Energy and information are fundamental concepts that play crucial roles in the foundations of science, each contributing to different aspects of our understanding of the natural world. Here we encounter that Fisher' measure is able to link them. We also investigate some interesting relations between Fisher's information, Poisson's distribution, and thermal quantifiers in a grand canonical ensemble. These connections permit one to visualize interesting novel artifacts in statistical mechanics. We clearly see them emerge in discussing the ideal gas. For example, we encounter that the Poisson distribution is intrinsically involved in the physics of the ideal gas, and we are able to give a Fisher-estimation of the difference in informational content between the grand canonical and canonical ensembles.
Energy and information are fundamental concepts that play crucial roles in the foundations of science, each contributing to different aspects of our understanding of the natural world. Here we encounter that Fisher' measure is able to link them. We also investigate some interesting relations between Fisher's information, Poisson's distribution, and thermal quantifiers in a grand canonical ensemble. These connections permit one to visualize interesting novel artifacts in statistical mechanics. We clearly see them emerge in discussing the ideal gas. For example, we encounter that the Poisson distribution is intrinsically involved in the physics of the ideal gas, and we are able to give a Fisher-estimation of the difference in informational content between the grand canonical and canonical ensembles.
Beginning with Mandelbrot’s insight that Fisher information may admit a thermodynamic interpretation, a growing body of work has connected this information-theoretic measure to fluctuation–dissipation relations, thermodynamic geometry, and phase transitions. Yet, these connections have largely remained at the level of formal analogies. In this work, we provide what is, to our knowledge, the first explicit realization of the epistemic-to-physical transition of Fisher information within a finite interacting quantum system. Specifically, we analyze a model of N fermions occupying two degenerate levels and coupled by a spin-flip interaction of strength V, treated in the grand canonical ensemble at inverse temperature β. We compute the Fisher information FN(V) associated with the sensitivity of the thermal state to changes in V, and show that it becomes an observer-independent, experimentally meaningful quantity: it encodes fluctuations, tracks entropy variations, and reveals structural transitions induced by interactions. Our findings thus demonstrate that Fisher information, originally conceived as an inferential and epistemic measure, can operate as a bona fide thermodynamic observable in quantum many-body physics, bridging the gap between information-theoretic foundations and measurable physical law.
We show that the mathematical form of the information measure of Fisher's I for a Gibbs' canonical probability distribution (the most important one in statistical mechanics) incorporates important features of the intrinsic structure of classical mechanics and has a universal form in terms of "forces" and "accelerations", i.e., one that is valid for all Hamiltonian of the form T+V. If the system of differential equations associated to Hamilton's canonical equations of motion is linear, one can easily ascertain that the Fisher information per degree of freedom is proportional to the inverse temperature and to the number of these degrees. This equipartition of I is also seen to hold in a simple example involving a non-linear system of differential equations.
We derive the quantum Fisher information for entropy estimation in a Gibbs state and show that it equals the inverse of the heat capacity, which is dual to the temperature Fisher information given by the heat capacity divided by the square of the temperature. Their product is independent of the Hamiltonian and depends only on the temperature, leading to a metrological uncertainty relation between the variances of entropy and temperature estimators in which all system-specific quantities cancel. This relation arises from the dually-flat structure of the Gibbs exponential family expressed in thermodynamic coordinates, and holds for all standard thermodynamically conjugate pairs. We identify energy measurement as the optimal protocol for entropy estimation, analyse critical-point scaling where the entropy Fisher information vanishes, and connect it to the Ruppeiner metric in entropy coordinates. We lastly examine the distinguished role of the von Neumann entropy within the Rényi family. Generalisations to the grand canonical and generalised Gibbs ensembles are given.
There are several links between information theory — between »information as such« whatever that might mean — and the foundations of nature, i e, basic physics. Well, since physics a science still evolving (despite propositions that »the end of science« now in sight), some of the links between information and physics are hypothetical. One linkage fairly obvious — thermodynamics depends upon entropy or some measure of order or disorder, and that clearly has something to do with information. The other possible relationship might be between the uncertainty that a key characteristic of quantum mechanics. We can never know everything on the quantum scale; the Heisenberg uncertainty principle tells that the more information we gain in one direction the less we know in another. If we, on that scale, obtain certainty as to one factor, then another will stay entirely undecided — try to affix a position to a quantum particle, and the momentum will be out of bounds entirely. (This the established dogma currently, though there are some deviating ideas.1) Then the question arises: who does the actual measuring, with what — what particles, at what level, are involved in this act of, as it pronounced, forcing the wave function to collapse? This wave function is the indeterminate particle: the various statistics describing the probabilities that the particle (which simultaneously takes on aspects of a wave) may be in one state or another. We will now attempt to discuss these different — or are they really different? — interdependencies, starting with black holes.
A suggestive relation links Fisher' information measure (FIM)Iand Schrödinger equation (SE). The connection is based upon the fact that the constrained minimization of I leads to a SE. This, in turn, is the origin of intriguing relationships between various aspects of SE, on the one hand, and the formalism of statistical mechanics derived from Jaynes's maximum entropy principle (MaxEnt), on the other one. The link entails the existence of a Legendre transform structure underlying the SE, which allows for the emergence of two first-order differential equations that must, respectively, be satisfied by i) the Fisher measure and ii) the SE energy eigenvalues. The complete A) I-solution and B) energy-solution are both obtained bypassing the SE and, furthermore, linked by the Legendre structure.
We review here the difference between quantum statistical treatments and semiclassical ones, using as the main concomitant tool a semiclassical, shift-invariant Fisher information measure built up with Husimi distributions. Its semiclassical character notwithstanding, this measure also contains abundant information of a purely quantal nature. Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies and to discover thermodynamic-like relations that involve the degree of delocalization. Fisher-related thermal uncertainty relations are developed and the degree of purity of canonical distributions, regarded as mixed states, is connected to this Fisher measure as well.