Mathematical Framework

Local Structure Descriptors

The local environment of each particle is characterized by radial structure functions, angular structure functions, and bond-orientational order parameters computed via spherical harmonics.

1. Radial Structure Functions
   Capture local density variations at different distances:

$$ G_\mu(i) = \sum_{j \in \mathrm{neighbors}} e^{-(r_{ij} - \mu)^2 / L^2} $$

where $\mu$ are radial distance parameters and $L$ is a characteristic length scale.

2. Angular Structure Functions
   Capture three-body correlations:

$$ \Psi_{\xi,\lambda,\zeta}(i) = \sum_{j,k \in \text{neighbors}} e^{-\xi^2 (r_{ij}^2 + r_{ik}^2 + r_{jk}^2)} (1 + \lambda \cos \theta_{jik})^\zeta $$

where $\theta_{jik}$ is the angle at particle $i$ formed by neighbors $j$ and $k$, and $\xi$, $\lambda$, $\zeta$ control radial decay and angular sensitivity.

3. Bond-Orientational Order Parameters (Steinhardt Parameters)
   These quantify the degree of local rotational symmetry using spherical harmonics. For a central particle $i$, neighbors within annular shells (defined by inner radii $r_{\text{inner}}$ and width $\Delta r = 0.5$) are considered.

   For each shell and each even angular momentum $l$ (typically $l = 2, 4, 6, 8, 10, 12, 14$):

$$ q_{lm}(i) = \frac{1}{N_b(i)} \sum_{j \in \text{shell}} Y_{lm}(\theta_{ij}, \phi_{ij}) $$

$$ q_l(i) = \sqrt{\frac{4\pi}{2l + 1} \sum_{m=-l}^{l} |q_{lm}(i)|^2} $$

where:

• $N_b(i)$ is the number of neighbors in the shell
• $Y_{lm}(\theta, \phi)$ are spherical harmonics
• $(\theta_{ij}, \phi_{ij})$ are the polar and azimuthal angles of the vector from particle $i$ to neighbor $j$

The rotationally invariant $q_l(i)$ measures the strength of $l$-fold symmetry in that shell. In practice, the implementation computes the average of $|Y_{lm}|$ over neighbors and then applies the normalization (equivalent under magnitude).

Softness Calculation (SVM)

Softness is computed as a linear combination of the structural descriptors using a trained Support Vector Machine: Softness is

$$ S_i = \mathbf{w} \cdot \mathbf{x}_i + b $$

where:

• $\mathbf{x}$ is the concatenated feature vector (radial $G_\mu$ and bond-orientational $q_l$ or angular $\Psi_{\xi,\lambda,\zeta}$ for multiple shells) for particle $i$
• $\mathbf{w}$ is the learned weight vector
• $b$ is the bias term

The SVM is trained to separate particles that undergo significant motion. To quantify particle motion, we use the hop parameter $p_{\text{hop}}$, following the activated‑dynamics framework of Candelier et al. This metric measures particle displacement over a fixed observation window.

For a window of 10 Lennard‑Jones time units:

$$ p_{\text{hop}}(i,t) = \sqrt{ \left\langle \left(\vec{r}_i(t) - \langle \vec{r}_i \rangle_{w_2}\right)^2 \right\rangle_{w_1} \left\langle \left(\vec{r}_i(t) - \langle \vec{r}_i \rangle_{w_1}\right)^2 \right\rangle_{w_2} } $$

where the time windows are $w_1 = [t-5, t]$ and $w_2 = [t, t+5]$, and $\langle \cdot \rangle_{w_i}$ denotes an average over the corresponding half‑window.