how can random noise help us - purdue universityyipn/talks/cmumathclub0309.pdf · one of the ﬁve...

How Can Random Noise Help UsGlobal Transports from Thermal Fluctuations

Aaron N. K. YipDepartment of Mathematics

Purdue University

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How Can Random Noise Help Us– p.1/32

Basics of Random Walki.e. Drunken WalkerStart from the origin, to decide where to go next, youtoss a coin, if the outcome is ahead, you move onestep to theright , if it is a tail , you move to theleft.

5. . . . .....

−1−2−3

q p

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Let Xi = 1 or−1 if the i-th toss is a head or tail.Then yourposition after timen is given by:

Sn = X1 + X2 + · · ·Xn


Basics of Random Walk – 2

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1Coin Tossing Experiment

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Supposeheadcomes up withprobability p andtailwith probability q = 1 − p. (If p = q = 1

2, the coin is

calledfair .)

What is thestatisticsof your position,Xn (at timen)?How Can Random Noise Help Us– p.3/32

Binomial Distribution

Prob(Sn = j) =

n

n+j

2

pn+j2 q

n−j2

where

0

@

n

m

1

A =n(n − 1)(n − 2) · · · (n − m + 1)

m(m − 1)(m − 2) · · · (2)(1)=

n!

m!(n − m)!

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Position

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Probability Distribution of the Position of an Unbiased Random Walk after 20 Steps


Gaussian (Normal) Distribution

Gµ,σ(x) =1√

2πσ2e−

(x−µ)2

2σ2

Mean = µ, Variance = σ2

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0.4Standard Gaussian (Normal) Distribution, Mean = 0, Variance = 1

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Binomial converges to Gaussian Distribution

As n → ∞,

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x

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n

What is thisTheoremcalled?


Binomial converges to Gaussian Distribution

As n → ∞,

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0.14Comparison between Binomial and Normial Distribution

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Central Limit Theorem


Gaussian Distribution and Heat DiffusionHeat Equation – describes thediffusion of heat:

∂u

∂t=

1

2

∂2u

∂x2; u(x, 0) = δ(x)

The solution is given by:u(x, t) =1√2πt

e−x2

2t

— Gaussian Distribution, mean zero and variance√

t

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0.4Solution for Heat Diffusion

x

Hea

t Dis

trib

utio

n

t=1

t=2

t=3

t=10


Random Walk and Brownian Motion

Let thetime step△t andspatial step△x converge tozero in such a way that

(△x)2

△t−→ Constant

thenRandom Walk can be described byBrownianMotion .

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Random Walk and Brownian Motion – 2


Brownian Motion – Some Historical Notes

Robert Brown observed the random motion ofpollens in water.


Brownian Motion – Some Historical Notes – 2

Albert Einstein made use of the concept of Brownianmotion to estimate thesizeof solute molecules in asolvent.

one of thefive fundamental papers Einstein wrote in1905:

Einstein’s Miraculous Year:Five Papers that Changed the Face of Physics.

Princeton University Press, 2005, John Stachel (ed.)


Brownian Motion – Some Historical Notes – 3• The 1926 Nobel Prize in physics was received by

J. Perrin for his precise determination of theAvogadro’s number. He compared theexperimentally measured diffusivity of sphericalgranules in water with theStokes-Einsteinformula for thediffusion coefficient:

D =kBT

6πηR, and D = lim

t→∞〈X(t)2〉

2t

• Other contributors to the understanding ofBrownian Motion: Smoluchowski; Langevin;Ornstein-Ohlenback; Wiener; Levy; ....


Gambler Ruin ProblemGiven initial capitalC, between0 andA, you keep on playing

until your capital becomes0 or A.

If p = 12

(i.e. the game is fair), then

Prob(Win) =C

A, Prob(Lose)= 1 − C

A

Theexpected durationof the game, either lose or win is:

C(A − C)

If p 6= 12

, the above formulas become (z = qp

)

Prob(Win) =zC − 1

zA − 1, Prob(Lose)=

zA − zC

zA − 1

Expected duration =

„

C

q − p

«

−

„

A

q − p

« „

1 − zC

1 − zA

«


Winning by Losing Strategies

Parrondo’s Paradox. G. P. Harmer, D. Abbott,Nature, 1999

Now you can choose two different games to play:

• Game A.One slightly biased coin is used:

Prob(Head) = 0.495

• Game B.Two different coins are used, depending on your

current sum (capital).

1. If your current sum is divisible by three i.e.

0, 3, 6, 9 . . ., you toss the coin with

Prob(Head)) = 0.095;

2. otherwise, you toss the coin withProb(Head) = 0.745.


Playing Game A Only

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Playing Game B only

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Playing Games of A and B in sequence:

AABBAABBAABB....

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Playing Games A and B at Random:

AABABAABBBBABB...

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Random Walk in Periodic Environment• Homogeneous Environment.

(No global drift )

.• Inhomogeneous (Symmetric) Environment.

(No global drift )

.How Can Random Noise Help Us– p.20/32

Random Walk in Periodic Environment – 2• Inhomogeneous (Asymmetric, Unbiased)

Environment. (Global drift? )

.• Inhomogeneous (Biased) Environment.

(Global drift? )


Random Walk in Periodic Environment – 2• Inhomogeneous (Asymmetric, Unbiased)

Environment. (Still no global drift )

.• Inhomogeneous (Biased) Environment.

(Global drift )


Extraction of useful workfrom random noiseandun-biased butnon-equilibrium fluctuations

• (I) Potential U(x).

. . .• (II) Potential ≡ 0.

. ..(I) and (II) individually does not giveglobal drift .


Extraction of useful work – 2nowalternate (I) and (II) with certainfrequency.

• (I) Potential U(x) is ON.

. . .• (II) Potential U(x) is OFF.

. . .How Can Random Noise Help Us– p.24/32

Extraction of useful work – 3• (I) Potential U(x) is ON again.

.• Soeffectively, there is aglobal drift .

...How Can Random Noise Help Us– p.25/32

Optimal Frequency – Stochastic Resonance

optim

al d

rift

frequence of turning U onand off

optimal frequency

Key ingredients:• Spatially asymmetric environment;• Non-equilibrium fluctuations;• Noise!


Back to Parrondo’s ParadoxPlaying Game A.

.winning prob = 0.495

Playing Game B.

winning prob = 0.75

.winning prob= 0.1

Soeffectively, there is awinning tendency.


Applications of Stochastic Resonance

Astumian, Science, Vol 276, 1997


Applications of Stochastic Resonance –2

Astumian, Science, Vol 276, 1997


Applications of Stochastic Resonance –3

Hanggi, et. al., Ann. Phys., Vol 14, 2005


Applications of Stochastic Resonance?

Feynman’s Lecture Notes in Physics


THANK YOU


how can random noise help us - purdue universityyipn/talks/cmumathclub0309.pdf · one of the ﬁve...

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