Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles Page: 27
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quantity whose possible values are (within the range of validity of the approximation):
(46) S=nl n + .
We can extend this line of reasoning to multidimensional systems whose dynamics is
ergodic. In this general case the volume entropy is given by Eq. (8). Again using
the quasi-classical viewpoint [26] the integral in Eq. (8) approximately counts the
number of quantum states not above a certain energy n = E. Since the levels are
nondegenerate this number is n+ 2, where one considers that the vacuum state counts
as a half state. The levels are nondegenerate because the corresponding classical
dynamics is ergodic. This can be understood by noticing that ergodicity implies that
the Hamiltonian is the only integral of motion. This, translated into the language of
quantum mechanics, says that the Hamiltonian alone constitutes a complete set of
commuting observables, so that the only quantum number is n.
At this point, it is quite easy to construct the quantum version of volume entropy.
Consider a finite (i.e., not necessarily macroscopic) nondegenerate quantum system.
Let N be the quantum number operator, i.e.,:
K
(47) N Z kk)(k
k=0
where { k) } is the complete orthonormal set of the Hamiltonian's eigenstates. K, the
total number of energy levels, can be infinite. The eigenvectors of N are the energy
eingenvectors, and the eigenvalues are the corresponding quantum numbers. Then
the quantum volume entropy operator can be defined as:
(48) S - In (N +
We adopt a system of units where kB, the Boltzmann constant, is equal to 1.
4.2. Proof of the Entropy Principle
Armed with a quantum mechanical analogue of thermodynamic entropy (48), we
can now study its evolution under a time-dependent perturbation. As prescribed by27
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Campisi, Michele. Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles, dissertation, May 2008; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc6128/m1/37/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .