Transcript What happens to detailed balance away from equilibrium? R. M. L. Evans University of Leeds School of Physics & Astronomy.

What happens to detailed
balance away from equilibrium?
R. M. L. Evans
University of Leeds
School of Physics & Astronomy
Driven steady states
Driven steady states are not at thermodynamic equilibrium
because, by definition, equilibrium states have no net fluxes.
Examples: Fluids under shear
g
Particularly interesting for complex fluids,
where typical relaxation times t are long,
~
so can reach the regime gt  1
Semi- dilute “living” polymers
A phase of
s
amphiphiles...
...self-assembled into
“worm-like micelles”:
g
Stirring causes de-mixing!
Lamellar phase of amphiphiles
...re-structure under shear
... into the onion phase:
Useful for encapsulating drugs.
Any concentrated suspension
Re-structures, under sufficiently high
shear, into force-bearing chains...
+ spectator particles.
Traffic
e.g. O. O’Loan & M. R. Evans ’97:
Ubiquitous.
Well-defined statistically steady states in dynamic
systems with spatial and temporal fluctuations.
Have transitions between differently-structured states
at well-defined values of the control parameters.
Exhibit phase transition in 1 dimension.
We can use intuition and approximation to model
them mathematically.
We understand the universal statistical principles
underlying their behaviour, and respect those
principles when constructing models.
Equilib.
states
Driven
states
Compare equilibrium states
with driven states
Current approaches to
non-equilibrium systems
(i) Near-equilibrium approximations
Assume free energy F(f) can still be defined at non-equilibrium
values of order parameter f, and postulate dynamics,
e.g.
f  
dF
df
—“Model A” for non-conserved O.P.
Implies local coarse-graining.
(ii) Invent microscopic dynamics
...and derive the macroscopic consequences.
e.g. The bus route model:
This is fine, but it’s not how we do things at equilibrium, where
rates cannot be chosen entirely arbitrarily, but must obey the
principle of detailed balance,  a b  eEa Eb  k T to respect Boltzmann.
B
b a
i.e. Macroscopic thermodynamics informs the microscopic model.
Where does Detailed Balance
come from?
a b

E E  k T
e a b
B
b a
•If Ea=Eb then ab=ba.
So DB embodies reversibility.
•DB says something about how
likely it is for the reservoir to give
a particular kick to the system.
a
cb
i.e. It is making a statement
about the statistics of the
reservoir: that the reservoir is in
the most likely state for the given
energy.
Is it necessarily right?
NO! The reservoir might contain anything.
If :
(i) system and reservoir have reversible microscopic dynamics
(ii) system and reservoir are in an ergodic steady state (exploring
phase space thoroughly)
(iii) reservoir is characterised only by its macroscopic observables
(energy) i.e. is in the most likely statistical state with no surprises
then the rates are constrained to respect detailed balance
( N/2 constraints on the N rates ab ).
Take such a system+reservoir, and subject it to continuous shear.
All 3 above conditions continue to be true.
Are we free to choose rates arbitrarily in every non-equilib model?
No! Those 3 conditions lead to N/2 constraints. i.e. There must be
a non-equilibrium counterpart to the principle of detailed balance.
Driven steady-state ensemble
Imagine a large set of fluid systems stacked up and sheared:
(e.g. fluid is the onion phase of amphiphiles)
Over a long duration t,
system N
system i follows path i
through phase space.
Dg
We derive the path
distribution p following
Gibbs:
h >> lcorr
system 2
Number of systems with
trajectory  is n=N p
system 1
Statistical weight of ensemble is no. of ways of
permuting these differently-experienced systems:
N 
N!
 n!

By definition, most ensembles maximise the statistical weight.
This is achieved by maximising S  N 1 ln N   p ln p

(subject to constraints)
Recipe
To find distribution p of phase-space paths (drawn only
from the set of physically possible paths),
maximise ‘path entropy’ S with respect to p, subject to constraints:
• Normalization,
 p  1

• Known macroscopic observables,
 p E 

• Extra constraint on driven ensemble:
Result: pdriven  pequilib e  g 
E
 p g  
(equilib.)
g

NB This is a well known recipe (proposed by Jaynes), but:
• is normally wrapped up with unsatisfactory subjective
(Bayesian/information-theoretic) interpretation of probabilities;
defined here in terms of concrete countable quantities.
• is often considered unhelpful, or used only approximately,
because  is a high-dimensl. object. We’ll use it to find rates ab.
Implications for rates
We have the probability of an entire path :
state
Would like to know prob. of a particular
transition a→b, to find ab  Pr a  b | a Dt

b

a
time

By counting all paths that contain
eq
Pr
Flux g | a  b
eq
dr
this
transition,
the
relation  ab
  ab
eq
dr
Pr eq Flux g 
between p and p implies:

eq
 ab
We are not entirely free to choose since
detailed balance must be
dr
respected. Hence, are subject
to the same amount of constraint.
 ab
Example: Hopping in 1D
Macro-observable:
x(t )  vt
G vt  1,t  1
t 
G vt ,t 
R dr  R eq Lim
1
4
1
2
 R eq 2  v 2  v
L
t 
R
0
x
where G(x,t) is equilib Green’s fn.
and
Rdr Ldr  const.
Activated processes
Simple model:
Leq  R eq  1
U eqe E  D eq  

E2
  0.5
E6
  100
Overview
-Postulated some exact rules for:
-microscopically reversible
-ergodic steady states
-with uncorrelated (i.e. weakly coupled) reservoir.
-Like detailed balance, rules do not fully specify rates.
-Results are only as good as the chosen model: should include
momentum variables.
Further work
-Aditi Simha has found rules yield interactions, in many-particle
systems, that respect Newton’s 3rd law.
-Adrian Baule will test rules by modelling real systems.
Thanks to:
Peter Olmsted, Richard Blythe, Mike Cates, Alistair Bruce, Tom McLeish.