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The distribution of the moment of inertia for harmonically trapped noninteracting Bosons at finite temperature: large deviations

This paper computes the full probability distribution of the moment of inertia for a harmonically trapped noninteracting Bose gas in the thermodynamic limit, demonstrating that its large deviation rate function exhibits a singularity at a critical value that serves as a real-space diagnostic for the Bose-Einstein condensation transition in dimensions greater than two.

Original authors: Manas Kulkarni, Satya N. Majumdar, Gregory Schehr

Published 2026-08-27

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CC BY 4.0

Original authors: Manas Kulkarni, Satya N. Majumdar, Gregory Schehr

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet world of ultra-cold atoms, scientists have long watched a peculiar transformation occur. When a cloud of certain particles, known as bosons, is cooled down enough, they stop acting like individual travelers and begin to move as a single, unified entity. This phenomenon, called Bose-Einstein condensation, is a fundamental state of matter where a large fraction of the particles collapse into the lowest possible energy state. For decades, physicists have understood this transition by looking at the energy levels of the particles, essentially counting how many are sitting in the ground floor versus the upper floors of a building. However, a different question has remained harder to answer: if you were to take a photograph of these particles in space, what would you see? Would the arrangement of the particles themselves reveal the moment this collective shift happens? While the energy changes are well known, the spatial signature of this transition in a trapped gas has been elusive, leaving a gap between the theoretical understanding of energy and the visual reality of the atoms' positions.

A team of researchers has now filled this gap by calculating exactly how the spatial arrangement of these atoms changes as they approach this critical point. They focused on a specific measurement called the moment of inertia, which in this context acts as a simple way to gauge how spread out the cloud of atoms is. Imagine the atoms as a swarm of bees trapped inside a bowl-shaped force field; the moment of inertia tells you whether the bees are huddled tightly near the center or spread out toward the edges. The researchers studied a gas of non-interacting bosons, meaning the atoms do not push or pull on each other, trapped in a harmonic potential, which is a smooth, bowl-shaped force field that gets stronger the further you move from the center. By analyzing the mathematics of this system at various temperatures and densities, they discovered that the way this "spread" behaves holds a secret signal about the phase transition, one that can be detected even before the transition fully occurs.

The study reveals that the probability of finding the atoms in a certain configuration follows a precise pattern that changes dramatically at a critical density. As the density of the gas increases, the system moves from a fluid-like state, where the atoms are more dispersed, to a condensed state where they crowd together. The researchers found that the mathematical function describing the likelihood of different spreads, known as the rate function, develops a sharp kink at a specific point. This kink is not a smooth curve but a place where the function's behavior changes abruptly, signaling the onset of the condensation. Remarkably, this kink appears in the mathematical description even when the gas is still in the fluid phase, meaning the system is technically not yet condensed. By measuring the distribution of the moment of inertia and looking for this specific irregularity in the data, an observer could detect the impending transition without ever having to push the density high enough to actually trigger it.

This finding offers a new way to diagnose the state of a quantum gas using real-space observations rather than just energy measurements. The researchers showed that once the system crosses the critical density and enters the condensed phase, the part of the distribution describing smaller spreads becomes "frozen." This means that adding more atoms to the gas no longer changes the shape of the distribution for those smaller values; the added atoms simply push the distribution outward, affecting only the larger spreads. This freezing effect is a direct consequence of the condensation and serves as a clear fingerprint of the new phase. The work provides a rigorous theoretical prediction for what experimentalists might see if they were to take many snapshots of a trapped Bose gas and measure how far the atoms are from the center in each image.

The significance of this result lies in its ability to bridge the gap between abstract energy states and tangible spatial patterns. While the transition was previously understood through the lens of energy levels, this study proves that the spatial arrangement of the particles carries the same information. The researchers demonstrated that the singularity in the distribution of the moment of inertia is a direct consequence of the Bose-Einstein condensation. If the system were in a dimension where such a transition does not occur, this sharp kink would be absent, and the distribution would remain smooth. This distinction allows the moment of inertia to serve as a diagnostic tool: by analyzing the shape of the distribution, one can determine whether the system is capable of undergoing this transition, even if the current conditions keep it in the fluid phase.

