Microscopic origin and single-shot interpretation of the density-grain statistics in the exactly solved one-dimensional Bose gas

The exact Yang–Yang solution of the one-dimensional repulsive Bose gas (Lieb–Liniger model) yields closed expressions for all cumulants of the particle number in a mesoscopic interval, from which the statistics of 'density grains' — the independent density-fluctuation domains of the gas — are exactly computable. The resulting phase diagrams contain several unexplained structures: a triangular region of negative grain-occupation skewness at $30/\gamma\lesssim\tau\lesssim 10$ in the fermionized regime, accompanied by moderate leptokurtosis $\kappa_{\mathcal N}\approx 6$; a numerically found crossover line $\tau\approx 30/\gamma$ of unknown physical origin; deeply quantum regimes with mean grain occupation $\overline{\mathcal N}\ll 1$, where the typical independent fluctuation domain carries less than one particle; a platykurtic region near the $\mu=0$ degeneracy crossover; and an untested operational proposal that density lumps visible in single-shot images reproduce the ensemble grain statistics. The problem asks for a microscopic physical account of these anomalies, consistent with and ideally derived from the exact solution, together with a validation or refutation of the single-shot lump–grain correspondence. These questions are open: none of the cited work addresses them.

Background

The one-dimensional repulsive Bose gas — the Lieb–Liniger model, described by the Hamiltonian

$$H=\int dx\,\left[\frac{\hbar^2}{2m}\partial_x\psi^\dagger\partial_x\psi+\frac{g}{2}\psi^{\dagger 2}\psi^2\right]$$

with repulsive contact coupling $g$ — is the paradigmatic exactly solved interacting many-body system: its thermodynamics at any interaction strength and temperature follows from the Yang–Yang thermodynamic Bethe ansatz (TBA), a closed set of integral equations for the rapidity distribution of the bosons, where the rapidity is the quasi-momentum labeling each particle's Bethe-ansatz state. Throughout, the interaction strength is written $\gamma=mg/(\hbar^2 n)$ and the reduced temperature $\tau=2mk_BT/(\hbar^2n^2)=k_BT/T_d$, where $n$ is the particle density, $T_d=\hbar^2n^2/2m$ is the degeneracy temperature, and $\mu$ is the chemical potential. The regime $\gamma\gg1$ is the strongly interacting, 'fermionized' regime (closely related to the Tonks–Girardeau limit, in which impenetrable bosons map onto free fermions).

Full counting statistics (FCS) is the study of the probability distribution of the number $N$ of particles in a spatial interval. For intervals of many correlation lengths (mesoscopic intervals), the Yang–Yang solution gives all cumulants exactly in the grand canonical ensemble — for example the third cumulant obeys $\langle\delta N^3\rangle=(k_BT)^2\partial^2\langle N\rangle/\partial\mu^2$. Pietraszewicz and Deuar used these exact formulas to define the 'density grains' of the gas: the gas behaves as if divided into statistically independent domains of length of order the correlation length, each typically carrying $\overline{\mathcal N}$ particles, with the heuristic identification $\overline{\mathcal N}\approx S_0$, where $S_0=\langle\delta N^2\rangle/\langle N\rangle$ is the number variance per particle. The skewness $s_{\mathcal N}$ and kurtosis $\kappa_{\mathcal N}$ (standardized third and fourth moments of the grain occupation) then follow from the higher cumulants, and the full $(\gamma,\tau)$ phase diagrams of $(\overline{\mathcal N},s_{\mathcal N},\kappa_{\mathcal N})$ can be computed exactly from the TBA.

These exact diagrams contain structures whose physics is not understood. In the fermionized regime the grain-occupation skewness is negative over a triangular region $30/\gamma\lesssim\tau\lesssim 10$, with moderate leptokurtosis $\kappa_{\mathcal N}\approx 6$ (kurtosis above the Gaussian value 3), bounded by temperature in a way whose cause the authors state is presently unknown (they offer only a conjectural 'possible explanation' in terms of a Pauli-like exclusion ceiling on grain occupation). A crossover line $\tau\approx 30/\gamma$ — equivalently $k_BT\approx 15\hbar^4n^3/(m^2g)$ — has a coefficient the authors call 'rather strange', with no known microscopic meaning. In deeply quantum regimes the gas is strongly antibunched (density fluctuations suppressed below the Poissonian, shot-noise level) and $\overline{\mathcal N}\ll 1$: the typical independent density fluctuation carries far less than one particle, whose physical meaning the authors leave as a tentative conjecture. Near the $\mu=0$ crossover between the degenerate and nondegenerate gas the statistics are platykurtic ($\kappa_{\mathcal N}$ below 3, approaching the square-distribution value 1.8 as $\gamma\to\infty$), which also lacks explanation. Finally, the same authors proposed an operational reading of the grains: in a single-shot image (one experimental realization of the density profile), the visible density 'lumps' could be counted — with lump boundaries where the local density crosses its spatial mean — and their statistics might reproduce the thermodynamic grain statistics. Whether single-shot lumps correspond quantitatively to the ensemble-derived independent grains is a long-standing point of discussion in the field and was left to experimental test.

