主題瀏覽
- 學位論文
鑰匙鏈在喇叭平台上振動的統計行為
The Statistics and Dynamics of Chains on a Vibrating Plate
摘要
鑰匙鍊在喇叭平台上的扭動情形，和高分子在溶液中的行為類似。不過由於前者在表面上是個極度非平衡的系統，是否適合沿用後者來自環境的「溫度」概念，則尚未可知。如果可以，那麼該如何從喇叭的頻率和振幅來定義溫度？異於高分子的端點距離的機率分布和溫度無關，觀察到的鑰匙鍊情況卻隨頻率和振幅改變，甚至出現扭動變緩的共振現象，又該如何解釋？最後，我們還設計了兩套探討熵力的實驗，來直接面對「溫度」定義的正當性。 由於目前使用的鑰匙鍊段數有限，加上鄰近鍊段的張角不能太大，我們在理論的分析上，融合了數值模擬和模型的建立。主要的結論包括：1）發現頻率和振幅可以間接透過鍊條跳動的最高高度影響端點距離分布，就如同高分子在垂直方向受到限制，這個邊界條件相當於全反射面，可以利用鏡像原理來分析。2）由於數位攝影機的鏡頭有限，會在距離分布的機率上，給出額外的修正權重。3）當鍊條在達到共振前後，跳動的高度會變低，端點的水平距離分布除了變窄外，峰值還會往長距離偏移。這暗示頻率和振幅多半不是簡單地透過一個類似溫度的概念進到鍊條的動態行為，並且連同後來的熵力分析，都有可能需要擔心受到共振的影響。 當高分子部分受到限制時，由於裡外亂度的不同，會感受到向外的熵力。我們在鑰匙鍊的情況，也觀察到同樣的行為。不過由於溫度的概念有所爭議，我們另外設計了一個在高分子實驗無法實現的狀況，亦即在部分區域加上軟墊，使得鍊條在碰撞時損失較多的能量，藉以模擬較低的「溫度」。由於兩邊在空間上都沒有限制，熵力主要來自於鍊條兩端感受到的不同能量，到底它和高分子的溫度定義有何差異？我們目前推測，溫度可用鍊條的平均水平動能來取代。
並列摘要
Abstract When a chain wriggles on a plate vibrated vertically by a speaker, its behavior is similar to that of a polymer in a solution. Since the former is far from equilibrium, it is doubtful to coin the concept of a temperature, which is determined by the heat bath a polymer solution comes into contact with. Some energy scale is clearly required to describe the vigor of the chain motion. Then, how does this scale depends on the frequency and amplitude（f/A）of the speaker? In the end, two experiments on entropic force are designed and discussed to tackle the nature of this energy scale. Because our chain is of finite steps and the neighboring steps are constrained with a maximum bending angle, we use both numerical simulation and model-building in our theoretical analysis. The main conclusions are: i) By taking into account the fact that there is a maximum height a chain can jump, f/A of the speaker can affect the distribution function, p(r), of the end-to-end distance of our chain. It is equivalent to placing a totally-reflecting plate above the chain, which enables us to use the image method. ii) Since the digital video has a finite lens, discarding of the images whose end points move outside of the scope introduces a bias to the distribution function. iii) Resonances appear in most of the f/A ranges. When they occur, p(r) becomes much narrower with its peak moving toward larger r values. All the three above properties imply that the f/A of the speaker comes into the problem in a subtler way than a simple characteristic energy scale. When partial segments of a polymer are confined within two walls, it will experience an entropic force due to the different entropies. This is also observed for our chain. Since there is a dispute over the concept of temperature, we construct another experiment to clarify it. Part of the plate is covered with soft material such that more energy is lost during each collision. We simulate those segments over it as experiencing lower “temperature”. Since neither parts of the chain is confined, the entropic force comes from the different energies they experience. Is the energy of a chain the same as the thermodynamic temperature? We preliminarily expect the average energy of the chain to play such a role.