Monotone Stress-Response Property

• Monotone stress-response property is defined as the predictable, non-oscillatory decay of excess dynamical activity in systems under external drive.
• It employs the Donsker–Varadhan functional to quantify deviations from stationarity, capturing the evolution of occupation probabilities in Markov processes.
• Conditions like normal linear response and the sector condition enable this property to serve as a Lyapunov-like measure for stability in nonequilibrium dynamics.

The monotone stress-response property refers to the mathematical and physical principle that certain systems, when subjected to external drives or "stress," exhibit responses that evolve in an ordered, non-oscillatory, and typically non-increasing or non-decreasing fashion according to monotonicity criteria. In statistical mechanics, dynamical systems, probability theory, and continuum mechanics, this property underlies the predictable and robust relaxation of a system toward steady states, the conservation of physically meaningful features under stochastic or deterministic evolution, and the design of analytical and computational tools for assessing stability, dissipation, and large deviation behaviors.

1. The Donsker–Varadhan Functional and Dynamical Activity

A central result in nonequilibrium statistical mechanics is the characterization of occupation-time large deviations for ergodic Markov processes using the Donsker–Varadhan (DV) rate function. For a Markov jump process with transition rates $k(x, y)$ on a finite state space, if $p_T(x)$ denotes the empirical fraction of time in state $x$ over $[0, T]$, large deviations of $p_T$ away from the stationary distribution $\rho$ are quantified by the DV rate function $I(\mu)$: $$P_{\rho}\left[ p_T \approx \mu \right] \sim \exp\left( -T \cdot I(\mu) \right)$$ with

$$I(\mu) = \sup_{g > 0} \left\{ -\sum_x \frac{\mu(x)}{g(x)} \sum_y k(x, y)[g(y) - g(x)] \right\}$$

and, under the transformation $g = e^{W/2}$,

$$I(\mu) = \sup_W \sum_{x, y} \mu(x)\left[ k(x, y) - k_W(x, y) \right], \qquad k_W(x, y) = k(x, y) \exp\left( \frac{W(y) - W(x)}{2} \right).$$

Physically, $I(\mu)$ represents the excess dynamical activity (total escape or transition rate) relative to a process artificially modified to make $\mu$ stationary. When $\mu$ is made stationary for some tilted process $k_V$, the functional reduces to

$$I(\mu) = \sum_{x, y} \mu(x) [k(x, y) - k_{V}(x, y)].$$

This large deviation functional thus serves as a quantitative measure of "dynamical stress" in the space of occupation distributions.

2. Monotonicity Under Markov Evolution

Empirical and theoretical studies have shown that, under the Markov evolution of the empirical measure $\mu_t$ governed by the Master equation, $I(\mu_t)$ decays monotonically as the system relaxes to nonequilibrium steady state. Formally, there exists $t_0$ on the order of the relaxation time such that: $$\frac{d}{dt} I(\mu_t) \leq 0 \quad \text{for all} \quad t \geq t_0.$$ This property does not generally follow from entropy-like arguments outside the regime of detailed balance; the monotonicity of relative entropy (Kullback–Leibler divergence) is a special feature of equilibrium. The DV functional, by contrast, encodes monotonicity only under specific physical or mathematical conditions on the dynamics that go beyond equilibrium assumptions.

The paper rigorously establishes this asymptotic monotonicity under the hypothesis that the underlying evolution exhibits "normal linear response."

3. Normal Linear Response Condition

Monotonicity of $I(\mu_t)$ for large times is guaranteed when the system conforms to a sector condition on the Markov generator $L$. Defining the symmetrized generator $L_s = (L + L^*)/2$, the normal linear response condition requires that for all test functions $f$,

$(L_s f, L f) \geq c \|f\|^2$

with $\|f\|$ the maximum variation of $f$ and $(\cdot, \cdot)$ the inner product weighted by the stationary measure. Operationally, this translates to the requirement that the second derivative of the dynamic susceptibility is negative at the origin: $$\frac{d}{dt} \chi_{ff}(t)\Big|_{t=0} = - (L_s f, L f) < 0,$$ ensuring that response functions decay in the same fashion as in equilibrium systems. This condition is well-motivated in physical models where initial susceptibilities decrease monotonically, ruling out pathological or non-Lyapunov-like behavior in the system's approach to stationarity.

