Fluctuating hydrodynamics for nonequilibrium steady states

José M. Ortiz de ZárateComplutense University. Madrid, Spain

Outline

• Thermal fluctuations in nonequilibrium states

• Jan Sengers contributions to the field• The fluctuation-Dissipation Theorem, in

equilibrium and outside of equilibrium

Thermal fluctuations in nonequilibrium states

• One-component fluid under a temperature gradient:

B2

( ) ( ')( )Tk T

r r

r r B2

( ) ( ')( ) ( ) ( )

( ) Tk T

r rr r r r

r

In equilibrium Out of equilibrium

2B NE2

( ) ( ')( ) ( ) ( , )Tk T T G

r r

r r r r

Anisotropic and spatially long-rangedDepending on direction, enhancement can be very large

Thermal fluctuations in nonequilibrium states

0.00 0.25 0.50 0.75 1.000.00

0.05

<GNE(r||=0,z=L/4,z')>

Non-Equilibrium

Equilibrium

z'/L

In the direction of the gradient

First predicted by Kirkpatrick, Cohen and Dorfmann (1982) by kinetic theoryConfirmed by Ronis and Procaccia by Fluctuating HydrodynamicsIn Fourier space (spatial spectra), with no gravity and no boundaries

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.

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

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

Jan Sengers contributions to the field - Experiments

• A picture of that time (at Jan house in Baketon, WV)

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

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)

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

Fluctuating hydrodynamics

pTc Tt

v Q

2

TL TT

Q Q Q

Balance law + Fluctuating phenomenological equation

( , ) 0

( , ) ( , ) ( ) ( )i j ij

t

Q t Q t L t t

Q r

r r r r

Linear phenomenological laws are validonly “on average”+Fluctuation-dissipation

2T Q

a ba b

2

TST

Q

A short tutorial based on heat equation

The fluctuation-dissipation theorem

• In equilibrium

2* B( ) ( ') ( )

p

k TT T

c

r r r r

2( , ) ( , ) ( ) ( )i j ijQ t Q t T t t r r r r

22

B

( )

Prob[ ( )] exp( / )

p

V

cS T

TT S k

r

r

The three formulations may be referred to as the fluctuation-dissipation theorem

The fluctuation-dissipation theorem

• Out of equilibrium

2* B( ) ( ') ( )

p

k TT T

c

r r r r

2( , ) ( , ) ( ) ( )i j ijQ t Q t T t t r r r r

22

B

( )

Prob[ ( )] exp( / )

p

V

cS T

TT S k

r

r

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