Slicer Map: Deterministic Model for Anomalous Diffusion

• Slicer Map is a deterministic one-dimensional transport model using interval-exchange transformations to produce sub-, normal, and superdiffusive regimes.
• A single control parameter, α, modulates the transport behavior, generating heavy-tailed distributions similar to those in stochastic anomalous diffusion models.
• Analytic derivations of moments, mean-square displacements, and position correlations provide deep insights into non-chaotic diffusion and polygonal billiard dynamics.

Searching arXiv for the cited Slicer Map papers and closely related records.

arXiv Search Results

Query: "([1507.04242](/papers/1507.04242))" OR "([1709.04980](/papers/1709.04980))" OR "slicer map anomalous diffusion" Found 6 results

1. A simple non-chaotic map generating subdiffusive, diffusive and superdiffusive dynamics Authors: G. P. Lepage, R. Salari, L. Rondoni, R. Klages (Salari et al., 2015) Submitted: 2015-07-15 URL: http://arxiv.org/abs/([1507.04242](/papers/1507.04242))v1
2. Equivalence of position-position auto-correlations in the Slicer Map and the Lévy-Lorentz gas Authors: R. Salari, D. Davalos, G. F. de Arruda, R. Klages, L. Diambra (Giberti et al., 2017) Submitted: 2017-09-14 URL: http://arxiv.org/abs/([1709.04980](/papers/1709.04980))v1
3. Anomalous diffusion in deterministic dynamical systems Authors: R. Klages cond-mat/0612008 Submitted: 2006-12-01 URL: http://arxiv.org/abs/cond-mat/0612008v1
4. Weakly chaotic dynamics and anomalous diffusion in mushroom billiards Authors: C. P. Dettmann, O. Georgiou (Jentschura, 2010) Submitted: 2010-11-23 URL: http://arxiv.org/abs/([1011.5275](/papers/1011.5275))v1
5. One-dimensional Lévy-Lorentz gas: A numerical study Authors: P. V. Buonsante, R. Burioni, D. Cassi, A. Vezzani (0911.2583) Submitted: 2009-11-13 URL: http://arxiv.org/abs/(0911.2583)v1
6. Lévy walks in nonhomogeneous environments Authors: A. Falcón-Cortés, D. Boyer, M. Sandoval, V. H. Ponce, J. C. Fernández-Toledo, C. A. Vargas (Foster et al., 2016) Submitted: 2016-08-01 URL: http://arxiv.org/abs/([1608.00433](/papers/1608.00433))v1

The Slicer Map (SM) is a one-dimensional deterministic model of transport defined as an interval-exchange transformation on an infinite chain of unit cells. Introduced in "A simple non-chaotic map generating subdiffusive, diffusive and superdiffusive dynamics" (Salari et al., 2015), it was constructed as an analytically tractable non-chaotic toy model for anomalous diffusion in polygonal billiards. Its defining feature is that a single control parameter, $\alpha$, tunes the asymptotic transport law across superdiffusion, normal diffusion, subdiffusion, logarithmic subdiffusion, and localization. A later analysis derived exact position–position auto-correlations and compared their asymptotic scaling with that of the Lévy–Lorentz gas (Giberti et al., 2017).

1. Definition as a lifted interval-exchange map

The phase space is built from the unit interval $M=[0,1]$ and the lattice extension

$\widehat M=M\times\mathbb Z,$

with cells $\widehat M_m=[0,1]\times\{m\}$ indexed by $m\in\mathbb Z$. For each cell, the map introduces slicer points at

$x=\ell_m,\qquad x=\tfrac12,\qquad x=1-\ell_m,$

where

$$\ell_m(\alpha)=\frac{1}{\bigl(|m|+2^{1/\alpha}\bigr)^\alpha},\qquad \alpha>0,$$

so that $0<\ell_m<\tfrac12$. The slicer map $S_\alpha$ transports points to the left or right neighboring cell while preserving the $x$-coordinate: $M=[0,1]$0

By construction, all distances are preserved except at the countable set of slicer points $M=[0,1]$1, which has Lebesgue measure zero. This yields vanishing Lyapunov exponents and guarantees that $M=[0,1]$2 is non-chaotic (Salari et al., 2015). In the 2017 formulation, the same map is described as a piecewise-constant jump on $M=[0,1]$3, with the coarse-grained displacement defined by

$M=[0,1]$4

for an ensemble initially supported on the right half $M=[0,1]$5 (Giberti et al., 2017).

