Fluctuating hydrodynamics for
nonequilibrium steady states
José M. Ortiz de Zárate
Complutense 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:
Out of equilibrium
In equilibrium
 (r )   (r ')
 kBT T  (r  r )
2

 (r)   (r ')
 kBT (r)T (r)  (r  r)
2
 (r)
 (r)   (r ')
2



k
T


(
r

r
)

(

T
) GNE (r, r) 
B
T
2


Anisotropic and spatially long-ranged
Depending on direction, enhancement can be very large
Thermal fluctuations in
nonequilibrium states
In the direction of the gradient
<G
(r =0,z=L/4,z')>
NE ||
0.05
Non-Equilibrium
Equilibrium
0.00
0.00
0.25
0.50
0.75
1.00
z'/L
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
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 longranged, 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
A short tutorial based
on heat equation
T
Q  L 2   Q   T   Q
T
 T

 2TQ   Q
 c p    v    T   
 t

T
S  Q  2
T
Balance law + Fluctuating phenomenological equation
 Q(r, t )  0
Linear phenomenological laws are valid
only “on average”+Fluctuation-dissipation
 Qi (r, t )   Q j (r , t )  L  ij  (r  r )  (t  t )
The fluctuation-dissipation theorem
• In equilibrium
2
k
T
 T (r)  T * (r ')  B  (r  r)
cp
cp
T 2 V
Prob[ T (r )]  exp( S / k B )
S 
 T 2 (r )
The three formulations may
be referred to as the
fluctuation-dissipation
theorem
 Qi (r, t )   Q j (r, t )  T 2  ij  (r  r)  (t  t )
The fluctuation-dissipation theorem
• Out of equilibrium
We KNOW that the intensity of
fluctuations HAS contribution(s)
from the gradients
2
k
T
 T (r)  T * (r ')  B  (r  r)
cp
cp
T 2 V
Prob[ T (r )]  exp( S / k B )
S 
 T 2 (r )
This may be the most fundamental
formulation of the fluctuationdissipation theorem. Is local both at
equilibrium and out of equilibrium
 Qi (r, t )   Q j (r, t )  T 2  ij  (r  r)  (t  t )