Physical Model and Domain of Validity

Introduces the physical model as a tool to delimit the validity domain of a description.

Conceptual context

Physics does not describe reality directly: it builds models. A physical model is a simplified representation of a phenomenon that, under certain conditions, allows useful quantitative predictions. The central question is not whether the model is "true" but under what conditions its predictions are sufficiently precise. That region of conditions is the $V$: the territory where the model keeps its promise. Outside it, $\delta_{sist}$ grows and predictions become unreliable.

🟢 Essential level

A physical model is like a road map: it is useful within the mapped area and at the level of detail it was designed for, but it does not show every stone on the road and it is not reliable outside its boundaries.

The $H$ are the tacit agreements we make when using the map: "we ignore relief, we assume roads are straight at this scale, we do not show minor paths." As long as those agreements hold, the map works and $\epsilon_r$ is high. When we cross outside the mapped zone or demand a detail the map lacks, $\delta_{sist}$ appears reproducibly and with a consistent direction. Recognizing when we are near the boundary of the $V$ is the core skill of a physical modeler.

For example, the frictionless free-fall model correctly predicts the fall time of a stone dropped from two meters, but fails completely for a feather, because the hypothesis of negligible air resistance is violated in that particular case, even though the mathematical equation is identical.

🔵 Formal level

Basic formulas

A physical model is defined by three inseparable components according to

Definition of physical model

where S is the idealized system, H the set of $H$ that allow treating S mathematically, and V the $V$ that bounds the conditions of application.

Operational validity is quantified through $\epsilon_r$ according to

Model validity condition

where $y_M$ is the value predicted by the model and $y_R$ is the experimentally measured value, both expressed in the units of the physical quantity under study. The model is valid when $\epsilon_r$ stays below the acceptable threshold for the application. That threshold depends on context: 1 % may be sufficient in standard mechanical engineering, while precision metrology may require 0.01 %.

Real student questions

Why can a model give "correct" results outside its domain? Sometimes $\delta_{sist}$ is small even when a $H$ is weakly violated. Partial violation produces moderate errors, not catastrophes. The real danger is when the violation is severe and the student does not detect it because they did not check the $V$.

How do I know whether the error is from the model or the experiment? Systematic $\delta_{sist}$ is reproducible and directional; it varies predictably with the system conditions. Random experimental error fluctuates around zero without a pattern. If repeating the measurement with better instruments does not reduce the error, it is systematic.

Is a more complex model always better? No. A more complex model has a broader $V$ but also higher computational and analytical cost. If the $\epsilon_r$ of the simple model is sufficient for the application, adding complexity without need is poor scientific practice.

Final synthesis

A physical model is worth its $V$, not its complexity. The $H$ are not weaknesses: they are the agreements that make the model useful. $\epsilon_r$ and $\delta_{sist}$ are the tools to know when those agreements remain valid. Recognizing the limits of the model is as important as applying it correctly within them.