Finite-size effects with boundary conditions on Bose-Einstein condensation

We investigate the statistical distribution that governs an ideal Bose gas with fixed particle number density in the limited cubic box. By adjusting the spatial sizes and imposing the boundary conditions that can be manipulated by phase factor, we numerically calculate the critical temperature of Bose-Einstein condensation to analyse the statistical properties in these systems. We find that, when the space size of the systems is small enough, finite-size effects can have a significant impact on the critical temperature of Bose-Einstein condensation. Moreover, we predict that in the case of extremely small spatial sizes, less than a dozen of bosons can also achieve Bose-Einstein condensation below the critical temperature.

The theory of Bose-Einstein condensation, predicted by Einstein, reveals that a macroscopic population of Bose gas will fall into the lowest energy quantum mechanical state below the crtical temperature. It is known that the theory has been developed in the thermodynamic limit. However, the corresponding experiments can not meet the strict thermodynamic limit [9][10][11][12]35], and according to the Weyl-Courant principle, the quantum properties are insensitive to the shapes and the boundary conditions of macroscopic systems. A natural problem has arisen followed, what will happen to Bose-Einstein condensation in extremely small volume systems? In fact, non-equilibrium Bose-Einstein condensation of less than ten photons has been realized experimentally [36]. This inspires us to explore the physics of Bose-Einstein condensation in limited volume systems.
Historically, in 1938, London simply considered a system of free particles of mass M confined in a cubic volume of linear dimension L, without an external trapping potential [37,38]. Intuitively, one expects some influences of boundary conditions even in the large-system limit. Subsequently, the periodic boundary condition and the Dirichlet boundary condition are imposed on the systems [39]. In order to learn the Bose-Einstein condensation in limited volume systems clearly, in this paper, similarly, we take the systems consisting of an ideal Bose gas with fixed particle number density confined in the limited cubic box. Then we will investigate the finitesize effects with boundary conditions on T c . This paper is organized as follows. In Section II, the specific formulas of the statistical distribution of an ideal Bose gas with fixed particle number density confined in the limited cubic box are directly given. In Section III, finite-size effects on BEC in the limited cubic box are investigated. In Section IV, in the case of boundary conditions that can be manipulated by phase factor, numerical changes on critical temperatures are obtained. Finally, in Section V, conclusions and discussions are briefly given.

II. THE BOSE DISTRIBUTION IN THE LIMITED CUBIC BOX
In this section, the formulas of the statistical distribution of Bose gas with fixed particle number density, confined in the limited cubic box with linear dimension L and boundary conditions will be given. We start from the review of the Bose distribution in the thermodynamic limit. For the given particle number density n, the fundamental formulation can be expanded as where µ is the chemical potential, k denotes the Boltzmann's constant and { l } is the single-particle energy spectrum. Since both energy and momentum are quasicontinuous in the thermodynamic limit, it is easy to substitute the integral for the summation: arXiv:1907.07917v3 [cond-mat.quant-gas] 4 Aug 2019 whereh is the Plank constant divided by 2π, m is the mass of the boson. Here it should be noted that this formula is only applicable to the case where the thermodynamic limit or the energy level spacing is much smaller than kT . Just as shown in Eq. (2), when the temperature drops to a certain critical temperature T c , µ → 0 ( 0 is the ground state). Hence the critical temperature T c is determined by the following formula: That is to say, below a given temperature, the population of the lowest quantum state becomes macroscopic and this corresponds to the onset of Bose-Einstein condensation. Note that it is necessary to remove the ground state in the process of calculation. Therefore, the critical temperature T c can be calculated as: We now put bosons into a limited cubic box of linear dimension L, which leads to discretization of energy and momentum. Consequently, we should use summation as Eq.
(1) instead of the integral of the thermodynamic limit as Eq. (3) to calculate the critical temperature.
In addition, it is the finite small volume that we can introduce the boundary conditions. Most generally, the boundary conditions can be expressed as ψ( L 2 ) = e iα ψ(− L 2 ), where e iα denotes the phase factor [40], and due to the periodicity of the function, the phase α ranges from 0 to 2π. It is known that the phase factor is defined for any evolution of a quantum system. Particularly, the phase factor represents for a given projection of the evolution on the projective space of rays of the Hilbert space [41]. That is to say, each phase factor in quantum systems is not only associated with a new geometric phase factor, but also a particular boundary condition. So that we can take differernt boundary conditions by manipulating the phase α. As a consequence, the quantized energy levels of each boson of mass m with boundary conditions can be expressed as: where the quantum states are characterized by the quantum numbers (n 1 , n 2 , n 3 ), (n 1 , n 2 , n 3 = 0, ±1, ±2, · · · ). Combing the Eqs. (1) and (5), the Bose gas with fixed particle number density n confined in limited cubic box with linear dimension L can be expressed by: Here it should be noted that the ranges of these quantum numbers in the process of the summation are determined by the boundary conditions. For clarity, here we define α = kπ ( k∈ [0, 2] ).
(iii)k = 1, the ground state (−1, According to the Eq. (6), it is straightforward to notice that the Bose distribution in the limited cubic box is not only associates with the particle number density, but also depends on the spatial sizes and the boundary conditions. In the following, we will investigate how these factors influence the statistical law in these systems.

