Lyapunov spectra and fluctuation relations: Insights from the Galerkin-truncated Burgers equation

The imposition of a global constraint of the conservation of total kinetic energy on a forced-dissipative Burgers equation yields a governing equation that is invariant under the time-reversal symmetry operation, {T:t→-t;u→-u}, where u is the velocity field. Moreover, the dissipation term gets strongly modified, as the viscosity is no longer a constant, but a fluctuating, state dependent quantity, which can even become negative in certain dynamical regimes. Despite these differences, the statistical properties of different dynamical regimes of the time-reversible Burgers equation and the standard forced-dissipative Burgers equation are equivalent, à la Gallavotti's conjecture of equivalence of nonequilibrium ensembles. It is shown that the negative viscosity events occur only in the thermalized regime described by the time-reversible equation. This regime is further examined by calculating the local Lyapunov spectra and fluctuation relations. A pairing symmetry among the spectra is observed, indicating that the dynamics is chaotic, and has an attractor spanning the entire phase space of the system. The negative events are found to satisfy fluctuation relations, namely, the Gallavotti-Cohen relation based on the phase-space contraction rate and the Cohen-Searles fluctuation relation based on the energy injection rate. The results suggest that these events are associated with the effects of the Galerkin-truncation, the latter is responsible for the thermalization.