---
title: Nonaffine Mean-Squared Displacement
url: https://www.emergentmind.com/topics/nonaffine-mean-squared-displacement
type: topic
---

# Nonaffine Mean-Squared Displacement

Nonaffine mean-squared displacement (MSD) quantifies local deviations from affine (homogeneous, linear) transformations in crystalline and amorphous solids, granular media, and glasses. It is a central metric for probing microscopic rearrangements, plasticity, structural disorder, and critical dynamics in driven and thermalized many-body systems. Nonaffine MSD measures the residual particle or atomic motions after subtracting the best-fit local linear strain and rotation, providing insight into defect nucleation, yielding, jamming, and dynamical heterogeneity.

## 1. Mathematical Formulation and Definitions

Nonaffine displacement fields are constructed by decomposing particle or atomic motions into affine and nonaffine components. For a particle $i$ over a time interval $\Delta t$, its displacement is:
$$
\Delta \mathbf{r}_i(t; \Delta t) = \mathbf{r}_i(t) - \mathbf{r}_i(t-\Delta t)
$$
To isolate the nonaffine part in systems with flow or macroscopic deformation, the local affine motion, either as a spatially averaged field (granular flows) or a best-fit strain tensor (crystals/amorphous solids), is subtracted. Two principal definitions are used:

- **Background-subtracted (flow-local) nonaffinity:**  
  $$
  \Delta \mathbf{r}_{i,\mathrm{NA}}(t;\Delta t) = \Delta \mathbf{r}_i(t;\Delta t) - \Delta \mathbf{r}_{\mathrm{mean}}(\mathbf{x}_i; \Delta t)
  $$
  where $\Delta \mathbf{r}_{\mathrm{mean}}$ is the local mean (affine) displacement field [2502.01605].

- **Best-fit local (strain-minimized) nonaffinity:**  
  Following the Falk–Langer D$^2_{\min}$ construction [2502.01605], for reference particle $0$ with $N_\mathrm{nbs}$ neighbors:
  $$
  D^2_{\min}(t, \Delta t) = \sum_{n=1}^{N_\mathrm{nbs}} \sum_{i=1}^2 \left\{
    r^i_n(t) - r^i_0(t) - 
    \sum_{j=1}^2 [\delta_{ij} + \epsilon_{ij}]
    [r^j_n(t-\Delta t) - r^j_0(t-\Delta t)]
  \right\}^2
  $$
  where $\epsilon_{ij}$ is the best-fit strain tensor. D$^2_{\min}$ is a local measurable with dimensions of length squared.

In coarse-grained crystal settings, nonaffinity is more generally cast as the minimum residual over a region $\Omega$:
$$
\chi = \min_{D} \sum_{i\in\Omega} \left| [u_i-u_0] - D \cdot (R_i - R_0) \right|^2 = \Delta^T P \Delta
$$
with $P$ the projection operator onto the nonaffine subspace and $\Delta$ the vector of relative displacements [1212.6377, 1907.07434]. The mean nonaffine MSD is then $\langle\chi\rangle = \textrm{Tr}[P C]$, with $C$ the displacement covariance.

## 2. Quantitative Scaling, Exponents, and Systematics

Extensive studies reveal system-specific, universal, and polydispersity-dependent scalings for the nonaffine mean-squared displacement:

- **Crystalline Harmonic Solids:**  
  For $d$-dimensional lattices with volume $|\Omega|$ and temperature $T$, the nonaffine mean $\langle\chi\rangle \propto k_B T |\Omega|$ with lattice-specific prefactors. In explicit cases (e.g. 2D triangular lattice, six nearest neighbors):  
  $$
  \langle\chi\rangle = 6.865\,k_B T
  $$
  The spectrum is dominated by $d_\mathrm{dom}$ soft modes, associated with defect precursors and separated by a gap from harder modes:  
  | Lattice      | $d_\textrm{dom}$ | $\sigma_\textrm{dom}$ ($T=1$) | gap $\Delta\sigma$ |  
  |--------------|:---------------:|:----------------------------:|:------------------:|  
  | Triangular2D |   2             | 0.36                         | 0.18               |  
  | Square2D     |   2             | 0.28                         | 0.16               |  
  | fcc3D        |   3             | 0.30                         | 0.15               |  
  [1907.07434, 1212.6377]

