Dynamic Distortion in Evolving Systems

• Dynamic distortion is defined as trajectory- or condition-dependent deformation rather than a static input–output mapping.
• Methodologies reveal mechanisms such as corner-cutting in stochastic oscillators and guidance-modified reverse dynamics in high-dimensional models.
• Implications span diverse fields, influencing optimized encoding in imaging, adaptive hardware design, and sequential risk control through recursive distortion models.

Searching arXiv for the provided topic and papers to ground the article in current records. {"^^^^^^^^6query6 \6^^^^^^^^"dynamic distortion\"^^^^^^^^6 OR ti:\6^^^^^^^^"Noise-induced distortion of nonequilibrium oscillator mean limit cycle\"^^^^^^^^6 OR ti:\6^^^^^^^^"Emergence of Distortions in High-Dimensional Guided Diffusion Models\"^^^^^^^^6 OR ti:\6^^^^^^^^"Viewport-Driven Rate-Distortion Optimized ^^^^6 OR ti:\66query6^^^^° Video Streaming\"^^^^^^^^6 OR ti:\6^^^^^^^^"Optimization of Signal-to-Noise-and-Distortion Ratio for Dynamic Range Limited Nonlinearities\"","max_results":^^^^6all: \6query6^^^^,"sort_by":"relevance"} In the cited literature, dynamic distortion denotes several related phenomena in which distortion is generated, reshaped, or revealed by an evolving process rather than by a fixed static mapping. The term appears in nonequilibrium oscillators, where noise deforms the observed mean limit cycle; in guided diffusion models, where reverse-time sampling dynamics alter the endpoint law; in immersive imaging and volumetric video, where distortion varies with navigation, latitude, readout time, or temporal representation; in communication and sensing hardware, where distortion depends on allocation, finite dynamic range, or continuous-time memory; and in sequential information, risk, and control problems, where distortion is imposed recursively stage by stage (&&&^^^^6query6^^^^&&&, &&&^^^^6all: \6^^^^&&&, &&&^^^^6 OR ti:\6^^^^&&&). A plausible unifying characterization is that dynamic distortion is distortion whose observable effect depends on trajectory, conditioning, or temporal evolution rather than on a single time-independent input–output relation.

^^^^6all: \6^^^^. Cross-domain scope

The provided literature uses the term across several technically distinct settings. In each case, the “dynamic” qualifier refers to dependence on an evolving state, trajectory, or conditional law, while “distortion” may refer to geometric deformation, distributional mismatch, rate–distortion loss, nonlinear waveform corruption, or symmetry-lowering lattice motion.

Domain	Distorted object	Dynamic mechanism
Nonequilibrium oscillators	Mean limit cycle	Noise-induced corner-cutting
Guided diffusion	Conditional sampling law	Guidance-modified reverse dynamics
Immersive imaging and vision	Viewport quality, ERP geometry, RS imagery, rendered views	Navigation, latitude, readout timing, temporal latent updates
Communication and sensing hardware	Carrier quality, sampled waveform, sensor readout	Allocation, nonlinearity, memory, finite dynamic range
Sequential information and control	Distortion budget or risk-to-go	Stage-wise recursion and conditional evaluation
Condensed matter and defects	Valley transport, defect symmetry	Dynamic lattice distortion and anharmonic nuclear motion

This distribution of usage suggests that “dynamic distortion” is not a single universal formalism. Rather, it is a family resemblance across fields: a static description is inadequate because the distortion itself depends on how the system evolves.

^^^^6 OR ti:\6^^^^. Noise-driven deformation in nonequilibrium oscillators

In active stochastic oscillators, dynamic distortion refers to the fact that the trajectory reconstructed by averaging many stochastic oscillations is generally not the deterministic limit cycle of the underlying nonlinear system (&&&^^^^6query6^^^^&&&). The paper studies a two-dimensional active stochastic oscillator with state PRESERVED_PLACEHOLDER_^^^^6query6^^^^ and additive, Gaussian, white, isotropic noise. For the standard supercritical Hopf oscillator,

PRESERVED_PLACEHOLDER_^^^^6all: \6^^^^

the deterministic cycle for PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^ is a circle of radius

PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^

Because this case is radially symmetric and the noise is isotropic, the mean limit cycle of the stochastic system coincides with the deterministic one.

