Lyapunov function gradient: ∂V/∂Ω = 2(Ω - Ω*) · @brief Lyapunov function gradient: ∂V/∂Ω

Read the equation

This relation defines @brief Lyapunov function gradient: ∂V/∂Ω in the source section “Lyapunov function gradient: ∂V/∂Ω = 2(Ω - Ω*) · @brief Lyapunov function gradient: ∂V/∂Ω”. Uses a derivative or gradient to describe a rate of change or a direction of variation. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive meaning to them.

Terms & definitions

The recovered passage does not supply a separate symbol table. Read the source context for the notation used in this version; familiar letters may carry different meanings in other builds.

How it works

1. Read @brief Lyapunov function gradient: ∂V/∂Ω as the quantity being defined or updated by this relation.

2. Uses a derivative or gradient to describe a rate of change or a direction of variation.

3. Use this version’s variable definitions and update order. The same symbol may be assigned a different meaning in another build.

Assumptions & limits

• This source expression was recovered from the pinned code revision. Its presence does not establish that the associated program was built, run, or scientifically validated.

Original context

    double diff = x - 1.0;
    double abs_diff = std::abs(diff);
    double denom = (1.0 + abs_diff) * (1.0 + abs_diff);
    return -std::copysign(1.0, diff) / denom;
}

/**
 * @brief Lyapunov function gradient: ∂V/∂Ω = 2(Ω - Ω*)
 */
inline double lyapunov_gradient(double omega, double omega_target) {
    return 2.0 * (omega - omega_target);
}

/**
 * @brief Free energy gradient: ∂F/∂x = 2(x - μ) - T·∂S/∂x
 */
inline double free_energy_gradient(double x, double mu, double temperature, double entropy_grad) {
    return 2.0 * (x - mu) - temperature * entropy_grad;

@brief Lyapunov function gradient: ∂V/∂Ω = 2(Ω - Ω*)

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