We construct a formally time-reversible, one-dimensional forced Burgers
equation by imposing a global constraint of energy conservation, wherein the
constant viscosity is modified to a fluctuating state-dependent dissipation
coefficient. The new system exhibits dynamical properties which bear strong
similarity with those observed for the Burgers equation and can be understood
using the dynamics of the poles, shocks and truncation effects such as tygers.
A complex interplay of these give rise to interesting statistical regimes
ranging from hydrodynamic behaviour to a completely thermalized warm phase. The
end of the hydrodynamic regime is associated with the appearance of a shock in
the solution and a continuous transition leading to a truncation dependent
state. Beyond this, the truncation effects such as tygers and appearance of
secondary discontinuity at the resonance point in the solution strongly
influence the statistical properties. These disappear at the second transition,
at which the global quantities exhibit a jump and attain values that are
consistent with the establishment of a 'quasi-equilibrium' state characterized
by energy equipartition among the Fourier modes. Our comparative analysis shows
that the macroscopic statistical properties of the formally time-reversible
system and the Burgers equation are equivalent in all the regimes, irrespective
of the truncation effects, and this equivalence is not just limited to the
hydrodynamic regime, thereby further strengthening the Gallavotti's equivalence
conjecture. The properties of the system are further examined by inspecting the
complex space singularities in the velocity field of the Burgers equation.
Furthermore, an effective theory is proposed to describe the discontinuous
transition.Comment: 25 pages, 18 figure