Transcript Statistical mechanics of driven steady states Fluid under continuous shear: R. M.

Statistical mechanics of driven steady states
Fluid under continuous shear:
R. M. L. Evans
University of Leeds
e.g. Lamellar phase of amphiphiles...
shear rate

...re-structures
under shear into
onion phase:
•Reproducible phase behaviour.
[M. Buchanan]
A model fluid is defined by n rates {wab}.
If noise source is equilib heat bath, rates cannot be chosen freely,
but obey detailed balance (n/2 constraints): w w  e Ea  Eb  k BT
ab
ba
Should non-equilibrium noise have similar constraints?
Counting uncorrelated objects
A large set of fluid systems (e.g. onion phase)
is stacked up and sheared:
In long duration t, system i
follows phase-space path Gi.
system N
Take thermodynamic limit:

h >> xcorr
system 3
(except via conserved quantities)
system 2
N.B. Gi still describes non-trivial
spatial & temporal correlations.
system 1
Can find distribution p
G
Maximise S  
{Gi} → uncorrelated
following Gibbs (counting permutations):
 pG ln pG
paths
subject to:
 pG  1  pG EG 
G
G
E
 pG  G 
G

Probability of a path G
driven
G
p

equilib   G
G
p

e
Want prob. of a transition Pr a  b | a  wab t
state
Result
b
a
time
By counting all paths that contain the transition, obtain exact
theorem:
Coef. requiring

 J Phys A: Math Gen 38,
eq 

293 (2005);
wdr

w

knowledge
of
all
eqilib.
ab
ab 
 PRL 92, 150601 (2004).
 Green' s functions

eq
We are not entirely free to choose w ab
, due to detailed balance.
dr
One-to-one mapping  same amount of constraint on w ab
.
Testing the prediction: wab wba  wab wba eq
A “fluid”data
of rotors:
Prelim.
at fixed <U> [Aditi Simha]
(Deterministic, Hamiltonian)
w w
w w
rate of
 (twist
periodic BC)
Nearest-neighbour interaction:
U
q
w
w
p
q
0
If :
ww wwgaps
•Inter-rotor
= systems

1
•Potential wellsw= microstates

then the theorem applies here.
•Quasi-steady state
w
Equilib.
symmetries
eq
w
eq
w
eq 
 w 
eq 
 w 

 w w  w w  
+p
Overview
-Find exact rules for:
-ergodic steady states
-ensemble-driven by weak coupling
-Like detailed balance, rules do not fully specify rates.
N.B. Not a simulation tool, but a constraint on theories.
Thanks to:
Aditi Simha, Peter Olmsted, Richard Blythe, Mike Cates,
Alistair Bruce, Tom McLeish,