Dynamic Pinning Conditions in Complex Systems

Updated 12 July 2026

Dynamic Pinning Conditions are defined by criteria that determine when localized constraints modify a system’s time-dependent response through control variables like temperature, drive, or defect density.
They are applied across diverse fields—glass-forming liquids, ferromagnetic films, superconductors, and beyond—to reveal exponential responses, boundary conditions, or depinning thresholds via specific experimental and theoretical metrics.
These dynamic criteria enable practical insights into mode selection and relaxation behaviors, allowing researchers to diagnose system-specific attributes such as cooperativity, energy barriers, and topology changes.

Dynamic pinning conditions denote the parameter regimes under which a pinned system develops a characteristic dynamic response. In the cited literature, the term is explicitly context-dependent: in glass-forming liquids it is specified by the pinning concentration $c$ , temperature $T$ , and the resulting relaxation time and cooperativity; in ferromagnetic films with interface Dzyaloshinskii–Moriya interaction it appears as a wave-number-dependent boundary condition for dynamic magnetization; in superconductors it is encoded in force–velocity or current–voltage characteristics under strong or directional vortex pinning; in soft wetting it is set by a Gibbs-type depinning inequality on a dynamically evolving wetting ridge; and in networked or computational systems it refers to node, agent, or thread selection rules under pinning control or CPU affinity assignment (Kob et al., 2014, Kostylev, 2014, Thomann et al., 2011, Gorcum et al., 2018, Zhang, 2024, Chasparis et al., 2016).

1. General structure of the concept

Across the cited works, dynamic pinning conditions are formulated through a small set of control variables and a corresponding stability, transport, or synchronization criterion. In condensed-matter settings, these variables are typically field, drive, wave number, temperature, or defect density. In control and computing settings, they are coupling matrices, pinning indicators, triggering rules, or placement strategies. A common formal pattern is that pinning is not treated as a purely static constraint: it modifies the accessible modes of motion, relaxation, or synchronization in a way that depends on observation scale, drive, or switching rate.

Domain	Control variables	Characteristic condition
Glass-forming liquids	$c$ , $T$	$\tau(c,T)/\tau(0,T)=\exp[cB(T)/T]$
Ferromagnetic films with IDMI	$k$ , $D$ , $L$	$d_p=Dkb/A$ , $k/L$ contribution to dispersion
Type-II superconductors	$T$ 0, $T$ 1, $T$ 2, $T$ 3	$T$ 4
Soft wetting on gels	$T$ 5, $T$ 6, $T$ 7	$T$ 8
Fast-switching network control	$T$ 9, $c$ 0, pinning gains	average topology with spanning tree, sufficiently fast switching

This breadth is not terminological drift so much as a family resemblance. The cited works consistently use pinning to mean a localized or selective constraint, and dynamic pinning conditions to mean the conditions under which that constraint changes the actual time-dependent response rather than only the static configuration (Reichhardt et al., 2021, Han et al., 2014).

2. Glass-forming liquids: random pinning, crossover, and entropy reduction

In a supercooled glass-forming liquid, dynamic pinning conditions are defined by the fraction $c$ 1 of permanently immobilized particles, the temperature $c$ 2, and the resulting relaxation times, cooperativity measures, and dynamic length scales (Kob et al., 2014). The model studied is a 50:50 binary mixture of elastic spheres at density $c$ 3, with onset temperature for slow dynamics $c$ 4 and mode-coupling temperature $c$ 5. A fraction $c$ 6 of particles is chosen and permanently pinned, with the pinned particles arranged “uniformly with a well characterized distance between them.”

The primary dynamical observable is the self-intermediate scattering function

$c$ 7

at $c$ 8, and the $c$ 9-relaxation time is defined by $T$ 0. The central result is an exponential response to pinning,

$T$ 1

obtained from the activated form $T$ 2 together with a small- $T$ 3 expansion $T$ 4 (Kob et al., 2014). The same work reports that the infinite-time collective overlap is linear in $T$ 5, so the non-linear dynamic response occurs in a regime where the static response remains linear.

The paper further identifies a crossover near $T$ 6. At high and intermediate $T$ 7, $T$ 8 increases as $T$ 9 decreases. At lower $\tau(c,T)/\tau(0,T)=\exp[cB(T)/T]$ 0, the activation energy $\tau(c,T)/\tau(0,T)=\exp[cB(T)/T]$ 1 becomes essentially constant and largely independent of $\tau(c,T)/\tau(0,T)=\exp[cB(T)/T]$ 2, so the slowing down saturates. Four-point susceptibility data show that near $\tau(c,T)/\tau(0,T)=\exp[cB(T)/T]$ 3 the peak $\tau(c,T)/\tau(0,T)=\exp[cB(T)/T]$ 4 decreases with increasing $\tau(c,T)/\tau(0,T)=\exp[cB(T)/T]$ 5, which is interpreted as a reduction of cooperatively rearranging regions. The bound $\tau(c,T)/\tau(0,T)=\exp[cB(T)/T]$ 6, rather than $\tau(c,T)/\tau(0,T)=\exp[cB(T)/T]$ 7, implies a fractal-like structure of CRRs with $\tau(c,T)/\tau(0,T)=\exp[cB(T)/T]$ 8 in the studied regime (Kob et al., 2014).

