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 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.
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.
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.
Consider an isolated statistical system specified by a coordinate x and its probability density p(x). A functional of p(x) called ``Fisher information'' can be used to measure the degree of disorder of the system due to the spread in p(x). Fisher information may be minimized, subject to a physical constraint, to attain a temporal equilibrium solution p(x). When the constraint is linear in the mean kinetic energy of the system, the equilibrium solution p(x) often obeys the correct differential equation for the system. In this way, the Schr\"odinger (energy) wave equation, Klein-Gordon equation, Helmholtz wave equation, diffusion equation, Boltzmann law, and Maxwell-Boltzmann law may be derived from one classical principle of disorder. The convergence rate for Fisher information is about that for alternative use of maximum entropy (in problems where both have the same equilibrium solution). This suggests that Fisher information defines an arrow of time. The arrow points in the direction of decreasing accuracy for the determination of the mean, or ideal, value of a parameter.
In (1), a variational approach is applied to Fisher’s information (with constraints) to obtain a differential equation in P(x)=probability to be at x. It is suggested the equation may be simplified using P(x)=W(x)W(x) where W(x) is real. The result is the nonrelativistic Schrodinger equation with mass=1. As a first observation, we note that Fisher’s information, which is a purely statistical quantity, should not necessarily yield a nonrelativistic equation which is an approximation of relativistic results. In particular, we argue that Fisher’s information is used to find <pp> not necessarily 1/2m <pp>, but this is a minor point. A more important issue, we argue, is the meaning of P(x) in a bound system. A classical bound system is sometimes said to have a P(x)=C/v(x) i.e. P(x)dx is proportional to dt, the amount of time a particle spends in each equally sized dx. In the high energy limit, the envelope of a quantum bound probability P(x)=W(x)W(x) (W(x)=wavefunction) is said to roughly match C/v(x), yet this density does not follow from the extremization of Fisher’s information (together with two general constraints which yield E and V(x)=potential). This begs the question: What is P(x)? If one considers a particle moving at constant speed (i.e constant momentum) then P(x)=constant because each dx point carries the same probability. Classically a particle placed in a box with infinite potential walls would have a P(x) =constant and p and -p motion would be separated in time. Thus if P(x) is to have meaning in a box, one must conclude that the probabilities associated with p and -p are not constant, but rather functions of x. This then leads to the notion of a complex new kind of probability (which maps to P(x)=constant) for a free particle i.e. W(x) =cos(px) + i sin(px) so that the modulus W*(x)W(x)=P(x)= 1. These ideas seem to go beyond the notion of Fisher information. What then is the role of Fisher information? We argue it is related to two very specific moments of the distribution cos(px) or rather W(x) = Sum over p a(p)cos(px) with a(p)=a(-p), namely and <pp>. In classical physics there exists p and pp at each x point (but one is the square of the other), but with distributions <p does not equal to <pp> in general. Both and <pp> are linked to physical concepts which exist in Newtonian mechanics, namely impulse and work. With the presence of cos(px) one has a combination of Newtonian mechanics and a statistical distribution. We argue that the role of Fisher’s information is to allow one to transform from to <pp>. In particular, -idW/dx appears in Fisher’s information (and is linked to ), but the specific role of a variation is to transform it into <pp> which appears in a conservation of energy equation. We argue that this is why Fisher’s information may be used to derive the Schrodinger equation.
