Extreme Physical Information Principle Research Articles

The Separability Problem in Molecular Quantum Systems: Information-Theoretic Framework for Atoms in Molecules.

Even though molecules are fundamentally quantum entities, the concept of a molecule retains certain classical attributes concerning its constituents. This includes the empirical separability of a molecule into its three-dimensional, rigid structure in Euclidean space, a framework often obtained through experimental methods like X-Ray crystallography. In this work, we delve into the mathematical implications of partitioning a molecule into its constituent parts using the widely recognized Atoms-In-Molecules (AIM) schemes, aiming to establish their validity within the framework of Information Theory concepts. We have uncovered information-theoretical justifications for employing some of the most prevalent AIM schemes in the field of Chemistry, including Hirshfeld (stockholder partitioning), Bader's (topological dissection), and the quantum approach (Hilbert's space definition). In the first approach we have applied the generalized principle of minimum relative entropy derived from the Sharma-Mittal two-parameter functional, avoiding the need for an arbitrary selection of reference promolecular atoms. Within the ambit of topological-information partitioning, we have demonstrated that the Fisher information of Bader's atoms conform to a comprehensive theory based on the Principle of Extreme Physical Information avoiding the need of employing the Schwinger's principle, which has been proven to be problematic. For the quantum approach we have presented information-theoretic justifications for conducting Löwdin symmetric transformations on the density matrix to form atomic Hilbert spaces generating orthonormal atomic orbitals with maximum occupancy for a given wavefunction.

Viruses as Living Systems—A Metacybernetic View

The debate over whether viruses are living organisms tends to be paradigmatically determined. The metabolic paradigm denies that they are, while new research evidences the opposite. The purpose of this paper is to deliver a generic model for viral contexts that explains why viruses are alive. It will take a systems biology approach, with a qualitative part (using metacybernetics) to provide deeper explanations of viral contexts, and a quantitative part (using Fisher Information deriving from the variational principle of Extreme Physical Information) which is in principle able to take measurements and predict outcomes. The modelling process provides an extended view of the epigenetic processes of viruses. The generic systems biology model will depict viruses as autonomous entities with metaphysical processes of autopoietic self-organisation and adaptation, enabling them to maintain their physical viability and hence, within their populations, mutate and evolve. The autopoietic epigenetic processes are shown to describe their capability to change, and these are both qualitatively and quantitatively explored, the latter providing an approach to make measurements of physical phenomena under uncertainty. Viruses maintain their fitness when they are able to maintain their stability, and this is indicated by information flow efficacy. A brief case study is presented on the COVID-19 virus from the perspective that it is a living system, and this includes outcome predictions given Fisher Information conditions for known contexts.

Ion-Based Cellular Signal Transmission, Principles of Minimum Information Loss, and Evolution by Natural Selection

The Extreme Physical Information EPI principle states that maximum information transmission or, equivalently, a minimum information loss is a fundamental property of nature. Prior work has demonstrated the universal EPI principle allows derivation of nearly all physical laws. Here, we investigate whether EPI can similarly give rise to the fundamental law of life: Evolution. Living systems require information to survive and proliferate. Heritable information in the genome encodes the structure and function of cellular macromolecules but this information remains fixed over time. In contrast, a cell must rapidly and continuously access, analyze, and respond to a wide range of continuously changing spatial and temporal information in the environment. We propose these two information dynamics are linked because the genes encode the structure of the macromolecules that form information conduits necessary for the dynamical interactions with the external environment. However, because the genome does not have the capacity to precisely locate the time and location of external signals, we propose the cell membrane is the site at which most external information is received and processed. In our model, an external signal is detected by gates on transmembrane ion channel and transmitted into the cytoplasm through ions that flow along pre-existing concentration gradients when the gate opens. The resulting cytoplasmic ion “puff” is localized in both time and space, thus producing spatial and temporal information. Small, localized signals in the cytoplasm are “processed” through alterations in the function and location of peripheral membrane proteins. Larger perturbations produce prolonged or spatially extensive changes in cytoplasmic ion concentrations that can be transmitted to other organelles via ion flows along elements of the cytoskeleton. An evolutionary constraint to the ever-increasing acquisition of environmental information is the cost of doing so. One solution to this trade-off is the evolution of information conduits that minimize signal loss during transmission. Since the structures of these conduits are encoded in the genome, evolution of macromolecular conduits that minimize signal loss is linked to and, in fact, governed by a universal principle, termed extreme physical information (EPI). Mathematical analysis of information dynamics based on the flow of ions through membrane channels and along wire-like cytoskeleton macromolecules fulfills the EPI principle. Thus, the empirically derived model of evolution by natural selection, although uniquely applicable to living systems, is theoretically grounded in a universal principle that can also be used to derive the laws of physics. Finally, if minimization of signal loss is a mechanism to overcome energy constraints, the model predicts increasing information and associated complexity are closely linked to increased efficiency of energy production or improved substrate acquisition.