The calculations were performed for a gas in a harmonic trap, a setup common in modern cold atom experiments. The researchers considered the limit where the number of atoms becomes very large and the trap becomes very wide, a scenario that mimics the behavior of a bulk material. In this limit, they derived the exact probability distribution for the moment of inertia across all temperatures and dimensions. They found that for dimensions greater than one, the system exhibits the condensation transition, and the associated rate function displays the predicted singularity. For dimensions one and below, where no such transition occurs, the function remains smooth and analytic. This confirms that the sharp feature in the distribution is inextricably linked to the existence of the condensation itself.

One of the most striking aspects of the work is the idea that the system "remembers" the condensed phase even when it is not in it. The researchers showed that by probing the tails of the distribution—looking at rare events where the atoms are unusually spread out or unusually clustered—one can access information about the condensed phase while the system remains in the fluid phase. This is similar to how a tilted potential in other physical systems can reveal rare events that are not seen in normal conditions. In this case, the mathematical structure of the moment of inertia allows the system to reveal the signature of the condensed phase without the need to actually reach the critical density. This suggests that experimentalists might be able to detect the onset of condensation by analyzing the statistical fluctuations of the cloud's size, rather than waiting for the cloud to visibly collapse.

The study also clarifies the nature of the transition in different dimensions. In higher dimensions, the transition is "normal," characterized by a standard quadratic behavior near the critical point. In lower dimensions, the transition is "anomalous," involving more complex mathematical behaviors. The researchers focused their detailed analysis on the normal case to keep the explanation clear, but noted that the fundamental result—the existence of the singularity in the rate function—holds true regardless of the dimension, provided the transition exists. This universality reinforces the idea that the spatial distribution of the atoms is a robust indicator of the phase transition.

Ultimately, this work provides a concrete, calculable prediction for the spatial statistics of a trapped Bose gas. It moves beyond the traditional focus on energy and momentum to show that the positions of the particles themselves encode the physics of the transition. For experimentalists working with cold atoms, this offers a new avenue for investigation. By recording real-space images of the gas and computing the moment of inertia for each snapshot, they can construct a histogram of these values. If the system is near the critical density, the shape of this histogram should reveal the predicted singularity, offering a direct, real-space confirmation of the Bose-Einstein condensation. This approach could prove invaluable in systems where the bosons interact weakly, as the theoretical framework suggests that the underlying mechanism for this spatial signature might persist even when interactions are introduced.

The research stands as a testament to the power of exact theoretical calculations in guiding experimental observation. By solving the problem for non-interacting particles, the authors have established a baseline against which real-world experiments can be compared. While actual atoms in a lab do interact, the clear, sharp signal predicted here provides a target for what to look for. If future experiments can measure the distribution of the moment of inertia with sufficient precision, they may be able to observe this freezing of the distribution and the associated kink, confirming the theoretical picture of how a gas of bosons organizes itself in space as it cools. The work transforms an abstract mathematical concept into a tangible, measurable quantity, bringing the invisible transition of the quantum world into the realm of spatial statistics.

Problem Statement
The paper investigates the spatial statistical properties of a gas of $N$ noninteracting bosons trapped in a harmonic potential $V(r) = \frac{1}{2}m\omega^2 r^2$ at finite temperature. While the mechanism of Bose-Einstein condensation (BEC) for noninteracting bosons is well understood in energy space, the authors address the challenge of identifying the thermodynamic BEC transition through real-space configurations. Specifically, they focus on the probability distribution of the moment of inertia, defined as $I \propto \sum_{i=1}^N \vec{r}_i^2$ , where $\vec{r}_i$ is the position of the $i$ -th particle. The study aims to compute the full probability distribution of this observable in the thermodynamic limit across all dimensions $d$ and temperatures, determining how the BEC transition manifests in the statistics of $I$ .

Methodology
The authors employ the canonical ensemble, fixing the number of particles $N$ and the inverse temperature $\beta$ . The thermodynamic limit is defined by taking $N \to \infty$ and $\omega \to 0$ while keeping the product $\rho = N\omega^d$ fixed, where $\rho$ acts as an effective density.

