What happens to detailed What happens to detailed balance away from equilibrium?balance away from equilibrium?What happens to detailed What happens to detailed balance away from equilibrium?balance away from equilibrium?

R. M. L. Evans

University of Leeds

School of Physics & Astronomy

Driven steady statesDriven steady states

Driven steady states are not at thermodynamic equilibrium because, by definition, equilibrium states have no net fluxes.

Particularly interesting for complex fluids, where typical relaxation times are long, so can reach the regime 1~

Examples:Examples: Fluids under shear

Semi- dilute “living” polymers

A phase of amphiphiles...

...self-assembled into “worm-like micelles”:

Stirring causes de-mixing!

Lamellar phase of amphiphiles

...re-structure under shear

Useful for encapsulating drugs.

... into the onion phase:

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:

Compare equilibrium states with driven states

Compare equilibrium states with driven states

Equ

ilib.

st

ates

Driv

en

stat

es

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.

Current approaches to non-equilibrium systems

Current approaches to non-equilibrium systems

(i) Near-equilibrium approximationsAssume free energy F() can still be defined at non-equilibrium values of order parameter , and postulate dynamics,

e.g. —“Model A” for non-conserved O.P.Implies local coarse-graining.

d

dF

(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, to respect Boltzmann.

i.e. Macroscopic thermodynamics informs the microscopic model.

Tkb

Ea

E

ab

ba Be

a

Where does Detailed Balance come from?

Where does Detailed Balance come from?

Tkb

Ea

E

ab

ba Be

•If Ea=Eb then ab=ba.So DB embodies reversibility.

b•DB says something about how likely it is for the reservoir to give a particular kick to the system.

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.

c

Is it necessarily right? NO! The reservoir might contain anything.

c

If :

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.

(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 ).

Driven steady-state ensembleDriven steady-state ensemble

h >> lcorr

system 1

system 2

system N

...

Imagine a large set of fluid systems stacked up and sheared: (e.g. fluid is the onion phase of amphiphiles)

Over a long duration , system i follows path i through phase space. We derive the path distribution p following Gibbs:

Number of systems with trajectory is n=N p

Statistical weight of ensemble is no. of ways of permuting these differently-experienced systems:

!

!

n

NN

By definition, most ensembles maximise the statistical weight.This is achieved by maximising(subject to constraints)

ppS lnln1

NN

RecipeRecipeTo 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,

• Known macroscopic observables, (equilib.)

• Extra constraint on driven ensemble:

1

p

EEp

p

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.

Result: epp equilibdriven

Implications for ratesImplications for ratesWe have the probability of an entire path :

time

stat

e

a

b

Would like to know prob. of a particular transition a→b, to find tabaab |Pr

We are not entirely free to choose since detailed balance must be respected. Hence, are subject to the same amount of constraint.

drab

eqab

Flux Pr

Flux Preq

eqeqdr | baabab

By counting all paths that contain this transition, the relation between and implies:dr

p eqp

Example: Hopping in 1D

Macro-observable:

where G(x,t) is equilib Green’s fn.

x

RL

0

v)(

x

vv

,v

1,1vLim

2

124

12eq

eqdr

R

G

GRR

const.drdr LRand

5.0

2

E

Activated processesActivated processesSimple model:

eqeq

eqeq 1

DeU

RL

E

100

6

E

OverviewOverview-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 workFurther 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:Thanks to: Peter Olmsted, Richard Blythe, Mike Cates, Alistair Bruce, Tom McLeish.