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An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of
bosons In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
, which have an integer value of spin, and abide by
Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic ...
. The statistical mechanics of bosons were developed by Satyendra Nath Bose for a
photon gas In physics, a photon gas is a gas-like collection of photons, which has many of the same properties of a conventional gas like hydrogen or neon – including pressure, temperature, and entropy. The most common example of a photon gas in equilibri ...
, and extended to massive particles by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a
Bose–Einstein condensate In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&nb ...
.


Introduction and examples

Boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s are quantum mechanical particles that follow
Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic ...
, or equivalently, that possess integer
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
. These particles can be classified as elementary: these are the Higgs boson, the photon, the
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind q ...
, the W/Z and the hypothetical graviton; or composite like the atom of hydrogen, the atom of 16 O, the nucleus of deuterium, mesons etc. Additionally, some quasiparticles in more complex systems can also be considered bosons like the
plasmon In physics, a plasmon is a quantum of plasma oscillation. Just as light (an optical oscillation) consists of photons, the plasma oscillation consists of plasmons. The plasmon can be considered as a quasiparticle since it arises from the quantiz ...
s (quanta of charge density waves). The first model that treated a gas with several bosons, was the
photon gas In physics, a photon gas is a gas-like collection of photons, which has many of the same properties of a conventional gas like hydrogen or neon – including pressure, temperature, and entropy. The most common example of a photon gas in equilibri ...
, a gas of photons, developed by
Bose Bose may refer to: * Bose (crater), a lunar crater * ''Bose'' (film), a 2004 Indian Tamil film starring Srikanth and Sneha * Bose (surname), a surname (and list of people with the name) * Bose, Italy, a ''frazioni'' in Magnano, Province of Biell ...
. This model leads to a better understanding of Planck's law and the
black-body radiation Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spec ...
. The photon gas can be easily expanded to any kind of ensemble of massless non-interacting bosons. The ''
phonon In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical ...
gas'', also known as Debye model, is an example where the normal modes of vibration of the crystal lattice of a metal, can be treated as effective massless bosons. Peter Debye used the phonon gas model to explain the behaviour of
heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat capacity i ...
of metals at low temperature. An interesting example of a Bose gas is an ensemble of helium-4 atoms. When a system of 4He atoms is cooled down to temperature near
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibratio ...
, many quantum mechanical effects are present. Below 2.17 kelvins, the ensemble starts to behave as a
superfluid Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two i ...
, a fluid with almost zero viscosity. The Bose gas is the most simple quantitative model that explains this phase transition. Mainly when a gas of bosons is cooled down, it forms a
Bose–Einstein condensate In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&nb ...
, a state where a large number of bosons occupy the lowest energy, the ground state, and quantum effects are macroscopically visible like wave interference. The theory of Bose-Einstein condensates and Bose gases can also explain some features of superconductivity where
charge carrier In physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and holes. The term is used ...
s couple in pairs (
Cooper pair In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper ...
s) and behave like bosons. As a result, superconductors behave like having no electrical resistivity at low temperatures. The equivalent model for half-integer particles (like electrons or
helium-3 Helium-3 (3He see also helion) is a light, stable isotope of helium with two protons and one neutron (the most common isotope, helium-4, having two protons and two neutrons in contrast). Other than protium (ordinary hydrogen), helium-3 is the ...
atoms), that follow
Fermi–Dirac statistics Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac d ...
, is called the
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. ...
(an ensemble of non-interacting
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s). At low enough particle
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects ( particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number ...
and high temperature, both the Fermi gas and the Bose gas behave like a classical ideal gas.


Macroscopic limit

The thermodynamics of an ideal Bose gas is best calculated using the grand canonical ensemble. The
grand potential The grand potential is a quantity used in statistical mechanics, especially for irreversible processes in open systems. The grand potential is the characteristic state function for the grand canonical ensemble. Definition Grand potential is defi ...
for a Bose gas is given by: :\Omega=-\ln(\mathcal) = \sum_i g_i \ln\left(1-ze^\right). where each term in the sum corresponds to a particular single-particle energy level ''ε''; ''g'' is the number of states with energy ''ε''; ''z '' is the absolute activity (or "fugacity"), which may also be expressed in terms of the
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
''μ'' by defining: :z(\beta,\mu)= e^ and ''β'' defined as: :\beta = \frac where ''k''B'' '' is
Boltzmann's constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
and ''T '' is the temperature. All thermodynamic quantities may be derived from the grand potential and we will consider all thermodynamic quantities to be functions of only the three variables ''z '', ''β'' (or ''T ''), and ''V ''. All partial derivatives are taken with respect to one of these three variables while the other two are held constant. The permissible range of ''z'' is from negative infinity to +1, as any value beyond this would give an infinite number of particles to states with an energy level of 0 (it is assumed that the energy levels have been offset so that the lowest energy level is 0).


