Gas in a box






Basic statistical model

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.


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.




Contents






  • 1 Thomas–Fermi approximation for the degeneracy of states


  • 2 Energy distribution


  • 3 Specific examples


    • 3.1 Massive Maxwell–Boltzmann particles


    • 3.2 Massive Bose–Einstein particles


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


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




  • 4 See also


  • 5 References





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


p=h2Lnx2+ny2+nz2nx,ny,nz=1,2,3,…{displaystyle p={frac {h}{2L}}{sqrt {n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}}qquad qquad n_{x},n_{y},n_{z}=1,2,3,ldots }p={frac {h}{2L}}{sqrt {n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}}qquad qquad n_{x},n_{y},n_{z}=1,2,3,ldots

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


n=nx2+ny2+nz2=2Lph{displaystyle n={sqrt {n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}}={frac {2Lp}{h}}}n={sqrt {n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}}={frac {2Lp}{h}}

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 ½ 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


g=(f8)43πn3=4πf3(Lph)3{displaystyle g=left({frac {f}{8}}right){frac {4}{3}}pi n^{3}={frac {4pi f}{3}}left({frac {Lp}{h}}right)^{3}}g=left({frac {f}{8}}right){frac {4}{3}}pi n^{3}={frac {4pi f}{3}}left({frac {Lp}{h}}right)^{3}

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


dg=π2 fn2dn=4πfVh3 p2dp{displaystyle dg={frac {pi }{2}}~fn^{2},dn={frac {4pi fV}{h^{3}}}~p^{2},dp}dg={frac {pi }{2}}~fn^{2},dn={frac {4pi fV}{h^{3}}}~p^{2},dp

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


Ni=giΦ(ϵi){displaystyle N_{i}={frac {g_{i}}{Phi (epsilon _{i})}}}N_{i}={frac {g_{i}}{Phi (epsilon _{i})}}

where









gi{displaystyle g_{i}}{displaystyle g_{i}}, degeneracy of state i
 

Φ(ϵi)={eβ(ϵi−μ),for particles obeying Maxwell-Boltzmann statisticseβ(ϵi−μ)−1,for particles obeying Bose-Einstein statisticseβ(ϵi−μ)+1,for particles obeying Fermi-Dirac statistics{displaystyle Phi (epsilon _{i})={begin{cases}e^{beta (epsilon _{i}-mu )},&{text{for particles obeying Maxwell-Boltzmann statistics}}\e^{beta (epsilon _{i}-mu )}-1,&{text{for particles obeying Bose-Einstein statistics}}\e^{beta (epsilon _{i}-mu )}+1,&{text{for particles obeying Fermi-Dirac statistics}}\end{cases}}}{displaystyle Phi (epsilon _{i})={begin{cases}e^{beta (epsilon _{i}-mu )},&{text{for particles obeying Maxwell-Boltzmann statistics}}\e^{beta (epsilon _{i}-mu )}-1,&{text{for particles obeying Bose-Einstein statistics}}\e^{beta (epsilon _{i}-mu )}+1,&{text{for particles obeying Fermi-Dirac statistics}}\end{cases}}}
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:



dNE=dgEΦ(E){displaystyle dN_{E}={frac {dg_{E}}{Phi (E)}}}dN_{E}={frac {dg_{E}}{Phi (E)}}

where dgE{displaystyle dg_{E}}{displaystyle dg_{E}}  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 PA{displaystyle P_{A}}P_{A} for a variable A{displaystyle A}A is defined through the expression PAdA{displaystyle P_{A}dA}P_{A}dA which is the fraction of particles that have values for A{displaystyle A}A between A{displaystyle A}A and A+dA{displaystyle A+dA}A+dA


PA dA=dNAN=dgANΦA{displaystyle P_{A}~dA={frac {dN_{A}}{N}}={frac {dg_{A}}{NPhi _{A}}}}P_{A}~dA={frac {dN_{A}}{N}}={frac {dg_{A}}{NPhi _{A}}}

where




dNA{displaystyle dN_{A}}dN_{A},  number of particles which have values for A{displaystyle A}A between A{displaystyle A}A and A+dA{displaystyle A+dA}A+dA


dgA{displaystyle dg_{A}}dg_{A},  number of states which have values for A{displaystyle A}A between A{displaystyle A}A and A+dA{displaystyle A+dA}A+dA


ΦA−1{displaystyle Phi _{A}^{-1}}Phi _{A}^{{-1}}, probability that a state which has the value A{displaystyle A}A is occupied by a particle


N{displaystyle N}N,      total number of particles.


It follows that:


∫APA dA=1{displaystyle int _{A}P_{A}~dA=1}int _{A}P_{A}~dA=1

For a momentum distribution Pp{displaystyle P_{p}}P_{p}, the fraction of particles with magnitude of momentum between p{displaystyle p}p and p+dp{displaystyle p+dp}p+dp is:


Pp dp=VfN 4πh3Φp p2dp{displaystyle P_{p}~dp={frac {Vf}{N}}~{frac {4pi }{h^{3}Phi _{p}}}~p^{2}dp}P_{p}~dp={frac {Vf}{N}}~{frac {4pi }{h^{3}Phi _{p}}}~p^{2}dp

and for an energy distribution PE{displaystyle P_{E}}P_{E}, the fraction of particles with energy between E{displaystyle E}E and E+dE{displaystyle E+dE}E+dE is:


PE dE=PpdpdE dE{displaystyle P_{E}~dE=P_{p}{frac {dp}{dE}}~dE}P_{E}~dE=P_{p}{frac {dp}{dE}}~dE

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


E=p22m{displaystyle E={frac {p^{2}}{2m}}}E={frac {p^{2}}{2m}}

while for massless particles,


E=pc{displaystyle E=pc,}E=pc,

where m{displaystyle m}m is the mass of the particle and c{displaystyle c}c is the speed of light.
Using these relationships,


  • For massive particles

dgE= (VfΛ3)2π β3/2E1/2 dEPE dE=1N(VfΛ3)2π β3/2E1/2Φ(E) dE{displaystyle {begin{alignedat}{2}dg_{E}&=quad left({frac {Vf}{Lambda ^{3}}}right){frac {2}{sqrt {pi }}}~beta ^{3/2}E^{1/2}~dE\P_{E}~dE&={frac {1}{N}}left({frac {Vf}{Lambda ^{3}}}right){frac {2}{sqrt {pi }}}~{frac {beta ^{3/2}E^{1/2}}{Phi (E)}}~dE\end{alignedat}}}{begin{alignedat}{2}dg_{E}&=quad left({frac {Vf}{Lambda ^{3}}}right){frac {2}{{sqrt {pi }}}}~beta ^{{3/2}}E^{{1/2}}~dE\P_{E}~dE&={frac {1}{N}}left({frac {Vf}{Lambda ^{3}}}right){frac {2}{{sqrt {pi }}}}~{frac {beta ^{{3/2}}E^{{1/2}}}{Phi (E)}}~dE\end{alignedat}}

where Λ is the thermal wavelength of the gas.


Λ=h2β2πm{displaystyle Lambda ={sqrt {frac {h^{2}beta }{2pi m}}}}Lambda ={sqrt {{frac {h^{2}beta }{2pi m}}}}

This is an important quantity, since when Λ is on the order of the inter-particle distance (V/N){displaystyle (V/N)}(V/N)1/3, quantum effects begin to dominate and the gas can no longer be considered to be a Maxwell–Boltzmann gas.


  • For massless particles

dgE= (VfΛ3)12 β3E2 dEPE dE=1N(VfΛ3)12 β3E2Φ(E) dE{displaystyle {begin{alignedat}{2}dg_{E}&=quad left({frac {Vf}{Lambda ^{3}}}right){frac {1}{2}}~beta ^{3}E^{2}~dE\P_{E}~dE&={frac {1}{N}}left({frac {Vf}{Lambda ^{3}}}right){frac {1}{2}}~{frac {beta ^{3}E^{2}}{Phi (E)}}~dE\end{alignedat}}}{begin{alignedat}{2}dg_{E}&=quad left({frac {Vf}{Lambda ^{3}}}right){frac {1}{2}}~beta ^{3}E^{2}~dE\P_{E}~dE&={frac {1}{N}}left({frac {Vf}{Lambda ^{3}}}right){frac {1}{2}}~{frac {beta ^{3}E^{2}}{Phi (E)}}~dE\end{alignedat}}

where Λ is now the thermal wavelength for massless particles.


Λ=chβ2π1/3{displaystyle Lambda ={frac {chbeta }{2,pi ^{1/3}}}}Lambda ={frac {chbeta }{2,pi ^{{1/3}}}}


Specific examples


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



Massive Maxwell–Boltzmann particles


For this case:


Φ(E)=eβ(E−μ){displaystyle Phi (E)=e^{beta (E-mu )}}Phi (E)=e^{{beta (E-mu )}}

Integrating the energy distribution function and solving for N gives


N=(VfΛ3)eβμ{displaystyle N=left({frac {Vf}{Lambda ^{3}}}right),,e^{beta mu }}N=left({frac {Vf}{Lambda ^{3}}}right),,e^{{beta mu }}

Substituting into the original energy distribution function gives


PE dE=2β3Eπ e−βE dE{displaystyle P_{E}~dE=2{sqrt {frac {beta ^{3}E}{pi }}}~e^{-beta E}~dE}P_{E}~dE=2{sqrt {{frac {beta ^{3}E}{pi }}}}~e^{{-beta E}}~dE

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:


Φ(E)=eβEz−1{displaystyle Phi (E)={frac {e^{beta E}}{z}}-1,}Phi (E)={frac {e^{{beta E}}}{z}}-1,

where   z=eβμ.{displaystyle z=e^{beta mu }.,}z=e^{{beta mu }}.,

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


N=(VfΛ3)Li3/2(z){displaystyle N=left({frac {Vf}{Lambda ^{3}}}right){textrm {Li}}_{3/2}(z)}N=left({frac {Vf}{Lambda ^{3}}}right){textrm {Li}}_{{3/2}}(z)

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


N=(VfΛc3)ζ(3/2),{displaystyle N=left({frac {Vf}{Lambda _{rm {c}}^{3}}}right)zeta (3/2),}{displaystyle N=left({frac {Vf}{Lambda _{rm {c}}^{3}}}right)zeta (3/2),}

where ζ(z){displaystyle zeta (z)}zeta (z) 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:


N=g0z1−z+(VfΛ3)Li3/2(z){displaystyle N={frac {g_{0}z}{1-z}}+left({frac {Vf}{Lambda ^{3}}}right){textrm {Li}}_{3/2}(z)}N={frac {g_{0}z}{1-z}}+left({frac {Vf}{Lambda ^{3}}}right){textrm {Li}}_{{3/2}}(z)

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:


Pν dν=h3N(VfΛ3)12 β3ν2e(hν−μ)/kBT−1 dν{displaystyle P_{nu }~dnu ={frac {h^{3}}{N}}left({frac {Vf}{Lambda ^{3}}}right){frac {1}{2}}~{frac {beta ^{3}nu ^{2}}{e^{(hnu -mu )/k_{rm {B}}T}-1}}~dnu }{displaystyle P_{nu }~dnu ={frac {h^{3}}{N}}left({frac {Vf}{Lambda ^{3}}}right){frac {1}{2}}~{frac {beta ^{3}nu ^{2}}{e^{(hnu -mu )/k_{rm {B}}T}-1}}~dnu }

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


Uν dν=(NhνV)Pν dν=4πfhν3c3 1e(hν−μ)/kBT−1 dν.{displaystyle U_{nu }~dnu =left({frac {N,hnu }{V}}right)P_{nu }~dnu ={frac {4pi fhnu ^{3}}{c^{3}}}~{frac {1}{e^{(hnu -mu )/k_{rm {B}}T}-1}}~dnu .}{displaystyle U_{nu }~dnu =left({frac {N,hnu }{V}}right)P_{nu }~dnu ={frac {4pi fhnu ^{3}}{c^{3}}}~{frac {1}{e^{(hnu -mu )/k_{rm {B}}T}-1}}~dnu .}

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:


N=16πVc3h3β3Li3(eμ/kBT).{displaystyle N={frac {16,pi V}{c^{3}h^{3}beta ^{3}}},mathrm {Li} _{3}left(e^{mu /k_{rm {B}}T}right).}{displaystyle N={frac {16,pi V}{c^{3}h^{3}beta ^{3}}},mathrm {Li} _{3}left(e^{mu /k_{rm {B}}T}right).}

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


Uν dν=8πhν3c3 1ehν/kBT−1 dν{displaystyle U_{nu }~dnu ={frac {8pi hnu ^{3}}{c^{3}}}~{frac {1}{e^{hnu /k_{rm {B}}T}-1}}~dnu }{displaystyle U_{nu }~dnu ={frac {8pi hnu ^{3}}{c^{3}}}~{frac {1}{e^{hnu /k_{rm {B}}T}-1}}~dnu }

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:


Φ(E)=eβ(E−μ)+1.{displaystyle Phi (E)=e^{beta (E-mu )}+1.,}Phi (E)=e^{{beta (E-mu )}}+1.,

Integrating the energy distribution function gives


N=(VfΛ3)[−Li3/2(−z)]{displaystyle N=left({frac {Vf}{Lambda ^{3}}}right)left[-{textrm {Li}}_{3/2}(-z)right]}N=left({frac {Vf}{Lambda ^{3}}}right)left[-{textrm {Li}}_{{3/2}}(-z)right]

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


  • Gas in a harmonic trap


References




  • Herrmann, F.; Würfel, P. (August 2005). "Light with nonzero chemical potential". American Journal of Physics. 73 (8): 717–723. Bibcode:2005AmJPh..73..717H. doi:10.1119/1.1904623. Retrieved 2006-11-20..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  • Huang, Kerson (1967). Statistical Mechanics. New York: John Wiley & Sons.


  • Isihara, A. (1971). Statistical Physics. New York: Academic Press.


  • Landau, L. D.; E. M. Lifshitz (1996). Statistical Physics (3rd Edition Part 1 ed.). Oxford: Butterworth-Heinemann.


  • Yan, Zijun (2000). "General thermal wavelength and its applications" (PDF). Eur. J. Phys. 21 (6): 625–631. Bibcode:2000EJPh...21..625Y. doi:10.1088/0143-0807/21/6/314. Retrieved 2006-11-20.

  • Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28.




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