stp_chap4 - Chapter 4 The Methodology of Statistical...

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Unformatted text preview: Chapter 4 The Methodology of Statistical Mechanics c circlecopyrt 2010 by Harvey Gould and Jan Tobochnik 8 October 2010 We develop the basic methodology of statistical mechanics and provide a microscopic foundation for the concepts of temperature and entropy. 4.1 Introduction We now will apply the tools and concepts of thermodynamics and probability which we introduced in Chapters 2 and 3 to relate the microscopic and macroscopic descriptions of thermal systems. In so doing we will develop the formalism of statistical mechanics . To make explicit the probabilistic assumptions and the type of calculations that are done in statistical mechanics we first discuss an isolated system of noninteracting spins. The main ideas that we will need from Chapter 3 are the rules of probability, the calculation of averages, and the principle of least bias or maximum uncertainty. Consider an isolated system of N = 5 noninteracting spins or magnetic dipoles with magnetic moment and spin 1/2 in a magnetic field B . Each spin can be parallel or antiparallel to the magnetic field. The energy of a spin parallel to the magnetic field is = B , and the energy of a spin aligned opposite to the field is = + B . We will consider a specific case for which the total energy of the system is E = B . What is the mean magnetic moment of a given spin in this system? The essential steps needed to answer this question can be summarized as follows. 1. Specify the macrostate and accessible microstates of the system . The macroscopic state or macrostate of the system corresponds to the information that we know. In this example we know the total energy E and the number of spins N . The most complete specification of the system corresponds to the enumeration of the mi- crostates or configurations of the system. For N = 5 there are 2 5 = 32 microstates specified by the 172 CHAPTER 4. STATISTICAL MECHANICS 173 (a) (b) Figure 4.1: (a) Example of an inaccessible microstate corresponding to the macrostate specified by E = B and N = 5. (b) The ten accessible microstates. Spin 1 is the leftmost spin. orientation of each of the N spins. Not all of the 32 microstates are consistent with the information that E = B . For example, the microstate shown in Figure 4.1(a) is not allowed, that is, such a microstate is inaccessible, because its energy is E = 5 B . The accessible microstates of the system are those that are compatible with the macroscopic conditions. In this example ten of the thirty-two total microstates are accessible (see Figure 4.1(b)). 2. Choose the ensemble . We calculate averages by preparing a collection of identical systems all of which satisfy the macroscopic conditions E = B and N = 5. In this example the ensemble consists of ten systems each of which is in one of the ten accessible microstates....
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stp_chap4 - Chapter 4 The Methodology of Statistical...

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