Talk:Anderson localization

Mathematics

I've created a new section to separate the edit of 75.85.165.206. Anderson localization may have been studied by mathematicians before physicists (you could say that about much of physics), but this particular description appears to be original research. In any case, a good clean-up is necessary.

Cdion 10:08, 17 September 2007 (UTC)

Can anyone think of better titles: "A mathematicians point of view" and "A physicists point of view" to distinguish, what the different kind of people think about Anderson localization. ElMaison (talk) 06:23, 5 April 2008 (UTC)

I've seen it in both contexts. Physicists tend to talk about semiconductors and physical models. Mathematicians try to find exact solutions to the discrete Laplace operator or rather the discrete Schrodinger operator. Even the one-dimensional case appears to be a challenge: They have managed to find solutions for two cases: periodic potentials which have travelling waves; also bandgaps, and the localized case (which was explicitly called 'Anderson localization' in the colloqium I went to), but they are still struggling with certain cases in between (where the potential is pseudo-periodic). Short of a full solution, the goal is to characterize the spectral gap. linas (talk) 06:03, 8 September 2008 (UTC)

Below follows the long rant that was added by User:75.85.165.206 In Sept. 2007. I copied it here because its an interesting rant. linas (talk) 06:15, 8 September 2008 (UTC)

Localization however is a very old pure-mathematics (graph-theory and percolation-theory) prediction and phenomenon! [ref.: P. G. Doyle and J.-L. Snell, "Random-Walks and Electric-Networks", Mathematical Association of America(MAA)(1984)-esp. Rayleigh's "short-cut method"!!!]; R. Zahlen, "Physics of Amorphous-Solids",Wiley(1983)]; C. St. J. Nash-Williams[Proc. Camb. Philo. Soc. 55, 181 (1959)]; J. Cohen[Discr. Appl. Math. 19, 113 (1986)]; E. Palmer["Graphical-Evolution", Wiley (1985)]; B. Bolabas["Random-Graphs", Academic (1985)]; famous seminal P. Erdos and J. Renyi[Publ. Math. Inst. Hungar. Acad. Sci., 5, 17 (1960); Publ. Math. Debrecen, 6, 290 (1959); Bull. Inst. Intl. Stat. 36, 343 (1961); Acta.Math. Acad. Sci. Hungar. 12, 261 (1961)], very famous E. P. Wigner[Ann. Math. 62, 548 (1955)]; related to 1870 C. Jordan["Cours d'Analyse", Gauther-Villars(1887)]-J. Schoenflies[Deutsch. Math. Verein.,2 Leipzig(1908)] curve theorem [ref.: C. Stillwell, "Classical-Topology and Combinatorial Group-Theory", Springer (1981)]; originated by famous seminal George Polya [ref.: Math. Annalen 84, 149 (1921)] following on much much earlier much more famous seminal J. W. S. (Lord) Rayleigh [ref.: "Collected Works" (1899); Phil. Trans. CLXI(1870)], following on even much much much earlier even much much more famous seminal B. Riemann["Collected Works"(1851); Dover(1953)].

So, historically, it should be properly termed the Riemann-Rayleigh-Polya-Anderson (RRPA) localization and localization-delocalization phase-transition critical-phenomenon after Anderson's rediscovery in physics of a century earlier pure-mathematics localization!!!

Inclusion in Lawvere-Goguen-Derrida-Chomsky["The Minimalist Program", MIT Press(1995)]-Wierzbicka-Langacker-Lakoff-Nunez["Where Mathematics Comes From", Basic Books (2000)]-Berwick-Baez-Siegel[J. Noncrystalline Solids, 40, 453 (1980); "IBM Conf. on Computers and Mathematics", Stanford(1986); "Aristotle Birthday Symposium", Thessoloniki (1990); "Symposium on Fractals...", Materials Research Society Fall Mtg., Boston (1989)-5-papers!!!]-Hofstadter["Fluid-Concepts and Creative-Analogies", Basic Books(1995)]-Coulson["Semantic-Leaps", Cambridge(2001)]-Fauconnier-Turner["The Way We Think: Conceptual-Blending...", Basic Books (2002)]-Cohen-Stewart["The Collapse of Chaos: Discovering Simplicity in a Complex World", (1994)-call for science unification simplicity via both bottom-up induction "complic-ity" and top-down deduction "simple-xity" simultaneously, what "FUZZYICS" {like Altschuler-Tsurakov-Lewis "TRIZ"(Russian acronym: "the method of inventive problem-solving"] as embodied in softwares: "Invention-Machine" and "Ideation" does automatically!!!] categorical-semantics {of Cytowic-Ramachandran "Synaesthesia"} forms on asymptotic-limit antipode of (googleable) "FUZZYICS" tabular list-format analysis truth-table for doing meta-physics[E. Siegel, "Symposium on Fractals...", Materials Research Society Fall Mtg., Boston (1989)-5-papers!!!] and increasingly meta-mathematics "Millennium-Problems": Siegel: 1964(@CCNY) simple understandable physicist's proof of Fermat's last-theorem(starting from Pythagorean-theorem, via seminal Menger["Dimensiontheorie", Teubner (1928)] dimension-theory, via Noether's-theorem, equivalence to Fermat's(hence no need for separate FLT proof!!!) principle of least-action and vector-subtraction[E. Siegel et. al., Am. Math. Soc. National-Mtg., San Diego(2002)]; recently computer-"science" (so miscalled) "computational-complexity"("feet of Clay") trivial P = NP conjecture[E. Siegel et. al., Am. Math. Soc. National-Mtg., San Diego(2008); proved P =/= NP, by Demosthenes and Euclid (~ -300-350 B.C.E.), a "Millennium Problem", but hidden in "jargonial-obfuscation" requiring critical cognitive-semantics disabiguation/deconstruction, a "Three Millenia Ago Problem"!!!]; harder algebraic-number-theory Birch and Swinnerton-Dyer conjecture[E. Siegel et. al., Am. Math. Soc. National-Mtg., San Diego(2008)]; with some progress in Riemann-hypothesis proof ongoing.

Revision?

I think several important parts are missing:

• definition of Anderson Hamiltonian (note that the article "Anderson model" is about something different)
• some explanation (or ref.) to what are localised and extended states
• summary of mathematically rigorous results: localisation in 1D (Goldsheid-Molchanov-Pastur ...), and in arbitrary dimension for high energy/high disorder (Frohlich-Spencer ...)

This looks like a major revision, so it would be nice to have some discussion here before starting it. Sasha (talk) 00:27, 16 June 2011 (UTC)

I have added an introduction with the precise definition of the model. Still planning to add:

• summary of mathematically rigorous results
• spectral formulation

Sasha (talk) 00:10, 7 August 2011 (UTC)

New Result

The page should updated to include the result described in http://news.illinois.edu/news/11/1007waves_BrianDeMarco.html I am not qualified to make this update. — Preceding unsigned comment added by 76.115.88.202 (talk) 02:35, 10 December 2012 (UTC)

Dropoff of V(r)

This article says V must drop off as 1/r^2 for localisation to occur, but the original article by Anderson has V dropping off faster than 1/r^3 with special behaviour in the exact 1/r^3 case. Which is correct? There's not citation for the claim here. Jaredjeya (talk) 21:19, 27 April 2018 (UTC)

Newer Result

Apparently (defect-free) quasicrystals can localize light as well, as has just been shown experimentally by Sinelnik et al.. This is especially interesting, as we do not have randomness here, but rather aperiodic "pseudo-randomness". I feel just as unqualified to update the article as the IP from 2012 in the talk section above sadly. --2003:CC:370B:9636:D58A:F5E5:6F35:74C2 (talk) 09:03, 11 November 2020 (UTC)

Diffusion or dispersion

Spreading of waves is usually called dispersion, not diffusion. The waves travel at different speeds, and the wave packets widen. The equation is a wave equation (e.g. a 2nd order hyperbolic equation or the Schroedinger equation), for which single-frequency waves travel unchanged.

Diffusion refers to heat distributions or chemical concentrations that spread by repeated random collisions of particles, governed by a parabolic equation.

Why is the spreading called diffusion here?

If it is because the waves knock around in the random medium a bit like particles in Brownian motion, this should be explained somewhere.

"The waves are said to diffuse because..."

2001:171B:2274:7C21:6513:804D:119:3D43 (talk) 11:47, 23 June 2022 (UTC)

In quantum mechanics, particles are waves. As particles do under normal circumstances diffuse, you can talk about wave diffusion too. Phillip Anderson himself titled his paper "Absence of diffusion in certain random lattices". And when localised, all transport (over longer distances than the localisation length) is arrested. The same applies to classical waves that are localised through the same mechanism -- they do not travel through the lattice, so you could say they're failing to diffuse.

Absence of dispersion would be a very different phenomenon. That would be ballistic transport through a medium, which is almost the opposite of localisation. Jaredjeya (talk) 13:34, 31 July 2023 (UTC)

In Anderson localization, when a wave is localized, there can't be any dispersion relation. That's the whole point! The waves don't propagate off to infinity; there is no "single frequency", their momentum is necessarily zero (although I guess you can cook up some localized wave-vector if you're careful; note how the article avoids any talk of momenta or wave-vectors.) 67.198.37.16 (talk) 23:50, 16 May 2024 (UTC)

Definitions of variables

It would be good to have formal definitions of all the variables, like W, and D etc — Preceding unsigned comment added by 89.114.64.121 (talk) 21:10, 5 February 2023 (UTC)

and Ψ_i in the Introduction (localized wave function ? it is not said that i, j, k are sites of a lattice) Lau 00 (talk) 11:09, 22 June 2023 (UTC)

Solutions to the Schrodinger eqn are called wave functions. The j,k are lattice locations. I updated the article to state this. 67.198.37.16 (talk) 23:26, 16 May 2024 (UTC)

Cobus et al. 2023

The article references a "Cobus et al. (2023)" under experimental evidence, but it isn't listed in the further reading and I could find no evidence of such a paper on localisation of light in 3D. There is a paper by Laura Cobus published in 2022, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.106.014208, which matches the description in the article, but not one in 2023. I suspect there's a typo but I don't want to edit the article without consulting the community first. Is anyone aware of a Cobus et al. (2023)? Jaredjeya (talk) 13:17, 31 July 2023 (UTC)

@Cornicello could you clarify which article you meant please? Jaredjeya (talk) 13:19, 31 July 2023 (UTC)

Article says:

Recent experiments (Naraghi et al., 2016; Cobus et al., 2023) ...

but there is no Cobus in the references. 67.198.37.16 (talk) 23:59, 16 May 2024 (UTC)

Is light localized or not?

One section states:

Direct numerical solution of Maxwell equations to demonstrate Anderson localization of light has been implemented (Conti and Fratalocchi, 2008).

but a later section states:

The existence of Anderson localization for light in 3D was debated for years (Skipetrov et al., 2016) and remains unresolved today.

Perhaps the earlier result is not for 3D? The critical dimension is 2, so ??? I guess the point here is that light is a vector, not a scalar? Why would vector vs. scalar make a difference? ??? 67.198.37.16 (talk) 23:56, 16 May 2024 (UTC)