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arXiv:0803.2241

Comment on "on a new definition of quantum entropy"

Keyvan Sadri

Published Mar 14, 2008• 0 citations
arXivHTMLPDF

Abstract

In the article, on a new definition of quantum entropy, Campisi has explained an operator for entropy based on quantum number operator. It has been claimed that the expectation values for this operator increases for every non-quasi-static time dependent perturbations. We have found an example for a case that entropy with this definition does not increase through a spontaneous process.

DOI
10.48550/arxiv.0803.2241

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Reply to "Comment on "On a new definition of quantum entropy""

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The example provided in the comment [arXiv:0803.2241] concerns a situation where the system is initially at negative temperature. It is known that in such cases the Law of Entropy Decrease holds. Nevertheless, this does not challenge the validity of the Second Law of Thermodynamics.

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In the article, on a new definition of quantum entropy, Campisi has explained an operator for entropy based on quantum number operator. It has been claimed that the expectation values for this operator increases for every non-quasi-static time dependent perturbations. We have found an example for a case that entropy with this definition does not increase through a spontaneous process.
This work studies the relationship between parametric amplification (or particle creation), adiabaticity and irreversibility in the non-quasi-static regime of a time-dependent quantum harmonic oscillator (TDHO) that evolves unitarily. We provide analytical results for the evolution of the TDHO valid for any functional value of the frequency, which enables us to monitor the behavior of the thermodynamical magnitudes in the non-quasi-static regime. In the latter, the largest modes of the energy eigenstates commonly undergo a process of spontaneous thermalization, where the concept of temperature naturally arises from the unitary evolution of the oscillator, i.e. without relation to any external source of temperature or thermal bath. As the evolution is unitary, this thermalization process can be reversible. We extend the standard definitions of quantum heat and work to account for the change in the populations of the energy levels and relate it to the diagonal entropy to obtain an effective temperature from the associated second principle of thermodynamics. The qualitative behavior of this temperature is similar to the temperature of the largest modes and is only significant in the non quasi-static region, returning to the original value T→0T\rightarrow 0T→0 afterwards. This effect is briefly analyzed in the case of a periodically driven quantum system, where the energy eigenstate mixing and the temperature grow and return to the original value periodically.
The law of increase of entropy is one of the most important laws in science but no exact explanation has been given to the origin of it. Statistical thermodynamics only explains it by using Boltzmann’s statistical interpretation together with not sufficiently verified assumptions. This paper shows that the increase of entropy in an isolated macroscopic system can be reasonably explained with no assumption based on quantum mechanics if we take into account that microscopic particles such as electrons and atomic nuclei not only move in space but also have the basic quality of spontaneously spreading into every possible spatial position. The present work allows us to give a plausible definition to the entropy of an isolated system by using the Hamiltonian operator and a wave function of it and provides a firm basis for thermodynamics. It also gives a new physical meaning of entropy effective for further studies of irreversible processes in nature.
We derive some Quantum Central Limit Theorems for expectation values of macroscopically coarse-grained observables, which are functions of coarse-grained hermitean operators consisting of non-commuting variables. Thanks to the hermicity constraints, we obtain positive-definite dis- tribution for the expectation values of observables. These probability distributions open some pathway for an emergence of classical behaviours in the limit of innitely large number of identical and non-interacting quantum constituents. This is in contradistinction to other mechanisms of classicality emergence due to environmental decoherence and consistent histories. The probabil- ity distributions so derived also enable us to evaluate the nontrivial time-dependence of certain differential entropies.
The concept of entropy is central to the formulation of the quantum statistical mechanics, and it is linked to the definition of the density operator and the associated probabilities of occupation of quantum states. The extension of this scheme to accommodate for quantum decaying states is conceptually difficult, because of the nature of these states. Here we present a way to treat quantum unstable states in the context of statistical mechanics. We focuss on the definition of the entropy and avoid the use of complex temperatures.