Non-equilibrium Statistical Mechanics: The Physics of Fluctuations and Noise

Non-equilibrium Statistical Mechanics: The Physics of Fluctuations and Noise

SCI-B1-0910 Blok 1, 2009Aud M (NBI Blegdamsvej), kl 10-12 Mondays and Fridays

1

Non-equilibrium Statistical Mechanics: The Physics of Fluctuations and Noise

SCI-B1-0910 Blok 1, 2009Aud M (NBI Blegdamsvej), kl 10-12 Mondays and Fridays

John Hertz office: Kc-10 (NBI Blegdamsvej)email: [email protected] tel. 3532 5236 (office Kbh), +46 8 5537 8808 (office Sth), 2055 1874 (mobil)http://www.nbi.dk/~hertz/noisecourse/coursepage.html

2

Source material“text”: N G van Kampen, Stochastic Processes in Physics and Chemistry (North –

Holland) [very clear, good on general formal methods, but little recent stuff. I will not follow it slavishly in the lectures, but I recommend you buy and read it.]

These two are good for anomalous diffusion:L Vlahos et al, Normal and Anomalous Diffusion: a Tutorial arXiv.org/abs/0805.0419R Metzler and J Klafter, The Random Walker’s Guide to Anomalous Diffusion, Physics

Reports 339, 1-77 (2000)On first-passage-time problems:S Redner, A Guide to First-Passage Problems (Cambridge U Press) [library reserve]On finance-theoretical applications:J-P Bouchaud and M Potters, Theory of Financial Risks: From Statistical Physics to Risk

Management (Cambridge U Press) [library reserve](and more to be mentioned as we go along)

3

Lecture 1: A random walk through the course

4

Lecture 1: A random walk through the courseThe ubiquity of noise (especially in biology): Changing conditions does not changestates; rather, it changes the relative probabilities of states.

5


Example: protein conformational changePotential energy:

6



V1(x)

x 7



V1(x)

x 8



V1(x) V2(x)

x 9



V1(x) V2(x)

x 10



The real story:

P1(x)

V1(x) P2(x) V2(x)

x 11

€

P1,2(x)∝ exp[−βV1,2(x)]

From (equilibrium) stat mech:

This course: how P(x) changes from P1(x) to P2(x):

12

€

P1,2(x)∝ exp[−βV1,2(x)]


This course: how P(x) changes from P1(x) to P2(x): Dynamics of P(x,t)

www.nbi.dk/hertz/noisecourse/demos/Pseq.matwww.nbi.dk/hertz/noisecourse/demos/runseq.m

€

dP(x, t)

dt=L

13

€

P1,2(x)∝ exp[−βV1,2(x)]


This course: how P(x) changes from P1(x) to P2(x): Dynamics of P(x,t)

www.nbi.dk/hertz/noisecourse/demos/Pseq.matwww.nbi.dk/hertz/noisecourse/demos/runseq.m

or

etc.

€

dP(x, t)

dt=L

€

d x

dt=L

d x(t1)x(t)

dt=L

14

Random walkswww.nbi.dk/~hertz/noisecourse/demos/brown.m

15

Random walkswww.nbi.dk/~hertz/noisecourse/demos/brown.m

16

Random walks

€

XN = x i ;i=1

N

∑

x i = 0; x i

2= a2; x i ⋅x j = 0

17

Random walks

independent steps

€

XN = x i ;i=1

N

∑

x i = 0; x i

2= a2; x i ⋅x j = 0

18

Random walks

independent steps

€

XN = x i ;i=1

N

∑

x i = 0; x i

2= a2; x i ⋅x j = 0

€

XN ⋅XN = x i ⋅x i

i=1

N

∑ + x i ⋅x j

i≠ j

N

∑

= Na2

19

Random walks

independent steps

i.e, rms distance

€

XN = x i ;i=1

N

∑

x i = 0; x i

2= a2; x i ⋅x j = 0

€

XN ⋅XN = x i ⋅x i

i=1

N

∑ + x i ⋅x j

i≠ j

N

∑

= Na2

XN

2= N a

20

Random walks and diffusion

21

Step length distribution χ(y)

€

dy χ (y) =1;∫ dy yχ (y) = 0;∫ dy y 2χ (y) = a2∫


22


Change in P from one step:

