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answers-02-10-2011 - Answers to Physics 176 One-Minute...

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Answers to Physics 176 One-Minute Questionnaires Lecture date: February 10, 2011 Are Einstein solids related to Bose-Einstein condensates? No, they are not. Einstein’s model of a solid addresses the vibrational prop- erties of a crystal, for example how vibrations of atoms in crystals contribute to the heat capacity. A Bose-Einstein condensate is a quantum state of mat- ter that occurs only at extremely low temperatures. The condensate arises when a collection of identical massive bosons (a certain class of fundamental particles that have integer spin, such as a gas of rubidium atoms) can be described by a single “coherent” macroscopic wave function. We will discuss briefly the properties of Bose-Einstein condensates and why they occur later this semester. What is the purpose and motivation behind calculating Ω total ? The goal of Section 2.3 of Schroeder and of the related discussion in class was to explain why the multiplicity of some macroscopic object spontaneously increases toward a maximum value, and why the transfer of energy (heat) between subsystems is an irreversible probabilistic phenomenon. The first step in the discussion was to calculate the multiplicity Ω total of a thermally isolated Einstein solid that consists of two interacting macroscopic subsystems A and B. Our discussion led to several insights. One was to label a macrostate of the total solid by the amount of energy q A in subsystem A, which then determines the amount of energy q B = q - q A in subsystem B since energy is conserved in the isolated total system. A second important insight was that, generally, two subsystems interact weakly with one another (they affect each other only for atoms near the surface where they contact one another). This implies that the behavior of subsystem A is statistically uncorrelated with the behavior of subsystem B (at least on times short compared to a relaxation time) which in turn implies that the total multiplicity is approximately equal to the product of the multiplicities of the subsystems: Ω total Ω A × Ω B . (1) This insight allows us to calculate Ω total for any specified amount of energy q A in subsystem A which led to a third insight as discussed in Sec- tion 2.3 of Schroeder: the total multiplicity Ω total ( q A ) generally has a large narrow peak as a function of q A (especially if number of oscillators in each 1
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subsystem is large). This means that some macrostates have many more accessible microstates than other macrostates and so, if all accessible mi- crostates are equally likely, certain macrostates are much more likely to be observed than others. This explains why the multiplicity of a closed system will “spontaneously” increase toward a maximum value, corresponding to the macrostate with the most accessible microstates.
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