The thermodynamic interpretation is given in both Adam–Gibbs and RFOT language through a linear reduction of configurational entropy,

$\tau(c,T)/\tau(0,T)=\exp[cB(T)/T]$ 9

In the Adam–Gibbs framework this yields

$k$ 0

while in RFOT the generalized Rabochiy–Wolynes–Lubchenko estimate gives

$k$ 1

and

$k$ 2

The reported decrease of $k$ 3 on approaching $k$ 4, followed by saturation, is used to argue for a change from string-rich to core-dominated relaxing entities (Kob et al., 2014). This suggests that dynamic pinning conditions can diagnose both the size and the geometry of cooperative relaxation.

3. Ferromagnetic films: boundary pinning from interface Dzyaloshinskii–Moriya interaction

For ferromagnetic films with interface Dzyaloshinskii–Moriya interaction, dynamic pinning conditions are boundary conditions on the dynamic magnetization rather than static spin anchoring (Kostylev, 2014). Starting from the linearized Landau–Lifshitz equation and Rado–Weertman torque balance, the interfacial DMI effective field

$k$ 5

produces interface exchange boundary conditions

$k$ 6

at the interface $k$ 7 (Kostylev, 2014).

Introducing circular components diagonalizes the problem: $k$ 8 with pinning parameter

$k$ 9

The unusual feature is that the pinning scales as the spin-wave wave number. For $D$ 0, $D$ 1, so IDMI produces no pinning under FMR conditions; pinning appears only for traveling waves. The same paper emphasizes that $D$ 2, tying the effect directly to chirality and to frequency non-reciprocity (Kostylev, 2014).

In the Damon–Eshbach geometry, the resulting dispersion has an IDMI term proportional to $D$ 3,

$D$ 4

and the non-reciprocity scales approximately as $D$ 5 (Kostylev, 2014). Here dynamic pinning conditions are therefore boundary conditions on mode profiles, not static surface anisotropy. A plausible implication is that the relevant “pinning strength” is mode-dependent in a literal sense: the same interface is free for $D$ 6 and pinned for $D$ 7.

4. Superconducting vortices: strong pinning, directional pinning, and moving-state effectiveness

In type-II superconductors, dynamic pinning conditions are formulated through force balance,

$D$ 8

with $D$ 9 and $L$ 0 (Thomann et al., 2011). In the dilute strong-pinning regime, the Labusch criterion $L$ 1 produces bistable pinned and unpinned branches, and the current–voltage characteristic becomes an excess-current form,

$L$ 2

or equivalently $L$ 3 near onset (Thomann et al., 2011, Thomann et al., 2017). The onset is hysteretic for $L$ 4, while for $L$ 5 it is smooth with

$L$ 6

These papers identify two key velocity scales, $L$ 7 and $L$ 8, and emphasize that $L$ 9 in the dilute strong-pinning limit (Thomann et al., 2017).

Experimental vortex systems add anisotropy and defect geometry. In EuRbFe $d_p=Dkb/A$ 0As $d_p=Dkb/A$ 1, AC susceptibility and magnetization measurements for $d_p=Dkb/A$ 2 give a vortex solid–liquid line $d_p=Dkb/A$ 3 with $d_p=Dkb/A$ 4– $d_p=Dkb/A$ 5, activation energy $d_p=Dkb/A$ 6, and

$d_p=Dkb/A$ 7

above $d_p=Dkb/A$ 8, consistent with thermally activated plastic pinning or planar defects (Vlasenko et al., 2019). The same work finds $d_p=Dkb/A$ 9, consistent with strong pinning.

In YBa $k/L$ 0Cu $k/L$ 1O $k/L$ 2 with self-assembled BaZrO $k/L$ 3 nanorods, the flux-flow resistivity obeys effective-mass angular scaling, but the Labusch parameter $k/L$ 4 and $k/L$ 5 do not (Pompeo et al., 2012). The paper reports that the equivalent matching field of the nanorods is $k/L$ 6, that $k/L$ 7 shows a strong peak at $k/L$ 8, and that the dc critical current contains an additional enhancement near the nanorod direction beyond the microwave-derived short-range estimate. This extra enhancement is attributed to a dynamic effect associated with a vortex Mott-insulator-like regime below $k/L$ 9 (Pompeo et al., 2012).

Pinning effectiveness in the moving state also depends on defect geometry. For vortices driven over conformal crystal pinning arrays, pinning is enhanced over random arrays over a wide field range, but at high fields the situation can reverse and the velocity–force curves cross because vortices dynamically order earlier in the conformal array (Ray et al., 2013). This suggests that dynamic pinning conditions are not exhausted by the depinning threshold; they also include whether the moving state remains plastic or becomes ordered.