€


€

P(x, t + Δt) = dy χ (y)P(x − y, t)∫


23



P

€


€

P(x, t + Δt) = dy χ (y)P(x − y, t)∫

P χ


24



P

€


€

P(x, t + Δt) = dy χ (y)P(x − y, t)∫

= dy χ (y) P(x, t) + y∂P

∂x+

1

2y 2 ∂ 2P

∂x 2+L

⎡

⎣ ⎢

⎤

⎦ ⎥∫P χ


25



P

€


€

P(x, t + Δt) = dy χ (y)P(x − y, t)∫

= dy χ (y) P(x, t) + y∂P

∂x+

1

2y 2 ∂ 2P

∂x 2+L

⎡

⎣ ⎢

⎤

⎦ ⎥∫

= P(x, t) +1

2a2 ∂ 2P

∂x 2

P χ


26



P

Diffusion equation:

€


€

P(x, t + Δt) = dy χ (y)P(x − y, t)∫

= dy χ (y) P(x, t) + y∂P

∂x+

1

2y 2 ∂ 2P

∂x 2+L

⎡

⎣ ⎢

⎤

⎦ ⎥∫

= P(x, t) +1

2a2 ∂ 2P

∂x 2

€

∂P

∂t= D

∂ 2P

∂x 2D =

a2

2Δt

P χ

Random walks and diffusionStep length distribution χ(y)


P

Diffusion equation:

diffusion constant

€


€

P(x, t + Δt) = dy χ (y)P(x − y, t)∫

= dy χ (y) P(x, t) + y∂P

∂x+

1

2y 2 ∂ 2P

∂x 2+L

⎡

⎣ ⎢

⎤

⎦ ⎥∫

= P(x, t) +1

2a2 ∂ 2P

∂x 2

€

∂P

∂t= D

∂ 2P

∂x 2D =

a2

2Δt

P χ

Solution of the diffusion equation:

28


29

€

P(x, t) =1

4πDtexp −

x 2

4Dt

⎛

⎝ ⎜

⎞

⎠ ⎟


30

€

P(x, t) =1

4πDtexp −

x 2

4Dt

⎛

⎝ ⎜

⎞

⎠ ⎟

Gaussian, spreading with time, variance 2Dt

http;//www.nbi.dk/~hertz/noisecourse/gaussspread.m


31

€

P(x, t) =1

4πDtexp −

x 2

4Dt

⎛

⎝ ⎜

⎞

⎠ ⎟

Gaussian, spreading with time, variance 2Dt

http;//www.nbi.dk/~hertz/noisecourse/gaussspread.m

Distribution obtained by simulating 20000 random walks:

32

Anomalous diffusion

Normal diffusion:

€

x 2 = 2Dt

Anomalous diffusion

Normal diffusion:

An experimental counterexample: €

x 2 = 2Dt

Anomalous diffusion

Motion of lipid granules in yeast cells

Tolic-Nørrelykke et al, Phys Rev Lett 93, 078102 (2004)

Normal diffusion:


x 2 = 2Dt

Anomalous diffusion

Motion of lipid granules in yeast cells

Tolic-Nørrelykke et al, Phys Rev Lett 93, 078102 (2004)

Normal diffusion:


x 2 = 2Dt

€

x 2 ∝ t 0.75

Sub- and superdiffusion

37

€

x 2 ∝ t H


38

€

x 2 ∝ t HH: Hurst exponent

H < 1: subdiffusionH > 1: superdiffusion


39

€



One way to get superdiffusion: long-time correlations between steps


40

€



One way to get superdiffusion: long-time correlations between steps

One way to get subdiffusion: long-time anti-correlations between steps

Levy walks

41

Step length distribution χ(y):

_______________

€

dy χ (y) =1;∫ dy yχ (y) = 0;∫ dy y 2χ (y) = ∞∫

Levy walks

42


_______________

€

dy χ (y) =1;∫ dy yχ (y) = 0;∫ dy y 2χ (y) = ∞∫

power law tail in step length distribution:

€

χ(y)∝ y−a, a ≤ 3

Levy walks

43


_______________

€

dy χ (y) =1;∫ dy yχ (y) = 0;∫ dy y 2χ (y) = ∞∫


Example: Cauchy (Lorentz) distribution€

χ(y)∝ y−a, a ≤ 3

€

χ(y) =1/π

1+ y 2

Levy walks

44


_______________

€

dy χ (y) =1;∫ dy yχ (y) = 0;∫ dy y 2χ (y) = ∞∫


Example: Cauchy (Lorentz) distribution

http://www.nbi.dk/~hertz/noisecourse/levy.m (a = 5/2)

€

χ(y)∝ y−a, a ≤ 3

€

χ(y) =1/π

1+ y 2

Levy walks

45


_______________

€

dy χ (y) =1;∫ dy yχ (y) = 0;∫ dy y 2χ (y) = ∞∫




€

χ(y)∝ y−a, a ≤ 3

€

χ(y) =1/π

1+ y 2

Levy walks

46


_______________

€

dy χ (y) =1;∫ dy yχ (y) = 0;∫ dy y 2χ (y) = ∞∫




Note: <x2> = ∞ for all t

€

χ(y)∝ y−a, a ≤ 3

€

χ(y) =1/π

1+ y 2

Brown vs Levy

47

Ising model

48

(an example of a system with many degrees of freedom)

Ising model

49


Binary “spins” Si(t) = ±1

Ising model

50



Dynamics: at every time step,

Ising model

51



Dynamics: at every time step, (1) choose a spin (i) at random

Ising model

52



Dynamics: at every time step, (1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)

Ising model

53



Dynamics: at every time step, (1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)(3) Si(t + Δt) = +1 with probability

€

P(h) =1

1+ exp(−2hi)

Ising model

54



Dynamics: at every time step, (1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)(3) Si(t + Δt) = +1 with probability

€

P(h) =1

1+ exp(−2hi)

Changing the interaction strength:

55

http://www.nbi.dk/~hertz/noisecourse/ising.m


56


Varying the interaction strength: Jneighbors = 0.25:


57


Varying the interaction strength: Jneighbors = 0.25, 0.45:


58


Varying the interaction strength: Jneighbors = 0.25, 0.45, 0.65:

Some basic concepts in probability theory

59

Random variable x


60

Random variable x

Probability distribution (“density”) P(x)


61

Random variable x

Probability distribution (“density”) P(x)If x is discrete-valued: P(xn) or Pn


62

Random variable x


Normalization:

€

P(x)dx =1 P(xn ) =1n

∑∫


63

Random variable x


Normalization:

Averages:

€


∑∫

€

A(x) ≡ A(x)P(x)dx∫


64

Random variable x


Normalization:

Averages:

Moments:

€


∑∫

€


€

x n = x nP(x)dx∫


65

Random variable x


Normalization:

Averages:

Moments:

Mean:

€


∑∫

€


€

x n = x nP(x)dx∫

€

x1

Some common distributions

66

Gaussian (normal):

€

P(x) =1

2πe− 1

2 x 2


67

Gaussian (normal):

Cauchy (Lorentzian):

€

P(x) =1

2πe− 1

2 x 2

€

P(x) =1

π

1

1+ x 2


68

Gaussian (normal):


(one-sided) exponential:

€

P(x) =1

2πe− 1

2 x 2

€

P(x) =1

π

1

1+ x 2

€

P(x) = Θ(x)e−x


69

Gaussian (normal):



Levy:

€

P(x) =1

2πe− 1

2 x 2

€

P(x) =1

π

1

1+ x 2

€

P(x) = Θ(x)e−x

€

P(x) =Θ(x)

2π

1

x 3 / 2exp −

1

2x

⎛

⎝ ⎜

⎞

⎠ ⎟


70

Gaussian (normal):



Levy:

Poisson:

€

P(x) =1

2πe− 1

2 x 2

€

P(x) =1

π

1

1+ x 2

€

P(x) = Θ(x)e−x

€

P(x) =Θ(x)

2π

1

x 3 / 2exp −

1

2x

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Pn =an

n!e−a

Thin and fat tails

71

Characteristic functions

72


73

€

G(k) ≡ exp ikx( ) = e ikxP(x)dx∫


74

€


G(k) is the moment-generating function (expand the exponential):


75

€



€

G(k) =(ik)n

n!n

∑ x n


76

€



€

G(k) =(ik)n

n!n

∑ x n

Note: G(0) = 1 (normalization)