The distortion appears when the scalar potential is modulated azimuthally in a generalized Hopf oscillator. The paper studies an PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^-fold perturbation near the circular trough and focuses on PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^, producing a four-lobed or “daisy-like” landscape. Stable deterministic cycling requires the nonconservative rotational drive to exceed

$\omega^\star = n b \frac{\alpha}{\mu}.$

For $\omega>\omega^\star$, the deterministic orbit develops sharp loops and pinched high-curvature regions. At finite noise, the observed mean cycle departs from that deterministic path.

The central mechanism is corner-cutting. Stochastic trajectories occasionally leave the deterministic trough and cross higher-potential regions instead of following the full sharp excursion. Averaging then rounds off the corner, so distortion is localized rather than uniform. The strongest deformation occurs where the confining potential normal to the cycle is weak and where the vector-potential drive has a substantial normal component. The paper identifies these regions as deformation “hot spots” and visualizes the effect with a normal-displacement heat map along arclength.

The analysis also gives a first-passage estimate for the return of a fluctuating trajectory to the mean cycle. Modeling normal displacement $x$ by a one-dimensional Smoluchowski equation in a harmonic confining potential yields an integrated survival probability

$$N(t)= {\rm erf}\left[ \left( \frac{\kappa}{2k_{\rm B}T(1-e^{-2t\kappa B})} \right)^{1/2} x_0 e^{-t\kappa B} \right].$$

This directly links weak confinement PRESERVED_PLACEHOLDER_^^^^6all: \6query6^^^^ to long return times and large distorted arclength regions. The broader methodological consequence is that averaging in a noisy active oscillator converges to a finite-noise mean orbit, not to the zero-noise deterministic cycle. The paper explicitly connects this to hair-cell oscillations, arguing that sharp deterministic features may be fundamentally hidden by stochastic averaging.

A common misconception is that noise merely broadens trajectories around a fixed geometric orbit. The oscillator analysis shows a stronger statement: in nonequilibrium active systems, noise can systematically deform the mean orbit itself.

^^^^6 OR ti:\6^^^^. Dynamic distortion in guided diffusion models

In high-dimensional guided diffusion, dynamic distortion is the mismatch between the classifier-free-guidance sampling distribution and the true conditional target, but the paper’s central claim is that this mismatch is generated by the reverse-time dynamics themselves, not merely by a static score substitution (&&&^^^^6all: \6^^^^&&&). The formal definition is

PRESERVED_PLACEHOLDER_^^^^6all: \6all: \6^^^^

with reverse-time sampling under the guided score

PRESERVED_PLACEHOLDER_^^^^6all: \6 OR ti:\6^^^^

The analysis rewrites the guided reverse SDE as diffusion in a time-dependent effective potential. For the Gaussian-mixture setting, the effective potential splits into a conditional term and an extra guidance term that depends on all competing classes. This produces two dynamical regimes: a guided phase and a conditional phase. The transition time between them is the speciation time PRESERVED_PLACEHOLDER_^^^^6all: \6 OR ti:\6^^^^, defined by a sign change in the relevant free energy. Distortion persists when the reverse path spends too much time in the guided phase near sampling time; it vanishes when the conditional phase begins sufficiently early.

The high-dimensional asymptotics are organized by the class-count scaling

PRESERVED_PLACEHOLDER_^^^^6all: \6 OR ti:\6^^^^

If PRESERVED_PLACEHOLDER_^^^^6all: \6 OR ti:\6^^^^ is sub-exponential in dimension, then PRESERVED_PLACEHOLDER_^^^^6all: \66^^^^ and

PRESERVED_PLACEHOLDER_^^^^6all: \6^^^^7

so the process has an effectively long conditional-denoising window and distortion vanishes asymptotically. If PRESERVED_PLACEHOLDER_^^^^6all: \6^^^^8 with PRESERVED_PLACEHOLDER_^^^^6all: \6^^^^9, then PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6query6^^^^, the guided phase persists to finite time before sampling, and distortion survives in the high-dimensional limit.

