Some of the thermodynamic properties of weakly interacting Bose systems are derived from dimensional and heuristic arguments and thermodynamic relations, without resorting to statistical mechanics. Our approach assumes only the existence of a branch of acoustic phonons in the interacting system.

1.
T.
Nattermann
, “
A scaling approach to ideal quantum gases
,”
Am. J. Phys.
73
,
349
356
(
2005
). Note that the mean particle spacing was denoted in this article by a, but in this publication it is denoted by .
2.

We derive all expressions in d=3 dimensions. The generalization of our results to general d is straightforward.

3.
L. D.
Landau
and
E. M.
Lifshitz
,
Quantum Mechanics
, 3rd ed. (
Pergamon
,
Oxford
,
1991
).
4.

With [m]=M, []=ML2t1, [U]=ML2t2, and [U0]=ML5t2, where L, M, and t stand for the dimensions of length, mass, and time, respectively, we find from the ansatz [U0αmβγ]=[a]=Lα=β=γ2=1, and hence amU02.

5.
L. D.
Landau
and
E. M.
Lifshitz
,
Fluid Dynamics
, 2nd ed. (
Pergamon
,
Oxford
,
1987
).
6.
P. W.
Anderson
,
Basic Notions of Condensed Matter Physics
(
Benjamin/Cummings
,
Menlo Park, CA
,
1984
).
7.

Some of the formulas resemble relativistic counterparts, for example, ξ0 plays the role of the Compton wave length, mcs2 that of the rest energy, but the crossover from nonrelativistic to relativistic behavior is opposite to the conventional one: particles with low momenta behave as relativistic, those with large momenta as nonrelativistic.

8.
Most of the results for the weakly interacting Bose gas were originally obtained by
T. D.
Lee
,
K.
Huang
, and
C. N.
Yang
, in “
Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties
,”
Phys. Rev.
106
,
1135
1146
(
1957
).
9.
We refer here to
L. D.
Landau
and
E. M.
Lifshitz
,
Statistical Physics, Part 2
, 2nd ed. (
Pergamon
,
Oxford
,
1981
), Eq. (25.11).
10.
More results on the weakly interacting Bose gas can be found in
J. O.
Andersen
, “
The weakly interacting Bose gas
,”
Rev. Mod. Phys.
76
,
599
639
(
2004
).
11.
Reference 9, Eq. (25.19).
12.
Reference 9, Eq. (25.14).
13.
Reference 9, Eq. (25.15).
14.

We assume here sufficiently low temperatures kBTmc2, where c denotes the speed of light so that there is no particle antiparticle generation.

15.
L. D.
Landau
and
E. M.
Lifshitz
,
Statistical Physics, Part 1
, 3rd ed. (
Pergamon
,
Oxford
,
1980
), Eq. 62.9.
16.
Reference 15, Eq. (62.2).
17.
Reference 15, Eq. (63.13).
18.
Reference 9, Eq. (27.10).
19.
Reference 9, Eq. (22.4).
20.
V. N.
Popov
,
Functional Integrals in Quantum Field Theory and Statistical Physics
(
Reidel
,
Dortrecht
,
1983
).
21.
Reference 9, Sec. 23.
22.
See, for example,
I. M.
Khalatnikov
,
An Introduction to the Theory of Superfluidity
(
Benjamin
,
New York
,
1965
).
23.
Reference 9, Eq. (23.7).
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