NUMERICAL SIMULATIONS OF PROPAGATION IN DISORDERED SYSTEMS

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NUMERICAL SIMULATIONS OF PROPAGATION IN
DISORDERED SYSTEMS

• Jacob Yunger Gabriel Cwilich
• Yeshiva University
• New York, USA

Partially supported by the Office of the
Vice-President for Academic Affairs, Yeshiva
University
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OBJECTIVES OF THIS WORK

• Simulations of the propagation of rays on a two
  dimensional random system with spherical
  scatterers (performed in Mathematica)
• Investigated the statistics of the lengths of the
  paths that are transmitted or reflected through
  the system
• Wave pulses sent through a random medium can also
  be used as an imaging tool

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THE SYSTEM

• Density of the system (area of
  scatterers/total area) 46
• Mean free path (for hard
  spheres) 1.707, in units of scatterer
  diameter

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THE ALGORITHM

• The walkers are launched from a random position
  at the left boundary, and they move along the
  horizontal direction in a ballistic trajectory
  with steps of a preset size.
• This walk continues until the walker reaches a
  scatterer, at which point it changes direction
  also according to a preset algorithm that
  characterizes the process of scattering.
• The walker then continues- again ballistically-
  in a different direction. This procedure is
  repeated until the walker either returns back to
  the left boundary or arrives to the right
  boundary.
• We considered absorbing, reflecting or periodic,
  lateral (horizontal) boundary conditions.

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EARLY SIMULATIONS

• Angle of scattering, random variable uniform
  over 0 , 2?
• Filling fractions 46 and 23 .
• Lengths of the system 33, 67, 100, 133, 167
  (scatterer diameter 1)
• We recorded the statistics for length of
  transmitted paths, length of the reflected paths,
  number of scattering events (in transmission and
  in reflection), fraction of all the paths that
  transmitted.
• We studied the dependence of the average
  quantities with L and obtained the exponent of
  such dependence assuming algebraic decay.
• ltrans ? L?, lrefl ? L? , T ? L-? . ? 1.828,
  b? 1.024, ? ?0.609.
• Reasonable agreement with diffusive theory.

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LATER SIMULATIONS

• Explored different scattering mechanisms to see
  the effect on the diffusive properties of the
  system
• 1000 walkers to improve statistics, more lengths.
• Density 46. Lateral periodic boundary
  conditions.
• In the random-backwards or random-forwards
  mechanisms, the angle is chosen with a uniform
  distribution within a preset range.

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SOME ILLUSTRATIVE PATHS
Random-backwards scattering (range 0.35)
Specular scattering
Random-backwards scattering (range 0.35)
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SCALING OF THE LENGTH OF THE PATHS
Transmitted Paths
Reflected Paths
Transmission
Exponential Dependencies Experimental and
Theoretical Values
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MEAN FREE PATH
Transmitted Paths
Reflected Paths

• Independent of length.
• At short lengths statistics is dominated by the
  short paths.

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TRANSITION TO DIFFUSION

• Tuning the range of the angle of scattering we
  study the transition from the ballistic to the
  diffusive regime.
• We determined the dependence of ltl
  transmgt and lt lreflgt with L, for different values
  of the range of the angle of scattering.
• The onset of diffusion (Exp ? 2) is smooth (no
  evidence of an abrupt transition.
• In reflection, for small lengths, soft dependence
  with L, since the typical paths do not explore
  the bulk.

TRANSMITTED PATHS
REFLECTED PATHS
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• Path length in transmission for a fixed system
  size
• random-forward scattering algorithm
• We see clearly transition ballistic ? diffusive,
  at range 0.6
• For the diffusive behavior length ? (range)1.33

Slope 1.33
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IMAGING

• Use the distribution of exiting positions of the
  random walkers to get information about the
  position of an inclusion.
• Shadow effect in transmission and increased
  intensity in the reflection are combined to
  locate the inclusion.
• Scattering algorithm random- forward with a
  small range of randomness -0.35,
  0.35 (Quasi-ballistic). The
  inclusion scatters specularly
• Walkers launched from random positions at the
  left boundary. Lateral absorbing boundary
  conditions

• Inclusion of radius 5 placed at different
  positions in the system.
• Histograms of the distribution of the exiting
  positions were obtained.

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IMAGING RESULTS

• The maximum in the distribution tracks quite
  successfully the position of the impurity.
• Similar results were obtained for the
  distribution of transmitted rays when the
  impurity is close to the right boundary

Reflection histograms
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• The vertical position of the reflected spot does
  not change it widens and become less intense as
  the inclusion moves away from the boundary.
• Combining the histograms of the reflected and
  transmitted rays, we obtain a good quantitative
  determination of the vertical position of the
  inclusion and qualitative information on its
  horizontal position
• We would need to increase statistics to locate
  even smaller inclusions with greater quantitative
  accuracy

Width 27.5
Width 20
Width 15
Width 25
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CONCLUSIONS

• Classical propagating trajectories can be
  simulated in real space their random-walk like
  propagation describes correctly the statistical
  properties of path lengths in the diffusive
  regime
• The system can be tuned to go from the ballistic
  to the diffusive regime to study the transition
• This simulation can be used as a tool of
  numerical experimentation to discuss the
  statistical properties of the paths
• Simple imaging problems can be considered at the
  level of the classical walks with reasonable
  degree of exactitude