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].

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at – modified 26 nov 2020