1. Fluctuating hydrodynamics for nonequilibrium steady states José M. Ortiz de Zárate Complutense University. Madrid, Spain

2. Outline Thermal fluctuations in nonequilibrium states Jan Sengers contributions to the field The fluctuation-Dissipation Theorem, in equilibrium and outside of equilibrium

3. Thermal fluctuations in nonequilibrium states One-component fluid under a temperature gradient: In equilibrium Out of equilibrium Anisotropic and spatially long-ranged Depending on direction, enhancement can be very large

4. Thermal fluctuations in nonequilibrium states In the direction of the gradient First predicted by Kirkpatrick, Cohen and Dorfmann (1982) by kinetic theory Confirmed by Ronis and Procaccia by Fluctuating Hydrodynamics In Fourier space (spatial spectra), with no gravity and no boundaries

5. Jan Sengers contributions to the field - Experiments Experimental confirmation by small-angle light-scattering in one-component fluids. 1988, with Bruce Law and Bob Gammon. Motivated by Ted moving to Maryland.

6. Jan Sengers contributions to the field - Experiments Small-angle light-scattering experiments in binary mixtures. 1994, with Wenbin Li, Phil Segré and Bob Gammon. Only qualitative confirmation at that time Theory was previously extended to mixtures by Law and Nieuwoudt

7. Jan Sengers contributions to the field - Experiments Full experimental confirmation of theory in mixtures using polymer solutions (easy to measure and characterize). 1998, with Wenbin Li, Kaichang Zhang, Bob Gammon and Ortiz de Zárate

8. Jan Sengers contributions to the field - Experiments A picture of that time (at Jan house in Baketon, WV)

9. Jan Sengers contributions to the field - Theory Since non-equilibrium fluctuations are spatially long- ranged, their spatial spectrum at small q will be strongly affected by gravity and finite-size effects Gravity effects: 1993, with Phil Segré

10. Jan Sengers contributions to the field - Theory Finite-size effects for one-componnet fluids. 2001, with Ortiz de Zárate and Pérez Cordón Both gravity and finite-size effects have been confirmed experimentally by other groups: Milan (Giglio) and Santa Barbara (Cannell)

11. Jan Sengers contributions to the field - Book J.M. Ortiz de Zárate and J. V. Sengers: Hydrodynamic Fluctuations in Fluids and Fluid Mixtures. Elsevier, 2006

12. Fluctuating hydrodynamics Balance law+ Fluctuating phenomenological equation Linear phenomenological laws are valid only “on average”+Fluctuation-dissipation A short tutorial based on heat equation

13. The fluctuation-dissipation theorem In equilibrium The three formulations may be referred to as the fluctuation-dissipation theorem

14. The fluctuation-dissipation theorem Out of equilibrium This may be the most fundamental formulation of the fluctuation- dissipation theorem. Is local both at equilibrium and out of equilibrium We KNOW that the intensity of fluctuations HAS contribution(s) from the gradients