Limit theorems for statistics of combinatorial partitions with applications to mean field Bose gas

Benfatto, G.; Cassandro, M.; Merola, Immacolata; Presutti, Errico

doi:10.1063/1.1855933

Cited by 17 publications

(31 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The situation is then similar to the ideal gas, and we expect the formula (1) to hold with c n x given by (10), and c 1 x 1 for all x.…”

mentioning

confidence: 86%

Relation between Feynman Cycles and Off-Diagonal Long-Range Order

Ueltschi

2006

Phys. Rev. Lett.

View full text Add to dashboard Cite

The usual order parameter for Bose-Einstein condensation involves the off-diagonal correlation function of Penrose and Onsager, but an alternative is Feynman's notion of infinite cycles. We present a formula that relates both order parameters. We discuss its validity with the help of rigorous results and heuristic arguments. The conclusion is that infinite cycles do not always represent the Bose condensate.

show abstract

“…The situation is then similar to the ideal gas, and we expect the formula (1) to hold with c n x given by (10), and c 1 x 1 for all x.…”

mentioning

confidence: 86%

Relation between Feynman Cycles and Off-Diagonal Long-Range Order

Ueltschi

2006

Phys. Rev. Lett.

View full text Add to dashboard Cite

show abstract

“…This case then forces all nonzero permitted values of the cycle counts to diverge to infinity, and thus they are naturally unlikely, and the 'greater than m Λ ' states are not allowed to support any particle mass. Therefore, the spatial cycle HYL model in this case is essentially equal to the particle mean field model studied for example in [1,7].…”

Section: Theorem 22 (Large Deviation Principles and Pressure Representationmentioning

confidence: 99%

An Explicit Large Deviation Analysis of the Spatial Cycle Huang–Yang–Luttinger Model

Adams

Dickson

2021

Ann. Henri Poincaré

View full text Add to dashboard Cite

We introduce a family of ‘spatial’ random cycle Huang–Yang–Luttinger (HYL)-type models in which the counter-term only affects cycles longer than some cut-off that diverges in the thermodynamic limit. Here, spatial refers to the Poisson reference process of random cycle weights. We derive large deviation principles and explicit pressure expressions for these models, and use the zeroes of the rate functions to study Bose–Einstein condensation. The main focus is a large deviation analysis for the diverging counter term where we identify three different regimes depending on the scale of divergence with respect to the main large deviation scale. Our analysis derives explicit bounds in critical regimes using the Poisson nature of the random cycle distributions.

show abstract

“…See [20] and [21] for proofs of this coincidence in the ideal Bose gas and some mean-field models. A different line of research is studying the effect of the symmetrization in random permutation and random partition models (see [1][2][3][4]23], or in spatial random permutation models going back to [11] and extended in [5]).…”

mentioning

confidence: 99%

A variational formula for the free energy of an interacting many-particle system

Adams¹,

Collevecchio²,

König³

2011

Ann. Probab.

View full text Add to dashboard Cite

We consider N bosons in a box in R d with volume N/ρ under the influence of a mutually repellent pair potential. The particle density ρ ∈ (0, ∞) is kept fixed. Our main result is the identification of the limiting free energy, f (β, ρ), at positive temperature 1/β, in terms of an explicit variational formula, for any fixed ρ if β is sufficiently small, and for any fixed β if ρ is sufficiently small. The thermodynamic equilibrium is described by the symmetrized trace of e −βH N , where HN denotes the corresponding Hamilton operator. The well-known Feynman-Kac formula reformulates this trace in terms of N interacting Brownian bridges. Due to the symmetrization, the bridges are organized in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviations analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process.In our proof of the lower bound for the free energy, we drop all interaction involving "infinitely long" cycles, and their possible presence is signalled by a loss of mass of the "finitely long" cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the "finitely long" cycles. We expect that the precise relationship between these two bounds lies at the heart of Bose-Einstein condensation and intend to analyze it further in future.

show abstract

Limit theorems for statistics of combinatorial partitions with applications to mean field Bose gas

Cited by 17 publications

References 11 publications

Relation between Feynman Cycles and Off-Diagonal Long-Range Order

Relation between Feynman Cycles and Off-Diagonal Long-Range Order

An Explicit Large Deviation Analysis of the Spatial Cycle Huang–Yang–Luttinger Model

A variational formula for the free energy of an interacting many-particle system

Contact Info

Product

Resources

About