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