4. Contrasting DV Functional and Relative Entropy

Whereas relative entropy diminishes monotonically in systems with detailed balance (reflecting the Second Law and linking to static fluctuation theory), the DV functional instead governs dynamical fluctuations—specifically, the statistics of time-averaged occupations. Notably, $I(\mu)$ cannot generally be expressed as a relative entropy outside the detailed balance context and does not capture informational distance. Rather, it quantifies the excess dynamical activity (or "reactive flux") required to enforce a non-typical occupation measure.

This distinction is critical for the analysis of nonequilibrium steady states (NESS), glassy dynamics, and kinetically constrained models, where entropy production falls short of describing the core mechanisms of relaxation and persistent dynamical activity.

5. Lyapunov-Like Role in Nonequilibrium Relaxation and Stress–Response

Monotonic decrease of the DV functional serves as a nonequilibrium analogue of a Lyapunov function: it assures that, under "stressed" conditions—such as application of external drives or imposition of nonequilibrium boundary conditions—systems relax in a way that the excess dynamical activity decays. This can be interpreted as a signature of monotone stress–response: after a system is driven away from stationarity (stressed), its escape rates or reactive transitions decrease in a systematic, non-oscillatory fashion until a new steady state is reached.

Physically, in open driven systems, the dynamical activity is closely tied to current-carrying structures and reactivity. A monotonic decay of $I(\mu_t)$ indicates that stress imposed by nonequilibrium forcing is dissipated in an ordered, Lyapunov-like fashion—distinct from entropy-driven relaxation but providing analogous convergence guarantees.

Furthermore, the gradient of the DV functional with respect to macroscopic parameters—interpreted as a nonequilibrium statistical force—suggests a structural framework for analyzing relaxation and response even in scenarios where standard thermodynamic potentials are either ill-defined or non-informative.

6. Implications and Applications

The monotone stress–response property formalized via the DV functional is particularly relevant in models where dynamical activity is a primary observable—such as glassy materials, systems far from equilibrium, or models with complex constraints. Its rigorous characterization enables:

• Identification of Lyapunov-like functionals for NESS that serve an analogous role to free energy in equilibrium,
• Systematic quantification of relaxation and stress-dissipation protocols,
• Analytical tools for exploring stability and convergence in Markovian and stochastic dynamical models where entropy-based methods are insufficient,
• Development of statistical force-based descriptions of system response in the absence of conventional thermodynamic potentials.

The results, while technically restricted to Markov jump processes with normal linear response at large times, set the groundwork for further exploration of monotonicity and Lyapunov structures in broader classes of nonequilibrium dynamics, including continuous-space diffusions, stochastic field theories, and systems with non-Markovian memory.

7. Summary Table

Property	Equilibrium (Detailed Balance)	Nonequilibrium (No Detailed Balance)
Lyapunov Functional	Relative Entropy (monotonic)	Donsker-Varadhan Functional (sometimes monotonic)
Observable	Static Fluctuations (occupation, entropy)	Dynamical Activity (escape rates, occupation-time fluctuations)
Monotonicity Condition	Automatic	Requires normal linear response / sector condition
Physical Picture	Relaxation of informational distance	Relaxation of excess dynamical activity
Signature of Relaxation	Decrease of relative entropy	Decrease of DV functional I(μₜ)

The essential insight is that monotonic decay of the Donsker–Varadhan functional, under suitable dynamical conditions, provides a rigorous indicator of monotone stress–response in nonequilibrium statistical mechanics, with direct implications for relaxation, control, and large deviation analysis in driven stochastic systems.