Comments on Piero Quarati et al. Negentropy in Many-Body Quantum Systems. Entropy 2016, 18, 63

Abstract

The purpose of this note is to express my concerns about the published paper by Quarati et al. (Entropy 2016, 18, 63). It is hoped that these comments will stimulate a constructive discussion of the issues involved.

The above paper by Quarati et al. [1] argues that the concept of negentropy can be useful in understanding the observed enhancements of nuclear fusion rates in the solar plasma and in liquid metals - with respect to standard Saltpeter screening in the solar plasma case, and in a liquid metal host as compared with the solid host. It is argued that the host systems possess a certain level of organization, expressed in terms of negentropy, that is available for transferral to the fusing nuclei to enhance their fusion rates.

The definition of negentropy is provided in the first sentence, “Negative entropy, or negentropy [1–3] (Refs. in [1]), can be defined as the specific entropy deficit of an ordered subsystem with respect to surrounding chaos.” The concept is elaborated at a number of points later (first at the end of the fourth paragraph) as “stored mobilizable energy in an organized system ... which can be spontaneously transferred or exchanged among the elements or clusters of elements or from environment to elements and subsystems.”

Exploring the use of negentropy to better understand purely physical phenomena is an attractive idea, for its success in this arena is bound to have implications for its use in complex chemical and biological systems. While the definition of negentropy may be necessarily vague, as with many other subtle scientific concepts, one would hope that in a simpler physical system, a more precise definition of negentropy could be fleshed out, and the detailed mechanism for entropy or free energy transfer described and worked out. In this work a sharper definition leading to numerical estimates is sought, while the detailed mechanism is left for future work.

My concern in this case is that the sharper definition, or “operational definition” corresponding to what is actually calculated, is inconsistent with the definition given above. Briefly stated, the equilibrium entropy of the medium consists of a number of terms, at least one of which contributes negatively. Such a negative term (or terms) are identified as negentropy. This is first stated at the beginning of the fourth paragraph of the introduction and made more explicit through examples in later sections. The negative contributions come from interactions which produce correlations in the ions of the metal or in the solar plasma, or from quantum degeneracy of the conduction electrons in metals. These contributions are identified explicitly by equations in the text: For liquid metals, by Equations (16) and (19) (equations derived by Wallace [2]); for the solar plasma, by Equation (24) (written in terms of the free energy as in Ref. [59] of [1]). In solid metals, the analogous terms are small compared with the above, but they provide instructive textbook examples of equilibrium entropies: The ionic lattice contribution is found in Equation (6). The conduction electron contribution (Equations (10) and (13)) is due to quantum degeneracy, and makes a similar small contribution in both solid and liquid metals.

The problem with the above operational definition is that the total entropy expressions of which these negative contributions are a part describe systems in equilibrium (not metastable equilibrium, as argued in Section II—see below). Such a contribution (representing correlations or quantum degeneracy) cannot represent a “specific entropy deficit” as in the initial definition; or a “stored mobilizable energy” (or free energy) available for spontaneous transferral to the fusing nuclei. Put another way, there are no thermodynamic states into which these systems, so described, can relax spontaneously with the release of free energy.

The initial draft of these comments ended here, with the belief that the argument is complete. However, this would not have been fair to the readers or to the authors. The authors’ arguments are grounded in earlier work, and they deserve further discussion. The arguments leading to the above operational definition are given in Section II. The total entropy is written there as the sum,

$$S=S^{+}-S^{-}$$

(1)

where $S^{+}$ is the positive kinetic contribution and $S^{-}$ is the (magnitude of) the negative contribution due to correlations, or to quantum degeneracy. It is then stated that “At global thermal equilibrium $S=S^{+}$ and $S^{-}=0$.” If this statement were true, then yes, the nonzero correlation entropy $S^{-}$ in Equation (1) above is available to play the role of negentropy, and the thermodynamic states characterized by the standard expressions referred to above are at most metastable. But how can this possibly be the case? It seems that the system would have to turn off its interactions or suspend its quantum degeneracy.

