Introduction to Stochastic SimulationRan Zhou, Jing Mu, Tianyu Liang, Yunxiang Zhang, Zhan Zhang(Project Leader), Ikemefuna Agbanusi(Faculty Mentor)

Illinois Geometry LabIGL Open House, May 5th, 2016

Main Problem

In this project, we have studied the movements ofparticles undergoing Brownian Motion in a boundeddomain, specifically, we have worked on:

The average time needed for a moving particle toleave a given domain for the first timeThe end point that a particle will leave from of a givenregionThe path of particles’ movements in a given regionwith different boundary conditions

Theoretical Model

Let u(x, t;x0) be the probability density function of findinga diffusing particle at location x at time t given that it wasinitially at location x0, u(x, t;x0) satisfies the heat equation

ut −Duxx = 0; 0 < x < L; t > 0

u(0, x) = δ(x− x0)

where x0 ∈ (0, L).Survival Probability:The probability that the diffusing particle is in theinterval [0, L] at time t given that it started at locationx0

S(t, x0) =

∫ L

0

u(t, x;x0)dx

CDF of First Passage Distribution:The probability that the particle has left from eitherboundary point by time t

P (τ ≤ t) = 1− P (τ ≥ t) = 1− S(t, x0)

where τ (x0) is the first exit time of the particleNo Passage Location Distribution:The conditional probability that particle appears in[0, z] given it has not left [0, L] at time t

G(t, z;x0) =

∫ z0 u(t, x;x0)dx

S(t, x0)

Probability of Exiting from a specific endpointThe probability of exiting at x = 0 given the particle isexiting at time t

j0(t) =ux(t, 0;x0)

ux(t, 0;x0)− ux(t, L;x0)

Simulation Results: Diffusing Path and Mean First Exit TimeDiffusing Path: Simulating paths by generating location of the particles at each time stepMean First Exit Time: For each initial location, generate several paths and calculate the mean exittime

Boundary Condition I : Dirichlet-Dirichlet

Starting at the midpoint of an interval withDirichlet-Dirichlet boundary condition, theparticle has equal probabilities of exiting fromeither end point

The closer the initial location is to the endpoint, the faster, on average, the particle willleave the domain

Boundary Condition II : Dirichlet-Neumann

Starting at the midpoint of an interval, theparticle will bounce back from the end withNeumann condition and exit from the end withDirichlet condition

The closer the initial location is to the endpoint with Neumann condition, the more timeit takes for the particle to exit the interval

ComparisonIn case of Dirichlet-Neumann condition, once the particle reaches the Neumann side, it will bounceback. Therefore, it takes longer, on average, for a particle to exit the region with Dirichlet-NeumannBoundary condition than it takes for Dirichlet-Dirichlet case.The Mean First Exit Time graphs also show that the empirical mean exit time distributions follow ourtheoretical results

Simulation Algorithms

We want to generate the path of a moving particle startingfrom initial location x0 in an interval [a, b]. We divide [a, b]into some subintervals [xi−1 − δ

2, xi−1 + δ2] and sample the

exit time ∆t and location xi in each subinterval usingMonte Carlo SimulationThe Algorithm:dt← Time intervalδ ← Length of subintervalti← Current timexi← Current Locationwhile xi ∈ [a, b] do· Sample first exit time ∆t from First PassageDistribution in current subinterval· Sample corresponding exit locationx′

i ∼ Bernoulli(jxi−1−δ2(∆t))

if ∆t < dt thenti← ti−1 + ∆txi← x

′

i

else· Sample No Passage location x′′iti← ti−1 + dtxi← x

′′

i

end ifend while

Applications

The project has applications in studying the process ofchemical reactions and behaviors of moving molecules:

The time needed for aprotein molecule to finda specific targetHow long does it takefor a patient to feel theeffect of a medication

Future WorkOur further work includes applying the 1-D case to:

Other mixed boundary conditions like Robin-RobinMultiple particles on a single lineMultiple particles in higher dimensions

These posters are made with the support of University of Illinois at Urbana-Champaign Public Engagement Office