Homogeneity
(physics)

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In general, homogeneity is defined as the quality or state of being
homogeneous
(of the same or similar nature, from Greek ὀμός
meaning 'same'). It also means having a uniform structure throughout. For
instance, a uniform electric field (which has
the same strength and the same direction at each point) would be compatible with
homogeneity (all points experience the same physics). A material constructed
with different constituents can be described as effectively homogeneous in the
electromagnetic
materials domain, when interacting with a directed radiation
field (light, microwave frequencies, etc.)[1][2]
In physics, homogeneous usually means describing a material or system that has
the same properties at every point of the space; in other words, 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]
Another related definition is simply a substance that is uniform in
composition.[5]

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]

Contents

1 Context

1.1 Homogeneous
alloy

1.2 Homogeneous
cosmology

2 Translation
invariance

2.1 Consequences

3 Dimensional
homogeneity

4 See also

5 References

6 Further
reading

Context[edit]

The definition of homogeneous 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 homogeneous 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, a material is not homogeneous in so far as it composed of
atoms and molecules.
However, at the normal level of our everyday world, a block of wood, a pane of
glass, or a sheet of metal is described as wood, glass, or stainless steel. In
other words, these are each described as a homogeneous material.

A few other instances of context are: Dimensional homogeneity (see
below) 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.

Homogeneous alloy[edit]

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 gold-like color. Steel is harder than iron and
can even be made rust proof (stainless steel).[6]

Homogeneous
cosmology[edit]

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. This problem became known as Olbers' paradox. He
presented a technical paper in 1826 that attempted to answer this conundrum. 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.[7]
However, the weakening is not sufficient to actually explain Olbers' paradox.
Many cosmologist thing that the fact that the Universe is finite in time, that
is that the Universe has not been around forever, is the solution to the
pardox.[citation
needed] The fact that the night sky is dark is thus an
indication for the Big Bang.

Translation
invariance[edit]

Main article: translational
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 laws, 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[edit]

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[edit]

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[edit]

Translational
invariance

References[edit]

^ a
b
Homogeneity. Merriam-webster.com

^ a
b
Homogeneous. Merriam-webster.com

^
Rennie,
Richard, Science Online (2003.). "Homogeneous (physics)". The Facts On File Dictionary of
Atomic and Nuclear Physics. "Describing a material or system that has the same
properties in any direction; i.e. uniform without irregularities."  (accessed November 16,
2009).

^
Tanton, James. "homogeneous." Encyclopedia of
Mathematics. New York: Facts On File, Inc., 2005. Science Online. Facts On
File, Inc. "A polynomial in several variables p(x,y,z,…) is called homogeneous
[...] more generally, a function of several variables f(x,y,z,…) is
homogeneous [...] Identifying homogeneous functions can be helpful in solving
differential equations [and] any formula that represents the mean of a set of
numbers is required to be homogeneous. In physics, the term homogeneous
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." James. homogeneous (math). (accessed: 2009-11-16)

^
"Kristi Lew,
"homogenous"". Science Online. Database. Facts On File, Inc. 2008.
p. 1.  |accessdate= requires
|url= (help)

^
Rosen, Joe. "Alloy." Encyclopedia of Physics. New
York: Facts On File, Inc., 2004. Science Online. Facts On File, Inc. accessed
2009-11-16

^
Todd, Deborah, and Joseph A. Angelo Jr. "Olbers,
Heinrich Wilhelm Matthäus." A to Z of Scientists in Space and Astronomy.
New York: Facts on File, Inc., 2005. Science Online. Facts On File, Inc. Olbers, Heinrich Wilhelm Matthäus (accessed
2009-11-16)

Further reading[edit]

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

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