Fisher-based thermodynamics: Its Legendre transform and concavity properties

Abstract

It is shown that the Legendre-transform structure of thermodynamics can be replicated without any change if one replaces the entropy S by Fisher's information measure I. Also, the important thermodynamic property of concavity is shown to be obeyed by I. By this use of the Fisher information measure we develop a thermodynamics that seems to be able to treat equilibrium and nonequilibrium situations in a manner entirely similar to the conventional one.

Fisher info and thermodynamics’ first law

“…and, as stated above, recognize that T dI plays the role of a heat-change dQ F , as expected (but not explicitly discussed there) from the philosophy espoused in [8]. Thus, we have here shown that work is represented by changes in expectation values.…”

“…[8]. It was shown there that, if we are discussing a thermodynamical setting involving M extensive variables O i , we can recover the formal, Legendre-structure of thermodynamics if we minimize the Fisher information measure I subject to appropriate constraints.…”

“…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.…”

“…Fisher information I allows for such a construction. The desired concavity property of I, demonstrated in [8], further illustrates on its utility as a "generator" of an underlying microscopic explanation for thermodynamics. In another, totally disctint vein, it was also recently shown, starting from first principles and using the point of view of information theory, that thermodynamics' first law (TFL) can be microscopically obtained without…”

“…In particular, it was shown in [8] that the entire Legendre-transform (LT) of thermodynamics can be microscopically reproduced using Fisher's information measure in place of Boltzmenn's entropy. This abstract LT constitues an essential ingredient that permits one to build up a "statistical mechanics".…”