# A Lagrangian Fluctuation-Dissipation Relation for Scalar Turbulence

# A Lagrangian Fluctuation-Dissipation Relation for Scalar Turbulence

A common approach to calculate the solution of a scalar advection-diffusion equation is by a Feynman-Kac representation which averages over stochastic Lagrangian trajectories going backward in time to the initial conditions and boundary data. The trajectories are obtained by solving SDE's with the advecting velocity as drift and a backward Itō term representing the scalar diffusivity. In this framework, we present an exact formula for scalar dissipation in terms of the variance of the scalar values acquired along each random trajectory. As an important application, we study the connection between anomalous scalar dissipation in turbulent flows for large Reynolds and Péclet numbers and the spontaneous stochasticity of the Lagrangian particle trajectories. The latter property corresponds to the Lagrangian trajectories remaining random in the limit Re,Pe→∞, when the backward Itō term formally vanishes but the advecting velocity field becomes non-Lipschitz. For flows on domains without boundaries (e.g. tori, spheres) and for wall-bounded flows with no-flux Neumann conditions for the scalar, we prove that spontaneous stochasticity is necessary and sufficient for anomalous scalar dissipation. The fluctuation-dissipation relation provides a Lagrangian representation of scalar dissipation also in turbulent flows where present experiments suggest that dissipation is tending to zero as Re,Pe→∞. We discuss an illustrative example of Rayleigh-Bénard convection with imposed heat-flux at the top and bottom plates. Our formula here shows that the scalar dissipation is given by the variance of the local time densities of the stochastic particles at the heated boundaries. The ``ultimate regime'' of turbulent convection predicted by Kraichnan-Spiegel occurs when the near-wall particle densities are mixed to their asymptotic uniform values in a large-scale turnover time. The current observations of vanishing scalar dissipation require that fluid particles be trapped at the wall and remain unmixed for many, many large-scale turnover times. This talk presents joint work with Gregory Eyink.