The paper further shows that vanilla CFG dynamically expands the mean and contracts the covariance. In the exactly solvable jointly Gaussian model, the scalar weights satisfy

PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6all: \6^^^^

In the guided phase of the Gaussian-mixture model, the instantaneous attractor is shifted and narrowed: PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^ For PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^, this gives PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^ and PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^. The paper therefore argues that positive-only guidance schedules are fundamentally incapable of preventing variance shrinkage. Its mitigation is a linear schedule

PRESERVED_PLACEHOLDER_^^^^6 OR ti:\66^^^^

with a negative-guidance window near the end of sampling, so that late-time variance expansion can counteract earlier shrinkage.

A frequent misconception is that CFG distortion is just a static consequence of replacing one score with another. The paper explicitly rejects that view: the endpoint law is distorted because the reverse trajectory evolves through a guidance-modified effective landscape for a finite portion of time.

^^^^6 OR ti:\6^^^^. Imaging, video, and geometric vision

In immersive imaging and video, dynamic distortion usually refers to distortion that varies across time, viewpoint, or image position. In viewport-driven PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^7 streaming, the distortion of interest is not full-sphere error but expected viewport distortion weighted by dynamic head-motion statistics (&&&^^^^6 OR ti:\6^^^^&&&). The paper partitions each ^^^^6 OR ti:\6^^^^K equirectangular video into a PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^8 tile grid, models tile rate–distortion per GOP, and builds GOP-level navigation likelihoods from head traces. The objective is the expected distortion

PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^9

optimized under a bandwidth budget and QP bounds. The reported result is a consistent viewport luminance PSNR advantage of roughly PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6query6^^^^–PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6all: \6^^^^ over monolithic encoding, with about PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^ rate savings relative to the monolithic baseline. Here dynamic distortion is the mismatch between uniformly encoded panorama quality and the time-varying subset of tiles that actually enters the viewport.

In omnidirectional image super-resolution, dynamic distortion is spatial rather than temporal: ERP geometry introduces latitude-dependent stretching (Yang et al., 2024). For ERP,

PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^

and the row-wise distortion map is

PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^

GDGT-OSR turns this geometric prior into a learned distortion guidance PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^, uses it to modulate key and value tensors in rectangle-window attention, and combines that branch with distortion-aware deformable self-attention and dynamic feature aggregation. The paper’s interpretation is that the distortion prior is static once image size is fixed, but its exploitation is spatially varying and input-adaptive.

Rolling-shutter correction provides a different notion of dynamic distortion: each image row corresponds to a different capture time (Zhong et al., 2022). The paper models an RS image row as

PRESERVED_PLACEHOLDER_^^^^6 OR ti:\66^^^^

and addresses recovery of latent global-shutter frame sequences from two synchronized RS images with reversed scan directions. Its IFED network estimates dual optical-flow sequences via iterative learning of a velocity field and reconstructs PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^7, PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^8, PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^9, or PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6query6^^^^ GS frames. On the synthetic RS-GOPRO benchmark, the reported PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6all: \6^^^^ result is PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^ PSNR, PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^ SSIM, and PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^ LPIPS, substantially above the cascade baselines. The paper’s main technical claim is that dual reversed distortion reduces the ambiguity that limits adjacent-RS methods, especially under varying readout settings and mixed camera/object motion.

Volumetric video coding extends the same idea into dynamic NeRF compression (Zhang et al., 2024). The method updates basis fields incrementally,

PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^

and trains the representation under a joint objective

PRESERVED_PLACEHOLDER_^^^^6 OR ti:\66^^^^

The paper argues that distortion in dynamic scenes should not be treated as a post hoc consequence of compressing a trained representation; it should be co-optimized with the dynamic model. Reported BD-rate gains over ReRF range from PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^7 to PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^8.

