Chaos and the physics of non-equilibrium systems

Henk van Beijeren

Institute for Theoretical PhysicsUtrecht University

( ) ( ) ( ) ( , (0))d X t f X X t X t Xdt

d ( ) ( , )d (0)X t M X t X

Dynamical Systems Theory: For flows:

Lyapunov exponents:

max

1( (0)) lim log | ( (0)

, if

, ) |

Chao 0 s

i itX X t

t

0

log[ ( ) / (0)]inf limKS t

n t nh

t

Kolmogorov-Sinai entropy

Pesin’s theorem:

0i

KS ih

Gibbs assigns approach to equilibrium to mixing and coarse-graining.

Is chaos related to approach to equilibrium?

The KS entropy describes the average rate of spreading in the expanding directions. Suggests this may be a measure of the speed of mixing and thus of the approach to equilibrium (at least in ergodic systems).

The KS entropy describes the average rate of spreading in the expanding directions. Suggests this may be a measure of the speed of mixing and thus of the approach to equilibrium (at least in ergodic systems).

Perhaps one should use the smallest positive Lyapunov exponent as a measure for the slowest decay to equilibrium.

The KS entropy describes the average rate of spreading in the expanding directions. Suggests this may be a measure of the speed of mixing and thus of the approach to equilibrium (at least in ergodic systems).

Perhaps one should use the smallest Lyapunov exponent as a measure for the slowest decay to equilibrium.

20

But in classical transport theory decay to equilibrium typically is described by hydrodynamic equations. E.g. in a simple diffusive system this decay is of the form exp( ).Dk t

Can one somehow connect these concepts?

Twodimensional Lorentz gas

Regular Sinai-billiard

Random Lorentz gas

Typically the diffusion constant is of the order

with the mean free path between collisions, the density and the radius

v v /(

of the scatterers.

2 ) D l al

a

Typically the diffusion constant is of the order

with the mean free path between collisions, the density and the radius

v v /(

of the scatterers.

2 ) D l al

a

There is one positive Lyapunov exponent. It may be estimatedeasily:

2

log | v '/ v | log( / 2 cos )

log(1/ )coll coll l a

a

2v ' vcos

la

• Density dependences are very different.

• Density dependences are very different.

• Various other differences as well:– Diffusion coefficient diverges for Sinai billiard with infinite horizon.

• Density dependences are very different.

• Various other differences as well:– Diffusion coefficient diverges for Sinai billiard with infinite horizon.– Diffusion coefficient vanishes below percolation density.

• Density dependences are very different.

• Various other differences as well:– Diffusion coefficient diverges for Sinai billiard with infinite horizon.– Diffusion coefficient vanishes below percolation density.– Wind tree model has diffusive behavior on large time and length scales, but zero Lyapunov exponents.

Wind tree model

System behaves diffusively on large time and length scales. It shows mixing behavior, butpower law with time. So the KS entropy equals zero.

Perhaps a definition of weak, nonexponential chaos is needed to describe this.

( ) / (0)n t n only increases as a

• Density dependences are very different.

• Various other differences as well:– Diffusion coefficient diverges for Sinai billiard with infinite horizon.– Diffusion coefficient vanishes below percolation density.– Wind tree model has diffusive behavior on large time and length scales, but zero Lyapunov exponents.

No obvious connections between Lyapunov exponent and hydrodynamic decay!

Are smallest Lyapunov exponents of many-particle systems related to hydrodynamics?

Lyapunov spectrum for 750 hard disks (Posch and coworkers)

Like in hydrodynamics there are branches of k-dependent eigenvalues that approach zero in the limit 0.k

0k In the limit both sets of eigenvalues approach zero, because the corresponding eigenmodes appoach to a symmetry

transformation. But no connection between the eigenvalues appears.

Lyapunov “shear” mode.Average displacement inx-direction as a function ofy-coordinate. Growth rateis proportional to k (vs. decay rate ~ k2 for hydrodynamic shear mode).

What connections do exist?

What connections do exist?

Most of them consider changes in dynamical properties due to deviations from equilibrium.

1. Gaussian thermostat formalism of Evans and Hoover:

Systems under external driving forces are kept at constant kinetic(or total) energy by applying fictitious thermostat forces, such that

intv vextii i i

dm F Fdt

Here has to be chosen such that the kinetic energy(or the total energy) remains strictly constant.

For such and a few different fictitious thermostats, minus the sum of all Lyapunov exponents (the average rate of phase space contraction!) can be identified with the rate of irreversible entropy production.

What connections do exist?

Most of them consider changes in dynamical properties due to deviations from equilibrium.

1. Gaussian thermostat formalism of Evans and Hoover:

2. The escape rate formalism of Gaspard and Nicolis.

For finite systems with open boundaries, through which trajectories may escape, the KS entropy satisfies

0i

rep

KS ih

For diffusive systems

connects a transport coefficient with dynamical systems properties.

20Dk so this relationship

Survival rate of exp( )t

What connections do exist?

