Infinite Mobility: Unbounded Transport

• Infinite Mobility is a regime in which movement or transport occurs at unbounded rates or distances, defying conventional constraints in geometry, energy, and system size.
• It spans multiple fields such as quantum lattice models, solid-state materials, fractal and stochastic mobility, wireless networks, and geometric rigidity systems.
• Insights into infinite mobility contribute to designing advanced materials, optimizing network stability, and developing scalable generative algorithms for motion synthesis.

Infinite mobility denotes a physical, mathematical, or algorithmic regime in which movement, transport, or flexibility occurs at formally unbounded rates or extends over unlimited distances or durations, often diverging from standard constraints imposed by geometry, external fields, disorder, computational resources, or system size. The term arises in condensed matter physics, statistical mechanics, wireless communications, geometric rigidity theory, motion generation, and procedural synthesis, where it encapsulates phenomena from charge transport without an applied field to condensation-driven current fluctuations, extended fractal exploration, and scalable articulated object generation.

1. Quantum Systems: Mobility Edges and Many-Body Localization

Infinite mobility in quantum lattice models predominantly emerges when the single-particle spectrum displays a mobility edge, as in the generalized Aubry–Andre (GAA) model (Li et al., 2015). The single-particle mobility edge (SPME) analytically divides localized and extended states according to the threshold

$$\alpha \varepsilon = 2\, \mathrm{sgn}(\lambda)(|t| - |\lambda|),$$

where $\varepsilon$ is the single-particle energy, $t$ the hopping amplitude, $\lambda$ the incommensurate potential strength, and $\alpha$ a model parameter.

Introduction of interactions drives the system beyond single-particle physics: a many-body mobility edge $E_L$ arises, separating eigenstates by entanglement entropy (EE) scaling—area law (localized) for $E < E_L$ and volume law (extended) for $E > E_L$. Independently, an energy scale $E_T$ marks thermalization onset according to the eigenstate thermalization hypothesis (ETH). Notably, $E_L < E_T$ generically, generating an extended non-ergodic regime ($E_L < E < E_T$) characterized by delocalized but non-thermal states.

Experiments are feasible with 1D optical lattices for cold atoms, exploiting engineered incommensurate potentials to realize the necessary GAA structure. Observables include energy-resolved entanglement, subsystem occupation statistics, and transport measurements. Infinite temperature many-body localization transitions persist despite SPME, offering direct insight into nonstandard dynamical phases.

2. Solid-State Physics: Charge Mobility in Muscovite

In natural muscovite mica, infinite charge mobility is observed at room temperature (Russell et al., 2018). The mechanism relies on the unique layered crystal structure—flat, monatomic potassium sheets sandwiched between silicate layers—which supports localized, mobile, anharmonic lattice excitations known as "quodons." These are generated by nuclear recoil events from the radioactive decay of $^{40}$K. Positive charge, once trapped in a quodon, propagates across the crystal at sonic speeds ($\sim 3500$ m/s) even in the absence of an applied electric potential.

The formal mobility $\mu$—defined as $\mu = v/E$ for drift velocity $v$ and electric field $E$—thus diverges when $E=0$, yielding the notion of "infinite mobility." Experimentally, the charge moves distances over $1000\times$ the alpha-particle range, bounded only by the crystal size. The effect, termed "hyper-conduction," distinguishes itself from superconductivity, suggesting new paradigms for electronic transport in insulators and potential device applications at elevated temperatures. Its realization depends critically on crystal quality and the continuous replenishment of a charge reservoir via ongoing $^{40}$K decay.

3. Fractal and Stochastic Models for Mobility Patterns

Infinite mobility in human and animal movement is modeled by embedding the Poisson Weighted Infinite Tree (PWIT) into Euclidean space (Darling et al., 2018). Vertices are indexed by the Ulam–Harris tree $\mathcal{U}$ with time ($\tau$) and spatial ($\chi$) assignment:

• Each node $v$ at time $\tau(v)$ and location $\chi(v)$ generates children via a Poisson process on $\mathbb{R}^d \times (\tau(v),\infty)$ with intensity

$$F(y,t) = f\left[\left(\frac{t}{\beta}\right)^{1/d} |y|\right] \frac{dy dt}{\theta}.$$

• Parameter $\rho = c_d (\beta/\theta)$, with $c_d = \int f(|x|) dx$, fixes fractal scaling.

As $n \to \infty$, agent visitation saturates a bounded region with Hausdorff dimension $d\rho$, producing random fractals and bursty inter-visit statistics. Finite realizations are achieved via Markovian point processes, with empirical acceptance probabilities

$$P(\text{accept}) \approx \exp(-\Delta n^{1/\alpha}),$$

where $\Delta$ is the nearest neighbor distance. This framework accurately reproduces the observed bounded yet infinite exploratory range in empirical mobility datasets, reconciling endless site visitation with spatial concentration.

