An ideal Bose gas is a quantum-mechanical
phase of matter
In the physical sciences, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, magnetiza ...
, analogous to a 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 ...
. 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 ...
, 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
Satyendra Nath Bose (; 1 January 1894 – 4 February 1974) was a Bengali mathematician and physicist specializing in theoretical physics. He is best known for his work on quantum mechanics in the early 1920s, in developing the foundation for ...
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 equilibr ...
, 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 −459.6 ...
.
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 spi ...
s are
quantum mechanical
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, qua ...
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. These particles can be classified as elementary: these are the
Higgs boson, the
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
, the
gluon, the
W/Z and the hypothetical
graviton
In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathem ...
; or composite like the atom of
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
, the atom of
16 O, the nucleus of
deuterium
Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one ...
,
meson
In particle physics, a meson ( or ) is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, ...
s etc. Additionally, some
quasiparticle
In physics, quasiparticles and collective excitations are closely related emergent phenomena arising when a microscopically complicated system such as a solid behaves as if it contained different weakly interacting particles in vacuum.
For exa ...
s 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 equilibr ...
, a gas of photons, developed by
Bose. This model leads to a better understanding of
Planck's law and the
black-body radiation. The photon gas can be easily expanded to any kind of ensemble of massless non-interacting bosons. The ''
phonon gas'', also known as
Debye model
In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (Heat capacity) in a solid. It treats the vibrations of the atomic lattice (hea ...
, 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
Peter Joseph William Debye (; ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist and physical chemist, and Nobel laureate in Chemistry.
Biography
Early life
Born Petrus Josephus Wilhelmus Debije in Maastricht, Netherlands, D ...
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 ...
of metals at low temperature.
An interesting example of a Bose gas is an ensemble of
helium-4
Helium-4 () is a stable isotope of the element helium. It is by far the more abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on Earth. Its nucleus is identical to an alpha particle, and consis ...
atoms. When a system of
4He atoms is cooled down to temperature near
absolute zero, many quantum mechanical effects are present. Below 2.17
kelvin
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 phy ...
s, 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 ...
, a fluid with almost zero
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
. The Bose gas is the most simple quantitative model that explains this
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 o ...
. 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 −459.6 ...
, a state where a large number of bosons occupy the lowest energy, the
ground state, and quantum effects are macroscopically visible like
wave interference
In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
.
The theory of Bose-Einstein condensates and Bose gases can also explain some features of
superconductivity where
charge carriers 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 Coope ...
s) and behave like bosons. As a result, superconductors behave like having no
electrical resistivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
at low temperatures.
The equivalent model for half-integer particles (like
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 ...
s or
helium-3 atoms), that follow
Fermi–Dirac statistics, 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 sp ...
(an ensemble of non-interacting
fermions). At low enough particle
number density and high temperature, both the Fermi gas and the Bose gas behave like a 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 ...
.
Macroscopic limit
The thermodynamics of an ideal Bose gas is best calculated using the
grand canonical ensemble
In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibriu ...
. 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 de ...
for a Bose gas is given by:
:
where each term in the sum corresponds to a particular single-particle energy level ''ε''
i ; ''g''
i is the number of states with energy ''ε''
i ; ''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:
:
and ''β'' defined as:
:
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
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer.
Thermometers are calibrated in various Conversion of units of temperature, temp ...
. 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
![Quantum ideal gas pressure 3d](https://upload.wikimedia.org/wikipedia/commons/7/74/Quantum_ideal_gas_pressure_3d.svg)
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 exce ...
article, we can apply the
Thomas–Fermi approximation 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
, which is close to
:
:
The degeneracy ''dg '' may be expressed for many different situations by the general formula:
:
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 ...
. For example, for a massive Bose gas in a box, ''α''=3/2 and the critical energy is given by:
:
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 ...
, and ''f'' is a degeneracy factor (''f''=1 for simple spinless bosons). For a massive Bose
gas in a harmonic trap we will have ''α''=3 and the critical energy is given by:
:
where ''V(r)=mω
2r
2/2 '' is the harmonic potential. It is seen that ''E
c''
is a function of volume only.
This integral expression for the grand potential evaluates to:
:
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 −459.6 ...
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
:
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.,
is finite at ''z'' = 1:
:
where ''ζ''(''α'') is the
Riemann zeta function (using Li
''α''(''1'') = ''ζ''(''α'')). Thus, for a fixed number of particles
, 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:
:
For example, for the three-dimensional Bose gas in a box (
and using the above noted value of
) we get:
:
For ''α'' ≤ 1, there is no upper limit on the number of particles (
diverges as ''z'' approaches 1), and thus for example for a gas in a one- or two-dimensional box (
and
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:
:
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:
:
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
In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions:
* The probability distribution of the number ''X'' of Bernoulli trials needed to get one success, supported on the set \;
* ...
, 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 ...
, 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 −459.6 ...
.
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 micr ...
, 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:
:
which gives instead
. 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-N
0/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 ''N
0/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
:
It can be seen that each of these parameters become linear in τ
α in the limit of low temperature and, except for the chemical potential, linear in 1/τ
α 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:
:
For a given ''N '' and ''τ'', this equation can be solved for ''τ
α'' and then a series solution for ''z '' can be found by the method of
inversion of series, either in powers of ''τ
α'' or as an asymptotic expansion in inverse powers of ''τ
α''. 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:
:
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
is shown.
It is seen that all quantities approach the values for a 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 ...
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:
:
A similar situation holds for the specific heat at constant volume
:
The entropy is given by:
:
Note that in the limit of high temperature, we have
:
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 an ...
. 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 formulat ...
. 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 t ...
. 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 othe ...
.
See also
*
Tonks–Girardeau gas
References
General references
*
*
*
*
*
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Bose–Einstein statistics
Ideal gas
Quantum mechanics
Thermodynamics
Satyendra Nath Bose