- **Polydisperse Granular Flows:**  
  Average r.m.s. nonaffine motion, $\langle|\Delta r_\mathrm{NA}|\rangle/\langle R \rangle$, monotonically decreases with particle size, with smallest particles showing $\sim$15–20% more than the largest. The mean $\langle|\Delta r_\mathrm{NA}|\rangle/\langle R\rangle$ remains almost constant with polydispersity $\delta$ (0.37 ± 0.01), but $\langle D^2_{\min}\rangle/\langle R\rangle^2$ grows linearly with $\delta$:  
  $$
  \langle D^2_{\min}\rangle/\langle R\rangle^2 = (0.11\pm0.01)\,\delta + (0.09\pm0.01)
  $$
  Both nonaffinity metrics show power-law growth with time interval $\Delta t$:  
  $$
  \langle|\Delta r_\mathrm{NA}|\rangle/\langle R\rangle \sim (\Delta t/\Delta t_0)^{\alpha_1} \quad (\alpha_1 \simeq 0.74)
  $$
  $$
  \langle D^2_{\min}\rangle/\langle R\rangle^2 \sim (\Delta t/\Delta t_0)^{\alpha_2} \quad (\alpha_2 \simeq 1.47 \simeq 2\alpha_1)
  $$
  [2502.01605]

- **Jamming and Athermal Compression:**  
  As the packing fraction approaches jamming ($\phi\to\phi_J^-$), the nonaffine MSD diverges:  
  $$
  \Delta \equiv \langle|\delta r^{{NA}}|^2\rangle/\delta\ell^2 \sim (\delta\phi)^{-\beta},\quad \beta\simeq 2.7
  $$
  and exhibits finite-size scaling, power-law displacement statistics, and fractal spatial localization with $d_f\simeq 2.04$ in $d=3$ [2009.01409].

- **Amorphous Sheared Solids:**  
  Under athermal quasistatic shear, the mean-squared nonaffine displacement scales with strain window $\Delta\gamma$:
  $$
  \langle \Delta r_{na}^2\rangle \sim
  \begin{cases}
      (\Delta\gamma)^2 & \textrm{elastic} \\
      (\Delta\gamma)^\alpha,\,1.5\lesssim\alpha\lesssim 1.8 & \textrm{near yield} \\
      (\Delta\gamma)^1 & \textrm{steady flow} \\
  \end{cases}
  $$
  The distribution $P(\Delta r_{na})$ is heavy-tailed, $P(\Delta r_{na})\propto (\Delta r_{na})^{-\tau}$ with strain-dependent $\tau$ between 2.4 and 4.8 [2201.11797].

## 3. Physical Interpretation, Defects, and Rearrangements

Nonaffine MSD encodes the propensity for localized, defect-related, or collective particle rearrangements:

- In compact crystals, the dominant nonaffine modes correspond to precursor configurations for lattice defects (e.g., $5$–$7$ dislocation pairs in 2D triangular lattices, stacking faults in fcc) [1907.07434]. The statistics of nonaffine MSD reflect the underlying defect nucleation landscape.
- In amorphous materials, large nonaffine displacements cluster into shear transformation zones (STZs), mediate plastic instabilities, and, in the post-yield regime, evolve into system-spanning shear bands [2201.11797].
- For granular flows, nonaffinity is enhanced for small particles due to their inability to follow averaged directions set by large neighbors, creating short-range spikes in $\langle|\Delta r_\mathrm{NA}(d)|\rangle$. Large particles exhibit more affine collective motion, effectively 'screening' local flow fluctuations [2502.01605].
- Near jamming, the divergence of nonaffine MSD and concurrent localization on fractal sets indicate the approach to a rigidity transition, with scaling equivalence to viscosity divergence.

