Microscopic Foundations of Thermodynamics and Generalized Statistical Ensembles Page: 28
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the entropy principle (see Chap. 1) we shall assume that the system is thermally
isolated from the environment. As discussed previously, the system energy is not
known. This means that the system is assumed to be in a statistical mixture of
states, described by a density matrix pi, rather than a pure state k). As prescribed
by Statement 3 we shall also assume that the system is initially at equilibrium. We
will translate this thermodynamic notion into the quantal requirement that = -0.
at
So the system is at equilibrium whenever = 0 and it is out of equilibrium whenever
a- 0. At t = ti, we switch on a perturbation. This is implemented by changing the
value of some external parameter A during the course of time: A = A(t). A can be for
example the volume V of a vessel containing the system, or the value of some external
field like an electric or a magnetic field. At time t = toff, the perturbation is switched
off. We assume that at some time tf > toff any transient effect has vanished and the
system has attained a new equilibrium state described by some pf, such that Pf -0.
Thus, before time ti and after tf the system is at equilibrium, and for ti <t < tf it is
out of equilibrium. Due to the perturbation the Hamiltonian changes from the initial
value Hi to the final value Hf, and accordingly the quantum entropy operator will
change in time and move from Si to Sf. We introduce the following time-dependent
orthonormal basis set {k, t)}. The vectors Ik, t) are defined as the eigenvectors of the
"frozen" Hamiltonian H(t). That is:
K
(49) H(t) ek(t)k, t) (k, t.
k=O
Since at time t 2 0, then [ps, H] = 0. This means that pi is diagonal over the
at
initial basis { k, ti)}:
K
k=O
As anticipated in the introduction we shall assume that the initial probability distri-
bution pi is decreasing:
(51) Po > P1 > ...> Pi > ...28
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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/m1/38/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .