chapter 3 random process partial differential...

Chapter 3

Random Process

&

Partial Differential Equations

Deterministic model ( repeatable results )

Consider N particles with coordinates 𝐫𝐫1, 𝐫𝐫2,⋯𝐫𝐫𝑁𝑁 the interactions between particles modeled by potential

𝑉𝑉 𝐫𝐫1, 𝐫𝐫2,⋯𝐫𝐫𝑁𝑁 the dynamics of the system represented by ordinary differential

equations

2

2 , 1, 2,3i

i idm i Ndt

= =r f

These coupled equations can be solved using numerical method

( )1 2, , , ,ii Ni i i

V V VVx y z

∂ ∂ ∂= −∇ = − − − ∂ ∂ ∂

rf r r r

3.0 Introduction

Verlet algorithm

相加

相减 ( )2O t+ ∆

Verlet integrator is an order more accurate than integration by simple Taylor expansion alone, with the same term 𝛥𝛥𝑡𝑡2.

3.0 Introduction

Taylor expansion

Verlet algorithm is not self-starting，we will use velocity Verletalgorithm in molecular dynamics simulations.

( )3O t+ ∆

( ) ( ) ( ) ( ) ( )22t t t

t t t t O t+ + ∆

+ ∆ = + ∆ + ∆a a

v v

3.0 Introduction

velocity Verlet algorithm

Probabilistic model ( unrepeatable results )

Consider the same system of N particles as before

• when there is uncetainty, say 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑓𝑓𝑟𝑟𝑟𝑟𝑓𝑓𝑓𝑓 𝐑𝐑𝑖𝑖, which has

a Gaussian probability distribution with correlation function

( )2

2 , 1, 2,3i i

i i id dm t i Ndt dt

γ+ = + =r r f R

( )1 2, , , ,ii Ni i i

V V VVx y z

∂ ∂ ∂= −∇ = − − − ∂ ∂ ∂

rf r r r

Each particle's motion can be described by certain probabilities, derived from Fokker- Planck Equation.

3.0 Introduction

( ) ( ) ( )' 6 'i i Bt t k T t tγ δ⋅ = −R R 𝛾𝛾: friction coefficient

Langevin Equation:

Tamás Vicsek, Collective motion, Physics Reports, Vol 517, 2012, Pages 71-140

(a) Wingless Locusts marching in the field.

(b) A rotating colony of army ants.

(c) A three-dimensional array of golden rays.

(d) Fish are known to produce such vortices.

(e) Before roosting, thousands of starlings producing

a fascinating aerial display.

(f) A herd of zebra.

(g) People spontaneously ordered into traffic lanes

as they cross a pedestrian bridge in large

numbers.

(h) Although sheep are known to move very

coherently, just as the corresponding theory

predicts, when simply hanging around (no

motion), well developed orientational patterns

cannot emerge.

3.0 Introduction

3.0 Introduction

our primary goal:

• to investigate the connection between probabilistic and deterministic models of the same phenomenon.

probabilistic deterministic

Connection ?

Micro view:single random process

Macro view: (ensemble average)definite distribution function

described by partial differential equation

3.0 Introduction

• Looks paradoxical that a random process can be characterized by a definite equation • But we know it is true from a lot of daily experience, such as coin tossing.

~50% ~50%

coin tossing

For single toss: no idea whether head or tail turns up. After a large number of tosses, proportion of heads or tails is ~0.5. With this example, it is not so surprising that there is a determinable

distribution of probabilities which characterizes a random process.

many parameters unknown • the initial orientation, velocity, and spin; • the properties of the table surface; • Various atomic defects, dislocations, grains, voids…

3.0 Introduction

Probabilistic model originates from• Incompleteness of information

e.g. coin tossing• Parameters sensitivity --- tiny perturbation in input induces huge variation

in output.e.g. In kinetics of gases, a slight change in the initial conditions would result in a tremendous change after many collisions.

By an averaging of the solutions with varying initial conditions random processes can be modeled successfully.

3.0 Introduction

• Section 3.11-D Brownian motion. An explicit expression of the probability w(m, N).

• Section 3.2Simplified expression of w(m, N). Asymptotic Series, Laplace’s Method

• Section 3.3a difference equation for w(m, N) leads to a partial differential equation.

