Two-dimensional Gas
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A two-dimensional gas is a collection of objects constrained to move in a planar or other
two-dimensional space In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as s ...
in a
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
eous state. The objects can be: classical
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
elements such as rigid disks undergoing
elastic collision In physics, an elastic collision is an encounter (collision) between two bodies in which the total kinetic energy of the two bodies remains the same. In an ideal, perfectly elastic collision, there is no net conversion of kinetic energy into o ...
s;
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, an ...
s, or any ensemble of individual objects in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
which obeys laws of motion without binding interactions. The concept of a two-dimensional gas is used either because:
    molecular A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
phenomena); or,
  • the two-dimensional form of the problem is more tractable than the analogous mathematically more complex three-
    dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
    al problem.
    • While
    physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate caus ...
    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
    superconductivity Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
    , gas
    thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
    , certain solid state problems and several questions in
    quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
    .


    Classical mechanics

    Research at
    Princeton University Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...
    in the early 1960s posed the question of whether the
    Maxwell–Boltzmann statistics In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of Classical physics, classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the ...
    and other thermodynamic laws could be derived from
    Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...
    ian laws applied to multi-body systems rather than through the conventional methods of
    statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
    . While this question appears intractable from a three-dimensional
    closed form solution In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th root ...
    , 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 Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
    distribution given several arbitrary initial conditions of the ideal gas.
    Relaxation time In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium. Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time ' ...
    s were shown to be very fast: on the order of
    mean free time Molecules in a fluid constantly collide with each other. The mean free time for 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. These conce ...
    . In 1996 a computational approach was taken to the classical mechanics non-equilibrium problem of
    heat flow Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
    within a two-dimensional gas. This simulation work showed that for N>1500, good agreement with continuous systems is obtained.


    Electron gas

    While the principle of the
    cyclotron A cyclotron is a type of particle accelerator invented by Ernest O. Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: Janu ...
    to create a two-dimensional array of
    electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
    s has existed since 1934, the tool was originally not really used to analyze interactions among the electrons (e.g. two-dimensional
    gas dynamics Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the r ...
    ). An early research investigation explored
    cyclotron resonance Cyclotron resonance describes the interaction of external forces with charged particles experiencing a magnetic field, thus already moving on a circular path. It is named after the cyclotron, a cyclic particle accelerator that utilizes an oscillati ...
    behavior and the de Haas–van Alphen effect in a two-dimensional electron gas. The investigator was able to demonstrate that for a two-dimensional gas, the de Haas–van Alphen oscillation period is independent of the short-range electron interactions.


    Later applications to Bose gas

    In 1991 a theoretical proof was made that a
    Bose gas An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and abide by Bose–Einstein statistics. The statistical mechanics of bosons were develo ...
    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

    In general, 2D molecular gases are experimentally observed on weakly interacting surfaces such as metals,
    graphene Graphene () is an allotrope of carbon consisting of a single layer of atoms arranged in a hexagonal lattice nanostructure.
    etc. at a non-cryogenic temperature and a low surface coverage. As a direct observation of individual molecules is not possible due to fast diffusion of molecules on a surface, experiments are either indirect (observing an interaction of a 2D gas with surroundings, e.g. condensation of a 2D gas) or integral (measuring integral properties of 2D gases, e.g. by diffraction methods). An example of the indirect observation of a 2D gas is the study of Stranick et al. who used a
    scanning tunnelling microscope A scanning tunneling microscope (STM) is a type of microscope used for imaging surfaces at the atomic level. Its development in 1981 earned its inventors, Gerd Binnig and Heinrich Rohrer, then at IBM Zürich, the Nobel Prize in Physics in 1986. ...
    in ultrahigh vacuum (UHV) to image an interaction of a two-dimensional
    benzene Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen atoms, ...
    gas layer in contact with a planar solid interface at 77
    kelvins The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
    . The experimenters were able to observe mobile benzene molecules 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. Integral methods that are able to characterize a 2D gas usually fall into a category of
    diffraction Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
    (see for example study of Kroger et al.). The exception is the work of Matvija et al. who used a
    scanning tunneling microscope A scanning tunneling microscope (STM) is a type of microscope used for imaging surfaces at the atomic level. Its development in 1981 earned its inventors, Gerd Binnig and Heinrich Rohrer, then at IBM Zürich, the Nobel Prize in Physics in 1986. ...
    to directly visualize a local time-averaged density of molecules on a surface. This method is of special importance as it provides an opportunity to probe local properties of 2D gases; for instance it enables to directly visualize a
    pair correlation function In statistical mechanics, the radial distribution function, (or pair correlation function) g(r) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle. If ...
    of a 2D molecular gas in a real space. If the surface coverage of adsorbates is increased, a 2D liquid is formed, followed by a 2D solid. It was shown that the transition from a 2D gas to a 2D solid state can be controlled by a
    scanning tunneling microscope A scanning tunneling microscope (STM) is a type of microscope used for imaging surfaces at the atomic level. Its development in 1981 earned its inventors, Gerd Binnig and Heinrich Rohrer, then at IBM Zürich, the Nobel Prize in Physics in 1986. ...
    which can affect the local density of molecules via an electric field.{{Cite journal, last1=Matvija, first1=Peter, last2=Rozbořil, first2=Filip, last3=Sobotík, first3=Pavel, last4=Ošťádal, first4=Ivan, last5=Pieczyrak, first5=Barbara, last6=Jurczyszyn, first6=Leszek, last7=Kocán, first7=Pavel, title=Electric-field-controlled phase transition in a 2D molecular layer, journal=Scientific Reports, volume=7, issue=1, pages=7357, doi=10.1038/s41598-017-07277-7, pmid=28779091, bibcode=2017NatSR...7.7357M, year=2017, pmc=5544747


    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 mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
    phenomena, whose solutions may be more appropriate in a two-dimensional environment; *Studies of
    phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
    s (e.g.
    melting Melting, or fusion, is a physical process that results in the phase transition of a substance from a solid to a liquid. This occurs when the internal energy of the solid increases, typically by the application of heat or pressure, which incre ...
    phenomena at a
    planar surface In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as su ...
    ); *
    Thin film A thin film is a layer of material ranging from fractions of a nanometer (monolayer) to several micrometers in thickness. The controlled synthesis of materials as thin films (a process referred to as deposition) is a fundamental step in many ap ...
    phenomena such as
    chemical vapor deposition Chemical vapor deposition (CVD) is a vacuum deposition method used to produce high quality, and high-performance, solid materials. The process is often used in the semiconductor industry to produce thin films. In typical CVD, the wafer (substra ...
    ; *Surface excitations of a solid.


    See also

    *
    Bose gas An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and abide by Bose–Einstein statistics. The statistical mechanics of bosons were develo ...
    *
    Fermi gas An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. ...
    *
    Melting point The melting point (or, rarely, liquefaction point) of a substance 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 ...
    *
    Optical lattice An 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 congregat ...
    *
    Three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...


    References


    External links


    Riemann problems for a two-dimensional gas
    Gases Non-equilibrium thermodynamics Statistical mechanics