Gas in a box

Last updated

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 except for instantaneous thermalizing collisions. This simple model can be used to describe the classical ideal gas as well as the various quantum ideal gases such as the ideal massive Fermi gas, the ideal massive Bose gas as well as black body radiation (photon gas) which may be treated as a massless Bose gas, in which thermalization is usually assumed to be facilitated by the interaction of the photons with an equilibrated mass.

Contents

Using the results from either Maxwell–Boltzmann statistics, Bose–Einstein statistics or Fermi–Dirac statistics, and considering the limit of a very large box, the Thomas–Fermi approximation (named after Enrico Fermi and Llewellyn Thomas) is used to express the degeneracy of the energy states as a differential, and summations over states as integrals. This enables thermodynamic properties of the gas to be calculated with the use of the partition function or the grand partition function. These results will be applied to both massive and massless particles. More complete calculations will be left to separate articles, but some simple examples will be given in this article.

Thomas–Fermi approximation for the degeneracy of states

For both massive and massless particles in a box, the states of a particle are enumerated by a set of quantum numbers [nx, ny, nz]. The magnitude of the momentum is given by

where h is Planck's constant and L is the length of a side of the box. Each possible state of a particle can be thought of as a point on a 3-dimensional grid of positive integers. The distance from the origin to any point will be

Suppose each set of quantum numbers specify f states where f is the number of internal degrees of freedom of the particle that can be altered by collision. For example, a spin 12 particle would have f=2, one for each spin state. For large values of n, the number of states with magnitude of momentum less than or equal to p from the above equation is approximately

which is just f times the volume of a sphere of radius n divided by eight since only the octant with positive ni is considered. Using a continuum approximation, the number of states with magnitude of momentum between p and p+dp is therefore

where V=L3 is the volume of the box. Notice that in using this continuum approximation, also known as Thomas−Fermi approximation, the ability to characterize the low-energy states is lost, including the ground state where ni = 1. For most cases this will not be a problem, but when considering Bose–Einstein condensation, in which a large portion of the gas is in or near the ground state, the ability to deal with low energy states becomes important.

Without using any approximation, the number of particles with energy εi is given by

where is the degeneracy of state i and

with β = 1/kBT, Boltzmann's constant kB, temperature T, and chemical potential μ. (See Maxwell–Boltzmann statistics, Bose–Einstein statistics, and Fermi–Dirac statistics.)

Using the Thomas−Fermi approximation, the number of particles dNE with energy between E and E+dE is:

where is the number of states with energy between E and E+dE.

Energy distribution

Using the results derived from the previous sections of this article, some distributions for the gas in a box can now be determined. For a system of particles, the distribution for a variable is defined through the expression which is the fraction of particles that have values for between and

where

It follows that:

For a momentum distribution , the fraction of particles with magnitude of momentum between and is:

and for an energy distribution , the fraction of particles with energy between and is:

For a particle in a box (and for a free particle as well), the relationship between energy and momentum is different for massive and massless particles. For massive particles,

while for massless particles,

where is the mass of the particle and is the speed of light. Using these relationships,

Specific examples

The following sections give an example of results for some specific cases.

Massive Maxwell–Boltzmann particles

For this case:

Integrating the energy distribution function and solving for N gives

Substituting into the original energy distribution function gives

which are the same results obtained classically for the Maxwell–Boltzmann distribution. Further results can be found in the classical section of the article on the ideal gas.

Massive Bose–Einstein particles

For this case:

where

Integrating the energy distribution function and solving for N gives the particle number

where Lis(z) is the polylogarithm function. The polylogarithm term must always be positive and real, which means its value will go from 0 to ζ(3/2) as z goes from 0 to 1. As the temperature drops towards zero, Λ will become larger and larger, until finally Λ will reach a critical value Λc where z=1 and

where denotes the Riemann zeta function. The temperature at which Λ = Λc is the critical temperature. For temperatures below this critical temperature, the above equation for the particle number has no solution. The critical temperature is the temperature at which a Bose–Einstein condensate begins to form. The problem is, as mentioned above, that the ground state has been ignored in the continuum approximation. It turns out, however, that the above equation for particle number expresses the number of bosons in excited states rather well, and thus:

where the added term is the number of particles in the ground state. The ground state energy has been ignored. This equation will hold down to zero temperature. Further results can be found in the article on the ideal Bose gas.

Massless Bose–Einstein particles (e.g. black body radiation)

For the case of massless particles, the massless energy distribution function must be used. It is convenient to convert this function to a frequency distribution function:

where Λ is the thermal wavelength for massless particles. The spectral energy density (energy per unit volume per unit frequency) is then

Other thermodynamic parameters may be derived analogously to the case for massive particles. For example, integrating the frequency distribution function and solving for N gives the number of particles:

The most common massless Bose gas is a photon gas in a black body. Taking the "box" to be a black body cavity, the photons are continually being absorbed and re-emitted by the walls. When this is the case, the number of photons is not conserved. In the derivation of Bose–Einstein statistics, when the restraint on the number of particles is removed, this is effectively the same as setting the chemical potential (μ) to zero. Furthermore, since photons have two spin states, the value of f is 2. The spectral energy density is then

which is just the spectral energy density for Planck's law of black body radiation. Note that the Wien distribution is recovered if this procedure is carried out for massless Maxwell–Boltzmann particles, which approximates a Planck's distribution for high temperatures or low densities.

In certain situations, the reactions involving photons will result in the conservation of the number of photons (e.g. light-emitting diodes, "white" cavities). In these cases, the photon distribution function will involve a non-zero chemical potential. (Hermann 2005)

Another massless Bose gas is given by the Debye model for heat capacity. This model considers a gas of phonons in a box and differs from the development for photons in that the speed of the phonons is less than light speed, and there is a maximum allowed wavelength for each axis of the box. This means that the integration over phase space cannot be carried out to infinity, and instead of results being expressed in polylogarithms, they are expressed in the related Debye functions.

Massive Fermi–Dirac particles (e.g. electrons in a metal)

For this case:

Integrating the energy distribution function gives

where again, Lis(z) is the polylogarithm function and Λ is the thermal de Broglie wavelength. Further results can be found in the article on the ideal Fermi gas. Applications of the Fermi gas are found in the free electron model, the theory of white dwarfs and in degenerate matter in general.

See also

Related Research Articles

Lorentz transformation Family of linear transformations

In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.

Dirac equation Relativistic quantum mechanical wave equation

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-12 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way.

Four-vector Vector in special relativity well-behaved with respect to Lorentz transformations

In special relativity, a four-vector is an object with four components, which transform in a specific way under Lorentz transformation. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts.

Einstein field equations Field equations in general relativity

In the general theory of relativity, the Einstein field equations relate the geometry of spacetime to the distribution of matter within it.

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important. Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

Bose gas State of matter of many bosons

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 obey Bose–Einstein statistics. The statistical mechanics of bosons were developed by Satyendra Nath Bose for a photon gas, and extended to massive particles by Albert Einstein 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 physics, the thermal de Broglie wavelength 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 the gas to be approximately (V/N)1/3 where V is the volume and N is the number of particles. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. On the other hand, when the thermal de Broglie wavelength is on the order of or larger than the interparticle distance, quantum effects will dominate and the gas must be treated as a Fermi gas or a Bose gas, depending on the nature of the gas particles. The critical temperature is the transition point between these two regimes, and at this critical temperature, the thermal wavelength will be approximately equal to the interparticle distance. That is, the quantum nature of the gas will be evident for

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 except for instantaneous thermalizing collisions. This situation is of great practical importance since many experimental studies of Bose gases are conducted in such harmonic traps.

Geodesics in general relativity

In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.

Generalized inverse Gaussian distribution

In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function

In theoretical physics, massive gravity is a theory of gravity that modifies general relativity by endowing the graviton with a nonzero mass. In the classical theory, this means that gravitational waves obey a massive wave equation and hence travel at speeds below the speed of light.

Plasma parameters

Plasma parameters define various characteristics of a plasma, an electrically conductive collection of charged particles that responds collectively to electromagnetic forces. Plasma typically takes the form of neutral gas-like clouds or charged ion beams, but may also include dust and grains. The behaviour of such particle systems can be studied statistically.

Covariant formulation of classical electromagnetism

The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.

Maxwells equations in curved spacetime Electromagnetism in general relativity

In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime or where one uses an arbitrary coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation.

In theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of scalar fields. A scalar field is invariant under any Lorentz transformation.

In statistical mechanics of continuous systems, a potential for a many-body system is called H-stable if the potential energy per particle is bounded below by a constant that is independent of the total number of particles. In many circumstances, if a potential is not H-stable, it is not possible to define a grand canonical partition function in finite volume, because of catastrophic configurations with infinite particles located in a finite space.

In the Newman–Penrose (NP) formalism of general relativity, independent components of the Ricci tensors of a four-dimensional spacetime are encoded into seven Ricci scalars which consist of three real scalars , three complex scalars and the NP curvature scalar . Physically, Ricci-NP scalars are related with the energy–momentum distribution of the spacetime due to Einstein's field equation.

Relativistic Lagrangian mechanics Mathematical formulation of special and general relativity

In theoretical physics, relativistic Lagrangian mechanics is Lagrangian mechanics applied in the context of special relativity and general relativity.

In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of these irreducible representations are known abstractly. The name derives from the German mathematicians Alfred Clebsch (1833–1872) and Paul Gordan (1837–1912), who encountered an equivalent problem in invariant theory.

Dual graviton

In theoretical physics, the dual graviton is a hypothetical elementary particle that is a dual of the graviton under electric-magnetic duality, as an S-duality, predicted by some formulations of supergravity in eleven dimensions.

References