III. FINITE-SIZE EFFECTS ON BOSE-EINSTEIN CONDENSATION
In this section, we adjust the spatial sizes with the periodic boundary condition, the counter-periodic boundary condition and the Dirichlet boundary condition to analyse the BEC in the limited cubic box.
(10) Likewise, the critical temperature can be calculated by:    (c) Dirichlet boundary condition Here we take ψ( L 2 ) = ψ(− L 2 ) = 0 to meet the Dirichlet boundary condition. In this case, the single particle  energy is: It should be noted that under this boundary condition, the range of the quantum numbers (n 1 , n 2 , n 3 ) are from 1 to ∞. In the process of the summation, the quantum numbers of comprise the triples (n 1 , n 2 , n 3 ), only the ground state (1, 1, 1), 0 = 3h 2 π 2 2mL 2 excluded. The critical temperature can be derived by: Similarly, when T < T c , the ground state particle number density can be derived by:  By calculating numerically, the variations of the critical temperature confined in the limited cubic box of linear dimension L, with the periodic boundary condition, the counter-periodic boundary condition and the Dirichlet boundary condition are shown in Figs. 1, 3,and 5, respectively. As shown in these figures, no matter what the specific boundary conditions we take, at a fixed particle number density, the critical temperature T c considerably increases as the spatial size L decreases in the extremely small size region. And when L > 2 × 10 −3 cm, the values of T c gradually return to the thermodynamic limit results, as shown in Eq. (4). That is exactly the Weyl-Courant principle expects. Additionally, the variations of the number density of ground state particles with temperature under the periodic boundary condition, the counter-periodic boundary condition and the Dirichlet boundary condition are shown in Figs. 2, 4, and 6, respectively. It is straightforward to observe that when the spatial size and particle number density are fixed, below the critical temperature T c , the number density of ground state particles gradually increases with decreasing temperature. Particularly, combining the variations of critical temperature and ground state particle number density in the above figures, when the spatial size L reaches a sufficiently small magnitude, it is find that less than a dozen of bosons can form Bose-Einstein condensation and its critical temperature increases.

IV. INFLUENCES OF BOUNDARY CONDITIONS ON CRITICAL TEMPERATURE IN THE LIMITED CUBIC BOX
Another important consequence in the present paper is that the Bose statistical properties change with the boundary conditions. According to the Figs. 1, 3,and 5, it is easy to find that there are changes of the critical temperatures with different boundary conditions in the same spatial sizes. In order to investigate the impact of different boundary conditions on the Bose distribution in detail, we numerically calculate the critical temperature via Eq. (6) with L = 2×10 −4 cm, 2×10 −3 cm, 1×10 −2 cm, respectively. For the sake of expression, we consider the quantum numbers (n 1 , n 2 , n 3 ) from −∞ to ∞, and the phase α from 0 to 2π. The calculated results are plotted in Figs. 7, 8 and 9, respectively. Based on the Eq. (6) and the summation process, for the systems with fixed particle number density, in the [0, 2π] interval, the temperature varies with α should be symmetric about a = π. As shown in Figs. 7, 8 and 9, it is obvious that the variation of the critical temperature is symmetric during [0, 2π], which is in accord with our analyses. From these Figures, as the phase α changes,  the boundary conditions gradually change from periodic boundary condition to counter-periodic boundary condition, the critical temperature become lower. Particularly, as an important feature of the critical temperature, its maximum are located at α = π. That is to say, the counter-periodic boundary condition is more capable of changing the critical temperature in the small enough limited volume systems. In addition, as the spatial size L increases and getting close to the thermodynamic limit, the influence of the adjustable boundary conditions on critical temperature becomes smaller and smaller. As a consequence, the critical temperature is sensitive to the boundary conditions that can be manipulated by the phase factors in the small enough limited volume systems.

V. CONCLUSION AND DISCUSSION
In this paper, we have considered the systems consisting of Bose gas with fixed particle number density confined in the limited cubic box. In terms of the Bose distribution, as shown in Eq. (6), we characterize the statistical properties by the critical temperature T c and ground state particle number density n 0 in these systems. And we have investigated the corresponding variations by adjusting the spatial size and regulatable boundary conditions.
On the base of the theoretical analysis and numerical calculation, the spatial sizes and the boundary conditions that can be manipulated by phase factors contribute to influencing the critical temperature only in the extremely small limited volume systems. Especially, the critical temperature is more sensitive to the smaller spatial size and counter-periodic boundary condition in these systems. Particularly, we predict that the extremely small size is capable of allowing less than a dozen of bosons to form BEC below the critical temperature, which is subject to experimental testing. As a consequence, both the spatial sizes and the boundary conditions can be effective to influence Bose distribution in limited volume systems.
Superconductivity and superfluidity, like Bose-Einstein condensation (BEC), are among the most fascinating phenomena in nature [42]. Moreover, these have a great deal in common with BEC and can be described by similar theoretical ideas. Therefore, it is expected that finite-size effects can have important influences on superconductivity and superfluidity, which needs a further investigation.