Everything about Homogeneity Physics totally explained
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For other uses, see homogeneous.
In
physics, homogeneous mixtures are mixtures that have definite, consistent composition and properties. Particles are uniformly spread. For example, any amount of a given mixture has the same composition and properties. Examples are solutions and some alloys (but not all). A homogeneous mixture is a uniform mixture consisting of only one phase. Examples are gasoline and margarine.
Translation invariance
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Main article: translational invariance.
By
translation invariance one means, the independence of the (absolute) position, especially when referring to a law of physics, or to the evolution of a physical system.
Indeed, fundamental laws of physics shouldn't (explicitly) depend on the position in space, not only in view of the fact that this would make them quite useless. In some sense, this is also linked to the requirement that experiments should be
reproducible.
This principle is indeed true for all basic laws of mechanics (
Newton's law, ...), 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,...) which make the description of the evolution of the system depending on the position (
potential wells, ...). However, this only stems from the fact that the objects creating these external fields are not considered as ("dynamical") part of the system.
Translational invariance as described above is equivalent to
shift invariance in
system analysis, although here it's most commonly used in
linear systems, whereas in physics the distinction isn't usually made.
The notion of
isotropy, for properties independent of direction, isn't a consequence of homogeneity. For example, a uniform electric field (for example which has the same strength and the same direction in each point) would be compatible with homogeneity (in each point physics will be the same), but not with
isotropy, since the field singles out one "preferred" direction.
Consequences
In the
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 of having quantities of same units both sides. A valid equation in
physics must be homogeneous, since equality can't hold between quantities of different nature. This can be used to spot errors in formulae 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 don't combine to [mass]·[length]²/[time]², it can't be a valid expression for some energy.
However, if the equation is homogeneous, it doesn't necessarily mean the equation will be true, since it doesn't take into account numerical factors. For example,
E = m v² could be or couldn't be the correct formula for the energy of a particle of mass
m traveling at speed
v, and one can't 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.
Further Information
Get more info on 'Homogeneity Physics'.
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