Species-Resolved Point-to-Set Correlations

• Species-resolved point-to-set correlations are defined as measures quantifying the constraints on a species’ local configuration imposed by its static environment, yielding species-specific correlation lengths.
• They are computed using methodologies like lattice-based TASEP and cubic-cavity protocols in atomistic liquids, which reveal cooperative dynamics and dynamic disorder across species.
• Findings in glass-forming silica demonstrate that silicon exhibits increasing cooperative rearrangements with cooling, while oxygen maintains localized relaxation, explaining dynamic decoupling.

Species-resolved point-to-set (PTS) correlations quantify the extent to which the local configuration of a given species is constrained by the arrangement of its surrounding environment, providing a link between structural order and cooperative dynamics in both stochastic lattice models and network-forming liquids. The PTS methodology enables the extraction of a correlation length, ξPTS,α, specific to species α, characterizing the spatial range over which static boundary conditions impose amorphous order. In multispecies exclusion processes and supercooled glass-formers, the PTS framework provides insight into species-dependent constraints and the emergence of dynamic disorder.

1. Formal Definition and Measurement Protocols

In lattice systems such as the homogeneous multispecies TASEP on a ring, a species-resolved configuration η maps each site to its occupant's species label, with imposed constraints on species counts (composition m). The stationary measure, π_m, governs the steady-state statistics. Given an anchor site i and subset $S \subset \{1,\ldots,N\} \setminus \{i\}$, the point-to-set correlation is expressed as

$$C_{i;S}(α;β) = \mathbb{E}\Big[\mathbf{1}\{\eta(i)=α\} \prod_{j\in S} \mathbf{1}\{\eta(j)=β_j\}\Big]$$

which depends only on relative distance due to translational invariance (Ayyer et al., 2014).

For atomistic liquids such as viscous silica, species-resolved PTS measures are implemented by a cubic-cavity protocol: freeze all species outside a cube of half-edge d, allowing the central region to relax, and measuring an overlap field $n_i(t)$ that quantifies retention of original species occupancy in each grid cell over time. The long-time plateau overlap $Q_α^\infty(d)$ yields an excess overlap

$q_α(d) = Q_α^\infty(d) - Q_{\mathrm{rand}}$

with $Q_{\mathrm{rand}}$ removing the random occupancy contribution (Kumar et al., 22 Nov 2025).

2. Species-Resolved Correlations in Multispecies TASEP

In the homogeneous multispecies TASEP with one particle of each species, explicit formulas for joint probabilities of species at distinct sites provide the foundation for species-resolved PTS analysis. Two-point nearest-neighbor correlations satisfy

$$\mathbb{P}[\eta(1)=a, \eta(2)=b] = \frac{\binom{a+b-2}{a-1}\binom{2n-(a+b)}{n-a}}{\binom{2n}{n}} \quad \text{for } a < b$$

with symmetry for $(a, b) \leftrightarrow (b, a)$ (Ayyer et al., 2014). For arbitrary separation,

$$\mathbb{P}[\eta(1)=j, \eta(a)=i] = \frac{1}{n^2(n-1)} \left[y_{a-1}\binom{n-i}{n-j+1} - y_{a-1}\binom{n-i}{n-j} + \cdots\right]$$

where $y_{a-1}=SSYT_{n-i,n-j+1}(n-1)$ counts semistandard Young tableaux.

Three-point nearest-neighbor correlations admit rational expressions for descending and other orderings, e.g.

$$E_{a,b,c} = \frac{(a-1)(b-c)(a-b)}{6\,n\,(n-1)\,(n-2)} \quad \text{for } a > b > c$$

with explicit formulae for all orderings except the fully increasing case (conjectured to be $1/n^3$ but unproven) (Ayyer et al., 2014). Conjectures extend these results to higher-order and block-independence patterns.

3. Independence Properties and Higher-Order Conjectures

A salient phenomenon in the multispecies TASEP is the emergence of independence between points that are “closer in position than in value.” For two points,

$$\mathbb{P}[\eta(r)=j, \eta(r+1)=i] = \frac{1}{n^2} \quad \text{whenever } j-i > 1$$

and, more generally,

$\mathbb{P}[\eta(k)=i_k\;\forall k=1..r] = (1/n)^r$

if the species labels $i_1 < \ldots < i_r$ satisfy $\max_{a<b}(i_b - i_a) > r-1$. This independence property is rigorously established for the two-point case and conjectured for higher-order correlations, with a conceptual proof for general $m$ points remaining open (Ayyer et al., 2014). Vandermonde-type determinant formulas are established for fully descending $r$-tuples and conjectured for broader orderings.

