Two-dimensional gas
Encyclopedia
A two-dimensional gas is a collection of N objects which are constrained to move in a planar or other two-dimension
al space in a gas
eous state. The objects can be: ideal gas
elements such as rigid disks
undergoing elastic collision
s; elementary particle
s, or any object in physics
which obeys laws of motion
. The concept of a two-dimensional gas is used either because:
While physicist
s have studied simple two body interactions on a plane for centuries, the attention given to the two-dimensional gas (having many bodies in motion) is a 20th century pursuit. Applications have led to better understanding of gas thermodynamics
, certain solid state
problems and several questions in quantum mechanics
.
in the early 1960s posed the question of whether the Maxwell-Boltzmann statistics and other thermodynamic laws could be derived from Newton
ian laws applied to multi-body systems rather than through the conventional methods of statistical mechanics
. While this question appears intractable from a three-dimensional closed form solution, the problem behaves differently in two-dimensional space. In particular an ideal two-dimensional gas was examined from the standpoint of relaxation time to equilibrium
velocity
distribution given several arbitrary initial conditions of the ideal gas. Relaxation time
s were shown to be very fast: on the order of mean free time
.
In 1996 a computational approach was taken to the classical mechanics non-equilibrium problem of heat flow
within a two-dimensional gas. This simulation work showed that for N>1500, good agreement with continuous systems is obtained.
While the principle of the cyclotron
to create a two-dimensional array of electron
s has existed since 1934, the tool was originally not really used to analyze interactions among the electrons (e.g. two-dimensional gas dynamics
). An early research investigation to explore a two-dimensional electron gas with respect to cyclotron resonance behavior and the de Haas-van Alphen effect
. The investigator was able to demonstrate for a two-dimensional gas, the de Haas-van Alphen oscillation
period is independent of the short-range electron interaction
s.
can exist in two dimensions. In the same work an experimental recommendation was made that could verify the hypothesis.
scanning microscope
to image a two dimensional benzene
gas layer in contact with a planar solid interface at 77 kelvins. The experimenters were able to observe benzene molecules moving freely as a two-dimensional gas on the surface of Cu(111), to which a planar monomolecular film of solid benzene adhered. Thus the scientists could witness the equilibrium of the gas in contact with its solid state, even by observing transient migration and phase transition of individual benzene molecules.
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al space in a gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...
eous state. The objects can be: ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
elements such as rigid disks
Disk (mathematics)
In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...
undergoing elastic collision
Elastic collision
An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter...
s; elementary particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
s, or any object in physics
Physics
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...
which obeys laws of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
. The concept of a two-dimensional gas is used either because:
- (a) the issue being studied actually takes place in two dimensions (as certain surface molecularMoleculeA 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...
phenomena); or, - (b) the two-dimensional form of the problem is more tractable than the analogous mathematicallyMathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
more complex three-dimensionDimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al problem.
While physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
s have studied simple two body interactions on a plane for centuries, the attention given to the two-dimensional gas (having many bodies in motion) is a 20th century pursuit. Applications have led to better understanding of gas thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
, certain solid state
Solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from...
problems and several questions in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
.
Classical mechanics
Research at Princeton UniversityPrinceton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
in the early 1960s posed the question of whether the Maxwell-Boltzmann statistics and other thermodynamic laws could be derived from Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
ian laws applied to multi-body systems rather than through the conventional methods of statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
. While this question appears intractable from a three-dimensional closed form solution, the problem behaves differently in two-dimensional space. In particular an ideal two-dimensional gas was examined from the standpoint of relaxation time to equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...
velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
distribution given several arbitrary initial conditions of the ideal gas. Relaxation time
Relaxation time
In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium.Each relaxation process can be characterized by a relaxation time τ...
s were shown to be very fast: on the order of mean free time
Mean free time
Molecules in a fluid constantly collide off each other. The mean free time of a molecule in a fluid is the average time between collisions. The mean free path of the molecule is the product of the average speed and the mean free time...
.
In 1996 a computational approach was taken to the classical mechanics non-equilibrium problem of heat flow
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...
within a two-dimensional gas. This simulation work showed that for N>1500, good agreement with continuous systems is obtained.
