v_k⁰)

2 ∆tinteraction v_j⁰, −H v⁰_k



(26)

With #_R, #_L and H (v_k⁰) in hand, the remaining task is putting them together and calculating ¹₂∆T_j⁰. The calculation is done in Appendix A. The result is

1

2∆T_j⁰ = −r₀N v_j⁰+N

2

√ 1

aπ v_j⁰
2

Z ∞ 0

dv_k⁰

"

e⁻(^v0k−v0j)²

a − e⁻(^v0k+v0j)²

a

# v_k⁰
2

∆tinteraction 0, v⁰_k
(27)

3.2.3 Correction of the flying time period

Next we include the effect of the square well potential. For square well potential, in view of Eq.16, we see that ∆tinteraction(0, v_k⁰) is given by

∆tinteraction 0, v_k⁰
= Z d2

d1

dr r

_v0 k

2

2

+_m^ +

Z r0

d2

dr r

_v0 k

2

2 = d₂− d₁ r

_v0 k

2

2

+_m^

+r₀− d₂

| ^v₂^0k | (28)

Plugging Eq.28 into Eq.27, we have

1

2∆T_j⁰ = −r₀N v⁰_j + N

2

√ 1

aπ v⁰_j
2

Z ∞ 0

dv⁰_k

"

e⁻(^v0k−v0j)²

a − e⁻(^v0k+v0j)²

a

#

× v_k⁰
2









d₂− d₁ r_v0

k

2

2

+_m^

+ r₀− d₂

| ^v₂^0k |









= −N d2

v⁰_j + N (d2− d1)

√aπ v⁰_j
2

Z ∞ 0

dv_k⁰

"

e⁻(^v0k−v0 j)²

a − e⁻(^v0k+v0j)²

a

# (v_k⁰)² q

(v⁰_k)²+ ^4_m (29)

To expand a physical quantity, we should take something dimensionless as our per- turbation parameter. Thus, we define the dimensionless velocity u to be

u⁰_i ≡ s

1

2m (v_i⁰)²

k_BT = v_i⁰ q2kBT

m

= v⁰_i

√a. (30)

In passing, we would like to emphasize that the introduction of this dimensionless velocity is quite helpful when dealing with the population in velocity distribution. With this dimensionless velocity, there is no parameter like temperature in Maxwell’s velocity distribution, and the Boltzmann factor just reads e^−u2. We will see the advantage of this dimensionless velocity later when we need to figure out the order of magnitude of some populations.

With this dimensionless velocity, Eq.29 can be written as

1

2∆T_j⁰ = −N d₂

v_j⁰ +N (d₂− d₁)√

√ a

π v_j⁰
2

Z ∞ 0

du⁰_k h

e⁻(^u0k−u⁰_j)² − e(^u0k+u0j)²i (u⁰_k)² q

(u⁰_k)²+_k^2

BT

(31)

Expand the integrand in _k^

BT and keep up to order O

 kBT



, we have

Z ∞ 0

du⁰_k h

e⁻(^u0k−u⁰_j)² − e(^u0k+u0j)²i (u⁰_k)² q

(u⁰_k)²+ _k^2

BT

= Z ∞

0

du⁰_k h

e⁻(^u0k−u⁰_j)² − e(^u0k+u0j)²i

u⁰_k
− 1 u⁰_k

  k_BT



= Z ∞

0

du⁰_kh

e⁻(^u0k−u⁰_j)² − e(^u0k+u0j)²i u⁰_k−

Z ∞ 0

du⁰_ke⁻(^u0k−u⁰_j)² − e(^u0k+u0j)² u⁰_k

!  k_BT

=√

πu⁰_j − πe⁻(^u0j)²erfi(u⁰_j)  k_BT

=r π

av⁰_j − πe⁻(^v0j)²

a erfi( v⁰_j

√a) 

k_BT (32)

Plugging Eq.32 into Eq.31, we have

1

2∆T_j⁰ = −N d1

v_j⁰ −N (d2− d1)√ aπ

v⁰_j
2 e⁻(^v0j)²

a erfi( v⁰_j

√a) 

k_BT (33)

This is the flying time correction due to a square well potential in the particle-particle interaction.

Let’s try to figure out what Eq.33 is telling us. The first term is the effect of the hard core, whose meaning is as expected. The effect of the hard core is decreasing the space that a particle can move. The length of the box, namely L, is replaced by L − N d1, and the corresponding flying time correction is ¹₂∆T_j⁰ = −^{N d}_v0¹

j . The second term is the effect of the attraction potential, which can be seen from the feature that there are d₂− d₁ and _k^

BT in this term. The second term is always negative for any v⁰_j (recall that v_j⁰ ≥ 0 by construction). It means that the effect of the attraction tail in