Minimal work principle: Proof and counterexamples

Abstract

The minimal work principle states that work done on a thermally isolated equilibrium system is minimal for adiabatically slow (reversible) realization of a given process. This principle, one of the formulations of the second law, is studied here for finite (possibly large) quantum systems interacting with macroscopic sources of work. It is shown to be valid as long as the adiabatic energy levels do not cross. If level crossing does occur, counter examples are discussed, showing that the minimal work principle can be violated and that optimal processes are neither adiabatically slow nor reversible. The results are corroborated by an exactly solvable model.

Understanding quantum measurement from the solution of dynamical models

“…Due to this ambiguity, splitting the ensemble E described byD into subensembles E k , described either by pure states as in (10.3) or by mixed states as in (10.4), is physically meaningless (though mathematically correct) if no other information thanD is available. The above indetermination leads us to acknowledge an important difference between pure and mixed quantum states [9,28,31,48,85,28,324]. If a statistical ensemble E of systems is described by a pure state, any one of its subensembles is also described by the same pure state, since in this case (10.4) can include only a single term.…”

“…The solution of models of quantum measurements is therefore expected to enlighten the foundations of quantum mechanics, in the same way as the elucidation of the paradoxes of classical statistical mechanics has provided a deeper understanding of the Second Law of thermodynamics, either through an interpretation of entropy as missing information at the microscopic scale [57,58,74,73,81,71,289,290], or through a microscopic interpretation of the work and heat concepts [72,291,292,293,294,295,296,297,298]. In fact, the whole literature devoted to the quantum measurement problem has as a background the interpretation of quantum mechanics.…”

“…In the statistical interpretation, such underlying pure states have no physical meaning. More generally, decompositions of the type (10.4) can be given a meaning only if the knowledge ofD is completed with extra information, allowing one to identify, within the considered ensemble E described byD, subensembles E k that do have a physical meaning [31,48,324].…”

“…We have stressed, however, its ambiguity ( § 10.2.3 and § 11.1.3); as a consequence, the very collection of pure states |φ k among which each individual system is supposed to lie cannot even be imagined. It seems therefore difficult to imagine the existence of "underlying pure states" which would carry more "physical reality" thanD [324,329]. The distinction between the two types of probabilities on which decompositions (10.3) rely is artificial and meaningless [10,11].…”

Nonequilibrium work equalities in isolated quantum systems

“…(19) is about the correlation of the operators at two times. Additionally, this equation clearly explains why defining the work operatorŴ (t f )= t f 0 ∂ τ H H (τ )dτ alone [40,54] cannot lead into the correct prediction about the second moment of the quantum wok except for the specifical canonical initial density matrix. Finally, the expansion does not matter with the initial canonical density matrix.…”