Fractals in Physics |

**In General** > s.a. fractals [other applications];
phase transitions; stochastic processes.

* __History__: 1915, J Perrin,
Brownian motion and snowflakes; 1972, K Wilson, Scale invariance in phase
transitions and renormalization group; 1980s, Cosmology, diffusion-limited
aggregation, dielectric breakdown.

* __Kinematical description__:
Fractal geometry; Notice that physical fractals are usually random.

* __Models for fractal growth__:
Ising model; Dielectric breakdown; Critical percolation clusters.

* __Goals__: Develop new
concepts, related to self-organization, irreversibility and
non-ergodicity, to understand the origin of fractal structure.

* __Applications__: Aggregation;
Diffusion; Percolation; Chaotic dynamics [coordinate-independent indicator].

@ __General references__: Kadanoff PT(86)feb;
Amann et al ed-88;
Pietronero PRP(89) [growth];
Hurd AJP(88)nov [RL];
Gouyet 96; Addison 97;
Aguirre et al RMP(09) [fractal basins in non-linear dynamics];
Gulick 12 [and chaotic dynamics];
news pw(12)oct [growth].

@ __And quantum mechanics__:
Cannata & Ferrari AJP(88)aug [particle paths];
issue CSF(94)#3;
Kröger PRP(00);
Wójcik et al PRL(00)qp [fractal wave functions];
Jadczyk 14;
Miller pt(18)jan [surface-electron system].

@ __Random fractal networks__: Nakayama et al RMP(94) [dynamics and scaling].

@ __Related topics__: Ishikawa & Suzuki PhyA(04) [breaking of fractal distribution];
Liaw & Chiu PhyA(09) [fractal dimension of a time sequence];
Vitiello PLA(12) [from coherent states and self-similarity induced non-commutative geometry];
> s.a. Hofstadter's Butterfly.

> __Related topics__:
see classical particles; generalized quantum
field theories; ising models; quantum
mechanics in general backgrounds; quantum oscillators;
random walk; thermodynamical
systems; turbulence.

**Matter Distribution in Cosmology** > s.a. cosmology
in generalized theories; matter in cosmology.

* __Galaxy distribution__:
A fractal structure in the distribution of luminous matter up to about
5–15 Mpc has been seen and is accepted by most of the community;
1995, L Pietronero and collaborators have been claiming for years that at
all observed scales the pattern continues, and that there is no evidence of
homogeneity at any scale; Although Pietronero et al seem to have done the
best analysis of the galaxy distribution, there would remain the puzzle of
why the cosmic microwave background is so isotropic, and the dark-matter
distribution is unknown; 2004, The consensus is that the distribution does
become homogeneous above 100 Mpc or so.

* __Inside the Milky Way__:
There is some evidence that the interstellar matter in our galaxy is
fractally distributed as well.

@ __Galaxy distribution__:
Ribeiro in(94)-a0910;
Sylos Labini et al ap/97-proc,
PRP(98)ap/97,
ap/98/NA;
Gabrielli et al EPL(99)ap/98 [gravitational force];
Terazawa MPLA(98);
Combes ap/99-proc;
Pietronero & Sylos Labini ap/99-proc,
ap/00-proc;
Baryshev AAT(00)ap/99 [rev];
McCauley Frac(98)ap/00;
Ribeiro GRG(01)ap;
Baryshev & Teerikorpi 02 [I];
Rozgacheva & Agapov a1103.

**Fractal Spacetime**
> s.a. approaches to quantum gravity; fractals [and
fractional spaces]; quantum spacetime; spacetime dimensionality.

* __Idea__: One motivation is obtaining
a discrete spacetime that is invariant under the renormalization group.

@ __General references__:
Crane & Smolin NPB(86);
Englert FP(87);
Nottale IJMPA(89),
93, CSF(94);
Ambjørn & Watabiki NPB(95)ht [dimension from 2-point function];
Kobelev phy/00,
phy/00 [and modified Lorentz group];
Porter gq/02 ['fractafolds' as spacetime structure];
Modesto & Nicolini PRD(10)-a0912 [spectral dimension of quantum spacetime];
Nottale 11;
Smolyaninov PLA(12)-a1110 [with fractal timelike variable];
He IJTP(14) [tutorial];
Schonfeld EPJC(16)-a1612 [not Lorentz invariant?];
Svozil FS(19)-a1712 [and compensating modified metric].

