Torque-Induced Activity Term in Complex Systems

• Torque-induced activity term is the explicit representation of rotational forces that affect the evolution of physical systems across friction, spintronics, optics, and molecular assemblies.
• It is modeled by incorporating explicit torque terms into evolution equations and balance laws to capture angular momentum transfer and rotational dynamics.
• Applications include predicting micro-slip events in frictional systems, optimizing spintronic switching, and controlling optical and molecular rotational behaviors.

The torque-induced activity term denotes the explicit contribution of internal or external torques in the dynamic evolution of physical systems, manifesting in a wide array of contexts such as frictional sliding, spintronic devices, soft active matter, molecular assemblies, and even in gravitational or optical frameworks. Across these domains, this term typically quantifies the effect of angular momentum transfer—whether from friction-induced geometric asymmetry, charge-current interactions, many-body electromagnetic fluctuations, or field-induced torques—on the system's microscopic or macroscopic evolution.

1. Fundamental Concepts and Generalized Definitions

In physical systems, a torque-induced activity term summarizes all dynamic contributions that arise due to applied torques rather than purely from translational or conservative force fields. Unlike scalar or vector forces, torques generate rotational motions or internal reorientations, and their inclusion in dynamical models is essential whenever the system displays rotational or orientational degrees of freedom (e.g., fiber directions in composites, spins or magnetic moments, molecule orientations, or rotating macro- and nano-objects).

The mathematical representation of torque-induced activity typically enters evolution equations as either explicit torque terms in balance laws (such as for angular momentum, orientation, or remodelling), or through additional terms in the force balance reflecting the cross product of moment arms and forces or their appropriate analogues. In stochastic or statistical descriptions, torque contributions often reflect in nonzero terms in the Fokker–Planck or Langevin equations governing rotational (angular) degrees of freedom.

2. Frictional Systems and the Emergence of Friction-Induced Torque

In frictional systems, particularly in the transition from static to stick–slip motion, friction-induced torques emerge when a tangential force is applied at a height offset from the frictional interface, creating an explicit torque about the center of mass (Scheibert et al., 2010). Even with minuscule tilt angles (∼10⁻⁴ radians), this torque generates a pronounced linear asymmetry in the pressure field along the interface. The resultant local pressure deficit at the trailing edge ensures that micro-slip events, or precursors, always nucleate there first.

Within a minimal quasistatic 1D model, the local shear strength is parameterized as $\sigma_c(x) = \mu_s p(x)$ (static) and, after slip, as $\mu_d p(x)$ (kinetic, with $\mu_d < \mu_s$). The torque–induced pressure asymmetry is modelled by $$\tilde{p}(x, \tilde{F}) = \frac{1}{\mu_s} + \frac{2g}{\mu_d}\tilde{F}x$$ where $g = 6\mu_d(H/L)$ quantifies the dimensionless torque. The nucleation and arrest conditions of slip fronts, their propagation, and the sequence of relaxation events all derive directly from this torque-induced activity term, which controls both the kinematics of stick–slip and the location/extent of micro-slip events. This mechanism accurately predicts trailing edge precursor nucleation, the asymmetric contact area, and the disparity between macroscopic and microscopic friction coefficients.

3. Spintronic and Magnetic Systems: Spin-Orbit and Field-Like Torques

Torque-induced activity terms are critical in spintronics, where nonconservative torques govern the dynamical switching, oscillation, or reorientation of magnetization in thin films and devices:

• Spin-Orbit Coupling Effects: For materials with strong spin–orbit coupling, the usual relation between spin-transfer torque (STT) and the divergence of spin current generalizes to include the total angular momentum current (spin + orbital) and its mechanical exchange with the lattice (Haney et al., 2010). The conservation law

$$\frac{dJ}{dt} - \nabla\cdot Q_J = -\tau_{\text{STT}} - \tau_{\text{lat}}$$

explicitly shows that STT and mechanical torque (on the lattice through orbitals) together enforce angular momentum conservation. In such systems, STT may persist throughout the ferromagnetic bulk (not merely at interfaces), and mechanical torques on the lattice emerge at sharp boundaries.

• Field-Like Torques and Magnetic Tunnel Junctions: In current-induced switching of MTJs, the torque-induced activity arises from two terms, the Slonczewski (damping-like) STT and the field-like STT. The composite torque

$$\tau_{\text{STT}} = a_J\left(\mathbf{m} \times (\mathbf{m} \times \mathbf{m}_p)\right) + b_J(\mathbf{m} \times \mathbf{m}_p)$$

determines the switching time, initial response ("kick"), and final alignment. The alignment and initial torque effects of the field-like term directly reduce switching thresholds under suitable angular and current configurations (Tiwari et al., 2014).

• Temperature-Dependent Spin Torque: Incorporating thermal fluctuations leads to a thermally-activated, finite in-plane magnetization when spin-torque terms are present. The resulting in-plane component orientation depends on both the spin-torque parameters and temperature, enabling thermally controlled rotation of magnetization in nanomagnets (Chotorlishvili et al., 2013).

4. Emergent Torque Terms in Many-Body and Disordered Systems

Torque-induced activity can arise even in the absence of net external torques, due to internal fluctuations or many-body effects:

• Disordered Charge Systems: For randomly charged dielectric surfaces, while the mean torque vanishes (by statistical isotropy), the sample-to-sample fluctuations are nonzero and long-ranged (Naji et al., 2011). The torque variance obeys

$$\langle \tau^2 \rangle = \frac{g_{1s}g_{2s}A^2}{128\pi^2\epsilon_0^2\epsilon_m^2 l^2}$$

and scales with $A^2$ (area squared) rather than $A$. This means torque fluctuations can vastly exceed lateral force fluctuations and constitute a sensitive probe of surface disorder.

