Potentialset

Potentialset is a term used in mathematical modeling to denote a subset of a system’s state space that is considered reachable, feasible, or likely under a given potential or energy criterion. The concept is used across optimization, physics-inspired modeling, and machine learning to focus analysis on promising configurations.

Formal definition can vary by context. Let X be the state space and V: X → R be a potential function that assigns a scalar value to each configuration. For a threshold t ∈ R, the potentialset at t is P_t = { x ∈ X | V(x) ≤ t }. Other common definitions use alternative criteria such as low-gradient regions, e.g., P = { x ∈ X | ||∇V(x)|| ≤ ε } for some ε > 0, or regions with probabilistic weight above a given level in a Boltzmann-type model. The family of potentialsets defined by different thresholds or criteria reflects the landscape of the potential function.

Properties of a potentialset depend on the chosen V and the defining criterion. A potentialset can be empty or nonempty, finite or infinite, and its shape is shaped by the geometry of V and the threshold. Different V or thresholds can yield nested or disjoint potentialsets, and they may change as the system evolves or as parameters are adjusted.

Applications often involve using potentialsets to guide search, sampling, or decision-making. In optimization and search, potentialsets highlight low-energy or favorable regions to explore first. In robotics and physics, they correspond to feasible directions or states under an energy landscape.

Notes: the term is not universally standardized and may be defined variably in literature. See also potential energy, energy landscape, feasible set, and state space.