A related optical–geometric formulation appears in drive-through vehicle reconstruction, where dynamic-scene SfM is combined with distortion-aware Gaussian Splatting (Kulkarni et al., 27 Mar 2026). The pipeline works directly on raw distorted ^^^^6 OR ti:\6^^^^K fisheye imagery, masks rotating wheels to restore rigid epipolar geometry, uses a rig-aware BA objective with CAD-derived pose priors, and renders with ^^^^6 OR ti:\6^^^^DGUT plus MCMC densification. On PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^9 vehicles across PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6query6^^^^ dealerships, the full pipeline reports PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6all: \6^^^^ PSNR, PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^ SSIM, and PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^ LPIPS on held-out views, a PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^ gain over standard ^^^^6 OR ti:\6^^^^D-GS. A plausible general lesson across these vision papers is that dynamic distortion is often best handled natively—by retaining the true geometry or timing model—rather than by flattening it into a static preprocessing step.

^^^^6 OR ti:\6^^^^. Communication, sensing, and nonlinear hardware

In communication payloads, dynamic distortion often means that distortion depends on the current resource allocation rather than being an intrinsic constant of the device. For ultra high-throughput satellites, the paper on distortion-aware dynamic carrier allocation defines distortion as configuration-dependent linear and nonlinear interference that changes with carrier center frequencies, bandwidths, symbol rates, roll-off factors, and multicarrier loading (&&&^^^^6all: \6query6^^^^&&&). Its analytical carrier-to-interference ratio combines first-order, mixed, and cubic terms,

PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6 OR ti:\6^^^^

In the three-carrier optimization example, the best allocation PRESERVED_PLACEHOLDER_^^^^6 OR ti:\66^^^^ attains PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^7, the worst PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^8, for a PRESERVED_PLACEHOLDER_^^^^6 OR ti:\6^^^^9 gain. The paper’s conceptual move is to treat distortion as a dynamic optimization variable through a fast analytical metric.

For dynamic-range-limited nonlinearities, dynamic distortion is clipping and bias-induced distortion produced by finite output range (&&&^^^^6all: \6all: \6^^^^&&&). The system is

$\omega^\star = n b \frac{\alpha}{\mu}.$^^^^6query6^^^^

and the objective is to maximize

$\omega^\star = n b \frac{\alpha}{\mu}.$^^^^6all: \6^^^^

The paper proves that the SNDR-maximizing memoryless response is a double-sided limiter with an affine middle region, so the optimal gain and bias depend on the input PDF and noise power. This is a different sense of dynamic distortion: the distortion is controlled by the interaction of clipping, saturation, and bias placement under finite dynamic range.

The RF/baseband-equivalent paper makes the temporal aspect explicit (&&&^^^^6all: \6 OR ti:\6^^^^&&&). A continuous-time Volterra distortion block $\omega^\star = n b \frac{\alpha}{\mu}.$^^^^6 OR ti:\6^^^^ between ideal modulation and demodulation yields a discrete-time baseband equivalent

$\omega^\star = n b \frac{\alpha}{\mu}.$^^^^6 OR ti:\6^^^^

where $\omega^\star = n b \frac{\alpha}{\mu}.$^^^^6 OR ti:\6^^^^ is a short-memory DT Volterra system and $\omega^\star = n b \frac{\alpha}{\mu}.$^^^^6 OR ti:\6^^^^ is a special long-memory DT LTI system. The paper’s central claim is that a short-memory CT nonlinearity does not map to a plain short-memory DT Volterra model; modulation and filtering induce an additional long-memory linear reconstruction effect. This motivates a DPD architecture that separates nonlinear memory from long linear memory rather than enlarging a standalone Volterra model.

In quantum sensing, dynamic distortion is the saturation and harmonic generation caused by the conventional $\omega^\star = n b \frac{\alpha}{\mu}.$6 readout of interferometric quantum sensors (&&&^^^^6all: \6 OR ti:\6^^^^&&&). The proposed QPSD protocol replaces the static readout by

$\omega^\star = n b \frac{\alpha}{\mu}.$7

so the desired phase is extracted as the phase of a controlled modulation rather than from the local slope of a sinusoid. The paper reports a $\omega^\star = n b \frac{\alpha}{\mu}.$8 linear dynamic range, $\omega^\star = n b \frac{\alpha}{\mu}.$9 sensitivity, and arbitrary frequency resolution, and demonstrates low-distortion recovery of melody and speech encoded onto magnetic fields.

