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Andrea Zoia
CEA/Saclay DEN/DANS/DM2S/SERMA/LTSD

Residence time and collision number statistics: a Feynman-Kac approach
International Conference "Random Processes, Conformal Field Theory and Integrable Systems"
September 19 - 23, 2011 Poncelet Laboratory, Moscow, Russia


Outline

Introduction Stochastic transport: random flights Collision number statistics and diffusion limit Exponential flights and residence time statistics Conclusions

Foreword: joint work with Eric Dumonteil and Alain Mazzolo

at LTSD laboratory, CEA/Saclay

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Stochastic transport
Random flights (Pearson's random walk, 1905)
Straight line `flights' (random length) Collisions Scattered with probability p Absorbed with probability 1-p

r0 rn

Renewal process with reorientation and reward Deceivingly simple: many open questions...

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Examples of random flights

Charge transport Neutron/photon flux

Search strategies

Plasmas

Porous media

Finance

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General framework
Collision number nV and residence time tV in a region V

Mean: average particle concentration in V Variance: uncertainty

Applications in reactor physics: neutrons or photons
power deposition and/or atomic displacements in a volume theory of Monte Carlo estimators: "collision" (nV) and "track length" (tV)

Hypotheses: iid flights, single speed, isotropic scattering
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Collision statistics
Statistics of collision number nV n in a volume V

Distribution:

Key of our analysis: moments d-dimensional setup (dependence on r0)
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Key ingredients
Isotropic point source: (r-r0) The propagator (r,n|r0) is the probability density of finding a particle in r entering the n-th collision, starting from r0 (*) Let (r,r') be the probability density of performing a flight from r' to r Define the transport operator If we define the n-iterated transport operator

it follows then (*)
The propagator depends on boundary conditions

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Collision statistics
Define the collision density (r|r0) Equilibrium (limit) distribution Then we have the moments

Kac integrals Stirling numbers of the second kind Link between V and nV
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Applications
Moment formula Numerical integration for arbitrary (r,r') Analytical calculations for simple geometries and propagators Example. d-dimensional "Gamma flights" in spherical geometries: random flights with Gamma-distributed lengths -1exp(-)/()

n-1 n

d

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Example 1
Gamma flights: d=3, =2, p=1, "transparent" boundaries r0 V is a sphere with radius R; walks can start inside or outside Collision density rn V

Moments:

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Example 1
Gamma flights: d=3, =2, p=1 Monte Carlo simulation (symbols), analytical curves (solid lines)

Fixed r0, varying R

Fixed R, varying r0

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Example 2
Gamma flights: d=1, =1, p=0.5, transparent boundaries Exponential flights: =1 r0 rn V is a "sphere" with radius R; walks can start inside or outside Collision density V

Omitted formulas
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Example 3
Gamma flights: d=1, =1, p=1, leakage boundaries Exponential flights: =1 Vr 0 rn+
1

Collision density

Method of images Omitted formulas
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Why? Proof (1)
Formal relation Survival probability Recall that Then, from we have where Rational function: for non-negative integer -m
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Polylogarithm (JonquiХre function)


Proof (2)
Define the operator Recall that Apply formal Neumann series Then From we can rewrite as
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an d

an d


Proof (3)
We have We can identify with It follows finally

Remark the recursion property

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Large nV behavior
Discrete moment generating function G(z|r0) Relation to the distribution

Moments expansion

It follows the small-z behavior

Hence from Tauberian theorems
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Simulations
Exponential decay of the distribution: simulations (circles) and log-lin fit (solid line)

n 3d Gamma flights (=2)

v

n 3d exponential flights
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v


Discussion: existence of Ck(r0)
Convolution Kac integrals Recursion

Hence the existence of Ck(r0) depends on (r|r0) depends on boundary conditions, p, and dimension d Worst case (transparent boundaries and p=1): d>2
Recurrent and transient walks: Polya's theorem