77

€



€

G(k) =(ik)n

n!n

∑ x n

Note: G(0) = 1 (normalization)

Gaussian: Cauchy:

Exponential: Levy:

€

G(k) = e− 12 k 2

G(k) = e− k

G(k) =1

1− ikG(k) = e− −2ik

Cumulants

Expanding log G generates the cumulants κn:

Cumulants


€

logG(k) ≡(ik)n

n!n=1

∞

∑ κ n

Cumulants


€

logG(k) ≡(ik)n

n!n=1

∞

∑ κ n

€

κ1 = x

κ 2 = x 2 − x2

= x − x( )2

≡ σ 2

κ 3 = x 3 − 3 x 2 x + 2 x3

κ 4 = x 4 − 4 x 3 x − 3 x 2 2+12 x 2 x

2− 6 x

4

Cumulants


€

logG(k) ≡(ik)n

n!n=1

∞

∑ κ n

€

κ1 = x

κ 2 = x 2 − x2

= x − x( )2

≡ σ 2

κ 3 = x 3 − 3 x 2 x + 2 x3

κ 4 = x 4 − 4 x 3 x − 3 x 2 2+12 x 2 x

2− 6 x

4

(mean)

Cumulants


€

logG(k) ≡(ik)n

n!n=1

∞

∑ κ n

€

κ1 = x

κ 2 = x 2 − x2

= x − x( )2

≡ σ 2

κ 3 = x 3 − 3 x 2 x + 2 x3

κ 4 = x 4 − 4 x 3 x − 3 x 2 2+12 x 2 x

2− 6 x

4

(mean) variance

Cumulants


€

logG(k) ≡(ik)n

n!n=1

∞

∑ κ n

€

κ1 = x

κ 2 = x 2 − x2

= x − x( )2

≡ σ 2

κ 3 = x 3 − 3 x 2 x + 2 x3

κ 4 = x 4 − 4 x 3 x − 3 x 2 2+12 x 2 x

2− 6 x

4

(mean) variance

skewness: γ3 = κ3/(κ2)3/2 kurtosis: κ4/(κ2)2

Multivariate distributions

84

€

P(x1,L ,xn ), P(x)


85

€

P(x1,L ,xn ), P(x)

€

P(x1) = P(x1,L ,xn )dx2∫ L dxnmarginal distribution of x1:


86

€

P(x1,L ,xn ), P(x)

€

P(x1) = P(x1,L ,xn )dx2∫ L dxn

P(x1,x2) = P(x1,L ,xn )dx3∫ L dxn

marginal distribution of x1:

etc.


87

€

P(x1,L ,xn ), P(x)

€




etc.

Independence:

€

P(x,y) = P(x)P(y)


88

€

P(x1,L ,xn ), P(x)

€




etc.

Independence:

Conditional probabilities:

€

P(y | x) : P(x,y) = P(x | y)P(y) = P(y | x)P(x)€

P(x,y) = P(x)P(y)


89

€

P(x1,L ,xn ), P(x)

€




etc.

Independence:

Conditional probabilities:

Bayes’s (Bayes) rule:€

P(y | x) : P(x,y) = P(x | y)P(y) = P(y | x)P(x)€

P(x,y) = P(x)P(y)

€

P(y | x) =P(x | y)P(y)

P(x)=

P(x,y)

P(x)

Adding random variables

90

€

x : P1(x); y : P2(y)


91

€

x : P1(x); y : P2(y)

€

z = x + y :


92

€

x : P1(x); y : P2(y)

€

z = x + y :

€

P(z) = δ(z − x − y)P1(x)P2(y)dxdy∫


93

€

x : P1(x); y : P2(y)

€

z = x + y :

€

P(z) = δ(z − x − y)P1(x)P2(y)dxdy∫= P1(x)P2(z − x)dx∫


94

€

x : P1(x); y : P2(y)

€

z = x + y :

€

P(z) = δ(z − x − y)P1(x)P2(y)dxdy∫= P1(x)P2(z − x)dx∫

characteristic functions:

€

G(k) = G1(k)G2(k)

Change of variables:

€

x : Px (x)

y = f (x)


€

x : Px (x)

y = f (x)

Py (y) = Px∫ (x)δ[y − f (x)]dx


€

x : Px (x)

y = f (x)