The SOA-MZI photonic sampler paper uses a post-distortion linearization law derived from complementary outputs $\omega>\omega^\star$^^^^6query6^^^^ and $\omega>\omega^\star$^^^^6all: \6^^^^ (&&&^^^^6all: \6 OR ti:\6^^^^&&&). Its practical correction is

$\omega>\omega^\star$^^^^6 OR ti:\6^^^^

Although the model is derived under the static assumption $\omega>\omega^\star$^^^^6 OR ti:\6^^^^, it remains effective in the dynamic regime of sampled sinusoids. The best reported dynamic improvement is $\omega>\omega^\star$^^^^6 OR ti:\6^^^^ THD at $\omega>\omega^\star$^^^^6 OR ti:\6^^^^ and $\omega>\omega^\star$6 modulation index, with $\omega>\omega^\star$7 improvement at $\omega>\omega^\star$8 and $\omega>\omega^\star$9 modulation index. This shows that static-model inversion can suppress dynamic distortion up to the point where carrier-memory effects dominate.

A recurrent misconception in these hardware papers is that dynamic distortion is simply “more nonlinearity.” The cited results instead separate several mechanisms: configuration dependence, finite-range clipping, continuous-time memory, and time-varying carrier dynamics.

6. Sequential distortion, dynamic risk, and robust control

In information theory and stochastic control, dynamic distortion is not necessarily geometric or waveform-based. It can denote a stage-wise fidelity or risk transformation embedded in a recursive decision problem. For nonanticipative rate–distortion of finite-horizon Markov sources, the paper studies a source with per-stage single-letter distortions

$x$^^^^6query6^^^^

and shows that the finite-horizon NRDF can be written as a belief-state stochastic dynamic program (&&&^^^^6 OR ti:\6^^^^&&&). The one-step optimal reproduction kernel has the implicit exponential form

$x$^^^^6all: \6^^^^

with

$x$^^^^6 OR ti:\6^^^^

The distortion is therefore dynamic because current reproduction trades off immediate distortion, information rate, and future cost-to-go through the belief update.

A distinct use of the term appears in dynamic coherent risk measures generated by distortion functions (&&&^^^^6all: \66^^^^&&&). The conditional Choquet integral

$x$^^^^6 OR ti:\6^^^^

defines a law-invariant dynamic coherent risk measure, and the paper proves that it coincides with a dynamic weighted Value at Risk representation. Its time-consistency results are sharply asymmetric: these measures are sub-martingale time consistent, but not super-martingale time consistent, and they are not weakly acceptance time consistent. Here “distortion” refers to a distortion function on conditional probabilities, but the dynamic aspect is the recursive conditioning across time.

Robust reinforcement learning with dynamic robust distortion risk measures extends the same logic into finite-horizon RL (&&&^^^^6all: \6^^^^7&&&). The robust nested objective is

$x$^^^^6 OR ti:\6^^^^

with one-step robust distortion maps defined over Wasserstein ambiguity sets around the conditional law. The value recursion is

$x$^^^^6 OR ti:\6^^^^

and the paper derives policy-gradient formulae from a quantile representation of the distortion risk measure. A plausible synthesis of these three papers is that “dynamic distortion” in sequential problems means that distortion or distorted risk is assessed locally and recursively, so that present decisions remain aligned with future conditional re-evaluations.