The core methodological innovation is the utilization of a "duality relation" (derived in Appendix A) that connects the cumulant generating function of the moment of inertia, $\langle e^{-\lambda I} \rangle$ , to the canonical partition function $Z(N, \beta)$ . The relation is given by:
$\langle e^{-\lambda I} \rangle = \frac{Z(N, \tilde{\beta}(\lambda))}{Z(N, \beta)}$
where the effective inverse temperature $\tilde{\beta}(\lambda)$ is determined by $\cosh(\tilde{\beta}\hbar\omega) = \cosh(\beta\hbar\omega) + \lambda\hbar\omega\sinh(\beta\hbar\omega)$ . This identity allows the authors to bypass the difficult computation of many-body wavefunctions and multidimensional integrals over spatial coordinates. Instead, the problem reduces to analyzing the asymptotic behavior of the partition function $Z(N, \beta)$ in the large $N$ limit.

The analysis of $Z(N, \beta)$ is performed using a saddle-point approximation (Bromwich integral) in the complex plane. The authors distinguish between the "fluid phase" ( $\rho < \rho_c$ ), where a saddle point exists, and the "condensed phase" ( $\rho > \rho_c$ ), where the saddle point vanishes and the integral is dominated by the behavior near the origin. The paper restricts its detailed analysis of the transition region to dimensions $d > 2$ (the "normal" condensation case), noting that $1 < d \le 2$ involves "anomalous" condensation with different singular behaviors.

Key Contributions and Results

Exact Large Deviation Form: The authors derive the full probability distribution of the moment of inertia in the thermodynamic limit, showing it admits a large deviation form:
$P_\beta(I, N) \sim e^{-V \Phi(I/V)}$
where $V = \omega^{-d}$ is a large parameter and $\Phi(z)$ is the rate function.
Singularity in the Rate Function: The primary result is the characterization of the rate function $\Phi(z)$ . The authors demonstrate that for systems exhibiting a BEC transition ( $d > 1$ ), $\Phi(z)$ possesses a singularity at a critical value $z = z_c$ . At this point, the second derivative $\Phi''(z)$ undergoes a discontinuous jump, while $\Phi(z)$ and $\Phi'(z)$ remain continuous. This indicates a second-order phase transition in the large deviation function.
Phase Diagram and Freezing: The analysis reveals a complex phase structure in the $(\lambda, \rho)$ plane involving four phases: fluid-fluid (F-F), fluid-condensed (F-C), condensed-fluid (C-F), and condensed-condensed (C-C).
- Freezing Phenomenon: On the condensed side ( $\rho > \rho_c$ ), the rate function $\Phi(z)$ becomes independent of the density $\rho$ for values $z < z_c$ . As density increases, the "frozen" part of the distribution remains unchanged, while the distribution for $z > z_c$ shifts, resembling a traveling front scenario.
- Critical Point Alignment: The location of the singularity $z_c$ relative to the minimum of the rate function $z_{min}$ (which corresponds to the mean moment of inertia) depends on the phase. In the fluid phase ( $\rho < \rho_c$ ), $z_c < z_{min}$ ; in the condensed phase ( $\rho > \rho_c$ ), $z_c > z_{min}$ . At the critical density $\rho = \rho_c$ , these points coincide.
Real-Space Diagnostic: The authors show that the mean $\langle I \rangle$ and variance $\text{Var}(I)$ exhibit discontinuities in their first and second derivatives with respect to density at the critical point, respectively. Furthermore, the variance freezes in the condensed phase.

Significance and Claims
The paper claims that the moment of inertia serves as a "real space diagnostic" for the BEC transition in noninteracting bosons. A significant finding is that one can detect the existence of a BEC transition even if the system is strictly in the fluid phase ( $\rho < \rho_c$ ). By measuring the distribution of $I$ and analyzing the singularity in the associated rate function $\Phi(z)$ , the presence of a critical point $z_c$ signals the potential for condensation, effectively providing information about the condensed phase without needing to tune the system's density across the critical threshold.

The authors note that while the BEC transition is well-known in momentum/energy space, this work provides an exact finite-temperature result for a spatial observable, which is typically difficult to compute due to the complexity of many-body eigenfunctions. They suggest that these theoretical predictions could serve as benchmarks for future experiments using quantum microscopes to image bosons in real space, although they acknowledge that real systems typically involve interactions. The paper concludes by identifying the extension to interacting bosons and anisotropic traps as open problems, noting that the specific duality relation used here relies on the isotropic harmonic nature of the trap and the non-interacting assumption.

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