All order effective action for charge diffusion from Schwinger-Keldysh holography

Understand

An effective action for diffusion of a conserved $U(1)$ charge is derived to all orders in the derivative expansion within a holographic model dual to the Schwinger-Keldysh closed time path.

• A systematic approach to solution of the 5D Maxwell equations in a doubled Schwarzschild-AdS$_5$ black brane geometry is developed.
• Constitutive relation for the stochastic charge current is shown to have a term induced by thermal fluctuations (coloured noise).
• All transport coefficient functions parameterising the effective action and constitutive relations are computed analytically in the hydrodynamic expansion, and then numerically for finite momenta.