Generalization of the Aoki–Yoshikawa sectoral productivity model based on extreme physical information principle

This paper presents a continuous variable generalization of the Aoki–Yoshikawa sectoral productivity model. Information theoretical methods from the Frieden–Soffer extreme physical information statistical estimation methodology were used to construct exact solutions. Both approaches coincide in first order approximation. The approach proposed here can be successfully applied in other fields of research.

Extending the Extreme Physical Information to Universal Cognitive Models via a Confident Information First Principle

The principle of extreme physical information (EPI) can be used to derive many known laws and distributions in theoretical physics by extremizing the physical information loss K, i.e., the difference between the observed Fisher information I and the intrinsic information bound J of the physical phenomenon being measured. However, for complex cognitive systems of high dimensionality (e.g., human language processing and image recognition), the information bound J could be excessively larger than I (J ≫ I), due to insufficient observation, which would lead to serious over-fitting problems in the derivation of cognitive models. Moreover, there is a lack of an established exact invariance principle that gives rise to the bound information in universal cognitive systems. This limits the direct application of EPI. To narrow down the gap between I and J, in this paper, we propose a confident-information-first (CIF) principle to lower the information bound J by preserving confident parameters and ruling out unreliable or noisy parameters in the probability density function being measured. The confidence of each parameter can be assessed by its contribution to the expected Fisher information distance between the physical phenomenon and its observations. In addition, given a specific parametric representation, this contribution can often be directly assessed by the Fisher information, which establishes a connection with the inverse variance of any unbiased estimate for the parameter via the Cramér–Rao bound. We then consider the dimensionality reduction in the parameter spaces of binary multivariate distributions. We show that the single-layer Boltzmann machine without hidden units (SBM) can be derived using the CIF principle. An illustrative experiment is conducted to show how the CIF principle improves the density estimation performance.

Principle of maximum Fisher information from Hardy's axioms applied to statistical systems

Consider a finite-sized, multidimensional system in parameter state a. The system is either at statistical equilibrium or general nonequilibrium, and may obey either classical or quantum physics. L. Hardy's mathematical axioms provide a basis for the physics obeyed by any such system. One axiom is that the number N of distinguishable states a in the system obeys N=max. This assumes that N is known as deterministic prior knowledge. However, most observed systems suffer statistical fluctuations, for which N is therefore only known approximately. Then what happens if the scope of the axiom N=max is extended to include such observed systems? It is found that the state a of the system must obey a principle of maximum Fisher information, I=I(max). This is important because many physical laws have been derived, assuming as a working hypothesis that I=I(max). These derivations include uses of the principle of extreme physical information (EPI). Examples of such derivations were of the De Broglie wave hypothesis, quantum wave equations, Maxwell's equations, new laws of biology (e.g., of Coulomb force-directed cell development and of in situ cancer growth), and new laws of economic fluctuation and investment. That the principle I=I(max) itself derives from suitably extended Hardy axioms thereby eliminates its need to be assumed in these derivations. Thus, uses of I=I(max) and EPI express physics at its most fundamental level, its axiomatic basis in math.

Information in hierarchical modular system

This paper presents a study on the hierarchical modular system (HMS). It shows how to generate the HMS by using the principle of extreme physical information. It also discusses the equal temperament music scale that has many shared properties with the HMS.