• Section 3.4The connection between probability and differential equations.

in the following sections,

3.0 Introduction

In Brownian motion, small particles move about in liquid or gas.

3.1 Random Walk in One Dimension; Langevin’s Equation

• The Roman Lucretius's (卢克莱修) described Brownian motion of dust in his scientific poem "On the Nature of Things" (60 BC)

• Botanist Robert Brown in 1827 studied pollen grains suspended in water.

• Albert Einstein in 1905 solved this problem.

• no possibility and no interest of computing the trajectory of each molecule.• one wishes to have an average understanding of the phenomenon.

https://en.wikipedia.org/wiki/Robert_Brown_(Scottish_botanist_from_Montrose)

The minimum model of random walk: 1-D lattice model

Particle moves according to the following rules:• Move in steps of a fixed length dx in a fixed time interval dt.• The probability to the right p and to the left q=1-p.

1b a= − a

Probability = number of observed eventstotal number of events

Goal: to obtain the probability w(m, N)• m steps to the right of the origin• N the total steps

3.1.1 An one dimension random walk model

random walk model can be found in various situations

a drunk staggering down a street

a gambling game in which a coin is tossed

Polymer Physics: Freely jointed chain model

……


Head ~50% Tail ~50%

Fair coin tossing

0.5 0.5


Random walk


To find w(m, N), the probability that a particle at a point m∈[-N,N] steps to the right of its origin after total N steps.

Suppose that the particle• p steps to the right, p>0• N-p steps to the left

Displacment mm = p - (N-p) = 2p – Np = (N + m)/2

N is even → m is even. N is odd → m is oddFor example,if N=3, the possible values of m = -3, -1, 1, 3.if N=4, the possible values of m = -4, -2, 0, 2, 4.

m

pN p−

3.1.2 Explicit solution

e.g. N=12 N-p=5 p=7 m=2

m

pN p−

To find out the number of paths with p steps to the right and N - p to the left.

The number of choices with p indistinguishable pink ball in N boxes

rightleft

Step: 1 2 3 4 5 6 7 8 9 10 11 12

Step: 1 2 3 4 5 6 7 8 9 10 11 12

Path 1:

Path 2:


equivalent

equivalent

Consider p distinguishable balls, which can be placed in N boxes in the following number of ways:

( )( ) ( ) ( )!1 2 1

!NN N N N p

N p− − − + =

−

1 2 4 3 57 8

Interchanging distinguishable balls does not change the pattern.There are p! permutations of p balls.

排列

12 4 3 57 8


( )!

! !Np

NCp N p

=−

( )! !

!Np

N C pN p

= ×−

number of ways pdistinguishable balls =can be placed in Nboxes

number of fullbox empty box patterns

number of full permutations ofdistinguishableballs within apattern

×

binomial coefficient

( )0

NN N N p p

pp

x y C x y−=

+ =∑


( )0

0

1,2

1 12 2

1 1 12 2

NN NNp

m N p

N p pNNp

p

N

w m N C

C

=− =

−

=

=

=

= + =

∑ ∑

∑

The sum of all probabilities is unity

• The total number of possible path is 2N• the probability that a particle at a point m steps to the right of

its origin after total N steps.

( ),2

NpN

Cw m N = where p = (N + m)/2


( ), 20w m

m

Characteristic functions• the characteristic function of any real-valued random variable

defines its probability distribution.

( ) ( )22 2 2 0 2 22i i i i iae be a e abe b eθ θ θ θλ θ − −= + = + +

Let

m=2 m=0 m=-2

these coefficients are the probability

( )2 2Pa a= ⋅

( )2 02

Pa b= ⋅

( )2 2Pb b−

= ⋅

( )NP m

( ) i iae beθ θλ θ −= +a1b a= −


characteristic function ( ) i i i xae be eθ θ θλ θ −= + = 1x = ±mean

( )

( )

( )

