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AIP conference proceedings

Fisher’s information and the arrow of time

Angel Ricardo Plastino

Published Jan 1, 2001• 1 citation
Publisher

Abstract

The fundamental equations of physics are invariant under time reversal, but everyday phenomena are usually irreversible ones. This is, in a nutshell, the so-called “arrow of time” dilemma. We address it here in terms of a rather old information measure: that of Fisher’s (1925).
DOI
10.1063/1.1381869

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Everyday experience tells us that it is impossible to travel backwards in time, no matter what science-fiction writers and movie directors tell us. However, this seemingly natural impossibility is actually one of the greatest mysteries in physics. Currently most physicists associate the irreversibility of time with the production of entropy in the warm macroscopic world. One consequence of this irreversibility – often called the “arrow of time” in thermodynamics is the ageing process. In plain terms, one grows old because of the irreversible physical processes that occur during one's life. However, the fundamental classical laws of physics discovered by Galileo, Newton and Einstein do not distinguish between the future and the past. In other words, they appear to be symmetric in time.
The arrow-of-time dilemma states that the laws of physics are invariant for time inversion, whereas the familiar phenomena we see everyday are not (i.e., entropy increases). I show that, within a quantum mechanical framework, all phenomena which leave a trail of information behind (and hence can be studied by physics) are those where entropy necessarily increases or remains constant. All phenomena where the entropy decreases must not leave any information of their having happened. This situation is completely indistinguishable from their not having happened at all. In the light of this observation, the second law of thermodynamics is reduced to a mere tautology: physics cannot study those processes where entropy has decreased, even if they were commonplace.
Consider an isolated statistical system specified by a coordinate x and its probability density p(x). A functional of p(x) called ``Fisher information'' can be used to measure the degree of disorder of the system due to the spread in p(x). Fisher information may be minimized, subject to a physical constraint, to attain a temporal equilibrium solution p(x). When the constraint is linear in the mean kinetic energy of the system, the equilibrium solution p(x) often obeys the correct differential equation for the system. In this way, the Schr\"odinger (energy) wave equation, Klein-Gordon equation, Helmholtz wave equation, diffusion equation, Boltzmann law, and Maxwell-Boltzmann law may be derived from one classical principle of disorder. The convergence rate for Fisher information is about that for alternative use of maximum entropy (in problems where both have the same equilibrium solution). This suggests that Fisher information defines an arrow of time. The arrow points in the direction of decreasing accuracy for the determination of the mean, or ideal, value of a parameter.
It is brought forward that viable theories of the physical world that have no variable at all that can play the role of time, do not exist; some notion of time is one of the very first ingredients a candidate theory should possess. Almost by definition, time has an arrow. In contrast, time reversibility, or even the possibility to run the equations of motion backwards in time, is not at all a primary requirement. This means that the direction of the arrow of time may well be uniquely defined in the theory, even locally. It is explained that a rigorous definition of time, as well as a formulation of the causality and locality concepts, can only be given when one has a model for the physical phenomena described. The only viable causality condition is one that is symmetric under time reversal. We explain these statements in terms of the author’s favoured deterministic cellular automaton interpretation of quantum mechanics, also to be referred to as 'vector space analysis', and expand on these ideas. It is also summarised how our more rigorous causality condition affects Bell’s theorems. What distinguishes quantum systems from classical ones is our fundamental inability to control the microscopic details of the initial state when phenomena are studied in the light of some theoretical model.
The foundation of irreversible, probabilistic time -- the classical time of conscious observation -- is the reversible and deterministic quantum time of the wave function. The tendency in physics is to regard time in the abstract, a mere parameter devoid of inherent direction, implying that a concept of real time must begin with irreversibility. In reality time has no need for irreversibility, and every invocation of time implies becoming or flow. Neither symmetry under time reversal, of which Newton was well aware, nor the absence of an absolute parameter, as in relativity, negates temporal passage. Far from encapsulating time, irreversibility is a secondary property dependent on the emergence of distinct moments from the ceaseless presence charted by the wave function.