Bose-Einstein Condensation in 1, 2 and 3 Dimensions for Massive and Massless Bosons in a Box

Bose-Einstein Condensation in 1, 2 and 3 dimensions for massive and massless bosons in a box

I. MASSIVE BOSONS

Consider a gas of massive, non-interacting, non-relativistic, identical, spin-0 bosons. The total number of bosons in a given state with energy is given by Z ∞ Z ∞ N = n¯()dN = n¯()D()d (1) 0 0 where 1 n¯() = (2) eβ(−µ) − 1 is the quantum distribution function for bosons, and dN D() = (3) d is the ”density of states” function. For a particle in a box of side length L, the contained modes are quantized by the condition that the wave function vanish at the walls, Ψ(x,y,z=0)=Ψ(x,y,z=L)=0. Thus for each spatial dimension i, we have the condition n π k = i (4) i L In 3D: dN dN dn dn D () = = = 4πn2 (5) 3D d dn d d For a massive particle in the box the energy is quadratic in the momentum, 1 2 2π2 2π2 = (p 2 + p 2 + p 2) = ~ (k 2 + k 2 + k 2) = ~ (n 2 + n 2 + n 2) = ~ n2 (6) 2m x y z 2m x y z 2mL2 x y z 2mL2 thus L √ L r2m n = 2m, and dn = d (7) ~π 2~π Combining terms into the density of states,

2 r 2mL L 2m L 3 3 √ D3D() = 4π = 2π (2m) 2 (8) ~2π2 2~π ~π The total number of particles can now be expressed as

∞ √ 2π L 3 3 Z N = (2m) 2 d 3D β(−µ) 8 ~π 0 e − 1 ∞ √ 2πV 3 Z 1 = (2m) 2 d 3 β(−µ) −β(−µ) h 0 e 1 − e Z ∞ √ ∞ 2πV 3 X −βl(−µ) = (2m) 2 e d h3 eβ(−µ) 0 l=0 Z ∞ ∞ (9) 2πV 3 √ X −βl(−µ) = (2m) 2 e d h3 0 l=1 Z ∞ ∞ 2πV 3 √ X −βl βlµ = (2m) 2 e e d h3 0 l=1 ∞ 3 βlµ V 2mπ 2 X e = 3 3 h β 2 l=1 l 2

where the factor of 8 has been introduced since we are including only the positive values of n, and thus only the ﬁrst quadrant of the 3D sphere in n-space. The last sum on the right is a polylogarithm function, also called the the weighted Zeta function (weighted by the exponential factor). We have used the expansion condition e−β(−µ) < 1, which is validated by the physical mandate that we do not obtain negative values forn ¯(). It follows then that we posit the restriction > µ. From the expression it can be seen that N3D is a maximum at µ=0, which is therefore when the condensate occurs. The sum can be evaluated and we arrive at the condensate phase transition temperature,

∞ 2 2 3 3 3 V 2mπ 2 X 1 V 2mπ 2 h N3D N3D = 3 3 ≈ 3 (2.612) =⇒ Tc ≈ (10) h β 2 h β 2mπkb 2.612V l=1 l | {z } 3 ζ( 2 ) For massive bosons in 2D we follow the same procedure. The density of states is

dN dN dn dn D () = = = 2πn . (11) 2D d dn d d The energy has a similar form as previously 1 2π2 = (p 2 + p 2) = ~ n2 (12) 2m x y 2mL2

Plugging into the integral for N2D, we note that the density of states here does not depend on the energy. Consequently the integral is over only the distribution function, with a factor of 1/4 that comes from dealing with only the ﬁrst quadrant of the 2D sphere in n-space.

∞ 2π L 2 2m Z ∞ 1 2πmA X eβlµ N = d = (13) 2D 4 π 2 eβ(−µ) − 1 h2β l ~ 0 l=1 For the condensate to occur, µ=0 and the above expression diverges, ζ(1)→ ∞. Hence, the condensate does not occur for massive bosons in 2D. Lastly for the 1D case, the density of states is simply dN dN dn dn D () = = = (1) (14) 1D d dn d d The energy is, again, the same as above 1 2π2 = (p 2) = ~ n2 (15) 2m x 2mL2 So the total number is

∞ √ r ∞ βlµ L √ Z 1/ L 2πm X e N = 2m d = (16) 1D β(−µ) 1 2h e − 1 2h β 2 0 l=1 l There is a factor of 1/2 from dealing with only positive n values. Again, this expression is non-physical for µ=0, so the condensate for massive bosons in 1D does not occur.

II. MASSLESS BOSONS

For massless bosons by contrast, we must express their energy relativistically. Thus from the relation

= c|p| (17) it is evident that the energy is linear in momentum. This alters the conditions for the BEC to occur. In every case the energy is given as nπ L = c~|k| = c~ =⇒ n = (18) L c~π 3

For 3D the density of states is

3 2 dn L 2 D3D() = 4πn = 4π (19) d c~π so the total number is

3 Z ∞ 2 ∞ βlµ 4π L 8πV X e = 8πV 8πV N = d = µ→0 ζ(3) ≈ 1.1202 (20) 3D 8 c π eβ(−µ) − 1 (chβ)3 l3 (chβ)3 (chβ)3 ~ 0 l=1 Now we can calculate the phase transition temperature for massless bosons in 3D:

1 N3D 3 ch Tc ≈ . (21) 8πV (1.1202) kb In 2D the density of states is

dn L 2 D2D() = 2πn = 2π (22) d c~π Note that now the 2D density of states does depend on the energy. The total number of particles is

2 Z ∞ ∞ βlµ 2 2π L 2πA X e = 2πA 2πA π N = d = µ→0 ζ(2) = (23) 2D 4 c π eβ(−µ) − 1 (chβ)2 l2 (chβ)2 (chβ)2 6 ~ 0 l=1 This result shows that massless bosons in 2D do indeed form a condensate, whereas massive bosons in 2D do not. The temperature of condensation here is

1 3N2D 2 ch Tc = 3 (24) Aπ kb Finally, for the 1D system of massless bosons we have a density of states that is independent of energy, just like the 2D massive boson system. dn L D1D() = = . (25) d c~π Again, the integral diverges for µ=0,

Z ∞ ∞ βlµ L 1 L X e ⇒ L N = d = µ→0 ζ(1) ⇒ ∞ (26) 1D 2c π eβ(−µ) − 1 ch l chβ ~ 0 l=1 and thus in the 1D massless case we ﬁnd the same condition as the massive bosons in 1D, i.e., the condensate is forbidden in this geometry.

III. CONCLUSION

For the particle-in-a-box model, BEC’s occur for both massive and massless bosons in 3D. They occur only in the massless case for 2D, and never for 1D. This model can in principle be applied to higher spatial dimensions whereupon evaluation of the total number of particles would be an integral of the form

Z ∞ (q−2)/2 ∼ q N ∼ d µ→0 ζ( )(Massive) qD eβ(−µ) − 1 2 0 (27) Z ∞ (q−1) N ∼ d ∼ ζ(q)(Massless) qD β(−µ) µ→0 0 e − 1 where q is the dimension of the space.