7. Lattice dynamics, valley transport, and dynamic Jahn–Teller distortion

In condensed-matter applications, dynamic distortion is literally a moving lattice distortion or a dynamically averaged symmetry-lowering mode. In ^^^^6 OR ti:\6^^^^D Dirac materials driven by a standing Love wave, the displacement

$x$6

produces vorticity

$x$7

and, after a $x$8 and Schrieffer–Wolff reduction, the one-band Hamiltonian becomes

$x$9

The last term is the paper’s new valley-vorticity coupling, which couples the valley orbital angular momentum to lattice vorticity and enables valley transfer, valley-dependent localization, insulating behavior, and pulsed current generation (&&&^^^^6all: \6^^^^8&&&). The distortion is dynamic because the transport control comes from the time-dependent lattice motion itself, not from static strain alone.

The neutral vacancy in diamond provides a different archetype (&&&^^^^6all: \6^^^^9&&&). Static DFT and harmonic phonons favor a tetragonal $$N(t)= {\rm erf}\left[ \left( \frac{\kappa}{2k_{\rm B}T(1-e^{-2t\kappa B})} \right)^{1/2} x_0 e^{-t\kappa B} \right].$$^^^^6query6^^^^ Jahn–Teller distortion, while the tetrahedral $$N(t)= {\rm erf}\left[ \left( \frac{\kappa}{2k_{\rm B}T(1-e^{-2t\kappa B})} \right)^{1/2} x_0 e^{-t\kappa B} \right].$$^^^^6all: \6^^^^ structure is a saddle point of the Born–Oppenheimer surface. The anharmonic analysis maps the two soft-mode subspace and finds three equivalent minima at

$$N(t)= {\rm erf}\left[ \left( \frac{\kappa}{2k_{\rm B}T(1-e^{-2t\kappa B})} \right)^{1/2} x_0 e^{-t\kappa B} \right].$$^^^^6 OR ti:\6^^^^

Solving the anharmonic vibrational problem in that subspace shows that the nuclear ground-state probability density is spread across all three minima. At $$N(t)= {\rm erf}\left[ \left( \frac{\kappa}{2k_{\rm B}T(1-e^{-2t\kappa B})} \right)^{1/2} x_0 e^{-t\kappa B} \right].$$^^^^6 OR ti:\6^^^^, the anharmonic vacancy formation energies are $$N(t)= {\rm erf}\left[ \left( \frac{\kappa}{2k_{\rm B}T(1-e^{-2t\kappa B})} \right)^{1/2} x_0 e^{-t\kappa B} \right].$$^^^^6 OR ti:\6^^^^ for the dynamic $$N(t)= {\rm erf}\left[ \left( \frac{\kappa}{2k_{\rm B}T(1-e^{-2t\kappa B})} \right)^{1/2} x_0 e^{-t\kappa B} \right].$$^^^^6 OR ti:\6^^^^ state and $$N(t)= {\rm erf}\left[ \left( \frac{\kappa}{2k_{\rm B}T(1-e^{-2t\kappa B})} \right)^{1/2} x_0 e^{-t\kappa B} \right].$$6 for the static $$N(t)= {\rm erf}\left[ \left( \frac{\kappa}{2k_{\rm B}T(1-e^{-2t\kappa B})} \right)^{1/2} x_0 e^{-t\kappa B} \right].$$7 state, so the tetrahedral dynamic state is lower by $$N(t)= {\rm erf}\left[ \left( \frac{\kappa}{2k_{\rm B}T(1-e^{-2t\kappa B})} \right)^{1/2} x_0 e^{-t\kappa B} \right].$$8. The paper further reports that the tetrahedral state becomes more stable with increasing temperature, with the free-energy difference increasing to $$N(t)= {\rm erf}\left[ \left( \frac{\kappa}{2k_{\rm B}T(1-e^{-2t\kappa B})} \right)^{1/2} x_0 e^{-t\kappa B} \right].$$9 at PRESERVED_PLACEHOLDER_^^^^6all: \6query6query6^^^^. This is a canonical example of dynamic distortion as anharmonically stabilized symmetry restoration: the observed high symmetry is not the symmetry of a static BO minimum but of a delocalized vibrational state.

Taken together, these materials papers show that dynamic distortion can be either an externally driven time-dependent lattice field or an internally delocalized nuclear motion on a multiminimum potential surface. In both cases, the essential point is the same: static structural intuition is insufficient, because the physically realized state is determined by dynamics.