how can random noise help usyipn/talks/cmumathclub0309.pdf · binomial converges to gaussian...
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How Can Random Noise Help UsGlobal Transports from Thermal Fluctuations
Aaron N. K. YipDepartment of Mathematics
Purdue University
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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
0 1 2 3 4
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
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Basics of Random Walk – 2
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Time
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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
Pro
babi
lity
Probability Distribution of the Position of an Unbiased Random Walk after 20 Steps
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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
x
Pro
babi
lity
Dis
trib
utio
n F
unct
ion
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Binomial converges to Gaussian Distribution
As n → ∞,
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0.14Comparison between Binomial and Normial Distribution
x
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Dis
trib
utio
n
What is thisTheoremcalled?
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Binomial converges to Gaussian Distribution
As n → ∞,
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lity
Dis
trib
utio
n
Central Limit Theorem
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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
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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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1Coin Tossing Experiment
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Random Walk and Brownian Motion – 2
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Brownian Motion – Some Historical Notes
Robert Brown observed the random motion ofpollens in water.
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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.)
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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; ....
How Can Random Noise Help Us– p.13/32
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
«
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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.
How Can Random Noise Help Us– p.15/32
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? )
.How Can Random Noise Help Us– p.21/32
Random Walk in Periodic Environment – 2• Inhomogeneous (Asymmetric, Unbiased)
Environment. (Still no global drift )
.• Inhomogeneous (Biased) Environment.
(Global drift )
.How Can Random Noise Help Us– p.22/32
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 .
How Can Random Noise Help Us– p.23/32
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!
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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.
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Applications of Stochastic Resonance
Astumian, Science, Vol 276, 1997
How Can Random Noise Help Us– p.28/32
Applications of Stochastic Resonance –2
Astumian, Science, Vol 276, 1997
How Can Random Noise Help Us– p.29/32
Applications of Stochastic Resonance –3
Hanggi, et. al., Ann. Phys., Vol 14, 2005
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Applications of Stochastic Resonance?
Feynman’s Lecture Notes in Physics
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THANK YOU
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