Abstract
For a holographically defined finite temperature theory, we derive an off-shell constitutive relation for a global $U(1)$ current driven by a weak external non-dynamical electromagnetic field. The constitutive relation involves an all order gradient expansion resummed into three momenta-dependent transport coefficient functions: diffusion, electric conductivity, and "magnetic" conductivity. These transport functions are first computed analytically in the hydrodynamic limit, up to third order in the derivative expansion, and then numerically for generic values of momenta. We also compute a diffusion memory function, which, as a result of all order gradient resummation, is found to be causal.
Highlights
Where ρ is the corresponding conserved charge density
The constitutive relation involves an all order gradient expansion resummed into three momenta-dependent transport coefficient functions: diffusion, electric conductivity, and “magnetic” conductivity
The formalism of [19] provides a systematic framework to construct nonlinear fluid dynamics, order by order in the fluid velocity derivative expansion, with the transport coefficients determined from the gravity side
Summary
We provide a clarification about what we mean by all-order gradient resummation, which along spacial derivatives includes an infinite number of time derivatives This means that the effective dynamical equations require an infinite set of initial conditions or, equivalently, the dynamics at hand is a theory of infinite number of degrees of freedom. E is normally turned on at negative times, so to create an initial charge density profile at t = 0, and turned off at t = 0 (E(t > 0) = 0), letting the system to freely relax to its equilibrium at infinite future For such experimental setup, for positive times the current t.
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