The paper observes that consistent lateral motion of probability within a probability density does not affect the Shannon entropy, i.e. that such movement of probability is an adiabatic or entropy‐preserving operation. Next it is observed that if such motion removes “kinks” in a probability density, it decreases the Fisher information. Then it is found that a minimum of Fisher information due to this procedure must result if all kinks are removed by pushing probability laterally toward the origin on the plus side. Finally an algorithm is developed which will effect this operation, and calculate the resulting minimum laterally‐adiabatically reduced Fisher information. Examples are worked out to illustrate the algorithm. Applications may include many processes which occur adiabatically, i.e. without entropy increase. Variational problems are also worked out.
The paper observes that consistent lateral motion of probability within a probability density does not affect the Shannon entropy, i.e. that such movement of probability is an adiabatic or entropy‐preserving operation. Next it is observed that if such motion removes “kinks” in a probability density, it decreases the Fisher information. Then it is found that a minimum of Fisher information due to this procedure must result if all kinks are removed by pushing probability laterally toward the origin on the plus side. Finally an algorithm is developed which will effect this operation, and calculate the resulting minimum laterally‐adiabatically reduced Fisher information. Examples are worked out to illustrate the algorithm. Applications may include many processes which occur adiabatically, i.e. without entropy increase. Variational problems are also worked out.
It is shown that Shannon entropy, Fisher information, and quantum-mechanical kinetic energy may all be viewed as measures of correlation. We begin with a fundamental correlation-destroying map: a joint probability density ρ(x,y) is replaced by ρx(x)ρy(y), the product of its marginal distributions. In the discrete case, Shannon's entropy is nondecreasing under this map. For continuous distributions, this fundamental map leads to a well-behaved, coordinate-invariant correlation measure, analogous to the discrete Shannon entropy: I[ρ]=h[ρx]+h[ρy]−h[ρ]. In the continuous case, however, another measure appears: Fisher information. The relative Fisher information J(ρ∥ρxρy) behaves similarly under the fundamental map. For a normalized real quantum wavefunction, this decrease is exactly proportional to the decrease in mean kinetic energy, ⟨T⟩ψ−⟨T⟩Φ=8mℏ2J(ρ∥ρxρy),Φ=ρxρy. In Jaynes's language, the result supports a principle of minimum justified correlation: given physical constraints and a set of possible distributions or related amplitudes satisfying those constraints, select from that set those with the least correlation.
This is Part I of a two-part paper. Here we develop the entropy, Fisher-information, and kinetic-energy identities above, and state the principle they support. Part II addresses questions which Part I raises but leaves unanswered: multiple solutions, time dependence, the role of spin, a route to the Schrödinger equation itself, and a proposed experimental test. AI has been used.
Existing information-theoretic derivations of quantum mechanics from Fisher informationlack a thermodynamic embedding: they contain no temperature, no entropy, and no free-energy balance, and therefore cannot address the quantum–classical boundary within thesame variational structure. We show that this limitation is removed by a Helmholtz-typefree-energy functional on probability distributions in which Fisher information provides theenergy and relative Shannon entropy, weighted by an effective temperature ϑ, provides theentropic cost. Chentsov’s theorem and the Shore–Johnson axioms constrain this functionalup to two coefficients, one of which is fixed by consistency with the quantum kinetic energy.At ϑ= 0 the functional reproduces the Fisher-information variational principle of Reginattoand Hall–Reginatto; extremisation of the associated action recovers the Schr¨odinger equationvia the Madelung transformation. Quantum mechanics is thereby identified as the zero-temperature fixed point of the Helmholtz structure. The finite-temperature sector ϑ > 0,which has no counterpart in previous approaches, yields three results: a gauge-invariantbalance scale Lc = ℏ/2√mϑ at which both the free-energy descent and the gradient-flowspreading law change character — the spreading exponent crossing over from the quantum-pressure value 1/4 to the diffusive value 1/2 — together with, under harmonic confinement,a genuine variationally stable minimum interpolating between the exact quantum ground-state and classical equipartition widths; a dissipation bound for localisation transitions,derived from a Wasserstein gradient flow with Lyapunov structure, from which Landauer’sprinciple follows as a corollary in the entropy-dominated regime; and a thermodynamicstability property that singles out the Born weights as the unique stable basin fractionsunder the dissipative dynamics, presented as a consistency check.
Given an unbiased die, the probability is ⅙ for a result. This probability value may be obtained directly by imposing the condition of a lack of bias and noting there are six possible results. Alternatively, one may maximize Shannon’s entropy subject to the constraint Sum over i p(i)=1 where i runs from 1 to 6. Maximization of Shannon’s entropy seems to remove as much bias as possible with the constraint indicating what bias remains. For the die case the constraint has no bias, but for an average energy constraint Sum over i ei p(i) there is, namely ei. In previous notes we have argued that the constraint used in the maximization of entropy is “special”. Here we argue that it represents a conserved quantity if interactions occur. Thus there may be many constraints in a system, but choosing a conserved quantity in an interaction is rarer, but seems to be linked with equilibration. Finally, we suggest that maximization of entropy subject to a conserved quantity constraint removes as much bias as possible and it is this maximized entropy-probability which is subject to the second law of thermodynamics. It is possible, however, to have probabilities and to create entropies which are not based on maximization of entropy. For example, Shannon’s entropy may be applied to the spatial density of various n-slit interactions for a particle with constant momentum p. In this case, even though there is no maximization of entropy, the emerging particle has the same p magnitude (hence same uncertainty within a wavelength), but there is the added uncertainty of its direction. As a result, it seems that entropy increases in accordance with the second law. A free particle with momentum p, however, has a two dimensional probability distribution in space exp(ipx) characterized by a wavelength hbar/p. This may be taken as related to the entropy of the free particle, but a free particle may undergo spontaneous interactions which either increase or decrease p, hence increase or decrease hbar/p and entropy. In other words, the second law does not apply to such an entropy, but this entropy is associated with an unusual constraint. This conclusion seems to hold when one considers two body scattering. Thus we suggest that there are various kinds of entropy. Some are linked with maximization of entropy subject to a conserved quantity constraint (and are associated with equilibration) and others are not. The latter do not necessarily obey the second law of thermodynamics, we argue.
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).
Abstract In this paper, Shannon entropy and Fisher information is studied for the screened Kratzer potential model and compared with the screened Coulomb in three dimensions. Our results showed similar higher‐order characteristic behavior for position and momentum space. Our numerical results showed that increases in the accuracy of predicting particle location occurred in the position space. Our result shows that the sum of the position and momentum entropies satisfies the lower‐bound Berkner, Bialynicki‐Birula, and Mycieslki inequality. The Stam‐Cramer‐Rao inequalities relation for Fisher information and the expectation values were also satisfied for the different eigenstates.
Consider a system obeying conservation of flow, as in classical particle flow or in relativistic quantum mechanics. In such cases a probability density function p(r\ensuremath{\Vert}t) may be used to describe the system, where r is particle position and t is time. Let H(t) be the Shannon form of the Boltzmann entropy corresponding to p(r\ensuremath{\Vert}t). It is found that (dH/dt)max=1/6I(t)d/dt〈r2(t)〉, where I(t) is the Fisher information about the centroid of the system, and 〈r2(t)〉 is the time-dependent mean-square particle position. A corollary is that, for classical particle flow obeying 〈r〉=0, positional uncertainty \ensuremath{\sigma}(t) must ever increase with time.