Homogeneity (physics)
Encyclopedia
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 domain, when interacting with a directed radiation field (light, microwave frequencies, etc.) 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.
Another related definition is simply a substance that is uniform in composition.
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".
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 atom
s and molecule
s. 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
.
. From the perspective of 19th-century cosmology (and before), the universe
was infinite, unchanging, homogeneous, and therefore filled with star
s. 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 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 shift
ed, which weakens their apparent light and makes the night sky dark.
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 well
s, 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.
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
.
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.
The atomic number was also known as a proton number.
Nature (innate)
Nature is innate behavior , character or essence, especially of a human. This is a way of using the word nature which goes back to its earliest forms in Greek...
, 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 domain, when interacting with a directed radiation field (light, microwave frequencies, etc.) 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.
Another related definition is simply a substance that is uniform in composition.
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".
Context
The definition of homogeneous strongly depends on the context used. For example, a composite materialComposite material
Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...
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
Asphalt
Asphalt or , also known as bitumen, is a sticky, black and highly viscous liquid or semi-solid that is present in most crude petroleums and in some natural deposits, it is a substance classed as a pitch...
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 atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s and molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s. 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
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
; and homogeneity in time implies conservation of energy
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
.
Homogeneous 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 gold-like color. Steel is harder than iron and can even be made rust proof (stainless steel).Homogeneous cosmology
Homogeneity, in another context plays a role in cosmologyCosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...
. From the perspective of 19th-century cosmology (and before), the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
was infinite, unchanging, homogeneous, and therefore filled with star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s. 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 conundrum. The faulty premise, unknown in Olbers' time, was that the universe is not infinite, static, and homogeneous. The Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
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 shift
Red shift
-Science:* Redshift, the increase of wavelength of detected electromagnetic radiation with respect to the original wavelength of the emission* Red shift, an informal term for a bathochromic shift...
ed, which weakens their apparent light and makes the night sky dark.
Translation invariance
By translationTranslation (geometry)
In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
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 well
Potential well
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy because it is captured in the local minimum of a potential well...
s, 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
System analysis
System analysis in the field of electrical engineering characterizes electrical systems and their properties. System Analysis can be used to represent almost anything from population growth to audio speakers, electrical engineers often use it because of its direct relevance to many areas of their...
, although here it is most commonly used in linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
systems, whereas in physics the distinction is not usually made.
The notion of isotropy
Isotropy
Isotropy is uniformity in all orientations; it is derived from the Greek iso and tropos . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary...
, 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
Isotropy
Isotropy is uniformity in all orientations; it is derived from the Greek iso and tropos . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary...
, since the field singles out one "preferred" direction.
Consequences
In LagrangianLagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
formalism, homogeneity (in space) implies conservation of momentum, and homogeneity in time implies conservation of energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
. 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
Noether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
.
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 physicsPhysics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
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
Speed
In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...
, units must always combine to [length]/[time]; if one is calculating an energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
, 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
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
, 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
Dimensional analysis
In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
.
Theoretical physicists tend to express everything in natural units
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...
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.
The atomic number was also known as a proton number.
Further reading
- Landau - Lifschitz: "Theoretical Physics - I. Mechanics", Chapter One.