Random Barrier Model in Transport

• Random Barrier Model is a framework that describes transport and relaxation in disordered systems by modeling random energy or kinetic barriers.
• It employs specific probability distributions, like exponential or truncated Gaussian, to characterize barrier heights and thermally activated transition rates.
• The model has broad applications, predicting experimental phenomena in conductivity, branching processes, and biological diffusion through scaling laws and percolation theory.

A random barrier model describes transport, relaxation, or propagation phenomena in systems where local transitions are governed by randomly distributed energy barriers or kinetic constraints. It provides a unifying approach for disordered conductors, branching processes with selection or absorption, percolation problems, biological diffusion, and memory effects in complex materials. Barriers, which may represent energy obstacles, physical blockages, or selection-induced thresholds, are represented by random variables with specified distributions. This statistical structure leads to anomalous macroscopic behaviors, including non-Arrhenius transport, subdiffusion, power-law relaxation, thresholds for survival or propagation, and emergent universality in dynamic scaling.

1. Statistical and Physical Foundations

The random barrier model (RBM) is characterized by random variables representing transition obstacles (energy, conductance, or topological barriers) distributed across a network or spatial domain. In charge transport, such as in amorphous semiconductors, disorder is encoded in a distribution of energy barriers $\rho(E_b)$ placed between sites or bonds of a lattice. The canonical choice for $\rho(E_b)$ is an exponential or a (truncated) Gaussian: $$P(E) = \frac{1}{\delta\sqrt{2\pi}} \exp\left( -\frac{(E-\phi_0)^2}{2\delta^2} \right), \quad E \geq 0,$$ where $\phi_0$ is the mean barrier height and $\delta$ the standard deviation (Baranovskii et al., 2019).

The local transition (hopping) rate is thermally activated: $\nu(E) = \nu_0 \exp(-E / (k_B T)),$ where $\nu_0$ is the attempt frequency, $k_B$ Boltzmann constant, and $T$ the absolute temperature. In percolation-type RBMs, only links with $E_b < E_c$ are accessible, with $E_c$ determined by a global connectivity condition (Baranovskii et al., 2019).

Branching random walks with random barriers introduce thresholds—growing, flat, or random—on genealogical trees, with population survival controlled by whether particles remain below these barriers (Lv et al., 2024, Liu et al., 2018, Lv et al., 2022).

2. Barrier-Controlled Transport and Percolation

In disordered conductors, long-range charge or particle transport is dictated not by the average rate but by the highest (dominant) barriers encountered along percolation paths. The RBM's key insight is that for a given distribution $P(E)$, global conductivity is determined by the percolation threshold (Baranovskii et al., 2019, Lohmann et al., 12 Jan 2026): $\int_0^{E_c} P(E) \, dE = p_c,$ where $p_c$ is the critical site or bond occupation probability for percolation. The activation energy for macroscopic transport is thus $E_c$, not the mean of $P(E)$, and the RBM predicts Arrhenius conductivity at low $T$: $$\sigma(T) \approx \sigma_0 \exp\left( -\frac{E_c}{k_B T} \right).$$ The RBM corrects the mean-field approach, which can yield unphysical results (e.g., negative effective activation energies) and fails except when disorder is weak ($k_B T \gg \delta$).

In lattice-based propagation, such as models for disease or pest invasion, barriers correspond to reduced edge connectivity or physical blocks. Random allocation strategies are evaluated by increases in the percolation threshold $\chi_c$ with barrier density $p_d$, well described by $q$-exponential forms: $\chi_c(p_d) = p_{cs} \exp_q(-\lambda p_d),$ with specific $(\lambda, q)$ parameters depending on the allocation scheme (Prieto et al., 8 Sep 2025).

3. Barrier Models in Branching Processes and Survival Analysis

Branching random walks (BRW) with random or deterministic barriers describe population survival under selection, absorption, or environmental noise. Letting the barrier position at generation $n$ be $g_n$, the survival probability exhibits phase transitions and universal exponents depending on the barrier's asymptotic growth $g_n \sim a n^\alpha$:

Exponent $\alpha$	Survival Criterion	Extinction Rate	References
$\alpha < 1/3$	always extinct	$$P(\text{survival to } n) \sim \exp(-C n^{1-3\alpha})$$	(Lv et al., 2022, Lv, 2018)
$\alpha = 1/3$	$a<a_c$: extinct; $a>a_c$: survive	$$P(\text{survival to } n) \sim \exp(-\rho(a) n^{1/3})$$	(Lv et al., 2024, Lv et al., 2022)
$\alpha > 1/3$	always survive	--	(Lv et al., 2022)

Critical thresholds $a_c$ and extinction rates are determined by small deviation theory applied to the associated random walk in the environment, involving explicit expressions with constants from the environment, e.g., $\gamma(1)$ and Brownian small deviation rates $\theta(0)$ (Lv et al., 2024, Lv et al., 2022, Liu et al., 2018). For $\varepsilon \downarrow 0$, the quenched survival probability for a BRW with a random barrier behaves as

$$\varrho_{\mathcal{L}}(\varepsilon) \approx \exp\left\{ \frac{\gamma}{\sqrt{\varepsilon}} \right\}$$

with a negative constant $\gamma$ explicitly dependent on the environment (Lv et al., 2024).