5. Skyrmions: Magnus-force depinning, reentrant pinning, and boundary pinning energetics

For skyrmions, dynamic pinning conditions are governed by the competition between dissipative drag, Magnus force, inter-skyrmion repulsion, and pinning. Particle-based models use

$T$ 00

with intrinsic Hall angle $T$ 01 in the clean limit (Reichhardt et al., 2019, Reichhardt et al., 2018, Reichhardt et al., 2021). The review literature emphasizes that dynamic pinning in skyrmion systems differs from overdamped vortex matter because the Magnus force produces drive-dependent Hall angles, side jumps, more isotropic shaking, and dynamic ordering into moving crystals rather than moving smectics (Reichhardt et al., 2021).

Driven skyrmions on random pinning exhibit elastic depinning at weak pinning and plastic depinning at stronger pinning, with a sharp increase in depinning threshold and proliferation of topological defects at the elastic-to-plastic transition (Reichhardt et al., 2018). In the strong-pinning, strong-Magnus regime, the moving phase can form density phase separated states or dynamically phase separated banded states, and the longitudinal velocity–force curve can show negative differential conductivity; these effects are absent in the overdamped limit (Reichhardt et al., 2018).

In inhomogeneous pinning arrays with a strongly pinned stripe coexisting with a pin-free region, the skyrmion Hall effect produces accumulation at the pinned boundary, dynamic row reduction, and reentrant pinning (Reichhardt et al., 2019). The phase sequence is low-drive shear flow in the pin-free region, then for sufficiently large pinning and Hall angle a drive-induced fully pinned phase, followed at higher drive by plastic flow and then a moving lattice. The same paper reports that reentrant pinning becomes more pronounced for increasing intrinsic skyrmion Hall angle (Reichhardt et al., 2019).

A separate energetic perspective is provided by thermal skyrmion dynamics in thin films, where the pinning landscape is reconstructed from

$T$ 02

For large skyrmions, the pinning originates at the skyrmion boundary and not at its core, and the pinning depends strongly on skyrmion size and shape (Gruber et al., 2022). The work reports micrometer-scale skyrmions with $T$ 03– $T$ 04, field-tunable size changes of $T$ 05– $T$ 06, and dynamic switching of pinning sites as the preferred boundary contour changes. This indicates that dynamic pinning conditions in skyrmion systems can be set not only by drive and disorder but also by internal mode content and shape flexibility (Gruber et al., 2022). A plausible implication is that “pinning strength” in skyrmionics is state-dependent even for fixed material disorder.

6. Soft wetting, runtime placement, and network control

In soft wetting, dynamic pinning conditions are expressed by a depinning inequality on a moving wetting ridge. For water on a PDMS gel with $T$ 07, $T$ 08, $T$ 09, and elastocapillary length $T$ 10, the solid opening angle obeys

$T$ 11

with $T$ 12, and the contact line depins when

$T$ 13

The critical speed is $T$ 14, above which motion becomes stick-slip with slip velocities $T$ 15 (Gorcum et al., 2018). The paper argues that bulk viscoelasticity alone cannot explain the depinning and identifies a dynamical increase of the solid surface tensions as the relevant mechanism.

In runtime computing, dynamic pinning means dynamic placement of threads to processing units. Each thread is an agent with mixed strategy $T$ 16, actual selection rule

$T$ 17

and reinforcement update

$T$ 18

With utilities chosen as a common objective, the process approaches neighborhoods of Nash equilibria corresponding to locally optimal placements (Chasparis et al., 2016). Here the pinning condition is the current thread-to-core affinity profile together with the strategy updates that govern its evolution.

In network control, pinning means applying direct control to a subset of nodes. For fast-switching cluster synchronization, the cited literature requires controllability of $T$ 19, sufficiently robust coupling strength, sufficiently fast switching, and an average topology with spanning tree (Du et al., 2024). For event-triggered pinning impulses, the trigger is

$T$ 20

and a sufficient condition for synchronization of the unpinned subnetwork is

$T$ 21

with an adaptive law

$T$ 22

used to remove the need for a fixed coupling threshold (Zhang, 2024). Markovian switching couplings and controller-node sets admit two complementary regimes: slow switching when each subsystem is individually stabilizable, and fast switching when the averaged coupling and pinning gains stabilize the averaged system (Han et al., 2014). In multi-network vehicular systems with asymmetric coupling, the stability conditions take LMI form,

$T$ 23

for a single network and

$T$ 24

for multi-network systems; these are then coupled to a genetic search over pinning-node choices, with overlapping nodes naturally favored because one pinned node can stabilize multiple layers (Guo et al., 2024).

Taken together, these literatures indicate that dynamic pinning conditions are best understood not as a single universal law but as a family of mode-selection criteria. In every case, pinning becomes dynamically relevant only when it is coupled to a timescale, a drive, a switching process, or an internal degree of freedom. That is the common technical content linking pinned particles in glasses, vortices, spin textures, contact lines, and controlled agents.