Fluctuation relation and pairing rule for Lyapunov exponents of inertial particles in turbulence

We study the motion of small particles in a random turbulent flow assuming linear law of friction. We derive a symmetry relation obeyed by the large deviations of the finite time Lyapunov exponents in the phase space. The relation applies when either the statistics of the strain matrix is invariant under the transposition or when it is time-reversible. We show that, as a result, the Lyapunov exponents come in pairs which sum is equal to minus the inverse relaxation time of the particles. We use the pairing to consider the Kaplan-Yorke dimension of the particles attractor in the phase space. In particular, the results apply to case of the flow which is white noise in time.