Drift-Elasticity Relationship

• Drift-elasticity relationship is the quantitative link between systematic drift (from forces or flows) and elasticity parameters, providing a unified framework across diverse systems.
• It is applied to model phenomena in statistical physics, quantitative finance, and material science, such as currency arbitrage under Gresham’s law and anomalous diffusion in polymers.
• The relationship is analyzed using structural equations and fractional stochastic models that relate force, elasticity, and drift dynamics, enabling accurate predictions and statistical inference.

The drift-elasticity relationship describes how drift—defined as systematic or stochastic flows, displacements, or rates in physical or financial systems—is quantitatively controlled or modulated by elasticity parameters. Across statistical physics, stochastic processes, condensed matter, and quantitative finance, drift rates are frequently expressible in terms of elastic moduli, exponents, or elasticity coefficients, and in several paradigmatic stochastic models the connection is exact and foundational. This relationship provides unified insight into the material, dynamical, and estimation-theoretic behavior of a broad class of systems, including currency flow under Gresham’s law, time-fractional Langevin dynamics, plastic deformation under noise, elasticity estimation in financial diffusions, and spatio-temporal scaling in driven lattices.

1. Mathematical Structuring of Drift via Elasticity

In each domain, a core structural equation expresses drift as a product or functional of elasticity and forcing. For example, under Gresham’s Law, the net arbitrage flow of a currency ($dQ_1$) is governed by the price-demand elasticity ($E_d^1$) and overvaluation ($\Delta P_1/P_1$):

$dQ_1 = -Q_1 E_d^1 \frac{\Delta P_1}{P_1}$

This parallels the standard drift of charged particles: current density $J = qn\mu E$, mapping $E_d^1$ to mobility $\mu$ and overvaluation to field $E$. In the generalized elastic model, constant-force drift for a tagged probe yields (Taloni et al., 2012, Taloni et al., 2013):

$$\langle h(t) \rangle = \frac{F_0}{K^+ \Gamma(1+\beta)} t^\beta$$

where the exponent $\beta$ is given explicitly by the elastic and hydrodynamic indices $\beta = \frac{z-d}{z+\alpha-d}$. In an elasto-perfectly-plastic oscillator, the drift coefficient for variance growth is given by (Bensoussan et al., 2011):

$$\mu = \frac{\mathbb{E}\left(\int_{\tau_0}^{\tau_1} y(s)\, ds\right)^2}{\mathbb{E}[\tau_1 - \tau_0]}$$

with dependence on stiffness $k$ and plastic bound $Y$.

2. Drift-Elasticity in Statistical Physics Models

The generalized elastic model synthesizes features from polymers, membranes, and rough interfaces into a linear stochastic framework, typically governed by fractional Langevin equations (FLE). When a localized force $F(t)$ is applied, the tagged probe experiences a drift whose time-law (sublinear or ballistic/exponential regimes) directly reflects the elastic exponent $z$ and hydrodynamic kernel $\Lambda$:

• Sublinear drift: For long times, $\langle h(t) \rangle \sim t^{\beta}$, with $\beta = \frac{z-d}{z+\alpha-d}$, signifying anomalous diffusive propagation controlled by elasticity.
• Ballistic/exponential regimes: For short times and local hydrodynamics ($\alpha=d$), with $z\neq2m$, drift scales as $t^2 \operatorname{sgn}[\sin(z\pi/2)]$, and drift reversal can occur for certain values of $z$.
• Oscillatory forcing and spatial domains: The system splits into domains with different amplitudes and phase shifts, with region boundaries set by hydrodynamic correlation times $\tau(|x-x^*|)$ (Taloni et al., 2012).

