Empirical relationship

Phenomenological model

A phenomenological model is a scientific model derived from empirical relationships. It forgoes any attempt to explain why its variables interact the way they do, and simply attempts to describe how they relate, with the assumption that the relationship extends, at least within an acceptable error margin, past the measured values. Although not directly derived from first principles, an empirical observation -based model will only be able provide reliable and accurate correspondence to the real world inasmuch as it is consistent with the underlying fundamental theory behind the phenomena being described. Regression analysis is sometimes used to create statistical models that serve as phenomenological models.

Applicability

Phenomenological models have been characterized as being completely independent of theories, though many phenomenological models, while failing to be derivable from a theory, incorporate principles and laws associated with theories. The liquid drop model of the atomic nucleus, for instance, portrays the nucleus as a liquid drop and describes it as having several properties (surface tension and charge, among others) originating in different theories (hydrodynamics and electrodynamics, respectively). Certain paradigmatic features of these theories—though usually not the complete theory—are then used to determine both the static and dynamical properties of the nucleus.

Approximations

Some empirical relationships are merely approximations, often equivalent to the first few terms of the Taylor series of an analytical solution describing a phenomenon. Other relationships only hold under certain specific conditions, reducing them to special cases of more general relationship. Some approximations, in particular phenomenological models, may even contradict theory; they are employed because they are more mathematically tractable than some theories, and are able to yield results.