Bernard H. Soffer

Information-theoretic significance of the Wigner distribution

A coarse-grained Wigner distribution p(W)(x,mu) obeying positivity derives out of information-theoretic considerations. Let p(x,mu) be the unknown joint probability density function (PDF) on position and momentum fluctuations x, mu for a particle in a pure state psi(x). Suppose that the phase part Psi(x,z) of its Fourier transform T-F[p(x,mu)]equivalent to parallel to G(x,z)parallel to exp[i Psi(x,z)] is constructed as a hologram. (Such a hologram is often used in heterodyne interferometry.) Consider a particle randomly illuminating this phase hologram. Let its two position coordinates be measured. Require that the measurements contain an extreme amount of Fisher information about true position, through variation of the phase function Psi(x,z). The extremum solution gives an output PDF p(x,mu) that is the convolution of the Wigner p(W)(x,mu) with an instrument function defining uncertainty in either position x or momentum mu. The convolution arises naturally out of the approach, and is one dimensional, in comparison with the ad hoc two-dimensional convolutions usually proposed for coarse graining purposes. The output obeys positivity, as required of a PDF, if the one-dimensional instrument function is sufficiently wide. The result holds for a large class of systems: those whose amplitudes psi(x) are the same at their boundaries [examples: states psi(x) with positive parity; with periodic boundary conditions; free particle trapped in a box].

Fisher info and thermodynamics’ first law

We show, starting from first principles, that thermodynamics’ first law can be microscopically obtained for Fishers information measure without need of invoking the adiabatic theorem. Further, it is proved that enforcing the Fisher-first law requirements in a process in which the probability distribution is infinitesimally varied is equivalent to minimizing Fishers information measure subject to appropriate constraints.

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.

Parallel Information Phenomena of Biology and Astrophysics

The realms of biology and astrophysics are usually regarded as distinct, to be studied within individual frameworks. However, current searches for life in the universe, and the expectation of positive results, are guiding us toward a unification of biology and astrophysics called astrobiology. In this chapter the unifying aspect of Fisher information is shown to form two bridges of astrobiology: (i) In Section 5.1 quarter-power laws are found to both describe attributes of biology, such as metabolism rate, and attributes of the cosmos, in particular its universal constants, (ii) In Section 5.2 we find that the Lotka-Volterra growth equations of biology follow from quantum mechanics. Both these bridges follow, ultimately, from the extreme physical information EPI principle and, hence, are examples of the “cooperative” universe discussed in Chapter 1. That is, the universe cooperates with our goal of understanding it, through participatory observation. The participatory aspect of the effect (i) is the observation of biological and cosmological attributes obeying quarter-power laws. In the Lotka-Volterra quantum effect (ii) the participation is the observation of a general particle member that undergoes scattering by a complex potential. This potential causes the growth or depletion of the particle population levels to obey Lotka-Volterra equations. Effectively, the interaction potentials of a standard Hartree view of the scattering process become corresponding fitness coefficients of the L-V growth equations. The two ostensibly unrelated effects of scattering and biological growth are thereby intimately related; out of a common flow of Fisher information to the observer.