processespotentialization

processespotentialization is a theoretical framework in which a collection of dynamical processes is represented by an associated potential function, called the potential, so that the evolution tends to reduce potential over time. The idea extends ideas from potential theory and Lyapunov analysis to systems where multiple processes interact.

Constructing a potentialization involves finding a scalar function V(x,t) whose decrease along trajectories captures the system's drift. In deterministic systems dx/dt = f(x,t), one seeks V such that dV/dt ≤ 0 along solutions. In gradient-form conditions, f = -M(x,t)∇V for some positive semidefinite mobility M. For stochastic systems, stationary distributions pi ∝ e^{-βV} lead to an effective potential landscape V.

Applications of this framework appear in stability analysis, optimization, and the design of energy-based models in machine learning, where the potential guides learning dynamics; planning and control, where gradient flows yield efficient trajectories; and evolutionary dynamics, where potential landscapes summarize selective pressures.

Limitations include that not all systems admit a global potential function, and the choice of metric or mobility M is not unique. Non-conservative forces can complicate the construction, and high-dimensional landscapes can be difficult to interpret. The framework is typically most useful when a gradient-structure or near-gradient structure can be identified.

Relation to other concepts: processespotentialization shares goals with Lyapunov functions, gradient flows, and energy landscapes, while remaining distinct from standard potential theory in physics, though it is linked through variational principles and energy-based modeling.