# Is matter continuous or discrete

## Continuum (physics)

A continuum (Latin: continuus for "connected"[1][2], Plural: the continua[1][2] and continuums[1][2]) denotes something that follows one another continuously (without gaps).[3] In physics, a quantity is continuous if, with every possible value, all values ​​are possible in a sufficiently small environment. Such a set of values ​​is called continuum. In contrast to this, a value is discrete if, apart from it, no other value from a sufficiently small area is possible (see also the grid model). Nevertheless, discrete values ​​are often calculated as a continuum and vice versa, because the error from the approximation is often significantly smaller than the numerical error of the calculation.

Mathematically, the terms open set, connected set or continuous function correspond to physical language usage, depending on the context.

For example, the binding energies of the hydrogen atom

discrete, the energies of the ionized electron-proton pair, on the other hand, that for large distances with pulses and run out, non-negative and continuous,

Likewise, the places and times that a particle can pass through are continuous, while the places at which the atoms (more precisely their ion cores) are in a crystal lattice are discrete.

Since physical approximations and measured variables are prone to errors, it can depend on the accuracy of the observation or the measurement whether a variable is viewed as discrete or continuous. For example, the solar wind is considered continuous, whereas cosmic rays are considered discrete.

### Continuum in mechanics

Continuum mechanics often uses the model of a continuum. In the case of a crack that can occur in a continuum or between two continuums, only compressive forces, but not tensile forces, can generally be transmitted; Due to the toothing / friction, shear stresses can often also be transferred (to a lesser extent).

In research, multiscale models of continuum mechanics are often used, which assume the existence of a representative volume element which is so small that it has constant stresses; this is often already achieved with a size 3–5 times smaller than the object under consideration.

The continuum mechanics is described by Cauchy's stress tensor with 6 independent components.

### Cosserat continuum

The Cosserat continuum, named after the brothers Eugène and François Cosserat,[4] based on the micropolar[5][6] Elasticity and goes beyond the classical continuum theory: here one assumes a non-symmetrical stress tensor with 9 independent components and a moment stress tensor[4] also with 9 components.[5]

The Cosserat continuum is used when the inhomogeneities are of the same order of magnitude as the dimensions of the structure.[5]