Phase transitions and fluctuations in mesoscopic systems

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Phase transitions and fluctuations in mesoscopic
systems

• Laszlo P. Csernai University of Bergen, Norway
• Idea-Finding Symposium for the
• Frankfurt Institute for Advanced Studies
• April 15 - 17, 2003

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Contents

• Meso-scale physics, static dynamic problems
• Dynamic fluctuations in other fields
• Stochastic resonances
• Dynamics of Neural Networks (NN)
• Dynamics of financial markets
• Outlook Diagnostics, energetics and laws of
  signal noise in fluctuating mesoscopic systems

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Phase transition A New State of Matter
- QGP
ITP, U.Frankfurt
The combined data coming from the seven
experiments on CERN's Heavy Ion programme have
given a clear picture of a new state of matter. .
. . We now have evidence of a new state of
matter where quarks and gluons are not confined.
There is still an entirely new territory to be
explored concerning the physical properties of
quark-gluon matter. L. Maiani, 2000
Not too many particles !?!?
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Phase transition
STATIC
In macroscopic systems two phases of different
densities (e) are in phase equilibrium.
Negligible density fluctuations!
Csernai, Kapusta, Osnes, PRD 67 (03) 045003
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Small, Mesoscopic Systems
STATIC
If N100, fluctuations are getting strong
(red). Close to the critical point, the two
phases cannot be identified (green). gt Landaus
theory of fluctuations near the critical
point. Nuclear Liquid-Gas phase transition
(first order)
Goodman, Kapusta, Mekjian, PRC 30 (1984) 851
CRAY - 1
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Lattice Field Theory
STATIC
First order (EW) phase transition statistical
ensemble. Fluctuations of density decrease with
increasing Lattice volume !! For macroscopic EoS
extrapolation is needed! For small systems,
100-200 fermi3, fluctuations are REAL !!!
Csernai, Neda PL B337 (94) 25
Farakos, Kajantie, et al. (1995) hep-lat/
Supercomputers are needed !
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Large Fluctuations
STATIC gt DYNAMIC

• The system is (effectively)
  mesoscopic !!!
• Difficult to determine the EoS, matter properties
• In dynamical systems Non-linearities develop
• Landau theory is overly idealized
• Computational Physics Approach
• Create statistical ensembles numerically to
  study EoS Dynamics!

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Dynamics (Lattice Field Theory)
DYNAMIC
Chern-Simons number asymmetry / matter
antimatter asymmetry
Examples of single element trajectories from
statistical ensembles. Three different
parametrizations, saturation is reached in 4
fm/c Initial statistical ensemble gt time
development of the ensemble
START
( 8 fm/c )
TIME
Smit, Tranberg, (03) hep-ph/0211243
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DYNAMIC
Heavy Ion Coll. at RHIC - Transverse velocities
- b0.5
Strottman, Magas, Csernai, BCPL User Mtg.
Trento, 2003
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Event by event Fluid Dynamics
DYNAMIC
Ensemble of solutions of Eulerian FD with
Langevin forces (thin lines), Their average (open
circles) Smooth Shock Wave gt Viscosity arises
from fluctuations via the Fluctuation
Dissipation theorem for fluids Csernai, Jeon,
Kapusta, PR E 56 (1997) 6668.
More realistic for mesoscopic or relativistic
systems than the Navier-Stokes equation approach !
Csernai et al., PR E 61 (2000) 237.
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Dynamics of Phase Transitions
DYNAMIC

• Homogeneous nucleation Csernai, Kapusta
• PR D46(92)1379
  PRL 62(92)737
• Domain wall dynamics Csernai, Kapusta, Osnes,
• 
  PR D67 (03) 045003
• For ultra-relativistic energies, baryon free
  matter gt no heat conductivity !
• (!) Used for fragment formation in chemistry
  too.