4. Wireless Networks: Heavy Traffic and Queue Stability

Wireless queueing models underscore infinite mobility's impact on heavy traffic dynamics (Simatos et al., 2019, Ramesan et al., 2020). In Processor-Sharing queue systems with mobile and static users, conventional scaling at load $\rho \to 1$ is $\sim [1/(1-\rho)]$ for the mean queue length. Introducing even small mobility (users with positive escape rate $\theta > 0$) regularizes this divergence to $-\log(1-\rho)$, seen as mobile users "spatially diffuse" congestion (Simatos et al., 2019).

In wireless networks under Poisson field interference, static interferer configurations can cause infinite workload probability—a form of instability. For any nonzero mobility, strong mixing from independent displacements of interferers ensures ergodicity and universal stability, as verified by Loynes’ criterion ($\lambda < \delta \mathbb{E}[s(0)]$) (Ramesan et al., 2020). Increasing mobility iteratively reduces queue workload and delay (demonstrated via the increasing-convex stochastic order), trending to optimality at infinite mobility where service processes are temporally i.i.d.

Simulations confirm these analytic predictions, with decorrelation rates varying by mobility model (random direction, waypoint, Brownian motion). The stochastic ordering of queue metrics and positive correlation in SINR level crossings link practical network stability and responsiveness directly to the regime of infinite mobility.

5. Geometric Constraint Systems and Extrusion Symmetry

Extrusion symmetry introduces infinite mobility into geometric framework theory (Owen et al., 2023). By extruding a bar-joint or point-hyperplane structure $t$ times along specified vectors, one forms a graph with symmetry group $\mathbb{Z}_2^t$ (extruded Cartesian product), realizing redundancy and flexibility not present in conventional point-group-symmetric frameworks.

The rigidity matrix $R(G,p)$ transforms under block-diagonalization:

$$B^T R(G,p) A = \bigoplus_{j=0}^{2^t-1} \tilde{R}_j(G,p),$$

with $\tilde{R}_j$ representing irreducible components. Systematic Fowler–Guest–type character counts in each block quantify infinitesimal flexes (extra mobility) and self-stresses, refining Maxwell's count.

Regularity ensures that detected infinitesimal flexes extend to finite continuous motions, confirmed via a "linear push" algorithm in which points are displaced along velocity vectors associated with infinitesimal motions. Applications span CAD, reconfigurable meta-materials, and architected designs where analytic mobility prediction via extrusion symmetry facilitates both theoretical understanding and computational tractability.

6. Disorder-Induced Mobility Enhancement in Confined Geometries

Strong scale-free disorder under geometrical confinement can enhance, and even render effectively infinite, particle mobility in low-temperature regimes (Shafir et al., 4 Mar 2024). Using the quenched trap model (QTM) and double-subordination analysis, the average longitudinal displacement in a channel of width $w$ under external drive scales as

$$\langle x(t) \rangle \sim \frac{w^{\alpha-1}}{A \Gamma^2(1+\alpha)} \left(\frac{F}{4}\right)^\alpha t^\alpha,$$

where $0 < \alpha < 1$ encodes disorder strength. Thus, confinement (decreasing $w$) increases mobility (since $\mu \sim w^{\alpha-1}$), counter to the classical expectation.

This effect is rooted in the suppression of extreme waiting times ("big jump principle"): narrowing the channel reduces the probability of encountering deep energy traps that dominate transport in disordered media. The theoretical framework applies to diffusion in amorphous solids, porous structures, living cells, and nanofluidic devices. The deviation from Stokes–Einstein and Drude paradigm emphasizes the necessity of nonstandard scaling and transport models for such systems.

7. Extended Motion Generation and Procedural Synthesis

Infinite mobility is also operationalized in motion generation and articulated object synthesis (Li et al., 11 Jul 2024, Lian et al., 17 Mar 2025). The Infinite Motion framework utilizes timestamp-segmented textual instructions and a diffusion-based generator (operating in low-dimensional latent spaces) for the synthesis of arbitrarily long, high-quality human motion sequences. Timestamp "stitcher" modules interpolate between atomic segments for seamless, narrative-tailored mobility.

In procedural generation of articulated objects (Lian et al., 17 Mar 2025), Infinite Mobility pipelines construct scalable, high-fidelity URDF-like trees by recursively applying category-specific rules to node growth. Meshes are procedurally refined and textured, and joint properties assigned algorithmically. Generated datasets surpass prior data-driven and simulation-based methods in complexity and physical realism, as confirmed by GPT-4V metrics and user studies.

These approaches facilitate scalable training data construction, rapid sample generation, and composable mobility for embodied AI, design, and simulation research, demonstrating how algorithmic infinite mobility can be harnessed for diverse real-world applications.

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In sum, infinite mobility spans physical, statistical, algorithmic, and design domains, signifying regimes in which the boundaries on movement, flexibility, or transport are formally lifted. The phenomenon manifests across quantum lattice models, solid-state materials, stochastic fractal constructions, network queueing theory, geometric rigidity, disordered media, and generative modeling, unified by the mathematical or procedural removal of conventional constraints and the emergence of qualitatively novel behaviors.