3. Drift-Elasticity Estimation in Stochastic Financial Models

In continuous-time finance, the CKLS (Cox–Ingersoll–Ross with elasticity) model for interest rates features a drift which is a function of the underlying elasticity parameter $\gamma$:

$$d\lambda_t = (\alpha_\lambda - b \lambda_t) dt + \sigma \lambda_t^\gamma dW_t$$

A state-space transformation yields a CIR-type process whose drift parameter $\beta(\gamma) = 2b(1-\gamma)$ (Ning et al., 8 Dec 2025), making the elasticity estimable via high-frequency inference on drift:

$$\widehat{\gamma}_n = 1 - \frac{\widehat{\beta}_n}{2b}$$

This closed-form mapping allows direct statistical inference, with asymptotic variance $\frac{5(1-\gamma_0)}{b}$. The drift-elasticity structure makes statistical learning of elasticity tractable in a diffusion context with explicit convergence properties.

4. Drift-Induced Mode Coupling in Driven Lattices

In overdamped crystals or lattices drifting through a dissipative medium, elasticity interacts with drift to couple transverse and longitudinal modes. The continuum equations (Dolai et al., 2017):

$$\Gamma [\partial_t u_x + v_d \partial_x u_x] = \mu\, \partial_x^2 u_x + \Lambda_2\, \partial_x u_z + \eta_x(x,t)$$

$$\Gamma [\partial_t u_z + v_d \partial_x u_z] = B\, \partial_x^2 u_z + \Lambda_3\, \partial_x u_x + \eta_z(x,t)$$

Here, elastic moduli ($\mu, B$) fix diffusion constants, while drift velocity ($v_d$) induces off-diagonal mode couplings ($\lambda_2, \lambda_3$) scaling linearly with drift. The net effect is a drift-renormalized wave speed $c = \sqrt{\lambda_2 \lambda_3}$, underdamped propagation, and KPZ universality of fluctuations at the longest scales.

5. Boundary Conditions, Stability, and the Role of Elasticity

Several models exhibit critical dependencies of drift behavior on elasticity:

• In Gresham’s Law, as elasticity $E_d^1 \to 0$, mispricing has vanishing effect; for large elasticity, drift is immediate.
• In the FLE model, the existence and sign of drift (including drift reversal) depends on the value of $z$ via the sign of $\sin(z \pi/2)$ (Taloni et al., 2012).
• In drift-driven lattices, linear stability and propagative wave vs. diffusive mode transitions are set by the sign and magnitude of $\lambda_2 \lambda_3$, which are themselves determined by the drift-elasticity interplay.

6. Generalization to Multi-Agent, Multi-Species, and Network Systems

Extensions of the drift-elasticity relationship appear naturally in networked or multi-component settings:

• Currency arbitrage in multi-currency, multi-country systems is governed by elasticity matrices and local/global maxima of misvaluation, with network topology dictating flows and local equilibria (Smith, 2012).
• In spatially-resolved elastic models, the drift response propagates through the system with scaling laws established by exponents controlling elastic and hydrodynamic interactions, and the presence of multiple spatial domains under periodic forcings (Taloni et al., 2013).

7. Fluctuation-Dissipation and Linear Response Connections

The drift-elasticity connection is underpinned by generalized fluctuation-dissipation relations (FDR) and linear response theory. In the generalized elastic model, the averaged drift under a force $F_0$ is linked to equilibrium correlation functions by:

$$\langle h(x, t) \rangle_{F_0} = \frac{\langle \delta h(x, t)\, \delta h(x^*, t) \rangle}{2 k_B T} F_0$$

This establishes not only the scaling but also the universality of the drift-law ($t^\beta$) as a function of the elasticity exponents, and provides explicit forms for Kubo relations in stochastic elasticity (Taloni et al., 2013).

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The drift-elasticity relationship is thus a robust mathematical and physical structure, manifest in linear and nonlinear stochastic models, classical dynamical systems, quantitative finance, and condensed matter. Drift rates and dispersions are determined by elasticity coefficients, exponents, and mode-coupling parameters, enabling both prediction and inference of time-scaling, stability, and propagation properties across systems.