Abstract
This chapter presents a pedagogical introduction to the variational methods in statistical thermodynamics. We start with some general considerations of the variational nature of thermodynamics, which is rooted in the second law, and manifested in the maximum-term method in the evaluation of the partition function in statistical mechanics. We present two common mathematical variational techniques, one based on the Gibbs-Bogoliubov-Feynman (GBF) variational bound and one based on the saddle-point (or steepest-descent) method. We illustrate the use of these techniques in the derivation of the mean-field theory for Ising model and the Poisson-Boltzmann equation. We also show that the GBF method provides a self-consistent treatment of fluctuation effects in weakly correlated systems.
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References
The uncertainty principle also takes the form of an inequality, but it can be deduced from the fundamental equations of quantum mechanics, e.g., the Schrödinger equation
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Wang, ZG. (2017). Variational Methods in Statistical Thermodynamics—A Pedagogical Introduction. In: Wu, J. (eds) Variational Methods in Molecular Modeling. Molecular Modeling and Simulation. Springer, Singapore. https://doi.org/10.1007/978-981-10-2502-0_1
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DOI: https://doi.org/10.1007/978-981-10-2502-0_1
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