--- title: Channel-Asymmetric Anomalous Diffusion url: https://www.emergentmind.com/topics/channel-asymmetric-anomalous-diffusion-caad type: topic --- # Channel-Asymmetric Anomalous Diffusion to=arxiv_search สำนักเลขานุการองค์กร ￣色രം 北京赛车女json {"query":"2510.00153 Anomalous diffusion in multichannel systems without a Lévy distribution of disorder", "max_results": 5} RTLU Channel-Asymmetric Anomalous Diffusion (CAAD) denotes anomalous transport generated by asymmetry between propagation channels rather than by a heavy-tailed disorder distribution. In the quantum multichannel formulation, CAAD arises in systems with uncorrelated but asymmetric Anderson-type disorder, where weak interchannel coupling produces long-tailed transmission statistics through the coexistence of ballistic and localized channel segments [2510.00153]. In a distinct quasiperiodic setting, anomalous diffusion can also become channel-asymmetric because collision-free corridors and their orientations contribute unequally to different Cartesian components of the mean-squared displacement [1206.1103]. Across these settings, the term refers to transport laws that deviate from normal diffusion through channel imbalance, anisotropic channel geometry, or both. ## 1. Definition and scope In the multichannel quantum setting, CAAD is introduced as a mechanism by which anomalous diffusion appears *without* a Lévy distribution of disorder. The defining ingredients are uncorrelated short-tailed onsite disorder, a pronounced asymmetry between channels, and sufficiently weak interchannel mixing. Under those conditions, transport at intermediate lengths is dominated by a superposition of ballistic and localized behavior, yielding anomalous effective scaling and broad transmission statistics [2510.00153]. In the quasiperiodic Lorentz-gas setting, anomalous diffusion is tied to the existence and geometry of collision-free channels. There the asymmetry is not between disorder strengths in coupled quantum channels, but between families of open strips whose widths and orientations enter the transport coefficients differently along different directions. The resulting transport can be weakly super-diffusive, normal, or subdiffusive depending on obstacle radius and channel closure, with unequal prefactors in different Cartesian components constituting the channel-asymmetric feature [1206.1103]. | Setting | Control of asymmetry | Principal anomalous signature | |---|---|---| | Two-channel quantum system | \(W_1 \gg W_2\) with weak \(t_\perp\) | \(\alpha \approx 1\), broad \(P(T)\), ballistic/localized coexistence | | Quasiperiodic Lorentz gas | Unequal channel widths \(w_k(r)\) and orientations \(\phi_k\) | \(t\ln t\) scaling with anisotropic prefactors, or subdiffusion after channel closure | A common misconception is that anomalous diffusion in these models must be driven by heavy-tailed disorder or rare arbitrarily strong scatterers. In the two-channel quantum model, all onsite energies remain uniformly distributed, and the anomalous regime is instead attributed to quantum interference between channels of differing disorder strength [2510.00153]. In the quasiperiodic Lorentz gas, the anomalous behavior is controlled by the geometry of collision-free corridors and, later, by bottleneck-induced trapping [1206.1103]. ## 2. Minimal two-channel quantum formulation The minimal quantum model consists of two coupled one-dimensional chains labeled \(\nu=1,2\), with scattering-region Hamiltonian \[ H \;=\;\sum_{\nu=1}^2\sum_{j=1}^L \varepsilon_{\nu j}\,c_{\nu j}^\dagger c_{\nu j} \;+\;t_\parallel\sum_{\nu=1}^2\sum_{j=1}^{L-1}\bigl(c_{\nu,j+1}^\dagger c_{\nu j}+{\rm h.c.}\bigr) \;+\;t_\perp\sum_{j=1}^L\bigl(c_{1j}^\dagger c_{2j}+{\rm h.c.}\bigr)\,, \] where \(c_{\nu j}^\dagger\) and \(c_{\nu j}\) create and annihilate an electron on site \(j\) of channel \(\nu\), \(t_\parallel\) is the intra-chain hopping amplitude, \(t_\perp\) is the interchain coupling, and \(\varepsilon_{\nu j}\) is Anderson-type onsite disorder drawn from a uniform distribution of width \(W_\nu\) and zero mean: \[ \varepsilon_{\nu j}\in\Bigl[-\tfrac{W_\nu}{2},\tfrac{W_\nu}{2}\Bigr],\qquad \langle\varepsilon_{\nu j}\rangle=0. \] In the simulations summarized for this model, \(W_1=t_\parallel\), \(W_2=t_\parallel/10\), and the tunable control parameter is \[ g \;\equiv\; \frac{t_\perp}{t_\parallel}\,. \] Strong coupling corresponds to \(g\approx1\), while weak coupling corresponds to \(g\ll1\) [2510.00153]. Transport is analyzed in the Landauer–Büttiker picture through the ensemble-averaged two-terminal transmission \(T(L)\) as a function of system length \(L\). The fit form is the generalized diffusion expression \[ T(L) \;=\; \frac{T_0}{\,1+\bigl(L/sp\bigr)^{\alpha/2}\!