Neighboring exact results delimit, but do not answer, these questions. Kheruntsyan et al. mapped the pair correlations and thermodynamic regimes of the finite-temperature gas; more recent FCS work (Bastianello–Piroli and Bastianello–Piroli–Calabrese) gives exact counting statistics for short intervals, a regime complementary to the long-interval grain picture; Kerr, De Rosi and Kheruntsyan gave analytic thermodynamics across six regimes of the Lieb–Liniger gas but none of the higher-moment structures. What is missing is a microscopic physical account of the grain-statistics anomalies above and a validated single-shot interpretation of the grains.

Problem Statement

Provide a microscopic physical account of the mesoscopic density-grain statistics of the one-dimensional repulsive Bose gas, derived from the Lieb–Liniger model and consistent with the exact Yang–Yang solution, and establish (or refute) the operational identification of the grains with single-shot density lumps. The account should jointly explain, with quantitative control over the $(\gamma,\tau)$ plane, the following exact structures of the grain-occupation cumulants (with $\gamma=mg/(\hbar^2 n)$, $\tau=2mk_BT/(\hbar^2 n^2)$, grain occupation $\mathcal N$ identified through the heuristic relation $\overline{\mathcal N}\approx S_0$):

1. Why, in the fermionized regime $\gamma\gg1$, the skewness $s_{\mathcal N}$ is negative throughout the triangular region $30/\gamma\lesssim\tau\lesssim 10$, accompanied by moderate leptokurtosis $\kappa_{\mathcal N}\approx 6$, and what mechanism bounds this region — in particular, whether the authors' conjectured Pauli-like exclusion ceiling on grain occupation is correct;
2. The physical origin of the crossover line $\tau\approx 30/\gamma$, equivalently $k_BT\approx 15\hbar^4n^3/(m^2g)$: which competing energy or length scales produce the coefficient $C\approx 30$, and does this line bound a distinct regime of the grain statistics;
3. What the sub-unit mean grain occupation $\overline{\mathcal N}\ll 1$ means physically in the strongly antibunched regimes — how the gas organizes number fluctuations into independent domains carrying on average far less than one particle, and in what sense these domains are localized physical events;
4. Why the grain statistics become platykurtic ($\kappa_{\mathcal N}$ below the Gaussian value 3, approaching the square-distribution value 1.8 as $\gamma\to\infty$) near the $\mu=0$ crossover between the degenerate and nondegenerate gas;
5. Whether density lumps counted in a single-shot realization (lump boundaries where the local density crosses its spatial mean) correspond quantitatively to the ensemble-derived independent grains — i.e. whether the proposed lump-counting protocol measures $(\overline{\mathcal N},s_{\mathcal N},\kappa_{\mathcal N})$ within stated statistical uncertainties.

A complete answer demonstrates that the account reproduces the exact Yang–Yang phase diagrams of $(\overline{\mathcal N},s_{\mathcal N},\kappa_{\mathcal N})$ across the relevant $(\gamma,\tau)$ range, including the exceptional cases $\mu=0$ and $\gamma\to\infty$; the published moment formulas serve here only to recompute these diagrams as a benchmark, so reproducing them is not by itself an answer. Parts 1–4 are resolved only by a stated physical mechanism — with a derivation route from the Lieb–Liniger model or an effective theory for it — that reproduces the quoted $(\gamma,\tau)$ boundaries and the sign of $s_{\mathcal N}$ and magnitude of $\kappa_{\mathcal N}$ within a stated tolerance; part 5 is resolved by a stated lump-counting protocol together with a validated or refuted correspondence to the ensemble grains at a stated statistical significance. All five parts are jointly required for a complete answer, and the $\tau\approx30/\gamma$ feature is shared by parts 1 and 2, so a single account of it serves both. The common objective uniting parts 1–5 is a single microscopic-plus-operational theory of the density grains; the extension to still higher cumulants and to phase-domain observables is a secondary extensibility direction and is not part of the completion boundary.

