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5.3 Particle orbits
Slide : [
deterministic 
Taylor 
stochastic 
FeynmanKac 
VIDEO
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In the Monte Carlo method, the initial condition or indeed any function is
discretized using quasiparticles; differential
operators are therefore reformulated in terms of particle motions.
Remember that a quasiparticle is defined in (1.4.7#eq.1)
by a weight
, a position
and a shape function
; for
simplicity, we assume here unit weights
and point shaped particles
in the form of Dirac pulses
. The solution can then be
calculated from the particles density obtained from the positions
called particle orbits. By construction, the lowest
order moment is perfectly conserved, since no particle is lost.
To show the contrast between the Taylor and the FeynmanKac
theorems, which provide a rigorous foundation for the Monte Carlo method,
the deterministic and the stochastic motion are first
considered separately.

Deterministic orbits
 satisfy an ordinary differential equation for
the position
as a function of time

(1) 
or

(2) 
This is solved numerically using the methods discussed in section 1.2.1, using an explicit or implicit discretization of time
For an ensemble of
particles with orbits
, the particle
density distribution function
is obtained by superposition

(4) 
and is plotted in the applet below using the
projection (1.4.7#eq.3) from section 1.4.
JBONE applet: press Start/Stop
to simulate the deterministic motion of a particle or random walker
(red dot).
The contribution to the density distribution (black line) is obtained
from a linear assignment of the position on a coarse grid.

If a large number of particles approximate a smooth density function
and each individual orbit
evolves in a deterministic manner

(5) 
then the Taylor's transport theorem states that the evolution of
the particle density distribution function
satisfies the
advection equation

(6) 
The theorem makes it possible to study a PDE instead of an
particle system; the reverse is also true and will be used to
solve the advection equation with particles.

Stochastic particle orbits
 are defined as an ensemble of possible
orbits with different probabilities. Imagine a snowflake falling slowly
from the sky: the motion is unpredictable and each evolution can be
described with an orbit
following the stochastic differential
equation

(7) 
where
is a Wiener processes (Brownian motion). Averaging over
a large number of orbits, the average velocity and the broadening of
possible orbits are readily obtained (exercise 5.02) as
Stochastic orbits can therefore be parametrized with statistical averages

(10) 
and lead to a different result each time the JBONE applet is
executed.
JBONE applet: press Start/Stop
to simulate the random motion of a particle (red dot).
The contribution to the density distribution (black line) is obtained
from a linear assignment of the position on a grid.

In the same spirit as in Taylor's transport theorem, it is possible to
relate the evolution of a large number of stochastic particle orbits to
a PDE; the socalled FeynmanKac theorem tells that a smooth
density distribution function
in which individual particles
move according to

(11) 
evolves according to the FokkerPlanck equation

(12) 
which is in fact an advectiondiffusion equation.
For example, let
be a stochastic process with an evolution of the
probability density distribution
following the advectiondiffusion
equation

(13) 
From the FeynmanKac theorem, it is easy to show (exercise 5.01) that
satisfies the stochastic differential equation
showing exactly what evolution is required to reproduce statistically the
properties desired at the macroscopic scale.
SYLLABUS Previous: 5.2 Stochastic theory
Up: 5 MONTECARLO METHOD
Next: 5.4 A scheme for
