By recourse to i) the Hellmann-Feynman theorem and ii) the Virial one, the information-optimizing principle based on Fisher's information measure uncovers a Legendre-transform structure associated with Schr\"odinger's equation, in close analogy with the structure that lies behind the standard thermodynamical formalism. The present developments provide new evidence for the information theoretical links based on Fisher's measure that exist between Schr\"odinger's equation, on the one hand, and thermodynamics/thermostatistics on the other one

As recently shown, a constrained Fisher-information extremizing (CFIE) process is able to deal with both equilibrium and nonequilibrium thermodynamic processes, thus being able to reproduce results deduced by a recourse to Boltzmann's transport equation (BTE). Here, we discuss the propagation of sound waves in a dilute gas and compare the ensuing CFIE solutions with those obtained by a recourse to Grad's approach to the BTE. The final molecular distribution function arrived at is the same following two alternative routes, either (i) the BTE via the Grad approach or (ii) the constrained Fisher treatment that does not require the use of the BTE. The way the necessary a priori information is used in these two instances, is however, quite different.

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Schrödinger's equation (SE) and the information-optimizing principle based on Fisher's information measure (FIM) are intimately linked, which entails the existence of a Legendre transform structure underlying the SE. In this comunication we show that the existence of such an structure allows, via the virial theorem, for the formulation of a parameter-free ground state's SE-ansatz for a rather large family of potentials. The parameter-free nature of the ansatz derives from the structural information it incorporates through its Legendre properties.

It is well known that a suggestive relation exists that links Schr\"odinger's equation (SE) to the information-optimizing principle based on Fisher's information measure (FIM). We explore here an approach that will allow one to infer the optimal FIM compatible with a given amount of prior information without explicitly solving first the associated SE. This technique is based on the virial theorem and it provides analytic solutions for the physically relevant FIM, that which is minimal subject to the constraints posed by the prior information