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Laws of scaleLaws of scale transformations transformations

From scale invariance to scaleFrom scale invariance to scalecovariancecovariance

Laurent NottaleCNRS

LUTH. Paris-Meudon Observatory

http://www.luth.obspm.fr/~luthier/nottale/

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RRééfféérences:rences:*Nottale, L., 1997, in "Scale invariance and beyond", proceedings of Les Houches school,Ed. B. Dubrulle, F. Graner and D. Sornette, (EDP Sciences /Springer), p. 249"Scale relativity"http://www.luth.obspm.fr/~luthier/nottale/arhouche.pdf

*Nottale L., 2002, in Traité IC2, Traitement du Signal et de l'Image, "Lois d'échelle,fractales et ondelettes", sous la direction de P. Abry, P. Gonçalvès et J. Levy Vehel(Hermès Lavoisier 2002), Vol. 2, Chap. 7, pp. 233-265."Relativité d'échelle, nondifférentiabilité et espace-temps fractal"http://www.luth.obspm.fr/~luthier/nottale/arloidechelle.pdf

English translation:*Nottale L., 2009, in Scaling, Fractals and Wavelets (Digital Signal and Image processingSeries), edited by Patrice Abry, Paulo Gonçalves and Jacques Lévy Véhel (ISTE-Wiley, 2009)“Scale relativity, non-differentiability and fractal space-time”

*Nottale L., 2007, Special Issue (July 2007) on “Physics of Emergence and Organization”,Electronic Journal of Theoretical Physics 4, No. 16(II), 187–274.“Scale Relativity : A Fractal Matrix for Organization in Nature”.

*Auffray C. & Nottale L., 2008, Progress in Biophysics and Molecular Biology, 97, 79-114.“Scale relativity theory and integrative systems biology. 1. Founding principles and scalelaws”. http ://www.luth.obspm.fr/luthier/nottale/arPBMB08AN.pdf

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General methodGeneral method

*Introduce scale variables in an explicit way

-experimental, observational: resolution ε -theoretical: differential element dx

*Write laws of scale transformation in terms of differentialequations (acting in « scale space », i.e. in the space of thezooms on these scale variables)

*Constrain these differential equations by the principle of scalerelativity

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Fundamental consequences of the explicitFundamental consequences of the explicitscale dependencescale dependence

f(r,dr) = f(r) + a dr + b/drExpressions like

become possibleWhile in standard calculus,

lim [ f(r) + a dr]dr-->0 = f(r)

and lim [b/dr]dr-->0 = ∞excluded

In the new calculus, dr --> 0, but without taking the limit

Then f2(r,dr) = f2(r) + 2 a b + …

New finite term generated as « 0 x ∞ »

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Solution of theSolution of thenondifferentiability problemnondifferentiability problem

*Since continuity is kept, « nondifferentiability » does notmean here that we cannot define differentials, e.g., dx, dt, butthat the derivative lim [dx/dt]dt-->0become undefined (either infinite, or fluctuating, or both)

*In the scale-relativistic new calculus, this problem is solved:the new derivative is defined as a fractal function, explicitlydepending on the scale variable dt,

v(t,dt)= [dx/dt]dt-->0

which may be divergent when dt-->0.

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Dilatation operator, Gell-Mann-Levy’s method

Simplest scale differential equationSimplest scale differential equation

Taylor expansion

Solution: fractal of constant dimension + transition

First order equation

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ln L

ln ε

trans

ition

fractal

scale -independent

ln ε

trans

ition

fractal

delta

variation of the length variation of the scale dimension

"scale inertia"

scale -independent

Scale dependence of the length and of the effective scale-dimension in the case of ‘scale-inertial' laws (which aresolutions of a first order linear scale-differential equation).