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Mesoscopic Systems - Fluctuations
DYNAMIC

• Can be studied accurately by models in
  microscopic approaches
• Few general features are known yet, so we are
  still far from the level of knowledge like the
  Laws of macroscopic Thermodynamics !
• Physics is a good test field Theory Exper.
• Many fields are relevant for these studies
• neural networks, process technology,
  meteorology, chemistry, fuel cells, surface
  reactions, radio noise, financial markets,

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General Phenomena or Laws in Mesoscopic systems
DYNAMIC

• Are there common features in these fields?
• Perceptive learning in Neural Networks
• Surface phenomena
• Process technology
• Etc.
• Some general features or phenomena may exist.
  E.g.
• Stochastic Resonances

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Stochastic Resonances Two state system
DYNAMIC
All states with a phase transition have a
potential with two metastable state.
Potential barrier

• Incoming signals
• Stochastic noise - (!) TUNABLE
• Harmonic modulation, weak
• 2nd law of thermodynamics
• Out
• (1) Oscillating signal (resonating)
• (2) Lower T noise

R. Benzi, et al 1981, 82, 83 .
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Stochastic Resonance
Periodic System Response
Weak periodic, harmonic forcing, 0.4 x 0.2 x 0.1
x of the potential barrier, ?V.
- Noise Strength ?V
No response without noise! Harmonic forcing is
too weak!
Gammaitoni et al, Rev Mod Phys 70 (98) 223
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Phase synchronization by SRs
System response
D Strong noise Medium noise (res.) gt
synchronization! Weak noise
Signal in
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Neural Networks

• Anatomy physiology of NN gt adv. in modelling,
  MCP., J von Neumann,,,
• Csernai, Zimanyi, Bio.Cyb. 34 (79) 49 Self
  organized perceptive learning in NNs with lateral
  inhibition.
  -
• Anatomy, physiology dynamics are related and
  vital for the function of NNs
  Increased synchronicity W. Singer, et
  al., (2002)

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Noise Pattern Recognition - Neural Networks
Are NN meso- or nano- scale systems? - YES!
Fluctuations are dominant in many cases. These
are important to achieve a good Signal to Noise
(S/N) ratio !
Frank Moss, C. for Neurodynamics, U of
Missouri at St. Louis
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Recent application of wavelet theory
What is S N ?
The International Standards Organization's
JPEG2000 committee has finalized specs for a new
algorithm that compresses images up to 200 times
with no appreciable degradation in quality. The
JPEG2000 spec, which will become ISO 15444 when
it's officially approved in 2001, uses wavelet
transformations instead of Fourier transforms to
achieve the performance gain. JPEG2000 also
strives to overcome the worst effects of current
JPEG compression, which processes images in 8 x
8-pixel blocks, leaving coarsely spaced artifacts
that spoil fine details. JPEG2000 promises to
compress images 200 times with better resulting
quality than current JPEG images compressed
fivefold.
J. Pipek, BCPL User Mtg. (2002)
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Financial Markets Econophysics (?)
Dow Jones Industrial Avg. 1 year (Apr. 2000
Mar. 2001) 50 stocks
max
close
Variation 10 (1 day 4)
min
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Financial Markets
Is this a mesoscopic system ?
Ford Motor Company (1 year)
Variation 19 (1 day 4 )
Subtract 3 month moving avg. gt
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Simulation
Mesoscopic ? YES ! How many?
20 large investors 150 brokers 5000 small invtrs.
Let us change some conditions gt
Variation 19 (1 day 4 ) SAME as Ford M.
Co.
Csernai, Pipek (2003) in prep.
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Simulation
Introduce a tax gt 20 large investors 150
brokers 200 small invtrs. Fluctuation Increases
!!!!
Variation 25 (1 day 8 )
Csernai, Pipek (2003) in prep.
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Conclusions

• Fluctuations indicate a mesoscopic system
  sophisticated and variable DIGNOSTIC tools are
  needed to analyse and understand such systems.
• Many tools exist already (e.g. wavelets analyses,
  Alfred Haar (1910) Martin Greiner or Ingrid
  Daubechies (1988) Janos Pipek).
  Supercomputers help and very much needed!

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Conclusion 2 and Outlook

• Stochastic Resonance(s) ? many varieties
  ____________________ similar to heat
  engines (e.g. Carnot Machine)

Strong Noise (T1)
Strong Signal
SR
Weak Signal
( or Maxwell Demon ? )
??? Weak Noise (T2)
Mesoscale experiments and Supercomputer modelling
!
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