References

1. J. Pietraszewicz and P. Deuar, 'Mesoscopic density grains in a 1D interacting Bose gas from the exact Yang–Yang solution', New J. Phys. 19, 123010 (2017). DOI: 10.1088/1367-2630/aa91c5; arXiv:1708.00031. https://doi.org/10.1088/1367-2630/aa91c5
2. E. H. Lieb and W. Liniger, 'Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State', Phys. Rev. 130, 1605 (1963). DOI: 10.1103/PhysRev.130.1605. https://doi.org/10.1103/PhysRev.130.1605
3. C. N. Yang and C. P. Yang, 'Thermodynamics of a One-Dimensional System of Bosons with Repulsive Delta-Function Interaction', J. Math. Phys. 10, 1115 (1969). DOI: 10.1063/1.1664947. https://doi.org/10.1063/1.1664947
4. K. V. Kheruntsyan, D. M. Gangardt, P. D. Drummond and G. V. Shlyapnikov, 'Pair Correlations in a Finite-Temperature 1D Bose Gas', Phys. Rev. Lett. 91, 040403 (2003). DOI: 10.1103/PhysRevLett.91.040403. https://doi.org/10.1103/PhysRevLett.91.040403
5. J. Pietraszewicz and P. Deuar, 'Classical fields in the one-dimensional Bose gas: Applicability and determination of the optimal cutoff', Phys. Rev. A 98, 023622 (2018). DOI: 10.1103/PhysRevA.98.023622. https://doi.org/10.1103/PhysRevA.98.023622
6. J. Pietraszewicz and P. Deuar, 'Complex wave fields in the interacting one-dimensional Bose gas', Phys. Rev. A 97, 053607 (2018). DOI: 10.1103/PhysRevA.97.053607. https://doi.org/10.1103/PhysRevA.97.053607
7. M. Arzamasovs and D. M. Gangardt, 'Full Counting Statistics and Large Deviations in a Thermal 1D Bose Gas', Phys. Rev. Lett. 122, 120401 (2019). DOI: 10.1103/PhysRevLett.122.120401. https://doi.org/10.1103/PhysRevLett.122.120401
8. J. De Nardis and M. Panfil, 'Edge Singularities and Quasilong-Range Order in Nonequilibrium Steady States', Phys. Rev. Lett. 120, 217206 (2018). DOI: 10.1103/PhysRevLett.120.217206. https://doi.org/10.1103/PhysRevLett.120.217206
9. G. Perfetto and A. Gambassi, 'Dynamics of large deviations in the hydrodynamic limit: Noninteracting systems', Phys. Rev. E 102, 042128 (2020). DOI: 10.1103/PhysRevE.102.042128. https://doi.org/10.1103/PhysRevE.102.042128
10. M. B. Kruk, P. Kulik, M. F. Andersen, P. Deuar, M. Gajda, K. Pawłowski, E. Witkowska, J. J. Arlt and K. Rzążewski, 'On the fluctuations of the number of atoms in the condensate', Rep. Prog. Phys. 88, 106401 (2025). DOI: 10.1088/1361-6633/ae0e33. https://doi.org/10.1088/1361-6633/ae0e33
11. A. Bastianello and L. Piroli, 'From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and full counting statistics', J. Stat. Mech. (2018) 113104. DOI: 10.1088/1742-5468/aaeb48; arXiv:1807.06869. https://doi.org/10.1088/1742-5468/aaeb48
12. A. Bastianello, L. Piroli and P. Calabrese, 'Exact local correlations and full counting statistics for arbitrary states of the one-dimensional interacting Bose gas', Phys. Rev. Lett. 120, 190601 (2018). DOI: 10.1103/PhysRevLett.120.190601; arXiv:1802.02115. https://doi.org/10.1103/PhysRevLett.120.190601
13. M. L. Kerr, G. De Rosi and K. V. Kheruntsyan, 'Analytic thermodynamic properties of the Lieb-Liniger gas', SciPost Phys. Core 7, 047 (2024). DOI: 10.21468/SciPostPhysCore.7.3.047; arXiv:2404.06092. https://doi.org/10.21468/SciPostPhysCore.7.3.047