  Turbulence is a paradigm for far-from-equilibrium systems without time
reversal symmetry. To capture the nonequilibrium irreversible nature of
turbulence and investigate its implications, we develop a potential landscape
and flux field theory for turbulent flow and more general nonequilibrium fluid
systems governed by stochastic Navier-Stokes equations. We find that
equilibrium fluid systems with time reversibility are characterized by a
detailed balance constraint that quantifies the detailed balance condition. In
nonequilibrium fluid systems with nonequilibrium steady states, detailed
balance breaking leads directly to a pair of interconnected consequences,
namely, the non-Gaussian potential landscape and the irreversible probability
flux, forming a 'nonequilibrium trinity'. The nonequilibrium trinity
characterizes the nonequilibrium irreversible essence of fluid systems with
intrinsic time irreversibility and is manifested in various aspects of these
systems. The nonequilibrium stochastic dynamics of fluid systems including
turbulence with detailed balance breaking is shown to be driven by both the
non-Gaussian potential landscape gradient and the irreversible probability
flux, together with the reversible convective force and the stochastic stirring
force. We reveal an underlying connection of the energy flux essential for
turbulence energy cascade to the irreversible probability flux and the
non-Gaussian potential landscape generated by detailed balance breaking. Using
the energy flux as a center of connection, we demonstrate that the four-fifths
law in fully developed turbulence is a consequence and reflection of the
nonequilibrium trinity. We also show how the nonequilibrium trinity can affect
the scaling laws in turbulence.