The RBM also arises in approximate models of branching Brownian motion with particle number selection via random or adaptive barriers, where the barrier's location approaches a Lévy process due to rare, large barrier upward jumps associated with "breakouts" (Maillard, 2011).

4. Anomalous Relaxation, Subdiffusion, and Non-Arrhenius Phenomena

Distributions of barrier heights with fat tails (e.g., exponential, power law) induce slow, non-exponential relaxations observable in both classical and quantum transport, as well as in resistive random barrier models for tunnel junctions. For a broad exponential tail $\rho(W) \propto \exp(-W/W_0)$, the resistive relaxation follows a power law (Bertin et al., 2010): $$R(t) - R(\infty) \propto t^{-\alpha}, \quad \alpha = \frac{k_B T}{W_0},$$ with similar scaling in the hysteresis amplitude under a.c.\ signals, $\Delta I \propto f^{\alpha}$.

In random walks on 1D lattices with i.i.d.\ heavy-tailed barriers $U(x)$ in the domain of attraction of an $\alpha$-stable law ($0<\alpha<1$), spectral analysis reveals that relaxation times scale as $\tau \sim n^{1+\alpha}$, corresponding to subdiffusive dynamics with exponent $1/(1+\alpha)$ (Faggionato, 2009).

Random walks with an absorbing barrier and heavy-tailed increments exhibit logarithmic scaling in the number of suppressed jumps, with centered and normalized statistics converging to the Gaussian law, showing universal renewal-theoretic features (Marynych et al., 2014).

5. Scaling Laws and Universal Conductivity Relations

The RBM predicts universal time-temperature scaling in disordered solids. Conductivity spectra (complex $\sigma(\omega, T)$) collapse on a master curve when rescaled by $\sigma_{\mathrm{dc}}(T)$ and a characteristic time $\tau(T)$: $$\frac{\sigma(\omega, T)}{\sigma_{\mathrm{dc}}(T)} = F(\omega \tau(T))$$ with the scaling parameters determined by the percolation threshold barrier $E_{dc}$,

$$\sigma_{\mathrm{dc}}(T) = A T^{-1} \exp(-\beta E_{dc}), \quad \tau(T) = \tau_0 \exp(\beta E_{dc})$$

and an explicit master-curve form derived by Dyre–Schrøder for broad disorder (Lohmann et al., 12 Jan 2026).

Mapping complex many-particle random site energy landscapes to an associated RBM enables extending the scaling description to higher temperatures, explaining the breakdown of single-parameter scaling in systems with multiple mobile species (e.g., mixed-alkali glasses) due to distinct effective barrier distributions for each species (Lohmann et al., 12 Jan 2026).

6. Applications Beyond Electronic Transport

The RBM framework generalizes to epidemiological and ecological settings where infection or pest spread in spatially structured populations is inhibited by random placement of physical, biological, or regulatory barriers. Monte Carlo studies on lattices with randomly placed defective sites (edges removed) quantify increases in the epidemic percolation threshold as a function of barrier density, with specific "corner" barrier arrangements reducing the needed barrier fraction by $5-10\%$ over random placement, supporting cost-efficient interventions (Prieto et al., 8 Sep 2025).

In biological tissues, the "random walk with barrier model" provides a quantitative basis for interpreting diffusion-weighted MRI in terms of cellular structure (e.g., surface-to-volume ratio and membrane permeability), with fitting protocols and acquisition strategies optimized based on RBM predictions and degeneracy analysis in the parameter landscape (Zou et al., 13 Jun 2025).

7. Phenomenological and Mathematical Structure

In systems such as the partially asymmetric exclusion process with random-force disorder, the RBM provides a microscopic mechanism for coarsening and current decay: each realized "barrier" (interval of reversed local bias) determines a local bottleneck, and the overall steady state current is controlled by the minimum over these. The currents scale as

$J(U) \propto e^{-U/2}$

through a barrier of height $U$, and the typical stationary current in a system of length $L$ follows

$J_{\mathrm{typ}}(L) \sim L^{-1/(2\mu)},$

where $\mu$ is determined by large deviations of the random potential increments (Juhász, 2011). Temporal coarsening and spatial anti-shock separation obey scaling exponents $\beta = 1/(1+2\mu)$, $\delta = \mu/(1+2\mu)$.

Numerical mean-field simulations and analytic results confirm the universality of these exponents and scaling forms, further validating the RBM as a quantitative phenomenological tool for driven diffusive and coarsening systems.

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In summary, the random barrier model underpins a diverse spectrum of anomalous behaviors in disordered, inhomogeneous, and interacting systems. Its mathematical backbone—a landscape of random obstacles dictating global dynamics—enables predictive power and unifying explanations for transport, absorption, survival, and memory effects across condensed matter physics, statistical mechanics, mathematical biology, and information theory.