The influence of correlations of uniform Fermi systems (nuclear matter, electron gas and liquid 3He) on Shannon's information entropy, S, is studied. It is found that, for three different Fermi systems with different particle interactions, the correlated part of S (Soor) depends on the correlation parameter of the systems or on the discontinuity gap of the momentum distribution through two parameter expressions. The values of the parameters characterize the strength of the correlations. A two parameter expression also holds between Scor and the mean kinetic energy (K) of the Fermi system. The study of thermal effects on the uncorrelated electron gas leads to a relation between the thermal part of S (Sthermai) and the fundamental quantities of temperature, thermodynamical entropy and the mean kinetic energy. It is found that, in the case of low temperature limit, the expression connecting Sthermai with Κ is the same to the one which connects Scor with K. Thus, regardless of the reason (correlations or thermal) that changes K, S takes almost the same value.
The influence of correlations of uniform Fermi systems (nuclear matter, electron gas, and liquid
3He) on Shannon's information entropy,
S, is studied.
S is the sum of the information entropies in position and momentum spaces. It is found that, for three different Fermi systems with different particle interactions, the correlated part of
S (Scor) depends on the correlation parameter of the systems or on the discontinuity gap of the momentum distribution through two parameter expressions. The values of the parameters characterize the strength of the correlations. A two parameter expression also holds between
Scor and the mean kinetic energy
(K) of the Fermi system. The study of thermal effects on the uncorrelated electron gas leads to a relation between the thermal part of
S (Sthermal) and the fundamental quantities of temperature, thermodynamical entropy, and the mean kinetic energy. It is found that, in the case of low temperature limit, the expression connecting
Sthermal with
K is the same to the one which connects
Scor with
K. There are only some small differences on the values of the parameters. Thus, regardless of the reason (correlations or thermal) that changes
K,
S takes almost the same value.
In this paper, a three parameter (α, λ, μ) ‘Hartree and Ingman (1933)’ type correlated wave function has been used to derive an analytical model to quantify the two electron systems. The model is used to construct the expressions for the single particle wave functions [ 𝜓(𝑟) , 𝜙(𝑝)] in coordinate space and momentum space respectively with the subscripts marked as correlated (c) and uncorrelated (uc) ones. Further, the correlated and uncorrelated wave functions are used to form the single particle charge densities [𝜌(𝑟) , 𝛾(𝑝)] in coordinate space and momentum space. These wave functions and the single particle charge densities are used to compute the numerical values of the Shannon (S) and Fisher (I) information entropies for both the correlated and uncorrelated systems in coordinate space and momentum spaces and furthermore to examine their correlation effects in coordinate and momentum spaces. The uncorrelated and correlated values of Shannon information entropies indicate the reciprocity between the coordinate and momentum spaces such that low values of the Shannon information entropy in coordinate space ( 𝑆𝜌), are associated with the high values of Shannon information entropy in momentum space (𝑆𝛾) . The uncorrelated and correlated values of Shannon information entropies (S) also validate the entropic uncertainty relations Bialynicki-Birula and Mycielski (BBM) inequality which reads as (𝑆𝜌+𝑆𝛾)≥3(1+𝑙𝑛𝜋), as well as another stronger version of the uncertainty relation that is (𝑆𝜌𝑐+𝑆𝛾𝑐)>(𝑆𝜌𝑢𝑐+𝑆𝛾𝑢𝑐). Further, both the products of uncorrelated and correlated values of Fisher information entropies 𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 and 𝐼𝜌𝑐𝐼𝛾𝑐 satisfy the inequality condition 𝐼𝜌𝑐𝐼𝛾𝑐>𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 in coordinate and momentum spaces. At the same time these products also satisfy the Fisher based uncertainty relation, 𝐼𝜌𝐼𝛾≥36. The consistency of the results satisfying the uncertainty relations can be checked from the data demonstrated in the Table 1 and Table 2.
We consider a nearly (or quasi) uniform gas of interacting electrons for which spin statistics play a crucial role. A previously developed procedure, based on the extension of the Levy-Lieb constrained search principle and Monte Carlo sampling of electron configurations in space, allows us to approximate the form of the kinetic-energy functional. For a spinless electron gas, this procedure led to a correlation term, which had the form of the Shannon entropy, but the resulting kinetic-energy functional does not satisfy the Lieb-Thirring inequality, which is rigorous and one of the most general relations regarding the kinetic energy. In this paper, we show that when the fermionic character of the electrons is included via a statistical spin approach, our procedure leads to correlation terms, which also have the form of the Shannon entropy and the resulting kinetic-energy functional does satisfy the Lieb-Thirring inequality. In this way we further strengthen the connection between Shannon entropy and electron correlation and, more generally, between information theory and quantum mechanics.
We will study fermionic systems like atomic nuclei and bosonic systems like the correlated atoms in a trap from an information-theoretical point of view. The Shannon and Onicescu information measures are calculated for the above systems by comparing the correlated and uncorrelated cases as functions of the strength of the short range correlations. One-body and two-body density and momentum distributions are employed. Thus, the effect of short-range correlations on the information content is evaluated. The magnitude of distinguishability between the correlated and uncorrelated densities is also discussed employing suitable measures for the distance of states i.e. the well known Kullback–Leibler relative entropy and the recently proposed Jensen–Shannon divergence entropy. We will see that the same information-theoretical properties hold for quantum many-body systems obeying Bose–Einstein and Fermi–Dirac (statistics).
The universal property for the information entropy S = a + h In Ζ is verified for atoms using a systematic study with Roothaan-Hartree-Fock (RHF) wave functions with atomic number Ζ — 2 — 54. The above relation was proposed previously for atoms, nuclei, atomic clusters and correlated atoms in a trap. Kullback-Leibler relative entropy Κ and Jensen-Shannon divergence J are employed to compare RHF with Thomas-Fermi (TF) density of atoms as well as another phenomenological density obtained by Sagar et al. Two-body density distributions in position- and momentum-space are used to calculate and compare the corresponding information entropies for correlated and uncorrelated nuclei and bosonic systems (correlated atoms in a trap). It is seen that short-range correlations (SRC) increase the values of S. One-body information entropy entropy S\ is compared with two-body information entropy and a conjecture is made for TV-body information entropy SN- The entropy Κ and the divergence J are also used to evaluate the information distance between correlated and uncorrelated densities both at the one- and the two-body levels for nuclei and trapped Bose gases.
In this paper, we have presented an analytical model of two electron systems consisting of a many particle correlated wave function with some variational parameters α, λ and µ and used it to quantify the electron-electron correlation described by the wave function containing explicitly r12 (inter atomic distance between two electrons) dependent term. The single particle wave functions and the charge densities have been extracted from the said correlated wave function both for the uncorrelated and correlated systems in coordinate space and its momentum analogs have been obtained by taking the Fourier transform of the coordinate analogs. We have computed and presented the results of the numerical values of the theoretic information entropies of the Shannon entropy, Fisher information entropy, Shannon power and the FisherShannon product. The numerical values are consistently found to satisfy the Beckner, Bialynicki-Birula and Mycielski (BBM ) inequality relation; Stam-Cramer-Rao inequalities or Fisher based uncertainty relation and Fisher-Shannon product relation for the uncorrelated and correlated systems in both the coordinate and momentum spaces.