We saw in section 1.5 how, given a Hamiltonian and using the tools of statistical mechanics, one can—in principle—completely describe a system’s thermodynamic behavior at any temperature or external field. In practice, however, this is extremely difficult; if it weren’t, statistical mechanics would be a closed book rather than a thriving field of active research. Very few Hamiltonians can be exactly “solved,” meaning that the corresponding free energy as a function of temperature, magnetic field, and so on, can be exactly calculated. In spite of this, the thermodynamic properties of a tremendous number of physical systems have been mapped out in detail, thanks to a powerful array of mathematical ...

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