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Log[

Abs

[v +

η (2

D)1/

2 Exp

[(-ln

dt)/2

]]]

lndt

Account of fractal fluctuations

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Local definition of the scale dimension

Asymptotic behavior

Scale transformation

Law of composition of dilations

Same mathematical structure as Galileo’s law of transformation ofmotion

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Length of a fractal curveTwo ways to measure the length of a fractal curve, in terms of:(1) Spatial resolution δX : static measurement(2) Time resolution δt  (= invariant): kinematic measurement(asymptotic formulae)

Relation between space-resolution-interval and time-resolution-interval:

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Galilean scale-relativity.1.Identification of standard scale transformations (constantfractal dimension) with a Galileo relativity group of scale

Scale transformation:

Same mathematical structure as Galileo group of inertial motion transformation:

x’ = x - V tt’ = tW = U + V

V = V (K’/K)

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Galilean scale-relativity. 2 .scale variable = invariant (proper) time resolution

Basic description of fractal fluctuation:*

(D = 2 ––> )

Scale transformation on the proper time differential interval:

The « scale-time »is now the Hölderexponent H=1/Dinstead of δ=D-1.

* See e.g. Célérier & Nottale, 2004, J. Phys. A37, 931

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Special scale-relativistic generalization

(1) Space resolution: 4 scale variables εk = δxk (L.N. 1992, IJMPA 7, 4899)

(2) Proper time resolution: 1 scale variable ε = δs

Transformation ds = λ= Compton length ––> ds, with D(λ) = 2

Planck length-scale,invariant under dilations,unreachable, impassable

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New mass-scale to length-scaletransformation in special scale-relativity

Leng

th S

cale

(in

Z sc

ale

unit)

91 10 10 10 10107 10 10 10

10

10

10

10

11 15 19 23 27 31 35

4

8

12

16

1λ

λ

Λ

Z

Zm m P

P

Mass Scale (GeV)

stand

ard re

lation new relation

Valid for both space-time resolution and proper time resolution

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Special Scale-RelativityGeneral law of linear scale transformation comingunder the principle of scale-relativity ?Find the four functions a(V), b(V), c(V), d(V)satisfying: *internal composition law *reflection invariance

Solution (LN, 1992, Int.J.Mod.Phys. A7, 4899): log-Lorentz transformation

New law of composition of dilations:

= length-scale invariant under dilations

Standard product of dilation (‘Galilean scale-relativity’) recovered in the limits

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SMALL SCALES:Identification of the invariant scale with the Planck length-time-scale

LARGE SCALES:Identification of the invariant scale with the length-scale of thecosmological constant

RATIO:

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ln L

ln ε

trans

itionfractal

ln ε

trans

ition

fractal

delta

special scale-relativity

Plan

ck s

cale scale

independentscaleindependent

Plan

ck s

cale

variation of the scale dimensionvariation of the length

(Simplified case : )

Scale dependence of the length and of the effective scaledimension in special scale-relativity (log-Lorentzian laws of

scale transformations)

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Scale dynamicsScale laws that are solutions of second order partial differential

equations in the scale space

Least action principle in scale space ––> Euler Lagrange scaleequations in terms of the « djinn »

Resolution identified as « scale velocity »:

Djinn (variable scale dimension) identified with « scale time »

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ln L

ln ε

trans

ition

fractal

ln ε

trans

ition

fractal

delta

constant "scale-force"

variation of the scale dimension

scaleindependent

scaleindependent

variation of the length

(asymptotic)

'Scale dynamics': scale dependence of the length and of theeffective scale-dimension in the case of a constant 'scale-force'

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ln L

ln ε

trans

itionfractal

ln ε

fractal

delta

harmonic oscillator scale-force

trans

ition

variation of the scale dimension

scaleindependent

scaleindependent

variation of the length

‘Scale dynamics’: scale dependence of the length and of theeffective scale-dimension in the case of an harmonicoscillator ‘scale-potential’

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Log-periodic law (discrete scaleLog-periodic law (discrete scaleinvariance) from scale covarianceinvariance) from scale covariance

« Scale covariant » generalization of fractals with constant dimension:

1. Scale differential equation, D = cste:

2. Introduction of a correction (2nd member):

3. Covariance = form invariance of equations––> one postulates that equations in Φ and χshould have the same form:

4. One sets , the eq.becomes:

5. One sets––> solution:

6. Part case –>log-per.

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ln L

ln ε

trans

ition

fractal

scaleindependent

ln ε

trans

itionfractal

delta

variation of the scale dimension

"log-periodic"

scaleindependent

variation of the length

Scale dependence of the length and of the scale dimension inthe case of a log-periodic behavior (discrete scale invariance)including a fractal / nonfractal transition.