## 4. Measurement Protocols and Methodological Considerations

Robust quantification of nonaffine mean-squared displacement requires careful specification of reference frames, averaging procedures, and spatial or temporal coarse-graining:

- **Reference frame selection:** For steady-state flows or sheared systems, background subtraction (local mean flow) is necessary to exclude global drift.
- **Best-fit locality:** The number and arrangement of neighboring particles, $N_\text{nbs}$, or the size of the coarse-graining region $\Omega$, directly controls sensitivity to local structural disorder versus extended collective modes [1212.6377, 1907.07434].
- **Thermal versus athermal protocols:** In thermal solids, harmonic fluctuations dominate nonaffine noise; in athermally driven (quasistatic) settings, plastic events and avalanches are primary contributors.
- **Temporal and ensemble averaging:** Nonaffine MSDs are typically averaged over steady-state cycles or over statistically independent samples; time-interval scaling reveals dynamic universality classes (ballistic, superdiffusive, diffusive).
- **Polydispersity control:** In granular media, size distributions characterized by polydispersity index $\delta$ are necessary for disentangling collective size effects from single-particle behavior [2502.01605].

## 5. Correlations, Criticality, and Distributions

Beyond the mean, the spatial and statistical correlations of nonaffine displacement provide deep insight into both collective dynamics and phase transitions:

- **Spatial correlations:**  
  In crystals and glasses, the two-point correlation function $G_\chi(r)=\langle\chi(0)\chi(r)\rangle-\langle\chi\rangle^2$ exhibits short-range, nearly isotropic exponential decay with a finite correlation length $\xi_\chi$, of a few lattice spacings [1212.6377].
  In amorphous materials, correlations in the elastic regime are exponential; post-yield, they transition abruptly to robust power laws $C_{D^2}(r)\sim r^{-\lambda}$ with $\lambda\sim1.3$, reflecting system-spanning avalanches [2201.11797].
- **Critical fields:**  
  Introduction of a conjugate field $h_\chi$ to the nonaffine measure induces a transition to a "maximally nonaffine" regime, with diverging mean and correlation length, mirroring elastic instabilities (e.g., Peierls) and amorphous melting [1212.6377].
- **Distribution tails:**  
  The distribution of single-particle nonaffine displacements $P(\Delta r_{na})$ is universally broad, with power-law tails whose exponents relate to underlying dynamical and geometric universality (fractal participation, power-law event sizes) [2009.01409, 2201.11797].

## 6. Material, Geometric, and Flow Specificities

Distinct materials and protocols yield different nonaffine MSD responses, which can guide the identification of reorganizational mechanisms:

- **Crystalline solids**: Soft nonaffine eigenmodes are isolated and well-separated from the phonon background in close-packed lattices, justifying a defect-centric view of plasticity [1907.07434].
- **Polydisperse granular media**: Nonaffine activity is highly size-dependent, and the role of polydispersity is primarily in enhancing the frequency and localization of rearrangements, not the global amplitude.
- **Amorphous solids under shear**: Transitioning from exponential to power-law spatial correlations and from quadratic to linear strain scaling in nonaffine MSD signifies yielding and the breakdown of affine elasticity [2201.11797].
- **Proximity to jamming**: Nonaffine MSD is the critical order parameter, diverging with the same exponents as viscosity, and becomes spatially intermittent on fractal structures [2009.01409].

## 7. Implications and Applications

Nonaffine mean-squared displacement serves as a diagnostic and predictive tool across condensed matter, soft matter, and materials science:

- Modulation of polydispersity can be employed to tune mixing, enhance flowability, or reduce effective viscosity in granular and suspension flows [2502.01605].
- Nonaffinity-based measures such as $\langle D^2_{\min}\rangle$ are robust predictors of the frequency and localization of plastic rearrangements.
- The critical divergence of nonaffine fluctuations allows the identification of rigidity/jamming transitions and the characterization of the underlying glass or defect landscape [1212.6377, 2009.01409].
- In simulations and experiments, projection and decomposition techniques furnish a detailed map of elastic versus plastic zones, supporting the design of materials with tailored mechanical response.

For all these reasons, nonaffine mean-squared displacement is now a foundational metric in the study of disordered, jammed, and driven non-equilibrium materials.

Source: https://www.emergentmind.com/topics/nonaffine-mean-squared-displacement