Abstract
In this chapter we shall see how the conservative character of the thermodynamic field, hence the existence of a thermodynamic potential, emerges from the very Hamiltonian nature of the equation of motion in the case of a particle in a U-shaped potential. This result is known as the Helmholtz theorem. We shall illustrate this with the most minimal mechanical model of a thermodynamic system, namely a particle in a 1D box, and other examples.
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Notes
- 1.
In classical mechanics (and as well in quantum mechanics), a quantity that remains “unvaried” (what we mean by that will become more clear below) in slow processes is called an adiabatic invariant [2].
- 2.
See Ref. [3].
- 3.
We shall come back to this later. This relation is discussed and derived in many textbooks of classical mechanics. For example Landau and Lifschitz classical mechanics textbook [2].
- 4.
See Landau and Lifschitz mechanics textbook [2] or any other textbook in classical mechanics.
- 5.
With this the solution of Exercise 2.4 is immediate.
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Campisi, M. (2021). Minimal Mechanical Model of Thermodynamics. In: Lectures on the Mechanical Foundations of Thermodynamics. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-87163-5_2
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