2. Exact transport probabilities and ensemble formulation

Transport is analyzed by evolving ensembles rather than individual trajectories. In the 2015 treatment, the initial ensemble is uniformly distributed in the central cell $M=[0,1]$6, and the coarse-grained cell occupation probability at time $M=[0,1]$7 is

$M=[0,1]$8

This distribution can be written in closed form. The traveling weights at $M=[0,1]$9 are

$\widehat M=M\times\mathbb Z,$0

while for $\widehat M=M\times\mathbb Z,$1 the sub-traveling weights are

$\widehat M=M\times\mathbb Z,$2

Hence

$\widehat M=M\times\mathbb Z,$3

For $\widehat M=M\times\mathbb Z,$4, the jump statistics have the heavy-tailed form

$\widehat M=M\times\mathbb Z,$5

which motivates comparison with stochastic anomalous-transport processes (Salari et al., 2015). The 2017 paper reformulates the same transport problem in terms of the integer-valued displacement $\widehat M=M\times\mathbb Z,$6 and derives correlation functions from exact integral representations over the initial ensemble (Giberti et al., 2017).

A structural point emphasized in both analyses is that the map generates long displacements without local stretching or folding. The transport is therefore deterministic and non-chaotic, but its coarse-grained statistics display power-law behavior normally associated with random-flight models.

3. Moments, mean-square displacement, and transport regimes

The $\widehat M=M\times\mathbb Z,$7-th moment of the position at discrete time $\widehat M=M\times\mathbb Z,$8 is

$\widehat M=M\times\mathbb Z,$9

A detailed asymptotic analysis gives, for even $\widehat M_m=[0,1]\times\{m\}$0,

$\widehat M_m=[0,1]\times\{m\}$1

while all odd moments vanish by symmetry. In particular,

$\widehat M_m=[0,1]\times\{m\}$2

so that the transport exponent is

$\widehat M_m=[0,1]\times\{m\}$3

(Salari et al., 2015).

The later correlation paper states the mean-square displacement more explicitly for the right-half initial ensemble: $\widehat M_m=[0,1]\times\{m\}$4 It also lists ballistic motion for $\widehat M_m=[0,1]\times\{m\}$5, with $\widehat M_m=[0,1]\times\{m\}$6 (Giberti et al., 2017).

Regime	Parameter range	Asymptotic MSD
Ballistic motion	$\widehat M_m=[0,1]\times\{m\}$7	$\widehat M_m=[0,1]\times\{m\}$8
Superdiffusion	$\widehat M_m=[0,1]\times\{m\}$9	$m\in\mathbb Z$0
Normal diffusion	$m\in\mathbb Z$1	$m\in\mathbb Z$2
Subdiffusion	$m\in\mathbb Z$3	$m\in\mathbb Z$4
Logarithmic subdiffusion	$m\in\mathbb Z$5	$m\in\mathbb Z$6
Localization / bounded variance	$m\in\mathbb Z$7	$m\in\mathbb Z$8

The phase diagram is therefore a straight line $m\in\mathbb Z$9 crossing the diffusive threshold $x=\ell_m,\qquad x=\tfrac12,\qquad x=1-\ell_m,$0 at $x=\ell_m,\qquad x=\tfrac12,\qquad x=1-\ell_m,$1 and vanishing at $x=\ell_m,\qquad x=\tfrac12,\qquad x=1-\ell_m,$2 (Salari et al., 2015). One consequence is that a single deterministic mechanism, controlled only by $x=\ell_m,\qquad x=\tfrac12,\qquad x=1-\ell_m,$3, spans the standard anomalous-transport regimes as well as the localized regime.