GRAVITY AND THE COMPLEXITY OF COORDINATES IN FISHER INFORMATION

The principle of extreme physical information is a tool for deriving various equations of physics. Physical information is defined to be the difference between the Fisher information and the bound information specific to every physical system. Using the extreme physical information, equations are obtained for the probability of finding the location of a physical entity (a system of particles or a field) in space–time. Those equations turn out to be the relevant physical equations (Dirac's, Maxwell's equations etc.). A peculiarity of the method is that this only works if some of the coordinates that specify the whereabout of the physical entity in space–time are chosen to be imaginary. I will argue that this peculiarity is a direct effect of gravity.

Restoration of a Poisson distributed image using a principle of extreme physical information

Suppose that an object which reflects or emits light is captured by a sensor. The image data are distorted in various ways. The aim is to restore the object. For this purpose, a new restoration method is devised. It uses the Fisher-information based principle of extreme physical information (EPI). This solves a variational problem I − J = extremum, expressing a flow of information J → I from source to output in the restoration problem. Here, the source of information is the Poissonian image data, and J is modeled as its Hartley information level. Also, I is the Fisher information in the output restored object. The approach dates from the 1980s. It has been already shown that many fundamental physical laws may be derived using EPI. In this paper, the EPI approach is applied to the restoration problem, wherein it regards the unknown restoration as another fundamental law. It models the source information J as obeying Poisson statistics, and assumes additional image distortions that are due to additive Gaussian noise and quantization of the image data. It is found that the quantization level determines the degree of smoothness (or sharpness) of the restored signal. An optimal level of image quantization can be accomplished by adjusting its cross entropy. Finally, the restoration method by EPI is validated by numerical simulations of the image forming and restoring system.

De Broglie’s wave hypothesis from Fisher information

Seeking the unknown dynamics obeyed by a particle gives rise to the de Broglie wave representation, without the need for physical assumptions specific to quantum mechanics. The only required physical assumption is conservation of momentum μ . The particle, of mass m , moves through free space from an unknown source-plane position a to an unknown coordinate x in an aperture plane of unknown probability density p X ( x ) , and then to an output plane of observed position y = a + z . There is no prior knowledge of the probability laws p ( a , M ) , p ( a ) or p ( M ) , with M the particle momentum at the source. It is desired to (i) optimally estimate a , in the sense of a maximum likelihood (ML) estimate. The estimate is further optimized, by minimizing its error through (ii) maximizing the Fisher information about a that is received at y . Forming the ML estimate requires (iii) estimation of the likelihood law p Z ( z ) , which (iv) must obey positivity. The relation p Z ( z ) ≡ | u ( z ) | 2 ≥ 0 satisfies this. The same u ( z ) conveniently defines the Fisher channel capacity, a concept central to the principle of Extreme physical information (EPI). Its output u ( z ) achieves aims (i)–(iv). The output is parametrized by a free parameter K . For a choice K = 0 , the result is u ( z ) = δ ( z ) , indicating classical motion. Or, for a finite, empirical choice K = ħ (Planck’s constant), u ( z ) obeys the familiar de Broglie representation as the Fourier transform of the particle’s probability amplitude function P ( μ ) on momentum μ . For a definite momentum μ , u ( z ) becomes a sinusoid of wavelength λ = h / μ , the de Broglie result.

Use of Fisher information in quantum chemistry

AbstractThe classical Fisher information concept is generalized to cover the complex probability amplitudes (wave functions) of quantum mechanics. Its contributions due to the probability and probability‐current densities are identified. General properties of this local information measure are examined and the Schrödinger functional for the kinetic energy is interpreted as the average Fisher information. The information current and source densities are introduced in terms of which the relevant continuity equation is formulated, expressing the local balance of the information content. The superposition principles for the particle probabilities, probability currents, and the Fisher information densities are examined. The interference (nonadditive) contributions to these quantities due to the quantum mechanical mixing of individual states are identified. An illustrative application of using the nonadditive Fisher information of the atomic orbital (AO) resolution in indexing the chemical bond within the 2‐AO model is presented. This AO‐phase sensitive index is capable of distinguishing the bonding, nonbonding, and antibonding electronic states, as do the familiar bond‐orders of quantum chemistry. It is interpreted as a measure of an extra delocalization of electrons due to the bond formation. This is in contrast to the so called quadratic bond‐multiplicity measures and the nonprojected entropy/information concepts of the communication theory of the chemical bond, which both loose the memory about the relative AO‐phases in molecular orbitals, thus failing to distinguish between the bonding and antibonding electron configurations. Finally, the Schrödinger variational principle of quantum mechanics and the related Kohn‐Sham principle of Density‐Functional theory are interpreted as constrained variational principles of the Fisher information contained in the electron probability distributions. They are examples of the Extreme Physical Information principle related to the measurement process. The Fisher‐information principle for the adiabatic (Born‐Oppenheimer) approximation of the molecular quantum mechanics is also examined. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

