General Driving-Force Expression

• General Driving-Force Expression is a unifying concept that quantifies the rate-controlling potential using thermodynamic, kinetic, and geometric variables.
• It underpins diverse applications including chemical reaction flux, actuator mechanics, grain boundary migration, phase-field fracture, and vortex dynamics.
• Its analytical and computational formulations guide design optimization and predictive modeling of irreversible system evolution.

A general driving-force expression relates macroscopic system evolution—be it mechanical motion, phase change, fracture, or chemical reaction—to the underlying thermodynamic, kinetic, or geometric variables determining the irreversible tendency of the system. In physical, engineering, and chemical contexts, the “driving force” quantifies the rate-controlling potential, affinity, or power that governs the system's response to external inputs, internal modifications, or spatial nonuniformities. This concept is universally applicable, as seen in mechanics (virtual work, actuator placement), condensed matter (vortex dynamics), materials migration (grain-boundary kinetics), and chemical network thermodynamics.

1. Thermodynamic and Kinetic Formalism of the Driving Force

In nonequilibrium chemical thermodynamics, the general driving-force expression is encapsulated by the relation between chemical affinities (generalized forces) and microscopic one-way fluxes. For a system with M elementary reversible reactions and internal/external species, the kinetic fluxes $J^{+\ell}, J^{-\ell}$ for the $\ell$-th reaction satisfy

$$k_BT\,\ln\left(\frac{J^{+\ell}}{J^{-\ell}}\right) = \Delta G_\ell$$

where $\Delta G_\ell$ is the Gibbs free-energy difference across the reaction. In arbitrary networks with reaction cycles $\tilde{c}$, the generalized force–flux relation becomes

$$k_B T \ln \left(\frac{J^+}{J^-}\right) = \Delta G_{cycle}$$

This relation extends to any composite or elementary cycle in complex reaction networks, providing a universal force–flux mapping (Peng et al., 2019). The chemical driving force $\Delta G$ thus sets the exponential bias between forward and backward transition events, unifying mesoscopic Markovian descriptions with macroscopic kinetics and serving as the foundation for defining entropy production.

2. Mechanical Systems: Actuator-Position-Independent Formulation

For discrete mechanical systems such as the scissor lift, the driving-force expression F required by the actuator to statically or quasi-statically maintain a load in equilibrium is derived via the virtual work principle:

$F = W_e \frac{dl}{dh}$

where $W_e = L + \frac{1}{2}$ (lift weight), $l$ is the actuator length, $h$ the lift height, and both are functions of link angle $\theta$ and normalized variables $(a, b, i)$ determined by actuator placement (Saxena, 2016). The explicit algebraic forms (for negatively and positively sloping actuator links) are:

• Negatively sloping:

$$F = \frac{W_e}{n} \frac{ \sqrt{A \cos^2\theta - 2B\cos\theta + C} }{ B \tan\theta - A \sin\theta }$$

with $A = (1-a)^2 - (i+a)^2$, $B = b(1-a)$, $C = b^2 + (i+a)^2$.

• Positively sloping:

$$F = \frac{W_e}{n} \frac{\sqrt{C' - 2B'\cos\theta + D \cos^2\theta}}{ (B'-b)\tan\theta - D\sin\theta }$$

with $B' = ba$, $C' = b^2 + (i+a)^2$, $D = i(2a+i)$.

This generalizes the classical screw-jack and vertical-jack formulas and enables systematic scanning over actuator positions to minimize driving force or optimize stroke.

3. Crystal Defect and Grain Boundary Migration

In materials physics, the driving-force (denoted $F$) for grain boundary migration or defect motion represents the thermodynamic potential bias per unit area or length. Microscopically, when a GB advances by distance $b$, the work done is $Y = C F$, leading to an asymmetric barrier for forward/backward migration events. The kinetic rate law is

$$v = 2Nb\omega \exp\left(-\frac{Q}{k_BT}\right) \sinh\left(\frac{Y}{k_BT}\right)$$

and the GB mobility is

$$M(F,T) = \frac{v}{F} = \frac{2Nb\omega}{F} \exp\left(-\frac{Q}{k_BT}\right) \sinh\left(\frac{C F}{k_B T}\right)$$

The activation barrier typically decreases as $Q(F) = Q_0 \exp(-C_1 F)$ with increasing $F$, and the transition temperature $T_{trans}$ for “anti-thermal” behavior is set by $Q(F) = k_B T_{trans}$ (Song et al., 2022). Nonmonotonic (“anti-driving-force”) dependencies in $M(F, T)$ arise from these forms.