Most of them consider changes in dynamical properties due to deviations from equilibrium.

1. Gaussian thermostat formalism of Evans and Hoover:

2. The escape rate formalism of Gaspard and Nicolis.

3. Relationships between Hausdorff dimensions of hydrodynamic modes, Lyapunov exponents and transport coefficients, obtained by Gaspard et al.

For two-dimensional diffusive systems, Gaspard, Claus, Gilbert andDorfman obtained the relationship

H20

D ( ) 1limk

kDk

This can probably be generalized to higher dimensions and generalclasses of transport coefficients.

Other connections between dynamical systems theory and nonequilibrium statistical mechanics involve:

1. Fluctuation theorems (Evans, Morriss, Searles, Cohen, Gallavotti and others) relate the probabilities of finding fluctuations in stationary systems with entropy changes of respectively andS S - .

Other connections between dynamical systems theory and nonequilibrium statistical mechanics involve:

1. Fluctuation theorems (Evans, Morriss, Searles, Cohen, Gallavotti, Kurchan, Lebowitz, Spohn and others) relate the probabilities of finding fluctuations in stationary systems with entropy changes of respectively

2. Work theorems (Jarzynski and others) allow calculations of free energy differences between different equilibrium states from work done in nonequilibrium processes.

andS S - .

Other connections between dynamical systems theory and nonequilibrium statistical mechanics involve:

1. Fluctuation theorems (Evans, Morriss, Searles, Cohen, Gallavotti, Kurchan, Lebowitz, Spohn and others) relate the probabilities of finding fluctuations in stationary systems with entropy changes of respectively

2. Work theorems (Jarzynski and others) allow calculations of free energy differences between different equilibrium states from work done in nonequilibrium processes.

3. Ruelle’s thermodynamic formalism.

andS S - .

Dynamical partition function:

Topological pressure:

In general,

1( , ) , ( , )Z t dX X t S X t

1( ) lim log ( , )t

P Z tt

1

(1)

[ ( )] KS

P

P h

4. SRB (Sinai-Ruelle-Bowen) measures may provide a general tool for describing stationary nonequilibrium states. These are the stationary distributions to which arbitrary initial distributions approach asymptotically. For ergodic Hamiltonian systems they coincide with the microcanonical distribution, for phase space contracting systems they are smooth in the expanding directions and have a highly fractal structure in the contracting directions.

For moving hard speres at low density the velocity deviations of two colliding particles are both upgraded to

a value of the order of .

Moving hard spheres and disks

max(| v | ,| v |)mfk l

la

and | v | | v |k l

Set

The distribution of these “clock values” approximately satisfies:

.

Can be solved for stationary profile of the form P(n,t)=P(n-vt) bylinearizing for large n. Then v log(lmf /a) is the largest Lyapunov exponent.

It is determined by .

1

( , ) { ( 1, )[ ( 1, ) 2 ( , )] ( , )}cm n

dP n t P n t P n t P m t P n tdt

(0)log( v / v )log( / )

kk

mf

nl a

2 1v min( )xex

Gives rise to largest Lyapunov exponent

Keeping account of velocity dependence of collision frequency onemay refine this to

Finite size corrections are found to behave as

May be compared to simulation results:

max *

max *

( 4.331 log n )

( 4.732 log n 11.73)-2 (log N)

c

c

Brownian motionWe consider a large sphere or disk of radius A and mass M in a dilute bath of disks/spheres of radius a and mass m. At collisions, the velocity deviations of the small particles change much more stronglythan those of the Brownian particle. But, because the collision frequency of the latter is much higher, it may still dominate the largest Lyapunov exponent. The process may be characterized by a stationary distribution of the variables

<x> can be identified as the largest Lyapunov exponent connected to the Brownian particle.

2 ; | | | | | |

vv vrx yr r r

These satisfy the Fokker-Planck equation,

Both “diffusion constants” are proportional to

Therefore <x> scales as

2 2

2 2

2 2][ 2

3 2 th wi1 mlog , =

M

( ) ( , , )

( ) ( , , )

x y xyt x y

x y

B

f x y t

D D f x y t

3/12 )log( Bv

0.5a and 1m 499.5,Afor , vs./ MBB

Maximal Lyapunov exponent for 2d system with 40 disks ofa=1/2 and m=1. Open squares: pure_fluid. Crosses: A=5 and M=100. Closed squares: A=1/(2√n).

Conclusions:There are several connections between dynamical systems theoryand nonequilibrium statistical mechanics,but none of them is particularly simple.

Dynamical properties of equilibrium systems seem unrelated to traditional properties of decay to equilibrium.

Fluctuation and work theorems look potentially useful.

SRB-measures may be the tool to use in stationary nonequilibrium states.

Thanks to many collaborators: Bob DorfmanRamses van ZonAstrid de WijnOliver MülkenHarald PoschChristoph DellagoArnulf LatzDebabrata PanjaEddie CohenCarl DettmannPierre GaspardIsabelle ClausCécile AppertMatthieu Ernst