Dynamical systems of eternal inflation: A possible solution to the problems of entropy, measure, observables and initial conditions

TL;DR

This paper proposes a dynamical systems approach to model eternal inflation, addressing key conceptual issues like entropy, measure, observables, and initial conditions by applying thermodynamic formalism and the chaotic hypothesis.

Contribution

It introduces a dynamical systems framework for eternal inflation, providing solutions to longstanding problems in entropy, measure, and initial conditions using equilibrium measures and the thermodynamic formalism.

Findings

The equilibrium measure replaces the Liouville measure, resolving the entropy problem.

The approach supports local trajectories on infinite paths, addressing the measure and initial condition issues.

Phenomenological implications of the fluctuation theorem are discussed.

Abstract

There are two main approaches to non-equlibrium statistical mechanics: one using stochastic processes and the other using dynamical systems. To model the dynamics during inflation one usually adopts a stochastic description, which is known to suffer from serious conceptual problems. To overcome the problems and/or to gain more insight, we develop a dynamical systems approach. A key assumption that goes into analysis is the chaotic hypothesis, which is a natural generalization of the ergodic hypothesis to non-Hamiltonian systems. The unfamiliar feature for gravitational systems is that the local phase space trajectories can either reproduce or escape due to the presence of cosmological and black hole horizons. We argue that the effect of horizons can be studied using dynamical systems and apply the so-called thermodynamic formalism to derive the equilibrium (or Sinai-Ruelle-Bowen) measure given by a variational principle. We show that the only physical measure is not the Liouville measure (i.e. no entropy problem), but the equilibrium measure (i.e. no measure problem) defined over local trajectories (i.e. no problem of observables) and supported on only infinite trajectories (i.e. no problem of initial conditions). Phenomenological aspects of the fluctuation theorem are discussed.