= Px∫ (x)δ[y − f (x)]dx

dydy


€

x : Px (x)

y = f (x)


= Px∫ (x)δ[y − f (x)]dx

dydy

= Px ( f −1(y))df −1(y)

dy


€

x : Px (x)

y = f (x)


= Px∫ (x)δ[y − f (x)]dx

dydy

= Px ( f −1(y))df −1(y)

dy(or use Py(y)dy = Px(x)dx)


€

x : Px (x)

y = f (x)


= Px∫ (x)δ[y − f (x)]dx

dydy

= Px ( f −1(y))df −1(y)

dy

Multivariate case:

€

Py (y) = Px (f−1(y))

∂f−1(y)

∂y

(or use Py(y)dy = Px(x)dx)


€

x : Px (x)

y = f (x)


= Px∫ (x)δ[y − f (x)]dx

dydy

= Px ( f −1(y))df −1(y)

dy

Multivariate case:

€

Py (y) = Px (f−1(y))

∂f−1(y)

∂y

inverse of Jacobian J

€

Jij =∂y i

∂x j

(or use Py(y)dy = Px(x)dx)

Gaussian (normal) distribution

102

€

P(x) =1

2πσ 2exp − 1

2 (x − μ)2 /σ 2( )


103

€

P(x) =1

2πσ 2exp − 1

2 (x − μ)2 /σ 2( )

characteristic function:

€

G(k) = exp ikμ − 12 k 2σ 2

( )


104

€

P(x) =1

2πσ 2exp − 1

2 (x − μ)2 /σ 2( )


cumulants:

€

G(k) = exp ikμ − 12 k 2σ 2

( )

€

κ1 = μ

κ 2 = σ 2

κ m = 0; m > 2


105

€

P(x) =1

2πσ 2exp − 1

2 (x − μ)2 /σ 2( )


cumulants:

moments (μ = 0 case):

€

G(k) = exp ikμ − 12 k 2σ 2

( )

€

κ1 = μ

κ 2 = σ 2

κ m = 0; m > 2

€

x 2n = (2n −1)!! x 2 ≡ (2n −1)(2n − 3)L 3⋅1 x 2

Multivariate Gaussian

106

correlation matrix

€

C jk ≡ x j − x j( ) xk − xk( )


107

correlation matrix

€

C jk ≡ x j − x j( ) xk − xk( )

€

P(x) =1

(2π )d / 2 detCexp −

1

2x j − x j( ) C

−1( )

jkxk − xk( )

jk

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


108

correlation matrix

€

C jk ≡ x j − x j( ) xk − xk( )

€

P(x) =1


1

2x j − x j( ) C

−1( )

jkxk − xk( )

jk

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


€

G(k) = exp ik j x j − 12

j

∑ k jC jkkk

jk

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


109

correlation matrix

€

C jk ≡ x j − x j( ) xk − xk( )

€

P(x) =1


1

2x j − x j( ) C

−1( )

jkxk − xk( )

jk

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


€


j

∑ k jC jkkk

jk

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

higher moments (Wick’s theorem):


110

correlation matrix

€

C jk ≡ x j − x j( ) xk − xk( )

€

P(x) =1


1

2x j − x j( ) C

−1( )

jkxk − xk( )

jk

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


€


j

∑ k jC jkkk

jk

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


€

x1x2x3x4 = x1x2 x3x4 + x1x3 x2x4 + x1x4 x2x3


111

correlation matrix

€

C jk ≡ x j − x j( ) xk − xk( )

€

P(x) =1


1

2x j − x j( ) C

−1( )

jkxk − xk( )

jk

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


€


j

∑ k jC jkkk

jk

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


(sum of all pairwise contractions)

€



112

correlation matrix

€

C jk ≡ x j − x j( ) xk − xk( )

€

P(x) =1


1

2x j − x j( ) C

−1( )

jkxk − xk( )

jk

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


€


j

∑ k jC jkkk

jk

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


(sum of all pairwise contractions)etc. for higher orders

€


Central limit theorem

sum of N iid random variables

€

y =1

Nx i

i=1

N

∑



distribution of xi:

€

y =1

Nx i

i=1

N

∑

p(x i)



distribution of xi:

