Billiard-like
systems and
different models of
diffusion
Математические бильярды
is a dynamical system in
which a particle moves
straight in free motion and
collides elastically with a
boundary.
Q
Q
0Q
Q
Bunimovich stadium
Q +
Sinai billiard
Correlations decay exponentially quickly
the angle of incidence is equal to the angle of reflection
Lyapunov exponent is a quantity that characterizes the rate of
separation of infinitesimally close trajectories
the flow of particles
is ergodic with mixing
The distribution of
particles coordinates is
homogeneous for all
regions of billiard
the particle motion is
characterized by two
coordinates x and y,
and the angle of the
direction of velocity
vector
the area of the accessible
billiard region
PDF is homogeneous
The probability of a collision per unit of time with a part of the scatterer
boundary with the length L at the angle of inclination
The probability of a collision per unit of time with any of the scatterers
scatterers perimeterThe mean
free path
The probability distribution for the angle of inclination does
not depend on the shape of the scatterer boundary
From the law of conservation
of momentum and energy:
boundary
velocityparticle
velocity
after n-th
collision
particle
velocity
before n-th
collision
periodic motion of the boundaries
Fermi acceleration for random oscillations
of the scatterers boundary
The particle speed as a function of time at diferent mean-free-paths. The
amplitude of the boundary velocity u0=0.3. The analytical results are presented by straight lines. The results of numerical simulation are presented
as fluctuating colored lines.
The Fermi acceleration as a function of radius of scatterers.
The boundary velocity u0 = 0.2; 0.4; 0.3.
velocity of the scatterer boundary is a harmonic function
Mean time
between
collision
The variation in the
size of the scatterers
is much less than
the mean free path
Sketch of particle acceleration as a random function of time with
complex statistics. Each pulse corresponds to a collision. The
heights of the pulses are modulated by cosines and is a random
variable, since the angle of incidence is random. Accelerating
positive pulses are more frequent than slowing negative ones.
The mean time interval between neighboring pulses decreases
with time, since a particle accelerates and collisions occur more
frequently.
The probability of a collision per unit of time
the velocity of the particle
relative to the moving boundary
the mean speed variation
only the terms with give nonzero output after integration
Fermi acceleration in the case of harmonically moving boundaries of
scatterers is four times greater than for stochastic oscillations.
The particle speed as a function of time at different amplitudes of the
boundary velocity u0. The analytical results are presented by black straight
lines for u0 = 0.2; 0.15; 0.1; 0.05. The results of numerical simulation with the
same values of u0 are presented as fluctuating colored lines in the clusters near the corresponding straight lines. Different lines in the cluster correspond
to different periods of oscillations 10
Fermi acceleration as a function of the oscillation period of the scatterers
with amplitudes of the boundary velocity u0, the same as in previous Fug.
Analytical results are presented by straight lines, the results of numerical
modeling are shown by dots. Their slopes from the analytical line are
random and independent of the period.
Random walk
Random walk
Velocity
Random walk in two dimensions
Random walk in one dimension
Fractal
Levy flight superdiffusion
Fermi acceleration strongly influences the diffusion of particles. A
linear increase in the particle velocity causes a linear growth in the
diffusion coefficient with time, which corresponds to one of the types
of superdiffusion in the form of ballistic diffusion
The mean square of displacement of the particle and root-mean-
square deviation (RMSD)
the time-dependent diffusion coefficient
the coeffcient of superdiusion
The Fermi acceleration mechanism is useful for explaining the extremely fast
diffusion of mass-selected gold clusters deposited on graphite
The anomalously high diffusion coefficients for different metallic
clusters on highly oriented pyrolytic graphite (HOPG) were
discovered in experiments. It was shown that the properties of the
graphite substrate are probably responsible for the effect, since
fast diffusion was observed on graphite for different metallic
clusters. The clusters interact with a part of the graphite layer
called the ‘flake’. These flakes are involved in thermal motion as a
whole.
The Fermi acceleration model gives an estimate of the diffusion
coefficient of clusters and resolves the contradiction between extremely
large diffusion coefficient and its dependence on temperature according
to the Arrhenius law
The energy of the chaotic motion of the cluster increases in
time. This fact can be explained using the Fermi acceleration
model
Superlubricity of graphite flakes
A slow moving flakeA fast moving flake
Life cycle of free clusters: the deposition, diffusion and
aggregation of clusters that results in the formation of
islands.
Tasks to solve
1 Prove that diffusion coefficient for the square
lattice of randomly moving scatterers is
2 Find more examples of anomalous diffusion