@ __Dynamics of gravity and fields__: Svozil JPA(87) [quantum electrodynamics and regularization];
Agop & Gottlieb JMP(06) [gravity];
Sadallah & Muslih IJTP(09) [Einstein gravity];
Calcagni PRL(10)-a0912 [scalar field];
Akkermans et al PRL(10)-a1012 [thermodynamics of massless scalar field];
Akkermans a1210-proc [quantum field theory and statistical mechanics];
Andrews et al JFAA(16)-a1505 [wave equation on 1D fractal, spectral decimation];
Aerts et al CSF(16)-a1506 [calculus, algebra and physics on a fractal];
> s.a. electromagnetic fields;
generalized quantum field theories;
thermodynamic systems;
types of field theories.

@ __In renormalization-group based theories__:
Lauscher & Reuter JHEP(05)ht [in asymptotically safe gravity, \(d^~_{\rm micro} = 2\)];
Reuter & Saueressig JHEP(11);
Calcagni IJMPA(13)-a1209 [asymptotic safety and Hořava-Lifshitz gravity].

@ __And non-commutative geometry__:
Castro et al ht/00;
Castro CSF(01)ht/00;
Arzano et al PRD(11)-a1107.

@ __Phenomenology__: Hill PRD(03)ht/02;
Iovane CSF(04)ap/03 [\(G(t)\), \(\ddot a\)];
Shapovalova G&C(03) [metric fluctuations];
Goldfain CSF(04) [and gauge hierarchy problem],
CSF(05) [and unified field theory];
Caruso & Ooguri ApJ(09) [bounds on dimensionality from cmb].

@ __Models__: Magliaro et al a0911 [Barrett-Crane spin-foam model];
Rotondo & Nojiri MPLA(17)-a1703 [2D fractal, based on square plaquettes].

> __Related topics__: see black-hole geometry
and quantum black holes; metric matching;
models in canonical quantum gravity; quantum
geometry; quantum groups.

**Multifractal Spacetime**

* __Idea__: Multifractal theories
are theories in which the dimensional flow related to the renormalization of
quantum gravity at short scales is implemented kinematically in the spacetime
measure, which admits a universal parametrization, and in the dynamics, via a
model-dependent kinetic operator; There are four types of multifractal scenarios,
which have ordinary, weighted, *q*- or fractional derivatives in the
dynamical action.

@ __ General references__:
Kobelev ht/00;
Calcagni AIP(12)-a1209 [intro];
Calcagni & Nardelli PRD(13)-a1210 [scalar field, symmetries and propagator];
Calcagni & Nardelli PRD(13)-a1304 [spectral dimension and diffusion];
Calcagni EPJC(16)-a1602 [in quantum gravity],
EPJC(17)-a1603 [Lorentz violations];
Calcagni PRD(17)-a1609 [from first principles], JHEP(17)-a1612 [rev];
Hu JPCS(17)-a1702 [and stochastic gravity];
Calcagni & Ronco NPB(17)-a1706 [dimensional flow and fuzziness];
Calcagni a2103-MPLA [rev];
> s.a. spacetime dimensionality.

@ __Cosmology__: Calcagni et al JCAP(16)-a1606 [cmb and inflation];
Calcagni JCAP(13)-a1307 [multi-scale gravity];
Calcagni & De Felice PRD(20)-a2004 [dark energy].

@ __ Other phenomenology__: Calcagni PLB(11)-a1012-proc,
PRD(11) [field theory on a multifractal spacetime],
JHEP(12)-a1107 [geometry and field theory];
Calcagni et al PRD(16)-a1512 [particle-physics constraints];
Calcagni & Ronco IJGMP(19)-a1709-proc [from the Standard Model to cosmology].

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