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

There are two main approaches to nonequilibrium 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 which 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.