Section II continues with supporting arguments which suggest a scenario. The arguments are based on the use of nonextensive statistical mechanics to describe a system which is imagined to be in a long-lived metastable state. Specifically, a Tsallis distribution is used for the one-particle energy distribution. It is characterized by the single parameter q, where $|q-1|$ is a measure of the departure from the Boltzmann-Gibbs distribution. (In liquid metals or the solar plasma, one can think of the velocity distribution of the ions, where $|1-q|$ measures the departure from Gaussian.) The corresponding entropy, $S_q$, satisfies $S_q\le S_1$ with equality if and only if $q=1$. Now $S_1$ is identified with the kinetic entropy, $S^{+}$, and $S_q$ with the total entropy $S^{+}-S^{-}$ (with q as a fitting parameter), so that $S_1-S_q$ represents the correlation entropy $S^{-}$, which is asserted to be the negentropy. That is, the Tsallis distribution represents the entropic effect of correlations without taking them into account explicitly, but rather through the single parameter q in the velocity distribution. But it appears to be taken more literally, as the third paragraph begins, “The negentropy can be written using several approximations and is a measure of the non-Gaussianity of the system [17,18] (Refs. in [1]).” If the actual velocity distribution conformed to this fitted Tsallis distribution, then indeed it would not be stable, and one could imagine it relaxing to a Gaussian. Of course, the relaxation would require some sort of interaction, if only between the ions of the medium and the fusing nuclei. But in order that the global equilibrium entropy be $S^{+}$ as claimed above, the interactions between the ions in this global equilibrium state would have to be weak enough for the correlation entropy to be negligible.

The problem with this scenario is that the Tsallis distribution described above is not close to the statistical distribution of the real system, which cannot be reduced to any one-particle distribution: Correlations are inescapable at liquid or plasma densities (keeping $S^{-}\ne 0$), because the ions cannot suspend their interactions, and the velocity distribution is much closer to a Gaussian than that of the fit (see paragraph below). The Tsallis distribution as used here represents a proxy system of noninteracting (or weakly interacting) particles placed initially in a Tsallis distribution (whose q is chosen to fit the total entropy $S^{+}-S^{-}$ of the real system). A good illustration that a Tsallis distribution with a chosen q (i) can indeed fit the total entropy; but (ii) misfits the actual distribution, consider two different analyses of melting entropies in real elements.

In the earlier (1991) analysis, Wallace (Ref. [56] in [1]) accounted for, and produced extensive tables of all of the important contributions to the melting entropies of 25 elements. This analysis made use of the earlier (1987) work [2] in which Wallace derived an expansion for the correlation contribution in the liquid state, showing that the ionic velocity distribution is a Gaussian, unaltered by the interactions in the liquid state. In the solid state, on the other hand, ionic motion is accounted for through the vibrational modes of the crystal lattice. This includes the effects of anharmonicity. Wallace found that most melting entropies fall into the range $0.63\le \Delta S\le 0.97$ (in k/at.), while a few were much larger ($1.48\le \Delta S\le 3.85$). He thus identified two groups, the 19 “normal elements” (all of them metals), and the remaining “anomalous elements,” whose melting entropies were attributed to changes in the electronic ground states. Most of the latter are nonmetals in their solid states, and metals in their liquid states.

A more recent (2013) analysis (Ref. [11] in [1]) by two authors of the present paper makes use of the Tsallis distribution for the liquid state, together with the canonical distribution (corresponding to $q=1$) in the solid state to fit the observed melting entropies, $\Delta S$. The parameters used in this fit ($\Delta S$, melting temperature and other parameters) were taken from Wallace’s numbers collected from many sources. This fit was done for a total of 18 elements (all included in Wallace’s analysis) with resulting q-values lying in the range of $q\approx$ 1.8 – 1.9. Furthermore, these values divide up according to the two groups—those near 1.9 representing the “normal elements,” and those near 1.8 the “anomalous elements.” The small spread (∼$2\%$ around each value) shows that q serves as a useful indicator. But the main point of relevance here is that all of the q values correspond to velocity distributions far from Maxwellian, and these are not representative of the actual distributions.

In conclusion, I do not argue with the starting definition given for negentropy, nor do I disagree that the negentropy concept might prove useful in purely physical systems. But I do argue that the calculated quantities reported in this paper do not conform to this starting definition, and they do not represent stored mobilizable energy available for spontaneous transfer to other subsystems. Moreover, the supporting arguments employing the Tsallis distribution, if correct, would imply that liquid metals, as described here and in the melting analysis [11], are not stable thermodynamic states.