The linear expansivity of a metal

Linear expansivity is a concept in the field of dynamical systems, which describes how the distance between two points evolving under a dynamical system changes over time. Specifically, a map (a function that maps elements from a set to itself) is called linearly expansive if there is a constant ε > 0 such that for any two distinct points x and y in the set, the distance between f(x) and f(y) is at least ε times the distance between x and y after some number of iterations n, provided n is sufficiently large.

Mathematically, this can be expressed as:

|f^n(x) - f^n(y)| ≥ ε |x - y|

where f is the map, x and y are points in the set, and ||.|| denotes the distance between the points. The constant ε is called the expansivity constant.

Linear expansivity implies that the map stretches the space uniformly along certain directions, without any contraction. This property is particularly useful in the study of hyperbolic dynamics, which often exhibit sensitive dependence on initial conditions, meaning that small changes in the starting point can lead to exponentially diverging trajectories over time. Linearly expansive maps are a type of uniformly hyperbolic system, which is a stronger form of chaos than simple sensitivity to initial conditions.

The concept of expansivity is closely related to the notion of topological mixing and ergodicity in dynamical systems. Topologically mixing systems have the property that given any two non-empty open sets, their images under the map will intersect after some finite number of iterations, while ergodic systems exhibit a form of statistical mixing, meaning that the time average of a function along a trajectory equals the space average of that function almost everywhere. Linear expansivity often implies both topological mixing and ergodicity, making it a strong form of chaotic behavior.

Examples of linearly expansive maps include the expanding circle map and the shift map on a symbolic space with two symbols (like the binary or ternary expansion of real numbers), where each symbol represents a stretching or expansion by a factor of 2 or 3, respectively. These maps are simple to define, yet they exhibit complex dynamics that can be studied using various mathematical tools from dynamical systems theory.


Linear expansivity, also known as linear coefficient of thermal expansion, is a measure of the fractional change in length of a material per degree of temperature change. For a metal, this property is crucial in understanding how its dimensions will change with temperature. The linear expansivity is typically denoted by the Greek letter alpha (α). It is defined as the ratio of the change in length (ΔL) to the original length (L₀) per degree of temperature change (ΔT), usually measured in Kelvins (K) or Celsius (°C):

α = (ΔL / L₀) / ΔT

The linear expansivity is a constant for a material at a given temperature range and is expressed in units of 1/K or 1/°C.

To find the linear expansivity of a metal, you can use the following steps:

1. Measure the original length of the metal sample (L₀) at a known temperature (usually room temperature or the temperature at which the material is expected to be used).
2. Heat or cool the metal sample by a known amount of temperature (ΔT) without applying any external forces or constraints that might prevent free expansion.
3. Measure the new length of the metal sample (ΔL) after the temperature change.
4. Calculate the linear expansivity (α) using the formula above.

The linear expansivity can also be determined from the material's coefficients of thermal expansion, which are often provided in material properties tables or databases. The coefficients of thermal expansion are generally determined experimentally by measuring the dimensional changes of a material as its temperature is varied.

Different metals have different linear expansivities due to variations in their atomic structures and bonding. For example, aluminum has a higher linear expansivity compared to steel, which means it will expand more for the same temperature increase. This property must be considered in engineering applications where precise measurements, thermal stresses, and material compatibility are important.