2 2

2 2 0 2 2 2

2 0 2 2 4

2 2 2 4

2

12

1 22

1 22

1 22

i

i i i i

i i i

i i

e d

a e abe b e e d

a e abe b e d

a d ab e d b e d

a

πθ

π

πθ θ θ

π

πθ θ θ

π

π π πθ θ

π π π

λ θ θπ

θπ

θπ

θ θ θπ

−

−

− −

−

− −

−

− −

− − −

=

= + +

= + +

= + +

=

∫

∫

∫

∫ ∫ ∫

To extract the coefficient analytically , for example as m=2

( ) ( )22 2 2 0 2 22i i i i iae be a e abe b eθ θ θ θλ θ − −= + = + +


Fourier transform( )2 2P

Generally, we extract the coefficient via Fourier transform

( ) ( )12

N i mNP m e d

πθ

π

λ θ θπ

−

−

= ∫

( )

( )

( )

( )

( )

0

0

0

2

0

12

12

12

12

12

Ni i i m

Ni N kN N k k i k i m

kk

Ni N kN N k k i k i m

N kk

Ni N pN p i p N p i m

pp

Ni p N mN p N p

pp

ae be e d

C a e b e e d

C a e b e e d

C a e b e e d

C a b e d

πθ θ θ

π

πθ θ θ

π

πθ θ θ

π

πθθ θ

π

πθ

π

θπ

θπ

θπ

θπ

θπ

− −

−

−− − −

=−

−− − −−

=−

− −− −

=−

− −−

= −

= +

=

=

=

=

∫

∑∫

∑∫

∑∫

∑ ∫


N Np N pC C −=

p N k= −

Thus, we have

( ) ( ), / 2N p N ppN C a b p N mP m −= = +

( ) ( ),2

when 1/ 2

Np

N N

CP m w m N

a b

= =

= =

( ) ( )( )

2

we notice0, / 211, / 22

i p N m if p N me dif p N m

πθ

π

θπ

− −

−

≠ += = +∫


Specifically,

General Characteristic function

( ) i i i xae be eθ θ θλ θ −= + =( ) ( )ikx ikxk e p x e dxλ∞

−∞

= = ∫Fourier transform of p(x).

• One important property of the characteristic function

( ) ( )0 1p x dxλ∞

−∞

= =∫


• probability density function (PDF) p(x). • displacement x is continuous. • the characteristic function is given by

General random walk model

• Consider a continuous 1-D random walk process of n steps• we have recursion relation:

( ) ( ) ( )1n nP x P y p x y dy∞

−−∞

= −∫

This means that the probability 𝑃𝑃𝑛𝑛 𝑥𝑥 of a particle at x after n steps is

• 𝑃𝑃𝑛𝑛−1 𝑦𝑦 the probability of arriving at y in n −1 steps

• 𝑝𝑝 𝑥𝑥 − 𝑦𝑦 the probability of displacements x-y in one step.

convolution


( ) ( ) ( )

( ) ( ) ( )