4. Exact position–position auto-correlations

"Equivalence of position-position auto-correlations in the Slicer Map and the Lévy-Lorentz gas" (Giberti et al., 2017) derives exact two-point position auto-correlations for the coarse-grained displacement: $x=\ell_m,\qquad x=\tfrac12,\qquad x=1-\ell_m,$4 The derivation partitions the integration domain into three regions according to the integer $x=\ell_m,\qquad x=\tfrac12,\qquad x=1-\ell_m,$5 and yields the asymptotic sum representation

$x=\ell_m,\qquad x=\tfrac12,\qquad x=1-\ell_m,$6

where

$x=\ell_m,\qquad x=\tfrac12,\qquad x=1-\ell_m,$7

Three asymptotic regimes are worked out explicitly.

For fixed $x=\ell_m,\qquad x=\tfrac12,\qquad x=1-\ell_m,$8 and $x=\ell_m,\qquad x=\tfrac12,\qquad x=1-\ell_m,$9,

$$\ell_m(\alpha)=\frac{1}{\bigl(|m|+2^{1/\alpha}\bigr)^\alpha},\qquad \alpha>0,$$0

For fixed time lag $$\ell_m(\alpha)=\frac{1}{\bigl(|m|+2^{1/\alpha}\bigr)^\alpha},\qquad \alpha>0,$$1 and $$\ell_m(\alpha)=\frac{1}{\bigl(|m|+2^{1/\alpha}\bigr)^\alpha},\qquad \alpha>0,$$2,

$$\ell_m(\alpha)=\frac{1}{\bigl(|m|+2^{1/\alpha}\bigr)^\alpha},\qquad \alpha>0,$$3

For linear separation $$\ell_m(\alpha)=\frac{1}{\bigl(|m|+2^{1/\alpha}\bigr)^\alpha},\qquad \alpha>0,$$4 and $$\ell_m(\alpha)=\frac{1}{\bigl(|m|+2^{1/\alpha}\bigr)^\alpha},\qquad \alpha>0,$$5,

$$\ell_m(\alpha)=\frac{1}{\bigl(|m|+2^{1/\alpha}\bigr)^\alpha},\qquad \alpha>0,$$6

These results are notable because fully analytic derivations of multipoint correlations are rare in anomalous-transport studies. The SM thus provides a deterministic setting in which both moments and two-point correlations can be obtained in closed asymptotic form.

5. Relation to Lévy walks, CTRW, gLE, and the Lévy–Lorentz gas

The heavy-tailed statistics of $$\ell_m(\alpha)=\frac{1}{\bigl(|m|+2^{1/\alpha}\bigr)^\alpha},\qquad \alpha>0,$$7 invite comparison with several stochastic models, but the comparison is explicitly limited to formal matching of moment-scaling exponents or asymptotic correlation laws rather than full dynamical equivalence (Salari et al., 2015).

In the superdiffusive regime $$\ell_m(\alpha)=\frac{1}{\bigl(|m|+2^{1/\alpha}\bigr)^\alpha},\qquad \alpha>0,$$8, the second moment $$\ell_m(\alpha)=\frac{1}{\bigl(|m|+2^{1/\alpha}\bigr)^\alpha},\qquad \alpha>0,$$9 formally matches that of certain Lévy-walk or Lévy-Lorentz models under an appropriate parameter mapping $0<\ell_m<\tfrac12$0. The 2015 analysis states that all even moments coincide with those of the one-dimensional Lévy-Lorentz gas for an equilibrium or non-equilibrium choice of initial condition, via a piecewise transformation between the Lévy index $0<\ell_m<\tfrac12$1 and $0<\ell_m<\tfrac12$2. It also stresses that the mechanisms differ: in the slicer map the heavy tails arise from deterministic slicing of phase space, whereas in a Lévy walk they arise from random flight times (Salari et al., 2015).

