Zentropy Versus the Subtraction Method of Temperature-Dependent Hartree-Fock

Zentropy Versus the Subtraction Method of Temperature-Dependent Hartree-Fock Francesco R. Ruggeri Hanwell, N.B. Nov. 12, 2025 In (1), it is noted that one may use the notion of zentropy to group over states linked in p(i), where p is probability. Shannon’s entropy (kb=1) is given by -Sum over i p(i) ln(p(i)). If for some reason there may be physical subsystems given by grouping various i values, then one may write: qk= probability for subsystem k = Sum over i (i’s in k) p(i). One may then obtain a set of qk’s for the various subsystems and calculate an entropy associated with not knowing in which subsystem a particle sits, but there also exist intrinsic entropies for each subsystem which use p(i)/qk for i within the subsystem k, so that each subsystem is normalized to 1. This then leads to the zentropy expression given in (1). In (2), we applied this approach to a quantum bound state based on x (position) as the variable i, and W*(x)W(x)=P(x) as the probability. We broke the supersystem into two subsystems, namely a classical bound region within classical turning points and a tunneling system outside, but this was based on spatial considerations alone. Here we note that it may be oversimplistic to define subsystems based on a single variable, namely x. In particular, we consider the temperature dependent Hartree-Fock analysis of a hot nucleus. This calculation leads to an overall Hartree-Fock wavefunction which contains the hot nucleus and the hot vapour outside. In order to find the properties of the hot nucleus, one does not simply create two systems based on space. In particular, some vapour particles may be in the hot nuclear region as quantum free particles have a wavefunction everywhere in space. Instead (3) suggests that one use a subtraction method. In other words, one performs two temperature-dependent Hartree Fock calculations, one for a hot nucleus with a vapour, and another simply for a vapour with the same temperature and subtracts according to the method in reference (5) in (3). (In (4) we have given a different method for subtraction based on phase shifts, but overall these cases show that it may be too simplistic to create subsystems based on a single parameter.) As a result, dealing with subsystems may be more complicated than the zentropy approach of (1) implies. Zentropy Zentropy as introduced in (1) considers p(i) probabilities of Shannon’s entropy, i.e. p is a probability and i a state. It is suggested that one may be able to group sets of i into subsystems which may have physical relevance. In such a case: qk = probability for subsystem k = Sum over i in k p(i) ((1)) Shannon’s entropy (kb=1): S= - Sum over i p(i) ln(pi)) ((2)) may then be written as: S (Shannon) = -Sum over k qk ln(qk) - Sum over k qk {Sum over i in k p(i)/qk ln(p(i)/qk) } ((3)) There is an entropy associated with not knowing in which subsystem a particle sits (first term on RHS of ((3))) and an average of intrinsic subsystem entropies (second term on RHS of ((3))). When calculating the intrinsic entropy of a subsystem, one must normalize the probability of each subsystem to 1, hence the use of p(i)/qk for i’s in k. In (2), we applied the above approach to a quantum one dimensional bound state, with q1 being the probability to find the particle within the classical turning points and 1-q1=q2, the probability to find it outside. Physically, however, a particle within the bound state may actually be beginning to tunnel and so it is not clear if it should be considered part of a bound subsystem.. Furthermore, a bound state is not necessarily marked strictly by the two classical turning points in quantum mechanics. Thus, separating space into the two regions and calling these two physical systems seems to be an approximation. It is possible that a more complicated approach may be necessary. To see this, one may consider that the thermodynamic treatment of a hot nucleus is often based on a subtraction method which goes beyond simply breaking space into bound and free regions. Temperature Dependent Hartree-Fock A temperature dependent Hartree-Fock approximation method is often used to find the single particle wavefunctions of a hot nucleus. The catch is that the overall temperature dependent HF solution includes vapour as well as nucleus material and the vapour material may be in the same spatial region as the bound single particle states. (The HF approach calculates single particle wavefunctions and some of these represent free particles.) In particular, a free quantum