Avalanche-Driven Self-Diffusion

• Self-diffusion is defined as the transport of individual particles in amorphous solids via collective, scale-free avalanche events.
• It shows that tracer diffusion emerges from correlated, Eshelby-like plastic events rather than uncorrelated local jumps.
• The framework establishes scaling relations between avalanche statistics, interevent stress, and diffusion coefficients, revealing finite-size effects.

A self-diffusion framework explicates the transport of individual particles within a system under various driving forces, without drift bias, and critically links microscopic kinematics to collective fluctuation-mediated mechanisms. In the context of amorphous solids under slow, dense shear, self-diffusion is governed not by uncorrelated random walks or local rearrangements, but by strongly correlated, scale-free burst dynamics—avalanches—whose hierarchical nature fundamentally determines the statistics of particle motion as well as resulting macroscopic transport properties (Karimi, 2019).

1. Avalanche-Driven Self-Diffusion in Dense Amorphous Solids

The framework is based on a particle-level numerical paper of amorphous solids in the dense slow flow regime, where deformation and flow occur as a sequence of discrete, collective stress relaxation events—avalanches—spanning a wide range of sizes. These avalanches correspond to correlated patterns of slip that relax internal stresses and collectively serve as mechanical noise on the motion of individual tracer particles. Tracer particle self-diffusion, observed at long time (Fickian) scales, emerges as the integral effect of many such avalanches, rather than from uncorrelated or purely local particle jumps.

Crucially, despite the system's apparent continuum behavior at large scales, the statistics of avalanche occurrence and spatial extent introduce strong finite-size effects in the diffusion coefficient, contrary to classical expectations.

2. Statistical Structure: Avalanches, Plastic Events, and Tracer Fluctuations

Plastic deformation in amorphous solids is mediated by “Eshelby-like” events—localized slip regions with long-range elastic quadrupolar signatures—which can cascade via stress transfer, producing scale-free avalanches. The statistics of these avalanches and their imprints on particle displacement distributions are central:

• Avalanche Size Statistics: The probability distribution for avalanche size $S$ follows

$$P(S) \propto S^{-\tau} \exp\left(-\frac{S}{L^{d_f}}\right)$$

with a measured exponent $\tau \approx 1.3$ and fractal dimension $d_f \approx 1.2$, $L$ the system size.

• Conditional Particle Displacement Variance: Tracer displacement $u$ during avalanches does not itself follow a power law, but its conditional variance, given avalanche size $S$, is

$$\langle u^2 \mid S \rangle \propto -\left(\frac{S}{L^{d_f}}\right)^2 \log\left(\frac{S}{L^{d_f}}\right)$$

for $S/L^{d_f} \ll 1$—highlighting the increasingly collective nature of fluctuations for larger avalanches.

• Unconditional Variance Scaling: The overall mean squared displacement per event (with event rates integrated out) scales as

$\langle u^2 \rangle \propto L^{-d_f(\tau-1)}$

so that as system size increases, the amplitude of individual particle fluctuations decreases, reflecting the decreasing probability of system-spanning avalanches.

3. Avalanche Timing and Poissonian Statistics

The occurrence of avalanches is described by the distribution of stress increments (or energy barriers to yielding) $f_y$: $$P(f_y) = \bar{f}_y^{-1} \exp\left(-\frac{f_y}{\bar{f}_y}\right)$$ This exponential distribution indicates a Poisson (memoryless) process for avalanche initiation. The mean interevent stress interval $\bar{f}_y$ exhibits system-size scaling: $\bar{f}_y \propto L^{-d + d_f(2-\tau)}$ where $d$ is the spatial dimension.

4. System-Size Scaling and the Diffusion Coefficient

At long time (Fickian) scales, the mean squared displacement of a tracer particle over total strain $\epsilon$ satisfies

$$\langle X^2 \rangle \propto D \epsilon, \quad D \propto L^{d - d_f}$$

This scaling for the diffusion coefficient $D$ is a direct consequence of the fractality and correlation range of avalanches. Notably, the critical exponent $\tau$ cancels in the ratio of amplitude to frequency (i.e., displacement fluctuations to event rate), making $D$ dependent only on the difference between the system dimension and avalanche fractal dimension. Thus, larger, more space-filling (higher $d_f$) avalanches or systems in lower dimensions exhibit suppressed diffusion coefficients.

5. Unified Scaling Framework: Connecting Mesoscale Plasticity and Microscale Transport

The self-diffusion framework synthesizes:

• Avalanche size distributions (parameters $d_f$, $\tau$)
• Interevent time/control parameter statistics (exponential waiting, system-size dependence)
• Tracer displacement statistics (conditional and unconditional variances)
• Effective (Fickian) diffusion coefficient scaling into a set of closed scaling relations. This unifies the mesoscale picture of plasticity (mechanical noise, avalanches) with the microscopic, observable diffusive transport of tracer particles.

Key summarized relations:

Quantity	Scaling/Equation
Avalanche distribution	$P(S) \propto S^{-\tau} \exp(-S/L^{d_f})$
Cond. disp. variance	$$\langle u^2|S\rangle \propto - (S/L^{d_f})^2 \log(S/L^{d_f})$$
Uncond. disp. variance	$\langle u^2 \rangle \propto L^{-d_f(\tau-1)}$
Mean interevent stress	$\bar{f}_y \propto L^{-d + d_f(2-\tau)}$
Diffusion coefficient	$D \propto L^{d-d_f}$

6. Physical Consequences and Interpretations

This framework demonstrates that in slowly-sheared amorphous solids, the effective self-diffusion coefficient is governed by collective, critical-like plastic fluctuations rather than local, uncorrelated motion. Mechanical noise, fundamentally rooted in the topology and statistics of stress relaxation avalanches, translates to diffusive single-particle motion only in a nontrivial, size-dependent way. The diffusion coefficient is not an intrinsic material parameter: it exhibits pronounced finite-size effects dictated by the correlated, fractal nature of avalanche dynamics.

This explains the observed breakdown of naive continuum scaling for bulk transport, justifies strong system-to-system variances in dynamic measurements, and provides quantitative predictions for how transport properties alter with system size, geometry, and deformation rate.

7. Significance for Theoretical and Experimental Studies

By providing a quantitative, scaling-based link between mesoscale collective plasticity and observable tracer diffusion, this self-diffusion framework enables robust predictions of transport in glasses and disordered solids. It reveals that attempts to extract “self-diffusion coefficients” (e.g., from long-time mean squared displacements) without accounting for avalanche statistics will misestimate their dependence on system size, geometry, and the statistics of underlying mechanical noise. This framework underpins current and future approaches to analyzing transport in glassy matter, supercooled liquids, and other densely packed, slowly relaxed amorphous media.