A continuous-time random walk with power-law waiting times can also produce $0<\ell_m<\tfrac12$3, but only for $0<\ell_m<\tfrac12$4 does the corresponding stable law apply. By contrast, the slicer map attains superdiffusion down to $0<\ell_m<\tfrac12$5, outside the CTRW domain of validity. In the subdiffusive regime $0<\ell_m<\tfrac12$6, the slicer MSD formally coincides with that of an overdamped generalized Langevin equation driven by power-law anti-persistent noise, yet the slicer probability distribution remains non-Gaussian and its memory/correlation structure is of purely geometric origin (Salari et al., 2015).

The 2017 paper sharpens the comparison with the Lévy–Lorentz gas, specifically LLg$0<\ell_m<\tfrac12$7, a discrete-time random walk with scatterers at iid distances distributed according to

$0<\ell_m<\tfrac12$8

together with equal reflection and transmission probabilities and a reflecting boundary at the origin. For LLg$0<\ell_m<\tfrac12$9 with non-equilibrium initial conditions,

$S_\alpha$0

where

$S_\alpha$1

Matching is imposed by choosing $S_\alpha$2 such that $S_\alpha$3, and three conjectured correlation scalings are then proposed for LLg$S_\alpha$4 directly from the SM formulas. The numerically estimated position-position correlations of the Lévy–Lorentz gas are reported to show remarkable agreement with the conjectured asymptotic scaling, over $S_\alpha$5, with no free fit parameters beyond the $S_\alpha$6 matching (Giberti et al., 2017).

A persistent misconception is that such agreements identify a unique stochastic surrogate for the SM. The source papers state the opposite: no single stochastic model reproduces the full $S_\alpha$7 line from $S_\alpha$8 up to localization, and the observed equivalences concern asymptotic scaling, not identical microscopic dynamics or full distributions (Salari et al., 2015).

6. Interpretation in non-chaotic diffusion and polygonal billiards

The slicer map was devised to capture what the 2015 paper identifies as the minimal geometric ingredient behind anomalous diffusion in non-chaotic polygonal billiards: repeated splitting of particle beams as they collide with polygonal corners, producing thinner and thinner sub-beams far from the origin (Salari et al., 2015). In polygonal channels, points diverge linearly almost everywhere, so Lyapunov exponents vanish, but corner collisions split trajectories. The one-dimensional slicer construction mimics that mechanism through progressively varying cuts.

This interpretation gives the SM a specific role as a reduced model. It can illustrate how subdiffusion, normal diffusion, and superdiffusion may all arise from a single mechanism under parameter variation; explain the extreme sensitivity of transport exponents to small geometrical changes in polygonal billiards; and serve as a test-bed for developing and comparing stochastic approximations of non-chaotic diffusion (Salari et al., 2015). A concrete example emphasized in the 2015 study is $S_\alpha$9, which yields

$x$0

in striking agreement with numerical experiments on certain rational-angle polygonal channels.

The 2017 correlation analysis adds a broader theoretical implication. It presents the agreement between SM and LLg$x$1 correlations as a form of universality in which the detailed microscopic dynamics—chaotic versus non-chaotic, deterministic versus random—do not affect the leading power-law behavior when the ballistic-flight lengths share the same exponent (Giberti et al., 2017). This suggests that the SM is most useful not as a literal reduction of a single physical system, but as a minimal analytic framework for isolating which geometric features are sufficient to generate given transport exponents and correlation laws.

Future work explicitly indicated in the 2015 paper includes correlation functions in the slicer map, stochastic embeddings, and rigorous derivations from explicit billiard geometries (Salari et al., 2015). The subsequent derivation of exact position–position correlations (Giberti et al., 2017) realizes part of that program while leaving the broader billiard-to-map correspondence as an open analytical problem.