space exists in all of space, including the region within the nucleus space. Furthermore, a free particle may form a resonance with the nucleus and it is not clear if this should be considered part of the hot nucleus or the vapour. Thus, the idea of separating space into a vapour region and hot nucleus region is considered too simplistic in this case. What is done in many cases is to apply a subtraction method based on reference (5) in (3). This subtraction method involves a calculation of a hot nucleus with vapor (temperature-dependent Hartree-Fock calculation) and a second temperature-dependent Hartree Fock calculation with only a vapour present. The two solutions are subtracted, leaving presumably a hot nucleus with no vapour. One may then calculate the entropy and other thermodynamics of this hot nucleus without using zentropy ideas. In (4), we have presented an alternative subtraction approach which considers scattering off the nucleus (in the case of vapour present) and is based on phase shifts. This approach includes free particle resonances as part of a hot nucleus and so is different from that of reference (5) in (3). Either way, one cannot simply separate the supersystem into two physical subsystems using space as the sole separating variable. Thus, caution must be applied when using the zentropy approach, we argue. Conclusion In conclusion, a zentropy approach is considered in (1) and is applied to thermodynamics. The approach is based on the Shannon probability p(i), where i represents a state, and grouping states into subsystems, prersumably physical ones. One may then define the probability to have state k, qk by Sum over i’s in k p(i). From this one may consider an entropy linked to the qk’s (one does not know in which subsystem a particle sits) as well entropies for each subsystem, based on probabilities normalized to 1 for each subsystem, i.e. using p(i)/qk for i’s within k. This leads to: S(Shannon) = - Sum over i p(i) ln(p(i)) = - Sum over k qk ln(qk) - Sum over k qk { Sum over i’ s in k p(i)/qk ln(p(i)/qk) }. In (2) , we applied this to the quantum single particle 1-D bound case, considering the region within classical boundary points as the bound subsystem and the region outside as the second subsystem. Here, we argue that this approach may be too simplistic because in quantum mechanics, the bound region cannot be precisely defined by sharp classical turning points, we argue. Furthermore, some particles within the turning points may already be considered to be tunneling out or in and so not officially be considered as part of the bound system. Thus, matters are complicated. We point out the existence of the similar problem of a hot nucleus with a surrounding vapour. Such a solution follows from a temperature dependent Hartree-Fock calculation which yields single particle wavefunctions.. In such a case, one does not simply argue that there exists a finite spatial region which represents the hot nucleus and the rest of the region is then considered vapour. Vapour, i.e. free particle solutions in quantum mechanics exist in all of space, including within the nuclear region and must somehow be removed. This removal is often done by using a subtraction method presented in reference (5) of (3). In such a case, one performs two temperature dependent Hartree-Fock calculations, one for a hot nucleus and vapour and another for a vapour alone (i..e not interacting forces in the second calculation). One then subtracts the two solutions to obtain the hot nucleus by itself. In (4), we presented an alternative subtraction method which includes resonance of a free particle as being part of the hot nucleus. This calculation is based on the idea of phase shifts and differs from that of reference (5) in (3). Nevertheless, both approaches show that it is too simplistic to simply create subsystems by separating regions of space. Thus, one cannot apply the zentropy approach of (1) (at least based on a single variable) to the hot nucleus case, and caution is needed when using zentropy, we argue. References 1.Myers, L., Hew, N, Shang, S. and Liu, Z. Recursive Entropy in Thermodynamics: Establishing the Statistical Physics of Zentropy Approach (2025) https://www.semanticscholar.org/reader/6099766d6f6a998fd60ac414b17be77ac437a0d 4 2. Ruggeri, Francesco R. Application of Zentropy to the Quantum Bound State (preprint, zenodo, 2025) 3. Zhang, Z. et al Symmetry Energy of Hot Nuclei in the Relativistic Thomas-Fermi approximation (2014) https://arxiv.org/pdf/1411.1584 4. Ruggeri, Franco thesis (physics department, U. of Wisconsin-Madison) 1996