View source for Two-dimensional gas

A '''two-dimensional gas''' is a collection of objects constrained to move in a planar or other [[Plane (mathematics)|two-dimensional space]] in a [[Physics:Gas|gas]]eous state. The objects can be: classical [[Physics:Ideal gas|ideal gas]] elements such as rigid [[Disk (mathematics)|disks]] undergoing [[Physics:Elastic collision|elastic collision]]s; [[Physics:Elementary particle|elementary particle]]s, or any ensemble of individual objects in [[Physics:Physics|physics]] which obeys [[Physics:Newton's laws of motion|laws of motion]] without binding interactions. The concept of a two-dimensional gas is used either because:
<ol style="list-style-type:lower-alpha;>
<li> the issue being studied actually takes place in two dimensions (as certain surface [[Physics:Molecule|molecular]] phenomena); or, </li>
<li> the two-dimensional form of the problem is more tractable than the analogous [[Mathematics|mathematically]] more complex three-[[Physics:Dimension|dimension]]al problem.</li>
</ol>

While [[Physicist|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 [[Physics:High-temperature superconductivity|superconductivity]],<ref>{{cite journal | last1 = Feld | display-authors=0 | year = 2011| title = Observation of a pairing pseudogap in a two-dimensional gas | journal = Nature | volume = 480 | issue = 7375 | pages = 75–78 | doi=10.1038/nature10627| pmid = 22129727 | arxiv = 1110.2418| bibcode = 2011Natur.480...75F | s2cid=4425050 }}</ref> gas [[Physics:Thermodynamics|thermodynamics]], certain [[Physics:Solid-state physics|solid state]] problems and several questions in [[Physics:Quantum mechanics|quantum mechanics]].

==Classical mechanics==

[[Image:Elastischer stoß 2D.gif|frame|Two-dimensional elastic collision|alt=]]

Research at [[Organization:Princeton University|Princeton University]] in the early 1960s<ref>C.M.Hogan, ''Non-equilibrium statistical mechanics of a two-dimensional gas'', Dissertation, Princeton University, Department of Physics, May 4, 1964</ref> posed the question of whether the [[Physics:Maxwell–Boltzmann statistics|Maxwell–Boltzmann statistics]] and other thermodynamic laws could be derived from [[Biography:Isaac Newton|Newton]]ian laws applied to multi-body systems rather than through the conventional methods of [[Physics:Statistical mechanics|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 [[Physics:Thermodynamic equilibrium|equilibrium]] [[Physics:Velocity|velocity]] distribution given several arbitrary initial conditions of the ideal gas. Relaxation times were shown to be very fast: on the order of [[Physics:Mean free time|mean free time]] .

In 1996 a computational approach was taken to the classical mechanics non-equilibrium problem of [[Physics:Heat transfer|heat flow]] within a two-dimensional gas.<ref>D. Risso and P. Cordero, ''Two-Dimensional Gas of Disks: Thermal Conductivity'', ''[[Physics:Journal of Statistical Physics|Journal of Statistical Physics]]'', volume 82, pages 1453–1466, (1996)</ref> This simulation work showed that for N>1500, good agreement with continuous systems is obtained.

==Electron gas==
{{See also|Two-dimensional electron gas|Fermi gas}}[[Image:Cyclotron patent.png|right|thumb|260px|Diagram of cyclotron operation from Lawrence's 1934 patent.]]

