Lecture-15-2023

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Simon Fraser University *

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Mechanical Engineering

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Oct 30, 2023

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Boltzmann Distribution ( N, V, T held constant) The probability a system will be found in the i th microstate is P i = 1 Z e - βE i , where Z = X i = all microstates e - βE i is the Partition Function , and β = 1 /k B T 141
Because the Boltzmann distribution is so important, we will derive it. You do not need to be able to derive the distribution for the exam. You need to understand what it represents and how to use it. It is important to understand every step in the derivation. This derivation will teach us how to understand the Boltzmann distribution how to relate the macroscopic observables with the partition func- tion So pay attention! 142
In order to derive the Boltzmann distribution, we first need to un- derstand the relation between the number of microstates and entropy What is the meaning of entropy from the microscopic point of view? 143
Consider systems A and B with the number of microstates W A and W B . The total entropy of the combined system is S A + S B . The total number of microstates in the combined system is W A W B . Therefore, the relation between the number of microstates W and entropy S must be logarithmic: S = k B ln W The proportionality constant k B is the Boltzmann constant. 144
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