Understanding Physical, Theoretical, and Mathematical Models: A Comprehensive Guide

A model is a structure that approximates or simulates the object or process under study. Models can be broadly divided into two categories: physical (concrete) models and theoretical (abstract) models.

Physical Models and Theoretical Models

Physical models can be further divided into simulation models and scale (reduced‑size) models.

Simulation models use other phenomena or processes to represent the studied phenomenon, with the model's properties standing in for the original ones. For example, electric current can simulate heat flow, fluid flow can simulate traffic, etc.

Simulation models can be classified as direct or indirect. Direct simulation means a one‑to‑one correspondence between variables of the model and the original phenomenon, such as using an electrical circuit to mimic a thermal conduction system. Indirect simulation lacks a direct correspondence but can still solve complex problems cleverly, as illustrated by the Steiner problem for minimizing total road length between construction sites.

Another example uses nails placed proportionally on a board to represent construction sites, immerses the board in soap solution, and then lifts it to let soap film form minimal‑length connections, solving the Steiner problem physically.

Scale models are reduced or enlarged replicas of real objects, such as aircraft models in wind tunnels or ship models in water tanks. They are valuable when precise mathematical models are difficult to build or when the real object is too large or small for direct experimentation.

Scale models share many properties of the original, including appearance, color, and sometimes function. However, they do not possess all properties of the full‑size object, which can make experiments more convenient but also introduces limitations and requires similarity theory to interpret results.

Theoretical models are abstract representations—either of physical objects, phenomena, or processes—expressed as a set of rules or laws. For instance, civil engineers may use a theoretical model of beam deflection under load, expressed by a formula involving beam length, load, material elastic modulus, and moment of inertia.

Beam length

Load

Elastic modulus of the material

Moment of inertia related to the cross‑section

Mathematical Models

Mathematical models are abstract, simplified, and approximate mathematical structures that describe a portion of the real world for a specific purpose.

Specifically, a mathematical model uses symbols, equations, inequalities, charts, or diagrams to represent the features and internal relationships of an objective phenomenon.

Mathematical models can be viewed broadly as abstractions of reality that approximate certain mathematical attributes of objects, such as Euler’s solution to the Seven Bridges of Königsberg problem.

Example 1: The gravitational attraction between two bodies can be expressed by Newton’s universal gravitation formula.

This formula constitutes a mathematical model of the attractive force.

Example 2: A linear spring’s deformation under force is described by Hooke’s law.

This equation is the mathematical model of the spring‑force relationship.

A system is a set of elements connected in a certain way; its mathematical model typically relates inputs and outputs, possibly including state variables. Different levels of approximation yield different models for the same system.

Example 3: For a mass subjected to a force, Newton’s second law (F = ma) provides a mathematical model of motion, with refinements for air resistance or relativistic effects as needed.

Example 4: Natural numbers form a simple mathematical model abstracted from counting real objects, supporting arithmetic operations and number theory.

Example 5: Euclidean geometry is a mathematical model that organizes points, lines, and shapes into a logical system based on axioms and theorems.

Mathematical models can be classified into formula models and graphical models. Formula models use equations (functions, algebraic equations, differential equations, etc.) to describe system characteristics, while graphical models employ diagrams such as flowcharts, state diagrams, or maps to represent structures and relationships.

In summary, any model provides an approximate description of reality, focusing on selected features relevant to the purpose, while ignoring others; the choice of model depends on the intended application and required level of detail.