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Now we use transfer again: in [MATH] [EQUATION] so the same holds in [MATH] and [MATH] is the wanted element. Uniqueness follows from Lemma 2.4 (c). |
(b) For [MATH] , the set of primes that divide [MATH] is clearly bounded ( [MATH] can not be divisible by primes greater than itself). It is also internal by the Internal Definition Principle, so it has the greatest element by Proposition 1.2 ; let [MATH] be this element. For [MATH] we define [MATH] to be the greatest ... |
To prove uniqueness, assume [MATH] for some [MATH] and some sequence [MATH] . If [MATH] , this would mean that [MATH] is divisible by [MATH] ; if [MATH] then [MATH] would not be divisible by [MATH] . Either way we reach a contradiction, so [MATH] . In a similar manner we get a contradiction if we assume that [MATH] for... |
Let [MATH] be the function calculating the level of each [MATH] in the [MATH] -hierarchy. More precisely, if [MATH] , let [MATH] . Its extension [MATH] does the same for elements in [MATH] , when represented as [MATH] , as in Theorem 2.5 . Namely, let ” [MATH] ” denote the formula |
[EQUATION] Then, by transfer, [MATH] is the unique function satisfying, for every [MATH] and every internal [MATH] , the formula [MATH] |
Here are some properties of the function [MATH] , proven easily by transfer. Lemma 2.6 Let [MATH] be such that [MATH] (a) Either [MATH] or [MATH] |
(b) If [MATH] , then there is [MATH] such that [MATH] [MATH] and [MATH] In the following two sections we will see that the nice structure of the [MATH] -hierarchy is mostly transferred to the first [MATH] -many levels of the [MATH] , but not above them. |
The connection with the Stone-Čech compactification We have already encountered several analogies between the divisibility relation [MATH] on [MATH] and the relation [MATH] on [MATH] . First, [MATH] is divisible by [MATH] if and only if [MATH] (Lemma 2.3 (a)) and [MATH] is divisible by [MATH] if and only if [MATH] |
, Lemma 5.1). Also, an ultrafilter [MATH] is prime if and only if it concentrates on the set of primes. By Lemma 2.2 the same thing holds in [MATH] for the relation [MATH] , so [MATH] is prime if and only if [MATH] is a prime ultrafilter. |
We will now establish a connection between the relation [MATH] and the divisibility in [MATH] , showing that these similarities are not coincidental. It also shows that [MATH] is, in some sense, ”the right” divisibility relation to investigate in [MATH] |
Theorem 3.1 The following conditions are equivalent for every two ultrafilters [MATH] (i) [MATH] (ii) in every enlargement [MATH] , there are [MATH] such that [MATH] [MATH] and [MATH] |
(iii) in some enlargement [MATH] , there are [MATH] such that [MATH] [MATH] and [MATH] Proof. (i) [MATH] (ii) Let [MATH] be an enlargement, and let [MATH] be such that [MATH] . We define a binary relation [MATH] |
[EQUATION] We prove that [MATH] is concurrent. Let a finite number of subsets of [MATH] be given; we need to find a pair [MATH] such that [MATH] for all given sets [MATH] . Since ultrafilters (and their complements) are closed for finite intersections, we may assume that we have at most four given sets: [MATH] [MATH] [... |
Now, since [MATH] is an enlargement, there is a pair [MATH] such that [MATH] for all [MATH] . By transfer, [MATH] holds if and only if [MATH] . Thus we get, for all [MATH] |
[EQUATION] This means that [MATH] and [MATH] (ii) [MATH] (iii) is obvious. (iii) [MATH] (i) In [MATH] we have, for every [MATH] [MATH] . Hence the same holds for [MATH] in any extension [MATH] i.e. [MATH] is closed upwards for [MATH] . This means that, if [MATH] for some [MATH] [MATH] , then [MATH] implies [MATH] for e... |
Note that the implication (iii) [MATH] (i) holds in any extension, not only in an enlargement. The next example shows that whether or not [MATH] holds is not independent from the choice of representatives [MATH] and [MATH] |
Example 3.2 Since [MATH] is reflexive (by transfer), it suffices to find [MATH] such that [MATH] and [MATH] . So assume [MATH] are such that [MATH] [MATH] and [MATH] . Let [MATH] be a function such that |
[EQUATION] [MATH] is easily constructed by recursion on [MATH] ). Now let [MATH] and [MATH] . Then [MATH] [EQUATION] (by Fact 1.3 (a)), but [MATH] , by ( ) and transfer. |
A chain (in [MATH] ) is a set [MATH] of elements linearly ordered by [MATH] ; an antichain is a set [MATH] of [MATH] -incomparable elements. The family of subsets of [MATH] containing the complements of all chains and the complements of all antichains has the finite intersection property, so there are ultrafilters cont... |
Lemma 3.3 Let [MATH] be an enlargement and [MATH] (a) [MATH] contains no infinite antichains as elements if and only if there are distinct [MATH] such that [MATH] |
(b) [MATH] contains no infinite chains as elements if and only if there are distinct [MATH] such that neither [MATH] nor [MATH] Proof. (a) First assume there is an infinite antichain [MATH] . In [MATH] we have [MATH] , so the same holds in [MATH] . Thus there are no [MATH] -comparable elements in [MATH] , so there are ... |
Now let [MATH] contain no infinite antichains. We define a binary relation [MATH] [EQUATION] [MATH] is concurrent: if we are given finitely many subsets of [MATH] , let [MATH] be the intersection of those in [MATH] , and [MATH] the intersection of those outside [MATH] . Then [MATH] , so it is not an antichain. Hence th... |
[MATH] is an enlargement, so there are distinct [MATH] such that [MATH] and [MATH] for all [MATH] ; but then [MATH] The proof for (b) is analogous. [MATH] |
Since ultrafilters on the first [MATH] -many levels of [MATH] -hierarchy contain antichains (an ultrafilter on the [MATH] -th level contains the set [MATH] ), using Lemmas 2.6 and 3.3 (a) by induction on [MATH] we can easily prove that, if [MATH] then [MATH] for every [MATH] . Hence such ultrafilters correspond precise... |
Lemma 3.4 Let [MATH] be any nonstandard extension. (a) [MATH] is of the form [MATH] for some [MATH] if and only if [MATH] for some prime ultrafilter [MATH] |
(b) [MATH] is of the form [MATH] for two distinct primes [MATH] such that [MATH] if and only if [MATH] (c) [MATH] is of the form [MATH] for two primes [MATH] such that [MATH] [MATH] and [MATH] if and only if [MATH] |
Proof. (a) Let [MATH] be the squaring function: [MATH] for [MATH] . Then [MATH] for some [MATH] implies [MATH] and [MATH] Now let [MATH] for a prime ultrafilter [MATH] . Then [MATH] (by Lemma 1.3 (b)) so [MATH] for some [MATH] |
(b) Let [MATH] for some [MATH] such that [MATH] . Let [MATH] . Since [MATH] , by transfer [EQUATION] so [MATH] . Thus [MATH] For the other direction let [MATH] be such that [MATH] for some prime ultrafilter [MATH] . Then [MATH] . In [MATH] we have [MATH] so, by transfer, [MATH] for some distinct [MATH] . To prove that ... |
(c) The proof is similar to the proof of (b). [MATH] Thus, ultrafilters containing families of the form [MATH] actually have two distinct ”ingredients”, but such that [MATH] can not distinguish between. |
Above finite levels There are, of course, also ultrafilters not concetrating on any [MATH] for [MATH] . The investigation of [MATH] becomes much more complicated at these higher levels. Limits of ultrafilters will prove useful for this purpose. |
Lemma 4.1 Let [MATH] be a [MATH] -increasing sequence in [MATH] (a) [MATH] for any nonprincipal ultrafilter [MATH] (b) For any two nonprincipal ultrafilters [MATH] and [MATH] |
[EQUATION] (c) Let [MATH] for some [MATH] . If [MATH] is such that [MATH] for all [MATH] , then [MATH] Proof. (a) If [MATH] for some [MATH] , then [MATH] for all [MATH] . Hence the set [MATH] is cofinite, so it belongs to [MATH] . It follows that [MATH] |
On the other hand, assume [MATH] is such that [MATH] for all [MATH] . Then [MATH] for all [MATH] , so [MATH] and [MATH] (b) Follows from (a). |
(c) [MATH] means that [MATH] . If this holds for all [MATH] , by (a) [MATH] , so [MATH] [MATH] In view of Lemma 4.1 (b), we will write [MATH] if [MATH] for some nonprincipal [MATH] |
Example 4.2 There are also infinite [MATH] -decreasing sequences in [MATH] . Let [MATH] be an enumeration of [MATH] . Let [MATH] , and [MATH] . Then each [MATH] is divisible by all [MATH] for [MATH] so, by Lemma 4.1 (c), [MATH] for [MATH] . Also, if [MATH] then [MATH] but [MATH] , so the sequence [MATH] is strictly dec... |
The following lemma is proved analogously to Lemma 4.1 Lemma 4.3 Let [MATH] be a [MATH] -decreasing sequence in [MATH] (a) [MATH] for any nonprincipal ultrafilter [MATH] |
(b) For any two nonprincipal ultrafilters [MATH] and [MATH] [EQUATION] (c) Let [MATH] for some [MATH] . If [MATH] is such that [MATH] for all [MATH] , then [MATH] |
Lemma 4.4 Let [MATH] [MATH] [MATH] and [MATH] . If there is [MATH] such that [MATH] , then [MATH] is an immediate successor of [MATH] in [MATH] |
Proof. By Theorem 3.1 [MATH] . Since [MATH] [MATH] . It remains to show that there is no [MATH] such that [MATH] [MATH] but [MATH] |
Claim. The set [MATH] is [MATH] . This means that every ultrafilter containing [MATH] is divisible by [MATH] . So assume [MATH] and let [MATH] . Clearly [MATH] . We can write [MATH] in the form [MATH] for some [MATH] such that [MATH] . By Lemma 2.3 (b), [MATH] . But [MATH] and [MATH] , so [MATH] must belong to every ul... |
Now assume [MATH] is as above. If [MATH] , by Claim we have [MATH] . Otherwise, if we assume [MATH] , then [MATH] . But [MATH] is disjoint from [MATH] , so [MATH] . This contradicts [MATH] [MATH] |
The following example shows that the condition of existence of [MATH] such that [MATH] can not be eliminated from the lemma above. |
Example 4.5 Let [MATH] for some [MATH] . We show that [MATH] . By Lemma 4.1 (a) [MATH] First, no elements divisible by any prime other than [MATH] can belong to [MATH] : if [MATH] and [MATH] are such that [MATH] , there is [MATH] such that [MATH] but [MATH] , so [MATH] , a contradiction. |
Now we prove that [MATH] for [MATH] . Assume the opposite, that there is [MATH] such that [MATH] for some [MATH] . Then [MATH] (otherwise [MATH] ). By the Transfer Principle [MATH] , a contradiction. |
It is easy to see that there is the [MATH] -maximal class in [MATH] (since [MATH] has the finite intersection property). Let [MATH] denote this maximal class. |
Lemma 4.6 For every [MATH] [MATH] if and only if [MATH] for all [MATH] Proof. If [MATH] , then [MATH] for all [MATH] so, for every [MATH] [MATH] , which is equivalent to [MATH] by Lemma 2.3 (a). |
Now assume [MATH] (i.e. [MATH] ) for all [MATH] . Let [MATH] be arbitrary and let [MATH] be a [MATH] -minimal element of [MATH] . Then [MATH] , so [MATH] . Hence [MATH] , which means that [MATH] [MATH] |
In particular, this means that [MATH] . Thus the distribution of ultrafilters by levels, described in previous section, fails for ultrafilters above finite levels by Lemma 4.1 (c): [MATH] would be on the [MATH] -th level, and at the same time has predecessors on infinite levels. |
Theorem 4.7 (a) Every [MATH] has an immediate successor in [MATH] (b) Every [MATH] such that there are [MATH] and [MATH] so that [MATH] has an immediate predecessor in [MATH] |
Proof. (a) Let [MATH] . Since [MATH] , by Lemma 4.6 there is [MATH] such that [MATH] . Thus there are [MATH] and [MATH] so that [MATH] . By Lemma 4.4 |
[MATH] is the immediate successor of [MATH] (b) Let [MATH] . As in (a) we can show that [MATH] is the immediate predecessor of [MATH] [MATH] |
Example 4.8 We show that the condition [MATH] of Theorem 4.7 (b) can not be omitted. Let [MATH] , as in Example 4.5 . Then [MATH] is divisible by all powers of [MATH] and not divisible by any other prime. Assume [MATH] has an immediate predecessor [MATH] . We consider two cases. |
[MATH] [MATH] for all [MATH] . Then by Lemma 4.1 (c) [MATH] , a contradiction. [MATH] [MATH] for some [MATH] . Let [MATH] and [MATH] be such that [MATH] . Then, by Lemma 4.4 [MATH] is a successor of [MATH] and, since [MATH] and [MATH] is the least common multiplier of [MATH] and [MATH] [MATH] as well, meaning that [MAT... |
Open problems and final remarks We mention several questions, the answers to which may shed some more light to the above results. |
Question 5.1 (a) Does [MATH] imply [MATH] for [MATH] (b) More generally, does [MATH] and [MATH] imply [MATH] for [MATH] Question 5.2 |
Does [MATH] imply [MATH] for [MATH] This is true if [MATH] are distinct. Namely, assume there is [MATH] . Let [MATH] [MATH] and [MATH] be subsets of [MATH] such that [MATH] and [MATH] are disjoint. By |
Lemma 3.7, [MATH] and [MATH] . Then [MATH] . If we define [MATH] by [MATH] (for [MATH] ) and [MATH] arbitrary if [MATH] , then [MATH] [MATH] [MATH] and [MATH] , a contradiction with [MATH] |
Question 5.3 Let [MATH] be an enlargement. (a) Does [MATH] imply [MATH] (b) Does [MATH] imply [MATH] Part (a) is true if the answer to Question 5.1 is ”yes”. Namely, [MATH] means that there are [MATH] and [MATH] such that [MATH] . But then it would follow that, for every [MATH] [MATH] as well. |
Question 5.4 Let us call a set [MATH] convex if for all [MATH] and [MATH] [MATH] and [MATH] implies [MATH] . Is [MATH] a convex set for every [MATH] |
Clearly, every [MATH] is convex: if [MATH] and [MATH] , then [MATH] would imply that [MATH] , so [MATH] and [MATH] would imply [MATH] , so [MATH] |
Question 5.5 Can we strengthen Example 4.5 in the following sense: if [MATH] for some [MATH] , does [MATH] The research was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (project 174006). |
# Source: arxiv 1806.06345 # Title: Universal Nonlinear Disordered Wave Packet Subdiffusion: 12 Decades # Sections: all # Downloaded: 2026-03-03T05:15:32.540584+00:00 |
Universal Nonlinear Disordered Wave Packet Subdiffusion: 12 Decades Abstract We use a novel unitary map toolbox – discrete time quantum walks originally designed for quantum computing – to implement ultrafast computer simulations of extremely slow dynamics in a nonlinear and disordered medium. Previous reports on wave ... |
Eigenstates of linear excitations in a one-dimensional medium exposed to an uncorrelated random external field are exponentially localized in space, due to the celebrated Anderson localization (AL) Anderson ( 1958 Thus any evolving compact wave packet in such a system will first spread, but then halt and not escape fro... |
Lifshits et al. 1988 Experimental verifications of AL with Bose-Einstein condensates of ultracold atomic gases loaded onto optical potentials were using precisely the above technique, i.e. the time evolution of a wave packet, to prove and quantitatively characterize the degree of AL Billy et al. 2008 ); Roati et al. 20... |
The interplay of disorder with many body interactions intrigued the minds of researchers ever since AL was established. Recent experimental attempts include granular chains Kim et al. 2018 , photonic waveguide lattices Lahini et al. 2008 , light propagation in fiber arrays Pertsch et al. 2004 and atomic Bose-Einstein c... |
Lucioni et al. 2011 . Different values of the exponent [MATH] were measured, which ranged between [MATH] and [MATH] . That imprecision is due to the slow dynamics of subdiffusion that did not allow to quantitatively assess the subdiffusion exponents. E.g. in the atomic gas case, the need to keep the condensate coherent... |
Computational studies of spreading wave packets in various nonlinear and disordered systems revealed interesting features. On times up to [MATH] , an initially compact wave packet expands up to the size of localization length [MATH] . After that a subdiffusive spreading of the wave packet Pikovsky and Shepelyansky ( 20... |
Qualitatively the subdiffusion can be explained as follows. The chaotic dynamics inside the wave packet leads to dephasing of the participating localized Anderson normal modes. With coherence lost, wave localization cannot be anymore sustained. The assumption of strong chaos (quick and complete dephasing of all modes) ... |
Flach et al. 2009 . An additional phenomenological estimate of the impact of finite but small probabilities of resonances between interacting normal modes |
finally leads to a substantial suppression of dephasing and the correct value [MATH] Flach et al. 2009 Interestingly the validity of this estimate was confirmed with tests of its predictions for larger system dimensions Laptyeva et al. 2012 , and different exponents of nonlinear terms which correspond to various [MATH]... |
Skokos and Flach ( 2010 . Successful tests of systems with quasiperiodic (instead of random) potentials Larcher et al. 2012 , and nonlinear versions of quantum kicked rotors Shepelyansky ( 1993 ); Gligorić et al. 2011 yielded subdiffusion with [MATH] as well, and revealed additional universality aspects of the observed... |
Skokos et al. 2009 . As a side note, in weakly nonlinear systems Anderson localization in the evolution of finite size wave packets is restored in a probabilistic manner |
Johansson et al. 2010 ); Ivanchenko et al. 2011 ); Basko ( 2011 To address the fundamental question whether wave packet spreading slows down or continues, we use a novel unitary map toolbox – Discrete Time Quantum Walks (DTQW). We peek beyond previous horizons set by the CPU time limits for systems of coupled ordinary ... |
DTQW were introduced as quantum generalizations of classical random walks by Aharonov et al. Aharonov et al. 1993 The DTQW evolution is a (discrete) sequence of unitary operators acting on a quantum state in a high dimensional Hilbert space. DTQW shows quantum interference/superposition Aharonov et al. 1993 , entanglem... |
Consider a single quantum particle with an internal spin degree of freedom, moving on a one-dimensional lattice. Its dynamics is determined by a time- and lattice-site-dependent two-component wave function [MATH] , which evolves under the influence of some periodic Floquet drive. Then, its evolution can be mapped onto ... |
[EQUATION] A schematic map flow is shown in Fig. This unitary matrix parametrization is a particular realization of the general case discussed in Ref. Vakulchyk et al. 2017 , with two angles [MATH] (kinetic energy) and [MATH] (site dependent internal synthetic flux). Such coin operators can be implemented by time–depen... |
The transfer operator [EQUATION] with [MATH] corresponding to the [MATH] components shifting either to the right or to the left. We will use [MATH] across the paper. The DTQW evolution follows as a sequence of successive [MATH] and [MATH] operators acting on the state: |
[EQUATION] where the matrices [MATH] are defined by the elements of [MATH] [EQUATION] Next, we consider a strongly disordered DTQW with random uncorrelated angles [MATH] being uniformly distributed over the entire existence domain [MATH] . The resulting unitary eigenvalue problem is solved by finding the orthonormal se... |
[MATH] with [MATH] and the eigenvalues [MATH] , where [MATH] is the quasienergy. All eigenvectors are exponentially localized on the chain Vakulchyk et al. 2017 . This is a manifestation of Anderson localization. Remarkably, for such strong disorder all eigenvectors [MATH] |
are characterized by one single localization length [MATH] which does not depend on the quasienergy [MATH] of a given state Vakulchyk et al. 2017 |
[EQUATION] Another remarkable feature is that for any value of the localization length – either small or large compared to the lattice spacing [MATH] – the spectrum of the quasienergies of an infinite chain is densely filling the compact space of angles of complex numbers on a unit circle Vakulchyk et al. 2017 Therefor... |
Anderson localization is manifested through the halt of spreading of an evolving wave packet. In our direct numerical simulations, we choose [MATH] , which results in a localization length [MATH] and a typical localized Anderson eigenstate occupying about 10 lattice sites. We choose the initial state to be localized on... |
[EQUATION] We evolve this state using Eq.( ) until [MATH] for a system of size [MATH] and [MATH] . The density distribution [MATH] observed for such time is presented in Fig. (a) (orange solid lines). The distribution is clearly localized, with the width of a few localization lengths. The tails are exponentially decayi... |
[EQUATION] The time dependence of [MATH] is plotted in Fig. (a). We observe a halt of the growth of [MATH] at [MATH] , which together with the profile of the halted wave packet (see Fig. ) is a clear demonstration of Anderson localization. |
We now leave the grounds of linear DTQW and generalize the DTQW to a nonlinear unitary map by adding a density-dependent renormalization to the angle |
[EQUATION] where [MATH] is the nonlinearity strength. We note that ( ) keeps the conservation of the total norm: [MATH] Our main goal is to measure the details of subdiffusive wave packet spreading on large time scales with [MATH] Therefore we use a low-density approximation for the quantum coin ( ) which approximates ... |
[EQUATION] The computational advantage of fast calculations of square roots as opposed to slow ones of trigonometric functions serves the purpose to further extend the simulation times. To guarantee unitarity of the evolution, we choose [MATH] such that [MATH] |
We evolve a wave packet with [MATH] and [MATH] and plot the density distribution at the final time [MATH] in Fig. (blue solid line). This is a new record evolution time, beating old horizons by a factor of [MATH] . We observe a familiar structure of the wave packet: a homogeneous wide central part with clean remnants o... |
However, we note that a straightforward fitting with a single power law can yield misleading results, since it is not evident where the asymptotic regime (if any) will start. To study the asymptotic regime in detail we quantitatively assess it by applying standard methods of simulations and data analysis sup . We calcu... |
In order to reduce the fluctuation amplitudes, we obtain [MATH] for [MATH] disorder realizations and compute the geometric average [MATH] In Fig. (a) the results are shown for various values of [MATH] up to times [MATH] (the corresponding values of [MATH] are [MATH] ). All curves approach the vicinity of [MATH] with fl... |
[MATH] It is instructive to rewrite the evolution equations in the basis of linear eigenmodes of the [MATH] case. Using the Taylor expansion of nonlinear terms valid at large times when [MATH] |
sup we rewrite the unitary evolution of the wave function in the [MATH] eigenmode basis [MATH] [EQUATION] with the overlap integral |
[EQUATION] The structure of these resulting equations is strikingly similar to the ones obtained from Hamiltonian dynamics Pikovsky and Shepelyansky ( 2008 ); Flach et al. 2009 . In particular, we obtain cubic nonlinear terms on teh rhs of the asymptotic expansion ( 17 ). Together with the one-dimensionality of the sys... |
To conclude, DTQW are very useful unitary map toolboxes which allow for extremely fast quantum evolution, in particular due to covering finite times with one step (jump), and due to the fast(est) realization of a transfer/hooping/interaction on a lattice. We used a disordered version to obtain Anderson localization wit... |
and the universal exponent [MATH] . The record time [MATH] is reached, which exceeds old horizons by 3-4 orders of magnitude. The size of the wave packet reaches [MATH] . The relative strength (or better weakness) of the nonlinear terms in the DTQW reaches [MATH] . No slowing down of the subdiffusive process was observ... |
We expect DTQW to be useful in the future also for exploring other hard computational tasks, e.g. subdiffusion in two-dimensional and even three-dimensional nonlinear disordered lattices, and many body localization in interacting quantum settings. |
Acknowledgements. This work was supported by the Institute for Basic Science, Project Code (IBS-R024-D1). Supplementary Material |
I.1 Asymptotic evolution equation The discrete-time evolution is defined by a nonlinear map operator, [EQUATION] In order to separate the nonlinear components of [MATH] in [MATH] , we consider a single coin operator on the site [MATH] . It can be factorized, |
[EQUATION] where [MATH] is the local coin operator under zero nonlinearity. Evaluating this to separate the linear part and consecutive nonlinear terms with different nonlinear exponent yields, |
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