4. Species-Resolved Point-to-Set Correlations in Glass-Forming Liquids

In amorphous silica, PTS correlations measured via the cubic-cavity protocol reveal species-dependent cooperativity. The long-time excess overlap $q_α(d)$ is well described by a compressed exponential

$$q_α(d) = A_α\exp\left[-(d/\xi_{\mathrm{PTS},α})^{\beta_α}\right]$$

with stretching exponent $\beta_α \in [1, 1.3]$ (Kumar et al., 22 Nov 2025). Extracted correlation lengths for silicon (Si) and oxygen (O) display marked asymmetry:

$T$ (CP units)	$\xi_{\mathrm{PTS},\mathrm{Si}}$	$\xi_{\mathrm{PTS},\mathrm{O}}$	$\xi_{\mathrm{PTS},\mathrm{Total}}$
0.60	1.00	0.90	0.95
0.45	1.05	0.92	1.00
0.36	1.15	0.90	1.05
0.29	1.30	0.88	1.10

These length scales are in units of the CP potential’s $\sigma_{\mathrm{Si–Si}}$. Notably, $\xi_{\mathrm{PTS},\mathrm{Si}}$ grows by ~25% with cooling, while $\xi_{\mathrm{PTS},\mathrm{O}}$ remains nearly constant; the total PTS length shows only weak growth (<20%), consistent with silica’s strong glass former character.

5. Microscopic Origin of Species Dependence in Cooperative Dynamics

The species-resolved PTS lengths are directly associated with the mechanism of atomic jumps. Silicon relaxation is controlled by the coordinated rearrangement of its fourth-nearest oxygen neighbors (rSiO$_4$) and, at lowest temperatures, fourth-nearest silicon neighbors (rSiSi$_4$). Oxygen relaxation is dictated by its second-nearest silicon neighbor (rOSi$_2$), a localized event. Freezing these slow variables within the cavity demonstrably blocks the associated local relaxation (Kumar et al., 22 Nov 2025). Thus, silicon dynamics are highly cooperative and spatially extended, whereas oxygen retains comparatively local and fast relaxation, resulting in the observed growth disparity between $\xi_{\mathrm{PTS},\mathrm{Si}}$ and $\xi_{\mathrm{PTS},\mathrm{O}}$. This asymmetry constitutes the microscopic origin of dynamical decoupling between Si and O in supercooled silica.

6. Linking Dynamic Disorder and Static Cooperativity

Species-resolved PTS correlations provide a static structural lens on dynamic heterogeneity, quantifying the spatial scale over which relaxation must be cooperative to escape boundary constraints. At elevated temperatures, jump statistics remain close to Poissonian, with PTS lengths constrained to the first coordination shell. Upon cooling, tendency toward dynamic disorder emerges: survival probabilities deviate from single exponentials, and static PTS lengths for silicon increase accordingly. Oxygen, remaining governed by localized constraints, does not show corresponding growth in PTS length. The aggregate PTS length thus underestimates underlying species-specific collective behavior. In silica, this explains the persistent disparity between the slowing of Si and O dynamics in the deeply supercooled regime (Kumar et al., 22 Nov 2025).

7. Open Problems and Conjectured Generalizations

In exclusion processes, conceptual proofs of higher-order independence properties and explicit formulas for increasing patterns remain open, although conjectured. For glass-forming networks, a plausible implication is that similar species-dependent PTS phenomena may occur in other multi-component systems, with collective rearrangements orchestrated by network connectivity and species-specific constraints. Theoretical frameworks unifying static PTS correlations and dynamic disorder provide promising avenues for further investigation into the spatial origins of glassy slowdown and dynamical decoupling.

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For explicit formulas, proof details, and additional conjectures in multispecies exclusion processes, refer to Ayyer & Linusson (Ayyer et al., 2014). For species-resolved PTS methodology and its connection to dynamic slowdown in silica, see (Kumar et al., 22 Nov 2025).