1

1

n n i ii

n n

P x P y p x y

P x P y p x y dy

−

∞

−−∞

= −

→ = −

∑

∫x1y 2y

( )1nP y− ( )1nP y−( )nP x

( )1p x y− ( )2p x y−

( ) ( ) ( ) ( )*f x g x f y g x y dy∞

−∞

= −∫

( ) ( )1 *nP x p x−=

( ) ( ) ikxn nP k P x e dx∞

−∞

= ∫

Let us define

( ) ( ) ( )1n nP k P k kλ−=

( ) ( ) ( ) ( ) ( ) ( )21 2 nn n nP k P k k P k k kλ λ λ− −= = = =

( ) ( ) ( )1 12 2

ikx n ikxn nP x P k e dx k e dxλπ π

∞ ∞− −

−∞ −∞

= =∫ ∫


( ) ( ) ( ) ( )*f x g x f y g x y dy∞

−∞

= −∫

( ) ( ) ( ) ( )1*2

F f x g x f k g kπ

= ⋅

( ) ( ) ( )1 *n nP x P x p x−=

( ) ( ) ikxk p x e dxλ∞

−∞

= ∫

convolution theorem

The expected value of function f is defined by

( ) ( ),N

m Nf f m w m N

=−

= ∑

( ) 1,2

NN Nn n n N

pm N m N

m m w m N m C=− =−

= =

∑ ∑ where p = (N + m)/2

( ) ( )0

1,2

NN NNp

m N pp p m w m N pC

=− =

= =

∑ ∑

mean displacementm

mean square displacement or variance2m

n-th moment

3.1.3 Mean, Variance, and the Generating function

In order to evaluate the various moments, we introduce the generating function

( ) ( )0

,N

p

pG u u w m N

=

=∑

( ) ( ) ( ) ( )10 0

' , ' 1 ,N N

p

p pG u pu w m N G pw m N p−

= =

= ⇒ = =∑ ∑

Example: to calculate 𝑟𝑟

0m =


( ) ( )0 0

1 1 1 112 2 2 2

N N p p NN NNp N N p

p pp p

G u u C C u u−

= =

= = = +

∑ ∑or

( ) ( ) ( )11 ' 1 / 22

NNG u u G N = + ⇒ =

since p = (N + m)/2

( ) ( )

( ) ( )

0

0 0

1 ,2

1 1, ,2 2 2 2

N

p

N N

p p

p N m w m N

mNNw m N mw m N

=

= =

= +

= + = +

∑

∑ ∑

/ 2p N=

( ) ( )10

' ,N

p

pG u pu w m N−

=

=∑

( ) ( ) ( ) ( ) 20

'' 1 1 , 1N

pG p p w m N p p p p

=

= − = − = −∑

Example：1/22 ?m =

( ) ( ) ( )20

'' 1 ,N

p

pG u p p u w m N−

=

= −∑

( ) ( ) 112

NNG u u = +

( ) ( )( ) 2 1'' 1 12

NNG u N N u − = − +

( ) ( )1'' 14

N NG

−=

( ) 2'' 1G p p= −

( ) 22 14 4 4

N N N Np p−

= + = +

m = 2p – N

( )22 2 2 2 22 4 4 42Nm p N p N N p N N N N N= − = + − = + + − =

1/22 1/2m N=


Its characteristic function

We obtain n-th moments using characteristic function

( ) ( )0

nnn

nk

d kx i

dkλ

=

= −

( ) ( ) ( )2 2 3 3

0

12! 3!

!

ikx ikx

n nn

n

k x k xk e p x e dx dx p x ikx i

i k xn

λ∞ ∞

−∞ −∞

∞

=

= = = + − + +

=

∫ ∫

∑

( )the th- moment n nn x p x x dx∞

−∞

= ∫Generally,


Theory by Einstein, experiment by Perrin

2 2 2 2 21 1 23 3

x x y z r Dt = + + = =

Einstein• Assume that the macroscopic resistance on the particle is

proportional to the velocity - by classical hydrodynamics• showed diffusion obey the statistical law

the diffusion coefficient D is given by

/D kT f=T : absolute temperature; K : Boltzmann’s constantf : the coefficient of resistance

6f aπµ=μ : viscosity coefficient; a : particle size

(Stokes’ law)

3.1.4 To determine Boltzmann’s constant from Brownian Motion

Verified by Perrin

Langevin’s equation ( )dm f tdt

= − +v v F

where v the velocity of the particle and m mass. The random force

follows Fluctuation-dissipation relation


( ) ( ) ( )' 6 'i i Bt t f k T t tδ⋅ = −F F

The modern theory of the Brownian motion

multiply with x, and take the ensemble average

( )dm f tdt⋅ = − ⋅ + ⋅

vx x v x F

( )2dm v f tdt

⋅ − = − ⋅ + ⋅

x vx v x F

2 0d f v

dt m⋅

+ ⋅ − =x v

x v

( ) 0t⋅ =x FNot correlated

2 2 stationary solutionf tm m mce v v

f f−

⋅ = + →x v


( )dm f tdt

= − +v v FTo solve

( )2 61 1 1 32 2 2

d rd d DtD

dt dt dt⋅

⋅ = = = =x x

x v

2 21 23

x r Dt= =

2m vf

⋅ =x v

2

2

1 1 32 2 21 32 2

m v f fD

m v kT

= ⋅ =

=

x v6f ak D D

T Tπµ

= =

6f aπµ=


Boltzmann constant

energy equipartition principle

作业： Ex. 4 Page 90

修正：Eq.(24) 应该为

( ) ( )( )

2

0

1! 1 2

k pk

pk

xJ xk k p

+∞

=

− = Γ + + ∑

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chapter 3 random process partial differential...

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