Macroscopic limit, result for uncondensed fraction

Following the procedure described in the
gas in a box In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other excep ...
article, we can apply the
Thomas–Fermi approximation In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other excep ...
which assumes that the average energy is large compared to the energy difference between levels so that the above sum may be replaced by an integral. This replacement gives the macroscopic grand potential function \Omega_m, which is close to \Omega: :\Omega_ = \int_0^\infty \ln\left(1-ze^\right)\,dg \approx \Omega. The degeneracy ''dg '' may be expressed for many different situations by the general formula: :dg = \frac\,\frac ~dE where ''α'' is a constant, ''E''c is a ''critical'' energy, and ''Γ'' is the
Gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except t ...
. For example, for a massive Bose gas in a box, ''α''=3/2 and the critical energy is given by: :\frac=\frac where ''Λ'' is the
thermal wavelength In physics, the thermal de Broglie wavelength (\lambda_, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. We can take the average interparticle spacing in t ...
, and ''f'' is a degeneracy factor (''f''=1 for simple spinless bosons). For a massive Bose
gas in a harmonic trap The results of the quantum harmonic oscillator can be used to look at the equilibrium situation for a quantum ideal gas in a harmonic trap, which is a harmonic potential containing a large number of particles that do not interact with each other ...
we will have ''α''=3 and the critical energy is given by: :\frac=\frac where ''V(r)=mω2r2/2 '' is the harmonic potential. It is seen that ''Ec''  is a function of volume only. This integral expression for the grand potential evaluates to: :\Omega_ = -\frac, where Li''s''(''x'') is the polylogarithm function. The problem with this continuum approximation for a Bose gas is that the ground state has been effectively ignored, giving a degeneracy of zero for zero energy. This inaccuracy becomes serious when dealing with the
Bose–Einstein condensate In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&nb ...
and will be dealt with in the next sections. As will be seen, even at low temperatures the above result is still useful for accurately describing the thermodynamics of just the un-condensed portion of the gas.


Limit on number of particles in uncondensed phase, critical temperature

The total
number of particles The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter ''N'', is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is ...
is found from the grand potential by :N_ = -z\frac = \frac. This increases monotonically with ''z'' (up to the maximum ''z'' = +1). The behaviour when approaching ''z'' = 1 is however crucially dependent on the value of ''α'' (i.e., dependent on whether the gas is 1D, 2D, 3D, whether it is in a flat or harmonic potential well). For ''α'' > 1, the number of particles only increases up to a finite maximum value, i.e., N_ is finite at ''z'' = 1: :N_ = \frac, where ''ζ''(''α'') is the Riemann zeta function (using Li''α''(''1'') = ''ζ''(''α'')). Thus, for a fixed number of particles N_, the largest possible value that ''β'' can have is a critical value ''β''c. This corresponds to a critical temperature ''T''c=1/''k''B''β''c, below which the Thomas–Fermi approximation breaks down (the continuum of states simply can no longer support this many particles, at lower temperatures). The above equation can be solved for the critical temperature: :T_=\left(\frac\right)^\frac For example, for the three-dimensional Bose gas in a box (\alpha=3/2 and using the above noted value of E_) we get: :T_=\left(\frac\right)^\frac For ''α'' ≤ 1, there is no upper limit on the number of particles (N_ diverges as ''z'' approaches 1), and thus for example for a gas in a one- or two-dimensional box (\alpha=1/2 and \alpha=1 respectively) there is no critical temperature.


Inclusion of the ground state

The above problem raises the question for ''α'' > 1: if a Bose gas with a fixed number of particles is lowered down below the critical temperature, what happens? The problem here is that the Thomas–Fermi approximation has set the degeneracy of the ground state to zero, which is wrong. There is no ground state to accept the condensate and so particles simply 'disappear' from the continuum of states. It turns out, however, that the macroscopic equation gives an accurate estimate of the number of particles in the excited states, and it is not a bad approximation to simply "tack on" a ground state term to accept the particles that fall out of the continuum: :N = N_0+ N_ = N_0 + \frac where ''N''0 is the number of particles in the ground state condensate. Thus in the macroscopic limit, when ''T'' < ''T''c, the value of ''z'' is pinned to 1 and ''N''0 takes up the remainder of particles. For ''T'' > ''T''c there is the normal behaviour, with ''N''0 = 0. This approach gives the fraction of condensed particles in the macroscopic limit: :\frac = \begin 1 - \left(\frac\right)^\alpha &\mbox \alpha > 1 \mbox T < T_, \\ 0 & \mbox. \end


Limitations of the macroscopic Bose gas model

The above standard treatment of a macroscopic Bose gas is straight-forward, but the inclusion of the ground state is somewhat inelegant. Another approach is to include the ground state explicitly (contributing a term in the grand potential, as in the section below), this gives rise to an unrealistic fluctuation catastrophe: the number of particles in any given state follow a geometric distribution, meaning that when condensation happens at ''T'' < ''T''c and most particles are in one state, there is a huge uncertainty in the total number of particles. This is related to the fact that the
compressibility In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a f ...
becomes unbounded for ''T'' < ''T''c. Calculations can instead be performed in the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
, which fixes the total particle number, however the calculations are not as easy. Practically however, the aforementioned theoretical flaw is a minor issue, as the most unrealistic assumption is that of non-interaction between bosons. Experimental realizations of boson gases always have significant interactions, i.e., they are non-ideal gases. The interactions significantly change the physics of how a condensate of bosons behaves: the ground state spreads out, the chemical potential saturates to a positive value even at zero temperature, and the fluctuation problem disappears (the compressibility becomes finite). See the article
Bose–Einstein condensate In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&nb ...
.


Approximate behaviour in small Bose gases

For smaller,
mesoscopic Mesoscopic physics is a subdiscipline of condensed matter physics that deals with materials of an intermediate size. These materials range in size between the nanoscale for a quantity of atoms (such as a molecule) and of materials measuring mic ...
, systems (for example, with only thousands of particles), the ground state term can be more explicitly approximated by adding in an actual discrete level at energy ''ε''=0 in the grand potential: :\Omega = g_0\ln(1-z) + \Omega_ which gives instead N_0 = \frac. Now, the behaviour is smooth when crossing the critical temperature, and ''z'' approaches 1 very closely but does not reach it. This can now be solved down to absolute zero in temperature. Figure 1 shows the results of the solution to this equation for ''α''=3/2, with ''k''=''ε''c=1 which corresponds to a gas of bosons in a box. The solid black line is the fraction of excited states ''1-N0/N '' for ''N ''=10,000 and the dotted black line is the solution for ''N ''=1000. The blue lines are the fraction of condensed particles ''N0/N '' The red lines plot values of the negative of the chemical potential μ and the green lines plot the corresponding values of ''z ''. The horizontal axis is the normalized temperature τ defined by :\tau=\frac It can be seen that each of these parameters become linear in τα in the limit of low temperature and, except for the chemical potential, linear in 1/τα in the limit of high temperature. As the number of particles increases, the condensed and excited fractions tend towards a discontinuity at the critical temperature. The equation for the number of particles can be written in terms of the normalized temperature as: :N = \frac+N~\frac~\tau^\alpha For a given ''N '' and ''τ'', this equation can be solved for ''τα'' and then a series solution for ''z '' can be found by the method of inversion of series, either in powers of ''τα'' or as an asymptotic expansion in inverse powers of ''τα''. From these expansions, we can find the behavior of the gas near ''T =0'' and in the Maxwell–Boltzmann as ''T '' approaches infinity. In particular, we are interested in the limit as ''N '' approaches infinity, which can be easily determined from these expansions. This approach to modelling small systems may in fact be unrealistic, however, since the variance in the number of particles in the ground state is very large, equal to the number of particles. In contrast, the variance of particle number in a normal gas is only the square-root of the particle number, which is why it can normally be ignored. This high variance is due to the choice of using the grand canonical ensemble for the entire system, including the condensate state.


Thermodynamics of small gases

Expanded out, the grand potential is: :\Omega = g_0\ln(1-z)-\frac All thermodynamic properties can be computed from this potential. The following table lists various thermodynamic quantities calculated in the limit of low temperature and high temperature, and in the limit of infinite particle number. An equal sign (=) indicates an exact result, while an approximation symbol indicates that only the first few terms of a series in \tau^\alpha is shown. It is seen that all quantities approach the values for a classical ideal gas in the limit of large temperature. The above values can be used to calculate other thermodynamic quantities. For example, the relationship between internal energy and the product of pressure and volume is the same as that for a classical ideal gas over all temperatures: :U=\frac=\alpha PV A similar situation holds for the specific heat at constant volume :C_V=\frac=k_(\alpha+1)\,U\beta The entropy is given by: :TS=U+PV-G\, Note that in the limit of high temperature, we have :TS=(\alpha+1)+\ln\left(\frac\right) which, for ''α''=3/2 is simply a restatement of the
Sackur–Tetrode equation The Sackur–Tetrode equation is an expression for the entropy of a monatomic ideal gas. It is named for Hugo Martin Tetrode (1895–1931) and Otto Sackur (1880–1914), who developed it independently as a solution of Boltzmann's gas statistics a ...
. In one dimension bosons with delta interaction behave as fermions, they obey
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formula ...
. In one dimension Bose gas with delta interaction can be solved exactly by Bethe ansatz. The bulk free energy and thermodynamic potentials were calculated by
Chen-Ning Yang Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese theoretical physicist who made significant contributions to statistical mechanics, integrable systems, gauge the ...
. In one dimensional case correlation functions also were evaluated. In one dimension Bose gas is equivalent to quantum
non-linear Schrödinger equation In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many ot ...
.


See also

*
Tonks–Girardeau gas In physics, a Tonks–Girardeau gas is a Bose gas in which the repulsive interactions between bosonic particles confined to one dimension dominate the system's physics. It is named after physicists Marvin D. Girardeau and Lewi Tonks. It is not a B ...


References


General references

* * * * * {{Authority control Bose–Einstein statistics Ideal gas Quantum mechanics Thermodynamics Satyendra Nath Bose