Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Particle number counting statistics in ideal Bose gases

Christoph Weiss and Martin Wilkens

Open Access Open Access

Abstract

We discuss the exact particle number counting statistics of degenerate ideal Bose gases in the microcanonical, canonical, and grand-canonical ensemble, respectively, for various trapping potentials. We then invoke the Maxwell’s Demon ensemble [Navez et el, Phys. Rev. Lett. (1997)] and show that for large total number of particles the root-mean-square fluctuation of the condensate occupation scales δn 0 α [T/Tc ] rNs with scaling exponents r = 3/2, s = 1/2 for the 3D harmonic oscillator trapping potential, and r = 1, s = 2/3 for the 3D box. We derive an explicit expression for r and s in terms of spatial dimension D and spectral index σ of the single-particle energy spectrum. Our predictions also apply to systems where Bose-Einstein condensation does not occur. We point out that the condensate fluctuations in the microcanonical and canonical ensemble respect the principle of thermodynamic equivalence.

©1997 Optical Society of America

Full Article  |  PDF Article
More Like This
Maxwell’s Demon at work: Two types of Bose condensate fluctuations in power-law traps

Siegfried Grossmann and Martin Holthaus
Opt. Express 1(10) 262-271 (1997)

Characterizing the coherence of Bose-Einstein condensates and atom lasers

R. J. Dodd, Charles W. Clark, Mark Edwards, and K. Burnett
Opt. Express 1(10) 284-292 (1997)

Multi-mode description of an interacting Bose-Einstein condensate

Krzysztof Góral, Mariusz Gajda, and Kazimierz Rzążewski
Opt. Express 8(2) 92-98 (2001)

View More...

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)
Figure 1.
Figure 1. Particle number counting statistics of the ground state occupation of an ideal Bose gas in a three-dimensional isotropic harmonic oscillator trapping potential for a total number of particles N = 200 in the microcanonical ensemble (dashed), canonical ensemble (solid), and grand-canonical ensemble (dotted). Temperatures are from left to right T/Tc = 1.252, 0.905, 0.557 (no grand-canonical curve for this case).

Download Full Size | PDF

Figure 2.
Figure 2. Exact data for the condensate root-mean-square fluctuations in a three-dimensional isotropic harmonic oscillator trapping potential for N = 10, 42 and 100 particles in the canonical ensemble (full) and microcanonical ensemble (dashed), respectively. Temperature is measured in units of the energy gap ∆ between the trap ground state and trap first excited state.

Download Full Size | PDF

Figure 3.
Figure 3. Exact data for a Bose gas of N = 400 particles in a one-dimensional box in the microcanonical ensemble (dashed), canonical ensemble (full) grand-canonical ensemble (dotted). Left: Ground state occupation counting statistics for T/Tc = 2.0, 0.3. Right: Root-mean-square fluctuations of ground state occupation with asymptotics (dotted-dashed) according to Eq. (28). Inset: Ground state mean occupation.

Download Full Size | PDF

Figure 4.
Figure 4. Exact data (full line) for the condensate root-mean-square fluctuations in a three-dimensional isotropic harmonic oscillator trapping potential (left) and three-dimensional box (right) for N = 104 particles (lower curves) and N = 105 particles (upper curves) in the canonical ensemble. Dotted-dashed line: predictions of the asymptotic formulas. Dashed line: asymptotic formula with finite-N corrections.

Download Full Size | PDF

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

P ν G ( n ) = 1 1 + n ν ¯ ( n ν ¯ 1 + n ν ¯ ) n ,
δ 2 n 0 = ( 1 − [ T T c ] η ) 2 N ¯ 2 ,
P ν C ( n ) = e − n β ε ν Z N − n Z N − e − ( n + 1 ) β ε ν Z N − n − 1 Z N .
Z N ( β ) = 1 N ∑ n = 1 N Z 1 ( n β ) Z N − n ( β ) , Z 0 = 1 ,
P ν M ( n ) = Ω N − n ( E − n ε ν ) Ω N ( E ) − Ω N − n − 1 ( E − ( n + 1 ) ε ν ) Ω N ( E ) ,
Ω N ( E ) = 1 N ∑ n = 1 N ∑ ν Ω N − n ( E − n ε ν ) .
Ω N ( E ) = 1 N ∑ n = 1 N ∑ ν = 0 ∞ f ( D ) ( ν ) Ω N − n ( E − n ħ ων ) ,
ϒ ( z ) ≡ Z N ∑ M = 0 N z M P 0 C ( N − M ) .
N − 〈 n 0 〉 = z ∂ ∂ z ln ϒ ( z ) | z = 1 ,
δ 2 n 0 = ( z ∂ ∂ z ) 2 ln ϒ ( z ) | z = 1 ,
〈 n 0 〉 ~ O ( N ) and δ n 0 ≪ 〈 n 0 〉 ,
ϒ ( z ) ≈ ( 1 − z ) ∑ M = 0 ∞ z M Z M ( β ) .
ln ϒ ≈ − ∑ ν ≠ 0 ln ( 1 − z e − β ε ν ) ,
ε ν = Δ ∑ i = 1 D c i ν i , σ 0 < σ ≤ 2 ,
k B T c Δ ~ [ ∏ i = 1 D c i ] 1 D [ Γ ( 1 σ + 1 ) ] σ [ N ζ ( D σ ) ] σ D ,
〈 n 0 〉 ~ ( 1 − [ T T c ] D σ ) N .
k B T c ħ ω ~ N ln N .
δ 2 n 0 ≈ ∑ ν ≠ 0 1 4 sinh 2 ( β ε ν 2 ) ,
δ n 0 2 ≈ ∑ k = 1 ∞ k [ Z ( k β ) − 1 ] ,
S ( β ) ≡ ∑ ν = 1 ∞ e − β ν σ ,
δ n 0 2 ≈ ∑ d = 1 D D d ∑ k = 1 ∞ k S ( k β ) d .
S ( k β ) ~ Γ ( 1 + 1 σ ) e − k β [ k β ] 1 σ .
δ 2 n 0 ~ ∑ d = 1 D D d Γ ( 1 + 1 σ ) d g d σ − 1 ( e − d β ) β d σ
β − η g η − 1 ( e − β ) ~ { ζ ( η − 1 ) β − η for η > 2 β − 2 ln ( β − 1 ) for η = 2 Γ ( 2 − η ) β − 2 for η < 2
δ 2 n 0 ~ C [ k B T Δ ] D σ ,
C = Γ ( 1 + 1 σ ) D [ ∏ i = 1 D c i ] 1 σ ζ ( D σ − 1 ) .
δ 2 n 0 ~ C [ k B T Δ ] 2 ln ( k B T Δ ) ,
δ n 0 2 ~ C ′ [ k B T Δ ] 2 ,
C ′ = ∑ ν ≠ 0 1 [ ν 1 σ + ν 2 σ + … + ν D σ ] 2 .
δ n 0 ~ A [ T T c ] r N s ,
r = { D 2 σ if D > 2 σ 1 if D < 2 σ , s = { 1 2 if D > 2 σ σ D if D < 2 σ .
T T c < 1 − A ′ N 1 − s .
H = ∑ ν = 0 ∞ ε ν n ν ,
Ω N ( E ) = ∑ n 0 = 0 ∞ ∑ n 1 = 0 ∞ ⋯ ∑ n ∞ = 0 ∞ δ H , E δ Σ n ν , N ,
P ( { n } ) = 1 Z N exp { − β ∑ ν = 0 ∞ ε ν n ν } δ Σ n ν , N ,
Z N ( β ) = ∑ n 0 = 0 ∞ ∑ n ν = 0 ∞ ⋯ ∑ n ∞ = 0 ∞ e − β H δ Σ n ν , N .
P ν M ( n ) = Ω N − n ( E − n ε ν ) Ω N ( E ) − Ω N − n − 1 ( E − ( n + 1 ) ε ν ) Ω N ( E ) ,
P ν C ( n ) = e − n β ε ν Z N − n Z N − e − ( n + 1 ) β ε ν Z N − n − 1 Z N ,
〈 n ν 〉 = 1 Z N ∑ n = 1 N e − n β ε ν Z N − n .
N = 1 Z N ∑ n = 1 N Z N − n ∑ ν e − n β ε ν ,
Z N ( β ) = 1 N ∑ n = 1 N Z 1 ( n β ) Z N − n ( β ) , Z 0 = 1
〈 n ν 〉 ≡ ∑ n = 1 N n P ν ( n | E , N ) .
〈 n ν 〉 = ∑ n = 1 N Ω N − n ( E − n ε ν ) Ω N ( E ) .
N ≡ ∑ ν 〈 n ν 〉
Ω N ( E ) = 1 N ∑ n = 1 N ∑ ν Ω N − n ( E − n ε ν ) .
Ω N ( E ) = 1 N ∑ n = 1 N ∑ ν = 0 ∞ f ( d ) ( ν ) Ω N − n ( E − n ħ ων ) .
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use