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。

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.

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

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