Energy Thermostats

📐 Definition

Given a field or state $u$ with energy $E(u)$ and target $E_{\text{target}} > 0$ , apply a thermostat scaling

u \leftarrow u \sqrt{\frac{E_{\text{target}}}{E(u)}}

to enforce the desired energy level. Variants add relaxation toward $E_{\text{target}}$ over multiple steps.

Domain and Codomain

Defined for states with positive finite energy $E(u)$ (e.g., $L^{2}$ or kinetic energy). Output remains in the same state space with rescaled magnitude.

⚙️ Key Properties

Preserves phase or direction of $u$ while adjusting magnitude. Instantaneously fixes energy in one step; relaxed versions converge geometrically toward the target.

If $E(u)=E_{\text{target}}$ , scaling is unity. When $E(u)$ is near zero, safeguards are required to avoid division blow-up (e.g., floor on $E(u)$ ).

🎯 Special Cases and Limits

$E(u)=E_{\text{target}}$ leaves $u$ unchanged.
$E(u)\approx 0$ requires regularization to avoid division blow-up.

Energy norms define $E(u)$ ; normalization is the unit-energy special case; gain control offers continuous compression instead of exact enforcement.

Usage in Oakfield

Oakfield implements energy/thermostat behavior as a dedicated operator:

thermostat operator regulates a field toward a target mean energy E_target = mean(|u|^2) with multiple modes (soft_lambda, add, mult).
Complex support uses $|u|^2$ and applies updates radially (ADD) or proportional to $u$ (MULT), preserving phase away from the origin.
Optional memory target: ADD/MULT modes can use a per-point target memory field M[i] instead of a global E_target.
Stability knobs include smoothing and clamp parameters for regulated scalars (e.g. lambda_smooth, min/max bounds).

Historical Foundations

📜 Thermostats and Controlled Energy

Thermostat-like rescalings appear in numerical dynamics as practical mechanisms to maintain or regulate energy-like invariants or target energy levels.

🌍 Modern Perspective

They are used as stabilization and constraint enforcement steps in iterative solvers and time-stepping pipelines.

📚 References

Hairer, Lubich, Wanner, Geometric Numerical Integration