A process exists in the literature (1) of creating an information term based on Fisher’s information using spatial density P and adding it to a classical Lagrangian based on a velocity field i.e. velocity=dS/dx. Maximization of S and P is performed and a differential equation in terms of spatial density and S is obtained. Finally, it is noted that a change of variables: W= sqrt(p) exp(iS/hbar) may simplify the differential equation so that it becomes the Schrodinger equation with W as the wavefunction. In this note, we consider the wavefunction as appearing “naturally” as a normalization constant when using conditional probability and argue that W is the velocity field. Given that the Schrodinger equation includes a term of the form -1/2m d/dx d/dx W (in 1-D) i.e. the first derivative is related to the average velocity and the second to the ensemble average of p*p. This follows because unnormalized conditional probability f(p,x) is the Fourier basis function i.e. exp(ipx). The second derivative term - 1/2m d/dx d/dx W may be obtained from a Lagrangian containing (d/dxW) (d/dx W), thus formally one may use maximization to obtain the Schrodinger equation at a level using W directly. The maximization form, however, is identical to the “classical translational” form (2) of Fisher’s information. Thus, it seems that the Fourier basis exp(ipx) and the maximization of classical translational Fisher’s information are linked. In this note, we wish to investigate this in further detail. We also use translational properties together with conditional probability to show that f(p,x) where P(p/x)= a(p) f(p/x) / [Sum over p a(p) f(p/x)] is exp(ipx). In particular, we obtain an equation taken from the translation of the conservation of energy equation at point x to x+delta x. This links df(p,x)/dx with -dV/dx (the force). If the force is stochastic and yields “k” momentum hits, -dV/dx= Sum over k kVk(x). We argue that df(p,x)/dx also brings out this k term i.e. is an eigenfunction of k i.e. -idf(p,x)/dx =k f(p,x) yields exp(ipx) as a solution. This f(p,x) is automatically an orthonormal basis (multiplied by an appropriate constant) and helps to show why the “momentum operator” is linked to a “translation operator” i.e. d/dx. The bound state Schrodinger equation is considered in this note.
Given a coin, one has two complementary pieces of information and a corresponding probability of .5 and .5. Similar considerations apply to a die. As a result, one may get the impression that there exists a unique probability distribution in a probabilistic system and that this single probability scheme is linked to the constrained maximization of Shannon’s entropy or Fisher’s information. We argue however, that there may be different probability schemes in the same problem associated with different pieces of information which are relevant to the physics which occurs in the system. Thus, we argue that it is the physics which selects the probability distribution to be examined. For example, one may loosely think in terms of a detailed probability scheme versus a kind-of coarse grained one in the same system. For example, for a quantum free particle, P(x)=1 i.e there is the same probability for a particle to be at any x point. This, however, is based on velocity, i.e. the relationship between dx and dt. There is another piece of information in the system, namely mo which together with v yields momentum (either relativistic or nonrelativistic) .This need not be measured using a clock (which velocity requires), but by the impact of an impulse on a target. Thus, nonrelativistically p=m1v1=m2v2 give the same impulse, but have different velocities. Quantum mechanics shows that a free particle is governed by exp(ipx) which contains two unnormalized probability distributions cos(px) and sin(px). These are key for impulse interactions (i.e. the physics of interactions such as with a 2-slit apparatus etc), yet exp(ipx)/sqrt(L) maps into the “coarse-grained” P(x)=exp(ipx)exp(-ipx) / L = constant. P(x) is one probability scheme in the problem and exp(ipx) another (for a free particle). Each answers some physical questions, but one needs to know which probability to use (i.e. what information is relevant) to a particular situation. P(x) is relevant to motion in space not involving interactions. exp(ipx) is relevant to interactions (even with a magnetic vector potential causing no magnetic field) or with a 2-slit apparatus. Furthermore, in a quantum bound state, one has W*(x)W(x) = P(x) (W(x)=wavefunction =Sum over p a(p)exp(ipx)) and a*(p)a(p).=P(p). Each probability scheme corresponds to the analysis of a different kind of measurement, but various schemes may also be linked. Thus one may have to start with one to create the others. For example, exp(ipx) is the base for creating W(x)=Sum over p a(p)exp(ipx), which ultimately yields P(x)=W*(x)W(x). As a result, we argue that if probability exists in a system, it is not necessarily the case that there is a unique probability distribution. There may be various probability schemes linked with different information and different measurements/interactions. Thus, physics leads one to focus on a specific probability scheme. As a result, we argue that one must be careful when discussing pure statistical quantities and their extremization, because these quantities, like Shannon’s entropy and Fisher’ s information, contain a classical probability distribution, but they do not indicate which distribution to use in the case that a system has more than one. . As a result, the analysis of a system requires knowledge of the physics (interactions) occurring.We argue against thinking in terms of physics following directly from statistical considerations. The constrained extremization of one statistical quantity may yield a probability distribution related to one piece of information, but not to another. Furthermore, Shannon’s entropy and Fisher’s information pertain to classical probability 0<=P(x)<1 and Integral P(x)dx =1. The wavefunction, exp(ipx)/sqrt(L), is comprised of two probability distributions and ADD and MULTIPLIES statistically (i.e. OR and AND ) situations, but is involved in relative average calculations because it is not normalized. It is also different because it allows for positive and negative values. These are properties which do not follow from statistics using a classical P(x).