Electron gas
Cyclotron
In technology, a cyclotron is a type of particle accelerator. In physics, the cyclotron frequency or gyrofrequency is the frequency of a charged particle moving perpendicularly to the direction of a uniform magnetic field, i.e. a magnetic field of constant magnitude and direction...
to create a two-dimensional array of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s has existed since 1934, the tool was originally not really used to analyze interactions among the electrons (e.g. two-dimensional gas dynamics
Gas dynamics
Gas dynamics is a branch of fluid dynamics concerned with studying the motion of gases and its consequent effects. Gas dynamics combines the principles of fluid mechanics and thermodynamics...
). An early research investigation to explore a two-dimensional electron gas with respect to cyclotron resonance behavior and the de Haas-van Alphen effect
De Haas-van Alphen effect
The de Haas–van Alphen effect, often abbreviated to dHvA, is a quantum mechanical effect in which the magnetic moment of a pure metal crystal oscillates as the intensity of an applied magnetic field B is increased. Other quantities also oscillate, such as the resistivity , specific heat, and sound...
. The investigator was able to demonstrate for a two-dimensional gas, the de Haas-van Alphen oscillation
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
period is independent of the short-range electron interaction
Interaction
Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect...
s.
Later applications to Bose gas
In 1991 a theoretical proof was made that a Bose gasBose gas
An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics...
can exist in two dimensions. In the same work an experimental recommendation was made that could verify the hypothesis.
Experimental research with a molecular gas
Stranick used an ultrahigh vacuumVacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
scanning microscope
Microscope
A microscope is an instrument used to see objects that are too small for the naked eye. The science of investigating small objects using such an instrument is called microscopy...
to image a two dimensional benzene
Benzene
Benzene is an organic chemical compound. It is composed of 6 carbon atoms in a ring, with 1 hydrogen atom attached to each carbon atom, with the molecular formula C6H6....
gas layer in contact with a planar solid interface at 77 kelvins. The experimenters were able to observe benzene molecules moving freely as a two-dimensional gas on the surface of Cu(111), to which a planar monomolecular film of solid benzene adhered. Thus the scientists could witness the equilibrium of the gas in contact with its solid state, even by observing transient migration and phase transition of individual benzene molecules.
Implications for future research
A multiplicity of theoretical physics research directions exist for study via a two-dimensional gas. Examples of these are- Complex quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
phenomena, whose solutions may be more appropriate in a two-dimensional environment; - Studies of phase transitionPhase transitionA phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
s (e.g. meltingMelting pointThe melting point of a solid is the temperature at which it changes state from solid to liquid. At the melting point the solid and liquid phase exist in equilibrium. The melting point of a substance depends on pressure and is usually specified at standard atmospheric pressure...
phenomena at a planar surface); - Thin filmThin filmA thin film is a layer of material ranging from fractions of a nanometer to several micrometers in thickness. Electronic semiconductor devices and optical coatings are the main applications benefiting from thin film construction....
phenomena such as chemical vapor depositionChemical vapor depositionChemical vapor deposition is a chemical process used to produce high-purity, high-performance solid materials. The process is often used in the semiconductor industry to produce thin films. In a typical CVD process, the wafer is exposed to one or more volatile precursors, which react and/or...
; - Surface excitationsExcited stateExcitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited state....
of a solid.
See also
- Bose gasBose gasAn ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics...
- Fermi gasFermi gasA Fermi gas is an ensemble of a large number of fermions. Fermions, named after Enrico Fermi, are particles that obey Fermi–Dirac statistics. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density,...
- Melting pointMelting pointThe melting point of a solid is the temperature at which it changes state from solid to liquid. At the melting point the solid and liquid phase exist in equilibrium. The melting point of a substance depends on pressure and is usually specified at standard atmospheric pressure...
- Optical latticeOptical latticeAn optical lattice is formed by the interference of counter-propagating laser beams, creating a spatially periodic polarization pattern. The resulting periodic potential may trap neutral atoms via the Stark shift. Atoms are cooled and congregate in the locations of potential minima...
- Three body problem