Entropy, Age and Time Operator

Ilias Gialampoukidis,Ioannis Antoniou

doi:10.3390/e17010407

Abstract

The time operator and internal age are intrinsic features of entropy producing innovation processes. The innovation spaces at each stage are the eigenspaces of the time operator. The internal age is the average innovation time, analogous to lifetime computation. Time operators were originally introduced for quantum systems and highly unstable dynamical systems. Extending the time operator theory to regular Markov chains allows one to relate internal age with norm distances from equilibrium. The goal of this work is to express the evolution of internal age in terms of Lyapunov functionals constructed from entropies. We selected the Boltzmann–Gibbs–Shannon entropy and more general entropy functions, namely the Tsallis entropies and the Kaniadakis entropies. Moreover, we compare the evolution of the distance of initial distributions from equilibrium to the evolution of the Lyapunov functionals constructed from norms with the evolution of Lyapunov functionals constructed from entropies. It is remarkable that the entropy functionals evolve, violating the second law of thermodynamics, while the norm functionals evolve thermodynamically.

Highlights

The idea to represent time as an operator goes back to Pauli, who remarked that the time-energy uncertainty relation is analogous to the position-momentum uncertainty relation, there is no self-adjoint time operator T canonically conjugate to the energy operator H (Hamiltonian) of a quantum system, as is the case with the position-momentum operators [1,2]
Time operator theory was extended to describe the statistical properties of complex dynamical systems [3,9,10,11,12,13], providing a new representation of non-equilibrium processes and the innovation of non-predictable stationary stochastic processes [14,15,16,17,18,19]
As entropies were historically the first Lyapunov functionals, we investigate in this work which entropies fit better the age formula and are compatible with the monotonic approach to equilibrium, as stated by the second law of thermodynamics (Section 4)

Summary

Introduction

The idea to represent time as an operator goes back to Pauli, who remarked that the time-energy uncertainty relation is analogous to the position-momentum uncertainty relation, there is no self-adjoint time operator T canonically conjugate to the energy operator H (Hamiltonian) of a quantum system, as is the case with the position-momentum operators [1,2]. After extending the time operator theory to more general processes, like Markov chains, we found non-canonical age formulas, where the correction to the canonical formula is proportional to a decaying exponential term [20]:. This rather unexpected new age formula resulted from the need to relate age with mixing time (time to approach equilibrium [22,23]). The internal age determines the mixing time of Markov chains [20].

Time Operator and Age

Lyapunov Functionals and Entropies

Age in Terms of Entropy

Concluding Remarks