• Van der Waals Torque and Many-Body Dispersion: In layered 2D materials with in-plane anisotropy, the quantum fluctuation–induced van der Waals torque depends on the many-body correlation of charge fluctuations. The MBD approach yields torques that can be an order of magnitude higher than atom-pairwise models, scale linearly with area, decay with separation, and vary sinusoidally with the disorientation angle (e.g., $M(\theta) \propto -\sin(2\theta)$) (Kou et al., 16 Dec 2024).

5. Torque Activity in Optical, Molecular, and Continuum Systems

Angular momentum transfer from electromagnetic, optical, and mechanical fields naturally gives rise to torque-induced activity, often in innovative or symmetry-breaking settings:

• Optical Torque via Nonlinearity and Multipolar Effects: In nonlinear light scattering, even non-absorbing or achiral nanostructures experience optical torques when symmetry is broken at the multipolar (pseudochiral) or harmonic generation level (Toftul et al., 2022, Achouri et al., 2023). The net torque,

$$T_z = \frac{1}{2} T_0/[\sigma_{\text{geom}}k^2(2\omega)] \sum_{j,m} (2m_\text{inc} - m)[W^{E}_{mj} + W^{M}_{mj}]$$

can switch sign ("negative optical torque") depending on the relative excitation of multipolar channels, e.g., via qBIC states. In achiral flat structures subjected to linearly polarized light, torque emerges from multipolar pseudochirality and nonlocal cross-polarized responses.

• Phase-Controlled Optical Torque: In BECs with closed-loop multi-level transitions, the optical torque induced by Laguerre-Gaussian beams depends sinusoidally on the relative phase ($\Phi$) of the applied fields, enabling direct phase control of the magnitude and direction of the induced rotational current (Kazemi et al., 2018).
• Continuum Mechanics and Eshelby Torque: In active fiber-reinforced materials, internal remodelling ("activity") arises from the Eshelby torque

$\tau_e = \frac{1}{2}[\mathbf{C},\mathbf{T}]$

in the evolution equation for the remodelling tensor. This torque drives the fiber orientation to minimize elastic energy by aligning the stress and strain tensors. External activity terms (e.g., from electric/magnetic fields) are incorporated through additional torque contributions (Ciambella et al., 2018).

6. Advanced Spintronic and Oscillator Systems: Resonant, Self-Induced, and Trapping Torques

Recent studies have highlighted previously overlooked or newly engineered torque-induced activity terms in spin torque oscillators and related devices:

• Resonant Enhancement and Unconventional β-Torque: For AC-induced interfacial spin-transfer torque, resonance occurs when the AC frequency approaches the exchange splitting energy, amplifying both the conventional and "β-term" torques:

$$\tau_e(\mathbf{r},\omega) = \frac{(j_s \cdot \nabla)m + i\omega\tau_{sd} m \times (j_s \cdot \nabla)m}{1-\omega^2\tau_{sd}^2}$$

where a time-derivative–linked β-term emerges even without spin relaxation, directly coupling to current transients (Fujimoto et al., 2019).

• Charge Conservation and Self-Induced Torque: In tunneling spin torque oscillators, explicit charge conservation mandates an additional self-induced torque—from pumped current flowing alongside magnetization precession—modifying the LLGS equation with a term

$$\mathbf{I}^p_s = (\hbar/4e)^2 (g_l^s)^2/(g_l(\theta) + g_r) [\mathbf{m}_{fix}\cdot(\mathbf{m}\times\dot{\mathbf{m}})] \mathbf{m}_{fix}$$

stabilizing frequency and amplitude in nanopillar oscillators (Gunnink et al., 2023).

• Current-Induced Magnon Trapping: In Ni₈₁Fe₁₉/Pt nanostructures, a trapping Oersted field generated by the current localizes magnon dynamics in the nano-gap. The locally suppressed resonance frequency creates a potential well, confining the oscillation mode and thus reducing both magnon radiation loss and the threshold current while narrowing the emission linewidth (Makiuchi et al., 18 Mar 2024).

7. Broader Implications, Experimental Probes, and Applications

The presence and design of torque-induced activity terms are central to interpreting and leveraging rotational phenomena in a broad variety of physical settings:

• Experimental Detection and Sensitivity: Techniques such as double torsion pendulum setups (Yasuda et al., 2021), rotational measurements of randomly charged surfaces (Naji et al., 2011), and lock-in detection of photon angular momentum transfer offer direct routes for measuring extremely weak or fluctuation-driven torques.
• Device and Material Design: Understanding and harnessing torque-induced activity underpins efficiency and function in spintronic devices (MRAM, STNOs), optical micro/nano-rotors, actuators in NEMS, and responsive biological or soft-matter systems.
• Theoretical and Modeling Advances: Many-body theoretical descriptions (e.g., MBD model for van der Waals torque), inclusion of activity terms in continuum and Fokker–Planck frameworks, and new modeling strategies for collective or resonantly enhanced torques provide a foundation for future research.

In summary, torque-induced activity terms constitute quantitatively and qualitatively essential elements across diverse systems, mediating transitions, instabilities, and steady-state behaviors in response to both internal fluctuations and external controllable stimuli. Their rigorous treatment enables precise modeling, measurement, and control of rotational and orientational dynamics across physics, materials science, nanotechnology, and engineering.