Lagrangians of physics and the game of Fisher-information transfer

Fisher info and thermodynamics’ first law

“…Here T will be a "Fisher-temperature" (FT), whose existence was conjectured in [10,11] and proved in [8]. Its properties have been extensively discussed in Ref.…”

“…One of the most important extant information measures was advanced by R.A. Fisher in the twenties -for details and discussions we refer to (for instance) [10][11][12]-. Fisher information (FI) arises as a measure of the expected error in the measurement of a parameter θ in a situation governed by a family of probability densities ρ(x, θ), where x is, of course, a random variable, and θ is the parameter characterizing the alluded mono-parameter family of probability densities [10].…”

Estimating a Repeatable Statistical Law by Requiring Its Stability During Observation

“…As usual, * denotes the complex conjugate. This is called the "Fisher channel capacity" [6], since it is an upper bound to the actual Fisher information [6,12]. However, to keep the analysis simple, we first merely assume the one-dimensional single-state case (2) of the information.…”

“…Principle (1) has been applied in [4][5][6][7][8][9][10] to derive wave equations of quantum mechanics, basic thermodynamics and modern cell biology. Also derived have been the phenomena of statistical physics and other sciences [6,[11][12][13][14][15][16][17][18][19]. This includes the laws of cancer growth [6,13], the near-ubiquitous occurrence of statistical power laws [19] in science, including the quarter-power laws of biological and cosmological allometry [6,19], the de Broglie wave hypothesis (now, no longer a mere hypothesis) [17], thermodynamics using Fisher information in place of entropy [6,7], the Euler equation [14] of the chemical density functional and the laws of optimum economic investment [11] and of population dynamics [6,8,18] of animate or inanimate systems.…”

“…In such cases, the expression (2) for I is easily generalized by the usual replacements x →x, dx → dx in the integrand for a vector x = x 1 , ..., x N of data fluctuations. Furthermore, the integrand (dq/dx) 2 is replaced [6,12] by the sum n (∇q n ) 2 , where ∇ is the gradient operator, for a system that can be in many different states n = 1, ..., N . Finally, as defined in Equation (12), in terms of complex amplitudes ψ n , n = 1, ..., N , the information becomes I = 4N dx n ∇ψ * n · ∇ψ n in problems where the x-space is continuous.…”