}\,, \] where \(T_0\) is the ballistic transmission in the absence of disorder, \(sp\) is the spread length at which \(T(L)=T_0/2\), and \(\alpha\) is the diffusion exponent. Within this convention, \(\alpha=2\) corresponds to normal diffusion, whereas \(\alpha<2\) indicates anomalous transport [2510.00153]. The asymmetry is built directly into the disorder strengths. Channel 1 is strongly disordered, Channel 2 is weakly disordered, and the role of \(t_\perp\) is to determine whether this asymmetry is effectively hybridized away or remains dynamically visible. ## 3. Crossover, transmission statistics, and segment lengths For strong coupling \(g\sim1\), the two chains hybridize fully and share the same mean-free-path \(\lambda\sim sp\). In that regime one recovers \(\alpha\approx2\) up to \(L\sim2sp\), followed by standard exponential localization for \(L\gg2sp\). For weak coupling \(g\ll1\), Channel 1 localizes quickly, \(T_1\to0\), while Channel 2 remains quasi-ballistic over very long distances, \(T_2\approx1\), so that \[ T(L)\;=\;T_1(L)+T_2(L) \] contains a mixture of localized and ballistic segments. Fitting the generalized diffusion form for \(L\gtrsim sp\) yields \(\alpha\approx1\) [2510.00153]. A semi-quantitative crossover criterion is \[ g \;\lesssim\;\frac{W_2}{W_1}\,, \] which marks the regime in which interchannel mixing is too weak to fully hybridize the disorder-strength asymmetry. This criterion places the onset of CAAD at weak transverse coupling and strong disorder asymmetry. The anomalous regime is also visible in the full transmission statistics. At fixed \(L\), the histogram of \(T\) over disorder realizations is neither the usual bimodal localized form nor the Dorokhov–Mello–Pereyra–Kumar curve associated with normal diffusion. Instead it shows a broad, almost uniform weight over a finite interval of \(T\), reflecting the superposition of one ballistic and one localized channel [2510.00153]. A second statistical diagnostic is obtained by “unfolding” the channel-resolved transmission along the chain and identifying ballistic intervals \(\ell\) over which Channel 2 remains essentially ballistic before a rare interchannel scattering event transfers amplitude into the localized channel. Numerically these segment lengths obey \[ P(\ell)\;\sim\;\ell^{-\mu},\qquad \ell\gg1, \] with exponent \(\mu\) related to the diffusion exponent by \[ \alpha \;=\; 3-\mu\quad(1<\mu<3). \] In the two-channel toy model, \(\mu\approx2\) is measured when \(\alpha\approx1\) [2510.00153]. This explicitly connects the transport fit exponent to the long-tail statistics of effective ballistic flights. ## 4. Quantum-interference mechanism and asymptotic localization The quantum formulation emphasizes that CAAD is intrinsically quantum. The long ballistic flights are attributed to interference between amplitude propagating in the weakly disordered channel and amplitude momentarily leaking into the strongly disordered channel. The “steps” are therefore quantum-coherent objects whose lengths are set by interchannel level splittings and weak hybridization, rather than by the statistics of rare, arbitrarily strong scatterers [2510.00153]. This point separates CAAD from classical Lévy-flight narratives. No heavy-tailed disorder distribution is introduced: all \(\varepsilon_{\nu j}\) remain short-tailed and uniform. The resemblance to Lévy flights arises at the level of long-tailed transmission or segment-length statistics, not at the level of the disorder ensemble itself. The anomalous regime is intermediate rather than asymptotically metallic. As \(L\to\infty\), all channels eventually localize. The localization length \(\xi\) is extracted from \[ \langle\ln T(L)\rangle\approx -L/\xi\,. \] However, the resulting \(\xi\) greatly exceeds the naive Thouless estimate \[ \xi_{\rm Thouless} \;=\; \frac{N_{\rm ch}+1}{2}\;\lambda, \] with \(N_{\rm ch}=2\) and \(\lambda\sim sp\) in the normal-diffusion regime. The dimensionless diagnostic \[ \eta\;\equiv\;\frac{2\,\xi}{(N_{\rm ch}+1)\,sp} \] satisfies \(\eta>1\) in the anomalous regime \((\alpha<2)\), while \(\eta\approx1\) once normal diffusion is recovered [2510.00153]. A common misunderstanding is therefore to equate CAAD with indefinite anomalous spreading. The summarized results state the opposite: CAAD governs transport at intermediate lengths, whereas conventional localization prevails asymptotically. The violation of the Thouless relation is not the absence of localization, but the persistence of a localization length enhancement beyond the normal-diffusion expectation. ## 5. Geometric CAAD in quasiperiodic Lorentz gases A distinct realization of channel-asymmetric anomalous diffusion appears in the two-dimensional quasiperiodic Lorentz gas through an embedding into a three-dimensional periodic billiard. Starting from the cubic lattice \(L=\mathbb{Z}^3\) with unit cell \[ C=[0,1]^3, \] one chooses a totally irrational plane \(E\subset\mathbb{R}^3\) with \(\dim E=2\) and \(E_\perp\oplus E=\mathbb{R}^3\). With \(\pi_E\) and \(\pi_{E_\perp}\) denoting orthogonal projections, the embedded obstacle in the unit cell is \[ K \;=\;\bigl\{\,x\in C:\|\pi_{E_\perp}(x)\|\le r\,\bigr\}, \] a circular cylinder of radius \(r\) whose axis is parallel to \(E\). Particle velocities are constrained to directions parallel to \(E\), and periodic boundary conditions in \(C\) reproduce motion in the quasiperiodic obstacle array [1206.1103]. Collision-free channels exist when a plane parallel to a face of the periodic cell does not intersect the cylinder \(K\). If \[ \delta \;=\;\min_{n\neq0}\bigl\|\pi_{E_\perp}(n)\bigr\|,\qquad r_c \;=\;\frac{\delta}{2}, \] then for any \(r