€

y =1

Nx i

i=1

N

∑

p(x i) assume finite variance



distribution of xi:

characteristic function

€

y =1

Nx i

i=1

N

∑

p(x i)

g(k) = exp − 12 k 2σ 2 − i

3! k 3κ 3 + 14! k 4κ 4 +L( )

assume finite variance



distribution of xi:


characteristic function of y:

€

y =1

Nx i

i=1

N

∑

p(x i)

g(k) = exp − 12 k 2σ 2 − i

3! k 3κ 3 + 14! k 4κ 4 +L( )

G(y) = g k N( )[ ]N




distribution of xi:



€

y =1

Nx i

i=1

N

∑

p(x i)

g(k) = exp − 12 k 2σ 2 − i

3! k 3κ 3 + 14! k 4κ 4 +L( )

G(y) = g k N( )[ ]N

logG(y) = N logg k N( )




distribution of xi:



€

y =1

Nx i

i=1

N

∑

p(x i)

g(k) = exp − 12 k 2σ 2 − i

3! k 3κ 3 + 14! k 4κ 4 +L( )

G(y) = g k N( )[ ]N


= N −k 2

2Nσ 2 −

ik 3

6N 3 / 2κ 3 +

k 4

24N 2κ 4 +L

⎛

⎝ ⎜

⎞

⎠ ⎟




distribution of xi:



€

y =1

Nx i

i=1

N

∑

p(x i)

g(k) = exp − 12 k 2σ 2 − i

3! k 3κ 3 + 14! k 4κ 4 +L( )

G(y) = g k N( )[ ]N


= N −k 2

2Nσ 2 −

ik 3

6N 3 / 2κ 3 +

k 4

24N 2κ 4 +L

⎛

⎝ ⎜

⎞

⎠ ⎟

N →∞ ⏐ → ⏐ ⏐ − 1

2 k 2σ 2




distribution of xi:



€

y =1

Nx i

i=1

N

∑

p(x i)

g(k) = exp − 12 k 2σ 2 − i

3! k 3κ 3 + 14! k 4κ 4 +L( )

G(y) = g k N( )[ ]N


= N −k 2

2Nσ 2 −

ik 3

6N 3 / 2κ 3 +

k 4

24N 2κ 4 +L

⎛

⎝ ⎜

⎞

⎠ ⎟

N →∞ ⏐ → ⏐ ⏐ − 1

2 k 2σ 2

y is Gaussian!


Stable distributions

122

N-1/2 x sum of N Gaussian variables has same distribution as the originalvariables: stable


123


Are there distributions which are stable but with a different scaling factor N-1/α (α ≠ 2) instead?


124



Require

€

g k N1/α( )[ ]

N= g(k)


125



Require

€

g k N1/α( )[ ]

N= g(k)

N logg k N1/α( ) = logg(k)


126



Require

Solution:

€

g k N1/α( )[ ]

N= g(k)


logg(k) = −ckα


127



Require

Solution:

or

€

g k N1/α( )[ ]

N= g(k)


logg(k) = −ckα

g(k) = exp −ckα( ) (α ≤ 2)


128



Require

Solution:

or

characteristic function for stable distribution of order α

€

g k N1/α( )[ ]

N= g(k)


logg(k) = −ckα

g(k) = exp −ckα( ) (α ≤ 2)


129

€

Pα (x) = gα (k)e−ikx∫ dk

2π= exp −ikx − ckα

( )∫ dk

2π

Stable distribution of order α:


130

€



( )∫ dk

2π


Asymptotic behaviour for large x: P(x) ~ 1/x1+α


131

€



( )∫ dk

2π



Note: Stable distributions have infinite variance for α < 2


132

€



( )∫ dk

2π



Note: Stable distributions have infinite variance for α < 2Stable distributions have infinite mean for α < 1


133

€



( )∫ dk

2π



Note: Stable distributions have infinite variance for α < 2Stable distributions have infinite mean for α < 1

For symmetric distributions, use

€

gα (k) = exp −c kα

( ); Pα (x) = exp −ikx − c kα

( )∫ dk

2π

Stable distributions: examples

134

Special cases (can do the Fourier inversion analytically):

α = 1/2: Levy

α = 1: Cauchy/Lorentzian

α = 3/2: Holtsmark

α = 2: Gaussian

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Non-equilibrium Statistical Mechanics: The Physics of Fluctuations and Noise

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