4. Fracture and Phase-Field Driving Force Expressions

In variational phase-field fracture formulations, the crack-driving force is the functional derivative of the free energy with respect to the phase- or damage-field. For a scalar field $d$ or $\phi$ and total energy $\mathcal{E}_\ell[u,\phi]$, the Euler–Lagrange equation yields:

$$\frac{\delta \mathcal{E}_\ell}{\delta \phi} = -2(1-\phi) \psi_a(\varepsilon) + G_c\left( \frac{\phi}{\ell} - \ell\Delta\phi \right) = 0$$

with the local driving term

$$g_{\mathrm{drive}}(\phi,\varepsilon) = 2(1-\phi)\psi_a(\varepsilon)$$

where $\psi_a$ is the "active" energy driving damage and is model-dependent (Bilgen et al., 2018, Navidtehrani et al., 2022). Models prescribing $\psi_a$ to match a Mohr–Coulomb or Drucker–Prager surface yield driving forces consistent with the corresponding strength envelope (Senthilnathan, 18 Dec 2024).

For arbitrary linear-in-parameter strength surfaces $F(\bar{\sigma}, \beta)$, the crack nucleation driving force is constructed analytically such that the phase-field strength locus matches the original surface at $n$ reference points and exhibits effective toughness $$\widehat{G}_c(\sigma) = -F(\bar{\sigma}, \beta^{\varepsilon})\, G_c$$ that varies with stress (Senthilnathan, 18 Dec 2024). This approach unifies strength and toughness in the crack nucleation law.

5. Curvature-Driven Flows with Inhomogeneous Driving Forces

For crystalline mean curvature flow, the driving-force function $f(x, t)$ enters additively in the geometric evolution equation:

$u_t + F(Du, \mathrm{div}(V_o(Du)) - f(x, t)) = 0$

with $F$ continuous and monotone, $f$ Lipschitz in $x$. The canonical PDE representation for a hypersurface with normal velocity $V_n$ is

$V_n = \kappa_{\mathrm{cryst}} + f(x, t)$

where $\kappa_{\mathrm{cryst}}$ is the crystalline curvature (Giga et al., 2020). The regularity of $f$ critically impacts well-posedness; a viscosity solution and comparison principle hold under these hypotheses.

6. Driving Force in Superconducting Vortex Dynamics

The driving force on an isolated moving vortex in the flux-flow state, derived from the time-dependent Ginzburg–Landau equation, splits into hydrodynamic and magnetic components. Integrating the local momentum balance over a region $S$ around the vortex yields

$${\bf F} = \oint_{\partial S} [\Pi_{\nu\mu} n_\nu - T_{\nu\mu} n_\nu] \, ds$$

where $\Pi_{\nu\mu}$ is the condensate momentum-flux tensor and $T_{\nu\mu}$ the Maxwell stress. For $\xi \ll R \ll \lambda$ (coherence length $\xi$, penetration depth $\lambda$), the driving force is dominated by the hydrodynamic term; for $R \gtrsim \lambda$, hydrodynamic and magnetic parts contribute equally. The macroscopic expression reduces to

$${\bf F} = \Phi_0 {\bf j}_{\rm tr}(0) \times \hat{\bf z}$$

with $\Phi_0$ the flux quantum (Kato et al., 2015).

7. General Principles and Application Strategies

General driving-force expressions serve as the central link between macroscopic observables (mobility, force, phase evolution, reaction rate) and the underlying energetic or kinetic bias imposed by geometry, network topology, or external/control parameters. Across domains:

• Thermodynamic force generates net flux; its explicit expression enables analytical and numerical exploration of nonequilibrium steady states (Peng et al., 2019).
• In mechanical systems, “position variables” parametrize actuator efficacy, allowing optimal design and force minimization for arbitrary linkage arrangements (Saxena, 2016).
• Migration and defect formation rates are quantitatively linked to driving-force-dependent barrier lowering and exhibit transitions in mobility and anti-driving-force phenomena (Song et al., 2022).
• Phase-field and fracture models translate external loads or strength criteria into driving terms that trigger damage or crack evolution, furnishing systematic phase-field construction for multiaxial strength (Bilgen et al., 2018, Senthilnathan, 18 Dec 2024).
• In geometric flows, spatially nonuniform or time-dependent driving-force terms modulate interface velocity, with regularity of $f$ controlling mathematical well-posedness (Giga et al., 2020).

The recurring structure—the driving force as the gradient, affinity, or bias conjugate to the system's rate of irreversible evolution—provides a unifying paradigm for analytical modeling and computational implementation across physical, chemical, and engineering systems.