Information principles in the theory of electronic structure

An alternative information-theoretic derivation of the stockholder principle is given, from the minimum principle of the local Kullback–Leibler missing information relative to the promolecule reference, in terms of the local conditional probabilities of finding an electron on molecular subsystems, and the asymptotic behavior of the Hirshfeld bonded atoms is examined. The extreme physical information principle of Frieden using the Fisher information measure is summarized and applied to derive the Kohn–Sham equations.

Population Genetics from an Information Perspective

Some basic effects of population genetics are derived governing the occurrences of allelesAi and genotypes AiAjamong its members. A principle ofextreme physical information (EPI) is used. These effects are (1) the equation of genetic change, (2) Fisher's theorem of partial change, (3) a new uncertainty principle, and (4) the monotonic decrease of Fisher information with time, indicating increased disorder for the population. General conditions of population change are allowed: fitness coefficientswij generally changing with time [except in effect (2)], population randomly or non-randomly mating, and a general number of loci present within each chromosome. EPI is a practical tool for deriving probability laws. It is an outgrowth of a physical process that occurs during any act of measurement. Here the measurement is the random observation of a genotype AiAj. This observation is to be used to estimate the time of the observation, called “evolutionary time”. The measurement activity incurs errors in the estimated observation time and fitness value of the observed genotype. By the Cramer–Rao inequality, the product of the two uncertainties must exceed unity [effect (3)]. The Fisher information I in data space is postulated to originate in the space of the genotype where it had some generally larger value J. The EPI principle extremizes the loss of information (I–J) withI =1/2 J. The solution gives rise to effects (1) and (2). Finally, it is shown that effect (4) holds when the population approaches an equilibrium state, e.g. for time values greater than a threshold if fitness coefficientswij are constant. EPI provides a common framework for deriving physical laws and laws of population genetics. The new effects (3) and (4) are confirmed through computer simulation.

A CRITICAL COMPARISON OF THREE INFORMATION-BASED APPROACHES TO PHYSICS

Many of the laws of physics are expressions that define probability distributions. These laws may be derived through variation of appropriate Lagrangians. We compare and contrast three Lagrangian approaches which are based on information-theoretic considerations: the Maximum Entropy (ME) principle, the Minimum Fisher Information (MFI) approach and the principle of Extreme Physical Information (EPI). (The latter also produces independent solutions by zeroing as well as varying the Lagrangian.) Though superficially similar, these three methods are markedly different in their world views and applicability to physics. Only the EPI principle applies broadly to all of physics, and we show that this is reasonable on the following grounds: Physics should not, depend upon arbitrary subjective choices, but ME and MFI, both intrinsically Bayesian approaches, reqiure the choice of arbitrary, subjectively defined inputs, such as prior probability laws and input constraints, for their implementation. EPI, in contrast, solves for its effective constraints, needs no prior distribution assumption and, hence, does not require any arbitrary subjective inputs.

The Frieden–Soffer extreme physical information principle in a non-extensive setting

We show that the Frieden and Soffer's Extreme Physical Information principle, applied to a non-extensive statistical scenario, yields solutions to several well-known classical dynamical problems.

Fisher information and semiclassical methods

Using the Frieden and Soffer principle of extreme physical information we present a method for the construction of WKB wave functions that avoids the matching problem between the classically allowed and forbidden regions. When this formalism is applied in conjunction with the semiclassical methods for N-fermion systems, it provides one with Thomas-Fermi and Wigner-Kirkwood densities, valid beyond the turning points that also include shell effects. The method is illustrated with reference to both the sextic bistable and the Morse potential.