While the principle of the [[Physics:Cyclotron|cyclotron]] to create a two-dimensional array of [[Physics:Electron|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 explored [[Physics:Cyclotron resonance|cyclotron resonance]] behavior and the [[Physics:De Haas–van Alphen effect|de Haas–van Alphen effect]] in a two-dimensional electron gas.<ref>{{cite journal | last1 = Kohn | first1 = Walter | year = 1961 | title = Cyclotron Resonance and de Haas–van Alphen Oscillations of an Interacting Electron Gas | journal = [[Physics:Physical Review|Physical Review]] | volume = 123 | issue = 4| pages = 1242–1244 | doi=10.1103/physrev.123.1242| bibcode = 1961PhRv..123.1242K| title-link = Cyclotron Resonance }}</ref> 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 [[Physics:Bose gas|Bose gas]] can exist in two dimensions.<ref>Vanderlei Bagnato and  Daniel Kleppner. ''[[Physics:Bose–Einstein condensation (network theory)|Bose–Einstein]] [[Physics:Condensation|condensation]] in low-dimensional traps'', [[Organization:American Physical Society|American Physical Society]], 8 April 1991</ref> 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, [[Physics:Graphene|graphene]] 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 in ultrahigh vacuum (UHV) to image an interaction of a two-dimensional [[Chemistry:Benzene|benzene]] gas layer in contact with a planar solid interface at 77 kelvins.<ref>Stranick, S. J. ; Kamna, M. M. ; Weiss, P. S, ''Atomic Scale Dynamics of a Two-Dimensional Gas-Solid Interface'',  Pennsylvania State University, Park Dept. of Chemistry, 3 June 1994</ref> 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 [[Physics:Diffraction|diffraction]] (see for example study of Kroger et al.<ref>{{Cite journal|last=Kroger|first=I.|title=Tuning intermolecular interaction in long-range-ordered submonolayer organic films|journal=Nature Physics|volume=5|issue=2|pages=153–158|doi=10.1038/nphys1176|year=2009|bibcode=2009NatPh...5..153S}}</ref>). The exception is the work of Matvija et al. who used a [[Engineering:Scanning tunneling microscope|scanning tunneling microscope]] to directly visualize a local time-averaged density of molecules on a surface.<ref>{{Cite journal|last1=Matvija|first1=Peter|last2=Rozbořil|first2=Filip|last3=Sobotík|first3=Pavel|last4=Ošťádal|first4=Ivan|last5=Kocán|first5=Pavel|title=Pair correlation function of a 2D molecular gas directly visualized by scanning tunneling microscopy|journal=The Journal of Physical Chemistry Letters|volume=8|issue=17|pages=4268–4272|doi=10.1021/acs.jpclett.7b01965|pmid=28830146|year=2017}}</ref> 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 of a 2D molecular gas in a real space.

If the surface coverage of adsorbates is increased, a [[Physics:Two-dimensional liquid|2D liquid]] is formed,<ref name="ReferenceA">{{citation|surname1=Thomas Waldmann|surname2=Jens Klein|surname3=Harry E. Hoster|surname4=R. Jürgen Behm|periodical=ChemPhysChem|title=Stabilization of Large Adsorbates by Rotational Entropy: A Time-Resolved Variable-Temperature STM Study |volume=14|pages=162–169|date=  2012|issue=1|language=de|doi=10.1002/cphc.201200531|pmid=23047526|s2cid=36848079 }}</ref> 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 [[Engineering:Scanning tunneling microscope|scanning tunneling microscope]] which can affect the local density of molecules via an electric field.<ref>{{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}}</ref>

==Implications for future research==
A multiplicity of theoretical physics research directions exist for study via a two-dimensional gas, such as:{{cn|date=November 2023}}

*Complex [[Physics:Quantum mechanics|quantum mechanics]] phenomena, whose solutions may be more appropriate in a two-dimensional environment;
*Studies of [[Physics:Phase transition|phase transition]]s (e.g. [[Physics:Melting point|melting]] phenomena at a [[Planar lamina|planar surface]]);
*[[Physics:Thin film|Thin film]] phenomena such as [[Chemistry:Chemical vapor deposition|chemical vapor deposition]];
*Surface [[Physics:Excited state|excitations]] of a solid.

==See also==
*[[Physics:Bose gas|Bose gas]]
*[[Physics:Fermi gas|Fermi gas]]
*[[Physics:Melting point|Melting point]]
*[[Physics:Optical lattice|Optical lattice]]
*[[Physics:Three-body problem|Three-body problem]]

== References==
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==External links==
*[http://citeseer.ist.psu.edu/lax95solution.html Riemann problems for a two-dimensional gas]
*[http://www.cec.uchile.cl/cinetica/papers/rc96.html Two-dimensional gas of disks]

[[Category:Gases]]
[[Category:Non-equilibrium thermodynamics]]
[[Category:Statistical mechanics]]
[[Category:Surfaces]]

{{Sourceattribution|Two-dimensional gas}}