Phase-space Rényi entropy, complexity and thermodynamic picture of density functional theory

Phase-space Rényi entropy and complexity are defined within the thermodynamic picture of density functional theory. The structural entropy defined by Pipek, Varga and Nagy, the LMC statistical complexity introduced by López-Ruiz, Mancini and Calbet and generalized complexity proposed by López-Ruiz, Nagy, Romera and Sanudo are extended to the phase space. It is shown that in case of constant local temperature the logarithm of the phase-space LMC complexity reduces to the position-space structural entropy defined by Pipek et al.

The structural entropy defined by Pipek, Varga and Nagy [48][49][50], the LMC statistical complexity introduced by López-Ruiz, Mancini and Calbet [3] and generalized complexity proposed by López-Ruiz, Nagy, Romera and Sanudo [9,10] are now extended to the phase space. It is shown that the logarithm of the phase-space LMC complexity reduces to the position-space structural entropy defined by Pipek et al. provided that constant local temperature is taken.

Rényi entropy and statistical complexity
Consider a D-dimensional density function f ( ) which is nonnegative and normalized to 1 ( ∫ f ( )d = 1 ). The Rényi entropy of order q is given by where stands for r 1 , ..., r D . Obviously, the limit q → 1 provides the Shannon entropy [51] The so-called LMC complexity [4,5] is defined by where and That is, it contains Rényi entropies of q → 1 and q = 2.
Replacing the Shannon entropy in Eq.(3) with the Rényi entropy of order q, a q-dependent measure of complexity is obtained [9]: f tends to the LMC complexity C LMC f in the limit q → 1. A further generalization leads to a two-parameter measure of complexity [10]: f . The generalized statistical complexity measure C (q 1 ,q 2 ) f has several important properties: inversion symmetry, monotonicity, universal bound, invariance under translations and rescaling transformations, near-continuity. These are inherent in the analysis based on Rényi entropies, therefore the localization map parameters of Pipek and coworkers already possess most of these properties. The generalized statistical complexity measure has been applied for studying different quantum systems [9,10].

Phase-space Rényi entropy and complexity
Phase-space Rényi entropy can be defined from the Wigner [52] and the Husimi [53] distribution functions. Phase-space Rényi entropy based on a special family of phasespace distribution functions has also been introduced and several theorems have been proved [29]. These distribution functions are nonnegative and produce the correct marginal distribution functions, that is, the integration of the distribution function with respect to the position variables leads to the momentum-space distribution function and the integration with respect to the momentum variables provides the position-space distribution function. We can remark here that the Wigner distribution function also yields the correct marginal distribution functions, but is not everywhere nonnegative. Wigner showed that bilinear distribution functions are not universally nonnegative. However, Cohen and Zaparovanny [54][55][56] proved that there exist distribution functions that are not bilinear but nonnegative and give the correct marginal distributions.
In this paper a phase-space distribution function derived by Ghosh, Berkowitz and Parr [33] is utilized. As we can see below, this distribution function provides only the correct marginal position-space distribution function. The marginal momentum-space distribution function differs from the correct one. Instead, in the Ghosh -Berkowitz -Parr (GBP) theory the kinetic energy density equals the correct one. Now the GBP theory [33] is summarized: an electronic system with density is taken and a phase-space distribution function g( , ) with the correct position-space marginal and the correct kinetic energy density t( ) is searched for. In DFT integrates to the number of electrons N: Obviously, the kinetic energy is given by GBP obtains g by maximizing the information with the conditions (10) and (11) (14) the well-known ideal gas expression is gained for the entropy. However, the kinetic energy density t( ) is not uniquely defined. Any term that integrates to zero can be added to the kinetic energy density resulting the same kinetic energy. Thus is not unique either. See Refs. [57][58][59] for the most frequently applied forms of t. Recently, another form has been proposed: the one for which phase-space Fisher/Shannon information is minimum/maximum [60,61]. It turned out that this condition provides a constant :

3
We add by passing that the derivation above is valid for excited states as well. Only the density and the kinetic energy density were used and these quantities can be excited-state density and kinetic energy density. We can observe that the derivation is valid both for interacting and non-interacting kinetic energy density. If the interacting kinetic energy is taken, the virial theorem can be utilized, i. e.
in Coulomb systems at equilibrium nuclear geometry, where E is the total energy. Then Eq. (18) can be rewritten as In the following instead of Eq. (17) the normalized phase-space distribution function is applied. Substituting f into Eq. (1) where is the density normalized to 1. is called shape function [62]. If q → 1 the phasespace Shannon entropy is obtained: On the other hand, Eq. (22) provides for q = 2.
If is constant the Rényi entropy can be written as is the position-space Rényi entropy for the normalized density and Using the kinetic energy instead of ( Eq. (18)) we are led to In a Coulomb system, using Eq. (20) Eq. (29) takes the form That is, the phase-space Rényi entropy is a sum of the position-space Rényi entropy and a term depending on the total energy and the order of q. Using Eq. (9) the logarithm of the generalized complexity takes the form where As a special case the phase-space generalization of the structural entropy is is obtained: If is constant we can apply Eq. (26) for the Rényi entropy, therefore It follows from Eq. (28) that disappears from lnC is the logarithm of the position-space generalized complexity. That is, the logarithm of the phase-space and the position-space generalized complexity measures differs only in a term depending only on q 1 and q 2 . Or, the phase-space generalized complexity is proportional to the position-space generalized complexity. If we take q 1 = 1 and q 2 = 2 , we obtain that the phase-space LMC complexity is proportional to the position-space LMC complexity. That is, the difference of the logarithm of the phase-space and the positionspace structural entropies is a constant : 3(1 − ln 2)∕2. This constant is due to the Gaussian form given in Eq. (15) being a direct consequence of the minimalization of Eq. (14) as Pipek and Varga has shown in their basic work [49].

Discussion
In DFT the density is the basic quantity determining the external potential, the Hamitonian, thus any property of the system under investigation. So the phase-space distribution function f is also determined by the density. In Coulomb systems f can be immediatelly given if the density is known in case of constant temperature. It is because the density decays as where E N−1 0 − E is the vertical ionization potential of the N-electron system. E is the energy of the state considered and E N−1 0 is the ground-state energy of the N − 1 electron system [63][64][65]. Thus, the asymptotic decay of the density decides the energy E and via the virial theorem. Therefore, (or ) determines f. Moreover, (or ) also determines the phase-space Rényi entropy as R (q) f is a sum of the position-space Rényi entropy R (q) that can be easily calculated from and the term A (q) that can be obtained from E [Eqs. (26, 27 and 30)]. In case of a Coulomb system these statements are true not only for the ground, but excited states as well [63][64][65].
In summary, phase-space Rényi entropy and complexity have been defined utilizing the thermodynamic transcription of density functional theory. Phase-space structural entropy, phase-space LMC and generalized complexity have been defined. It has been shown that the logarithm of the phase-space LMC complexity reduces to the position- Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission