An Improved Calculation of Bose–Einstein Condensation Temperature

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors The author presents an improved calculation of the Bose-Einstein condensation (BEC) temperature, addressing inconsistencies in previous models by rigorously evaluating the thermodynamic balance between coherent (Bose-Einstein condensed) and incoherent (Fermi-Dirac) particle populations. The author use this method to calculate the BEC dynamics for three scenarios: isotropic (3D), stacked 2D, and anisotropic cases. Then various examples and discussions are shown, implying that the theory can be applied to both conventional and high-temperature superconductors. This paper is clear and well-written. It may advance the understanding of BEC in superconductors by integrating thermodynamic principles with quantum statistics, emphasizing the need for further experimental and theoretical exploration of the Fermi-Dirac to Bose-Einstein transition. Below are some comments and suggestion.   1) The reason why "If a stable Fermi-sea and Bose-Einstein condensate co-exist, the \mu = 0 condition represents their balance." should be clarified. 2) Line 357-358: why this is the only way and gives the Eq.(39)? 3) When \mu=0 (fulfilling Rule 1 on page 7), the well-known BEC temperature formulas (e.g., Eqs. 28, 29, and 32) are recovered. However, these results should be contextualized with foundational references, as they align with established BEC theory found in standard textbooks.  4) The reference details must be carefully verified and corrected, as most entries currently lack volume numbers, page ranges, or contain inaccuracies. 5) Line 118: the second “is” should be deleted. 6) Line 169: the second “must” should be deleted. 7) For the famous BEC temperature formula, more original references should be cited. The information of references should be carefully checked and revised. 8) Line 357-385: why this is the only way?

Author Response

Thank you very much for the useful comments. I agree with them, and revised the article accordingly.

1) The reason why "If a stable Fermi-sea and Bose-Einstein condensate co-exist, the \mu = 0 condition represents their balance." should be clarified.

Answer: a brief explanation has been added.

 

2) Line 357-358: why this is the only way and gives the Eq.(39)?

Answer: an explanation has been added.

 

3) When \mu=0 (fulfilling Rule 1 on page 7), the well-known BEC temperature formulas (e.g., Eqs. 28, 29, and 32) are recovered. However, these results should be contextualized with foundational references, as they align with established BEC theory found in standard textbooks. 

Answer: relevant references were added (highlighting the first publication of the critical temperature formula by F London).

 

4) The reference details must be carefully verified and corrected, as most entries currently lack volume numbers, page ranges, or contain inaccuracies.

Answer: done.

 

5) Line 118: the second “is” should be deleted.

Answer: done.

 

6) Line 169: the second “must” should be deleted.

Answer: done.

 

7) For the famous BEC temperature formula, more original references should be cited. The information of references should be carefully checked and revised.

Answer: done (see answers to comments 3 and 4).

 

8) Line 357-385: why this is the only way?

Answer: clarified (see comment 2)

Reviewer 2 Report

Comments and Suggestions for Authors

Bose-Einstein condensate (BEC) is recognized as a fundamental physical phenomenon (special matter state) demonstrating the manifestation of quantum laws at the macroscopic level due to the collective behavior of a coherent quantum ensemble of ultracold dilute gases, as well as quasiparticle modes in condensed matter. BEC research is one of the most important areas in modern physics, so the peer-reviewed work studying BEC models that take into account the influence of temperature effects is relevant.

In this paper, the author considers a statistical approach to the analysis of BEC and, in parallel, the phenomenon of superconductivity with an emphasis on high-temperature superconductivity. It is known that these two fundamental phenomena are different, but there are some connections between them that are traced in this study. We emphasize that this aspect enhances the significance of this study.

The paper discusses an approach to a more convenient estimate of the condensation temperature based on the analysis of the thermodynamic balance between the concentrations of coherent (condensed) and incoherent (non-condensed) particles in the vicinity of the condensation temperature. In the context of the considered approach to the analysis of Bose-Einstein statistics and Fermi-Dirac statistics and the phenomenon of particle condensation, the author notes the similarity of this statistics with the statistics of electrons and electron pairs in the phenomenon of superconductivity, as well as recent achievements in the problem of high-temperature superconductivity.

The method for calculating the condensation temperature proposed by the author is based on the analysis of the statistical equilibrium of populations of Fermi-Dirac particle pairs, unpared Fermi-Dirac particles, and Bose-Einstein condensed particles. Within the framework of the proposed approach, 3D and 2D scenarios are considered, and also the anisotropic case that lies in-between these two topologies.

In the Introduction, the author provides a brief overview of the literature and basic information related to BEC, including useful historical reminders of the discovery and experimental condensate obtaining. The role of temperature, both in the condensation state and in mixed condensate and non-condensate states at low temperatures, is specifically discussed, as well as the physical features of the manifestation of condensation in the phenomena of superfluidity and superconductivity. The presented review provides the necessary insight into the topic of study.

Section 2 presents the calculation of the quantitative characteristics (density) of states of pseudo-free electrons (delocalized in the terminology of this work) of non-interacting particles in a limited box (for example, in a crystal of finite dimensions), and in Section 3 the entropy of these states is calculated and the Bose-Einstein formula for the distribution of discrete levels is derived from the condition of maximum entropy. All these considerations are well known and are given in standard courses in statistical physics and serve as the initial background for the further formulation of the approach.

Section 4 examines the continuum representation of the obtained energy distribution densities of delocalized particles for 2d and 3d cases, from which the author presents an expression for the total number of particles and the particle densities for both cases. The model proposed by the author is essentially based on these expressions. The idea of the work is that non-condensed Fermi particles can transition to a condensed state and back. The author describes the dynamic equilibrium between the states of non-condensed Fermi particles and the states of condensate particles at a given temperature using the chemical potential, which is applied to the system in which the transition between phases occurs.

The balance between the phases in the condensate is described by the condition of the chemical potential being equal to zero, which allows one to estimate the particle density, formula (25) in the 3d case. This formula, as well as its 2d analogue, is the basis of the model considered in this paper. In the following section 5 thermodynamic estimates are derived for the densities of the condensate particles and for the corresponding temperature. In section 6 the relationship between the properties of the condensate and superconductivity is considered. The expressions obtained for the condensate parameters are applied to a number of superconductors, for which the author demonstrates quite satisfactory agreement with the experimental data.

My critical comments are below.

Point 1. In the introduction, the author should have more clearly formulated not only the ideas of the proposed approach, but also the formal methods used in his model. It is also worth highlighting more clearly the novelty of the model, since the calculations performed by the author are based on well-known, fairly simple methods of statistical physics.

Point 2. Some typos are found:

2.1. Comma and question mark in page 2, line 42.

2.2. It is considered that mathematical formulas, including those highlighted in a separate line, are part of a sen-tence and punctuation marks are applied to them as to words. In all formulas, in my opinion, there are not enough punctuation marks: periods, commas. For example, in formulas (1), (2) there are no periods, and in formula (3) there is no comma. The same is true for other formulas.

Point 3. In studies of BEC of diluted alkali metal vapors, a mean-field model is used based on the Gross-Pitaevskii/nonlinear Schrodinger equation, where the interaction of particles is taken into account. However, in this paper, no attention is paid to this well-known approach. Such a discussion seems appropriate to me, since the approach proposed by the author is based on well-known, fairly elementary formulas of equilibrium statistical mechanics and does not take into account explicitly the interactions of the condensate, which is important in the mentioned theories of BEC. In these approaches, estimates for the condensate densities are also obtained, and it would be interesting to compare both approaches.

Summarizing, I believe that the peer-reviewed work is written clearly, logically structured, the conducted research can be considered as a contribution to the description of the BEC phenomenon from the standpoint of thermodynamics and connections with the theory of superconductivity.

The merit of the work is the comparison of the results of BEC modeling with experimental results, which achieves good agreement. In addition, the author substantiates the applicability of the model proposed by him for achieving progress in the problem of high-temperature superconductivity.

Thus, I can recommend this work for acceptance taking into account the comments made.

Author Response

Thank you very much for these useful comments. I agree with them, and revised the article accordingly.

Point 1. In the introduction, the author should have more clearly formulated not only the ideas of the proposed approach, but also the formal methods used in his model. It is also worth highlighting more clearly the novelty of the model, since the calculations performed by the author are based on well-known, fairly simple methods of statistical physics.

Answer: the last part of Introduction section was extended accordingly.

 

Point 2. Some typos are found:

2.1. Comma and question mark in page 2, line 42.

2.2. It is considered that mathematical formulas, including those highlighted in a separate line, are part of a sen-tence and punctuation marks are applied to them as to words. In all formulas, in my opinion, there are not enough punctuation marks: periods, commas. For example, in formulas (1), (2) there are no periods, and in formula (3) there is no comma. The same is true for other formulas.

Answer: the above-mentioned issues were corrected.

 

Point 3. In studies of BEC of diluted alkali metal vapors, a mean-field model is used based on the Gross-Pitaevskii/nonlinear Schrodinger equation, where the interaction of particles is taken into account. However, in this paper, no attention is paid to this well-known approach. Such a discussion seems appropriate to me, since the approach proposed by the author is based on well-known, fairly elementary formulas of equilibrium statistical mechanics and does not take into account explicitly the interactions of the condensate, which is important in the mentioned theories of BEC. In these approaches, estimates for the condensate densities are also obtained, and it would be interesting to compare both approaches.

Answer: I clarified at the start of section 2 that my calculations' scope is non-interacting particle systems. I added a footnote explaining that the allowed quantum mechanical states are solutions of the Gross-Pitaevskii equation when particle interactions cannot be ignored. The density of states is then calculated according such set of allowed states, and the rest of article logic applies in the same way.