Remark that we have
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Use and abuse of the formula
+

Direct approach: from equilibrium distribution to moments
Knowledge of the process allows assessing collision statistics Example: neutron or photon transport

Inverse approach: from moments to equilibrium distribution
Knowledge of the moments allows assessing features of underlying process Example: biology or economics

Warning: it is a difficult problem!
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Diffusion limit

: flight length : domain size



Choose «

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Brownian functionals
Central Limit Theorem: when « , every "reasonable" random flight rn converges to Brownian motion Bn Analogously, F[rn] converges to F[Bn] What happens to F = When « , nV explodes We need a rescaling: the natural candidate is the time t = n ( / v) ?

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Diffusion limit (1)
Finite speed v (neglect absorption) Isotropy: Identically distributed flight times

Diffusion limit: small , which implies small We impose a finite ratio Residence time in V: Effects of boundary conditions r0

, an d

rn

r0 rn+
1

«
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Diffusion limit (2)
Introduce the distribution of

In Laplace space the distribution of the sum is

For any "reasonable" w(t), we have Then, which implies Hence,

when

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Diffusion limit (3)
We can combine with (p=1)

Rescale r by : each term in the sum gives Only the leading order m survives We finally obtain the celebrated Kac formula which is known for Brownian motion Moreover, we have the recursion property

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Exponential flights
Random flights with jump lengths

Physical meaning: homogeneous scattering centers Defining time: t=/v Markovian (memoryless) Chapman-Kolmogorov:
Forward transport operator

Collision density:
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Residence times of exponential flights
V Understanding the relation between tV Residence time nV tV V Diffusion limit "Transport"

«

Collisions: dots Residence time: solid line

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Kac moment formula
Moments z z
m-1

z Markovian: partition trajectory over z
i

z

4

0

z

1

z

2

z

3

Convolutions in phase space: (z

0

z) = (z

0

z1)*(z1 z2)*...*(z

m-1

z)

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Kac moment formula
We have then the convolution products

Fubini's theorem

Moment formula
Collision density

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Some calculations
Exponential flights in 1d ("rod model") Oversimplified model, but captures essential transport features Analytical results: compare the moments of nV and tV Two cases:
Leakage boundary conditions and pure scattering (homogeneous finite-size medium V surrounded by vacuum: first-passage problem) Transparent boundaries and absorption (homogeneous infinite medium: observe statistics on a finite-size domain V)

Set v=1, and rescale r=r/ (equal average flight time and flight length)
Directly compare the moments of nV and tV
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Leakage boundaries
Moments for isotropic source: depend on x0 and R

Average nV and tv unbiased to each other

... but have different variances

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Transparent boundaries
Moments for isotropic source: depend on x0, R and p

Average nV and tV unbiased (for any p)

... but have different variances

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Conclusions and perspectives
General mathematical framework for random flights Collision statistics and equilibrium distribution Exponential flights as a particular case: time statistics

Strong hypothesis: single space scale
Adequate model for `homogeneous' media In the diffusion limit, converges to Brownian motion

Transport in disordered (heterogeneous/anisotropic) media
Coexistence of many space scales Converges to anomalous diffusion (?)

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Thank you
Questions? References [ArXiv]
A. Zoia, E. Dumonteil, A. Mazzolo, Collision densities and mean residence times for d-dimensional exponential flights, Phys. Rev. E 83, 041137 (2011). A. Zoia, E. Dumonteil, A. Mazzolo, Collision number statistics for transport processes, Phys. Rev. Lett. 106, 220602 (2011). A. Zoia, E. Dumonteil, A. Mazzolo, Residence time and collision statistics for exponential flights: the rod problem revisited, Phys. Rev. E 84, 021139 (2011). A. Zoia, E. Dumonteil, A. Mazzolo, Collision statistics for random flights with anisotropic scattering and absorption, in preparation.

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Feynman-Kac formulae
Kac functional

Recursion for the moments

Infinite observation time:

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