Geen diatitel

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Chaos and the physics of non-equilibrium systems
Henk van Beijeren Institute for Theoretical
Physics Utrecht University
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CONTENTS Dynamical systems Lyapunov
exponents KS entropy Chaos and approach to
equilibrium The Lorentz gas Diffusion
coefficient Lyapunov exponents Connections? Coll
ective Lyapunov modes What connections DO
exist? Gaussian thermostat formalism Escape
rate formalism Hausdorff dimensions of
hydrodynamic modes Thermodynamic
formalism Other connections Calculations for
moving spheres or disks Conclusions
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Dynamical Systems Theory For flows

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For maps
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• Lyapunov exponents

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Kolmogorov-Sinai entropy
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• Pesins theorem

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Is chaos related to approach to equilibrium?

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

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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).
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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.
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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.
Can one somehow connect these concepts?
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Twodimensional Lorentz gas
Regular Sinai-billiard
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Random Lorentz gas
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There is one positive Lyapunov exponent. It may
be estimated easily
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• Density dependences are very different.

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• Density dependences are very different.
• Various other differences as well
• Diffusion coefficient diverges for Sinai billiard
  with infinite horizon.

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• 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.

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• 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.

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Wind tree model
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System behaves diffusively on large time and
length scales. It shows mixing behavior,
but power law with time. So the KS entropy equals
zero. Perhaps a definition of weak,
nonexponential chaos is needed to describe this.
only increases as a
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• 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!

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Are smallest Lyapunov exponents of many-particle
systems related to hydrodynamics?
Lyapunov spectrum for 750 hard disks (Posch and
coworkers)
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Like in hydrodynamics there are branches of
k-dependent eigenvalues that approach zero in
the limit
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 velocity
deviation in x-direction as a function of
y-coordinate. Growth rate is proportional to k
(vs. decay rate k2 for hydrodynamic shear
mode).
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What connections do exist?
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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
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Systems under external driving forces are kept at
constant kinetic (or total) energy by applying
fictitious thermostat forces, such that
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.
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What connections do exist?

• Most of them consider changes in dynamical
  properties due to
• deviations from equilibrium.
• Gaussian thermostat formalism of Evans and
  Hoover
• The escape rate formalism of Gaspard and Nicolis.

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For finite systems with open boundaries, through
which trajectories may escape, the KS entropy
satisfies
Survival rate of
so this relationship
For diffusive systems connects a transport
coefficient with dynamical systems properties.
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What connections do exist?

• Most of them consider changes in dynamical
  properties due to
• deviations from equilibrium.
• Gaussian thermostat formalism of Evans and
  Hoover
• The escape rate formalism of Gaspard and Nicolis.
• Relationships between Hausdorff dimensions of
• hydrodynamic modes, Lyapunov exponents and
• transport coefficients, obtained by Gaspard
  et al.

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For two-dimensional diffusive systems, Gaspard,
Claus, Gilbert and Dorfman obtained the
relationship
This can probably be generalized to higher
dimensions and general classes of transport
coefficients.
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• Other connections between dynamical systems
  theory and
• nonequilibrium statistical mechanics involve
• Fluctuation theorems (Evans, Morriss, Searles,
  Cohen, Gallavotti
• and others) relate the probabilities of
  finding fluctuations in
• stationary systems with entropy changes of
  respectively

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• Other connections between dynamical systems
  theory and
• nonequilibrium statistical mechanics involve
• 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.

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• Other connections between dynamical systems
  theory and
• nonequilibrium statistical mechanics involve
• 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. Ruelles thermodynamic formalism.

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Dynamical partition function Topological
pressure In general,
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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.
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Moving hard spheres and disks
For moving hard speres at low density the
velocity deviations of two
colliding particles are both upgraded to a
value of the order of .
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Set The distribution of these clock values
approximately satisfies . Can be solved
for stationary profile of the form P(n,t)P(n-vt)
by linearizing for large n. Then v log(lmf /a) is
the largest Lyapunov exponent. It is determined
by .
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Gives rise to largest Lyapunov exponent Keeping
account of velocity dependence of collision
frequency one may refine this to Finite size
corrections are found to behave as May be
compared to simulation results
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Brownian motion
We 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 strongly than 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 ltxgt can be
identified as the largest Lyapunov exponent
connected to the Brownian particle.
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These satisfy the Fokker-Planck
equation, Both diffusion constants are
proportional to Therefore ltxgt scales as
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Maximal Lyapunov exponent for 2d system with 40
disks of a1/2 and m1. Open squares pure_fluid.
Crosses A5 and M100. Closed squares
A1/(2vn).
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Conclusions
There are several connections between dynamical
systems theory and 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.
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Thanks to many collaborators
Bob Dorfman Ramses van Zon Astrid de Wijn Oliver
Mülken Harald Posch Christoph Dellago Arnulf
Latz Debabrata Panja Eddie Cohen Carl
Dettmann Pierre Gaspard Isabelle Claus Cécile
Appert Matthieu Ernst