Fluid dynamics in action

Abstract

In this thesis we formulate an effective field theory for nonlinear dissipative fluid dynamics. The formalism incorporates an action principle for the classical equations of motion as well as a systematic approach to thermal and quantum fluctuations around the classical motion of fluids. The dynamical degrees of freedom are Stuckelberg-like fields associated with diffeomorphisms and gauge transformations, and are related to the conservation of the stress tensor and a U(1) current if the fluid possesses a charge. This inherently geometric construction gives rise to an emergent "fluid space-time", similar to the Lagrangian description of fluids. We develop the variational formulation based on symmetry principles defined on such fluid space-time. Through a prescribed correspondence, the dynamical fields are mapped to the standard fluid variables, such as temperature, chemical potential and velocity. This allows to recover the standard equations of fluid dynamics in the limit where fluctuations are negligible. Demanding the action to be invariant under a discrete transformation, which we call local KMS, guarantees that the correlators of the stress tensor and the current satisfy the fluctuation-dissipation theorem. Local KMS invariance also automatically ensures that the constitutive relations of the conserved quantities satisfy the standard constraints implied e.g. by the second law of thermodynamics, and leads to a new set of constraints which we call generalized Onsager relations. Requiring the above properties to hold beyond tree-level leads to introducing fermionic partners of the original degrees of freedom, and to an emergent supersymmetry. We also outline a procedure for obtaining the effective field theory for fluid dynamics by applying the holographic Wilsonian renormalization group to systems with a gravity dual.