Homogeneity

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Homogeneity is defined as the quality or state of being homogeneous. Homogenous means - of the same or similar nature. It also means having a uniform structure throughout. For instance, a uniform electric field (i.e., which has the same strength and the same direction at each point) would be compatible with homogeneity (at each point physics will be the same). A material constructed with different constituents can be described as effectively homogenous in the electromagnetic domain, when interacting with a directed raditaion field (light, microwave frequencies, etc.)^[1][2] In physics homogeneous usually means describing a material or system that has the same properties in any direction; i.e. uniform without irregularities. In physics, it also describes a substance or an object whose properties do not vary with position. For example, an object of uniform density is sometimes described as homogeneous^[3][4]

Mathematically, homogeneity has the connotation of invariance, as all components of the equation have the same degree of value whether or not each of these components are scaled to different values, for example, by multiplication or addition. Cumulative distribution fits this description. "The state of having identical cumulative distribution function or values".^[1][2]

Context

The definition of homogenous strongly depends on the context used. For example, a composite material is made up of different individual materials, known as "constituents" of the material, but may be defined as a homogenous material when assigned a function. For example, asphalt paves our roads, but is a composite material consisting of asphalt binder and mineral aggregate, and then laid down in layers and compacted.

In another context any material is actually made up of atoms and molecules. At this level, then, any material is not homogenous. However, in our normal everyday world, a block of wood, a plane of glass, or a sheet of metal is described as wood, glass, or stainless steel. In other words, these are described as a homgenous material.

Dimensional homogeneity is the quality of an equation having quantities of same units on both sides. Homogeneity (in space) implies conservation of momentum, and homogeneity in time implies conservation of energy.

Homogenous alloy

In the context of composite metals is an alloy. A blend of a metal with one or more metallic or nonmetallic materials is an alloy. The components of an alloy do not combine chemically but, rather, are very finely mixed. An alloy might be homogeneous or might contain small particles of components that can be viewed with a microscope. Brass is an example of an alloy, being a homogeneous mixture of copper and zinc. Another example is steel, which is an alloy of iron with carbon and possibly other metals. The purpose of alloying is to produce desired properties in a metal that naturally lacks them. Brass, for example, is harder than copper and has a more goldlike color. Steel is harder than iron and can even be made rust proof (stainless steel).^[5]

Homogenous cosmology

Homogeneity, in another context plays a role in cosmology. From the perspective of 19th-century cosmology (and before), the universe was infinite, unchanging, homogeneous, and therefore filled with stars. In 1826, this being true according to the German astronomer Heinrich Olbers, then the entire night sky should be filled with light and as bright as daytime, but it is dark. He presented a technical paper in 1826 that attempted to answer this conondrum. The faulty premise, unknown in Olbers' time, was that the universe is not infinite, static, and homogeneous. The Big Bang cosmology replaced this model (expanding, finite, and inhomogeneous universe). However, modern astronomers supply reasonable explanations to answer this question. One of at least several explanations is that distant stars and galaxies are red shifted, which weakens their apparent light and makes the night sky dark.^[6]

Translation invariance

By translation invariance, one means independence of (absolute) position, especially when referring to a law of physics, or to the evolution of a physical system.

Fundamental laws of physics should not (explicitly) depend on position in space. That would make them quite useless. In some sense, this is also linked to the requirement that experiments should be reproducible. This principle is true for all laws of mechanics (Newton's law, etc.), electrodynamics, quantum mechanics, etc.

In practice, this principle is usually violated, since one studies only a small subsystem of the universe, which of course "feels" the influence of rest of the universe. This situation gives rise to "external fields" (electric, magnetic, gravitational, etc.) which make the description of the evolution of the system depending on the position (potential wells, etc.). This only stems from the fact that the objects creating these external fields are not considered as (a "dynamical") part of the system.

Translational invariance as described above is equivalent to shift invariance in system analysis, although here it is most commonly used in linear systems, whereas in physics the distinction is not usually made.

The notion of isotropy, for properties independent of direction, is not a consequence of homogeneity. For example, a uniform electric field (i.e., which has the same strength and the same direction at each point) would be compatible with homogeneity (at each point physics will be the same), but not with isotropy, since the field singles out one "preferred" direction.

Consequences

In Lagrangian formalism, homogeneity (in space) implies conservation of momentum, and homogeneity in time implies conservation of energy. This is shown, using variational calculus, in standard textbooks like the classical reference [Landau & Lifshitz] cited below. This is a particular application of Noether's theorem.

Dimensional homogeneity

As said in the introduction, dimensional homogeneity is the quality of an equation having quantities of same units on both sides. A valid equation in physics must be homogeneous, since equality cannot apply between quantities of different nature. This can be used to spot errors in formula or calculations. For example, if one is calculating a speed, units must always combine to [length]/[time]; if one is calculating an energy, units must always combine to [mass]·[length]²/[time]², etc. For example, the following formulae could be valid expressions for some energy:

if m is a mass, v and c are velocities, p is a momentum, h is Planck's constant, λ a length. On the other hand, if the units of the right hand side do not combine to [mass]·[length]²/[time]², it cannot be a valid expression for some energy.

Being homogeneous does not necessarily mean the equation will be true, since it does not take into account numerical factors. For example, E = m·v² could be or could not be the correct formula for the energy of a particle of mass m traveling at speed v, and one cannot know if h·c/λ should be divided or multiplied by 2π.

Nevertheless, this is a very powerful tool in finding characteristic units of a given problem, see dimensional analysis.

Theoretical physicists tend to express everything in natural units given by constants of nature, for example by taking c = ħ = k = 1; once this is done, one partly loses the possibility of the above checking.

See also

• Translational invariance
• Noether's theorem
• Dimensional analysis

References

Further reading

• Landau - Lifschitz: "Theoretical Physics - I. Mechanics", Chapter One.