Cluster Expansion in the Canonical Ensemble

Pulvirenti, Elena; Tsagkarogiannis, Dimitrios

doi:10.1007/s00220-012-1576-y

Cited by 42 publications

(79 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To prove Theorem 2.1, we follow the strategy presented in [42]. As a result, the radius of convergence or the value of c 0 can be determined in the same way as in [42].…”

Section: )mentioning

confidence: 99%

“…To prove Theorem 2.1, we follow the strategy presented in [42]. As a result, the radius of convergence or the value of c 0 can be determined in the same way as in [42]. However, one can easily obtain slightly better values by following the machinery developed in [13] and also the improvements on the tree-graph inequality in [41] and applied in the case of the canonical ensemble as in [30].…”

Section: )mentioning

confidence: 99%

“…Then, for all other polymers, A = ∅, we can have a connection structure as explained below (and as in [42]). Since it is always true that the total number of labels (m + …”

Section: Leading Order Terms Proof Of Theorem 27mentioning

confidence: 99%

“…In [42] the validity of the cluster expansion in the canonical ensemble has been established for the free energy combining the cluster expansion techniques for abstract polymer model and tree-graph estimates for particle systems. Because of the latter, no significant improvement for the radius of convergence for the virial expansion over the activity expansion was achieved.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, there is no known analogue of the tree-graph inequality for two-connected graphs. We bypass this by using the new technique of [42] developed for the canonical ensemble. However, the direct correlation function is an example of an object that is not a thermodynamical object in the canonical ensemble, but the technique is nevertheless applicable.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Convergence of Density Expansions of Correlation Functions and the Ornstein–Zernike Equation

Kuna

Tsagkarogiannis

2018

Ann. Henri Poincaré

Self Cite

View full text Add to dashboard Cite

Abstract.We prove absolute convergence of the multi-body correlation functions as a power series in the density uniformly in their arguments. This is done by working in the context of the cluster expansion in the canonical ensemble and by expressing the correlation functions as the derivative of the logarithm of an appropriately extended partition function. In the thermodynamic limit, due to combinatorial cancellations, we show that the coefficients of the above series are expressed by sums over some class of two-connected graphs. Furthermore, we prove the convergence of the density expansion of the "direct correlation function" which is based on a completely different approach and it is valid only for some integral norm. Precisely, this integral norm is suitable to derive the OrnsteinZernike equation. As a further outcome, we obtain a rigorous quantification of the error in the Percus-Yevick approximation.

show abstract

“…To prove Theorem 2.1, we follow the strategy presented in [42]. As a result, the radius of convergence or the value of c 0 can be determined in the same way as in [42].…”

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

“…Then, for all other polymers, A = ∅, we can have a connection structure as explained below (and as in [42]). Since it is always true that the total number of labels (m + …”

Section: Leading Order Terms Proof Of Theorem 27mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Convergence of Density Expansions of Correlation Functions and the Ornstein–Zernike Equation

Kuna

Tsagkarogiannis

2018

Ann. Henri Poincaré

Self Cite

View full text Add to dashboard Cite

show abstract

Phase Transitions and Coarse-Graining for a System of Particles in the Continuum

Pulvirenti

Tsagkarogiannis

2016

From Particle Systems to Partial Differential Equations III

Self Cite

View full text Add to dashboard Cite

We revisit the proof of the liquid-vapor phase transition for systems with finite-range interaction by Lebowitz, Mazel and Presutti [1] and extend it to the case where we additionally include a hardcore interaction to the Hamiltonian. We establish the phase transition for the mean field limit and then we also prove it when the interaction range is long but finite, by perturbing around the meanfield theory. A key step in this procedure is the construction of a density (coarse-grained) model via cluster expansion. In this note we present the overall result but we mainly focus on this last issue.

show abstract

Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

Basti,

Cenatiempo,

Giuliani

et al. 2024

Arch Rational Mech Anal

View full text Add to dashboard Cite

We consider a gas of bosons interacting through a hard-sphere potential with radius $$\mathfrak {a}$$ a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term $$4\pi \rho \mathfrak {a}$$ 4 π ρ a and shows that corrections are smaller than $$C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}$$ C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant $$C > 0$$ C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order $$\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}$$ ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.

show abstract

Cluster Expansion in the Canonical Ensemble

Cited by 42 publications

References 16 publications

Convergence of Density Expansions of Correlation Functions and the Ornstein–Zernike Equation

Convergence of Density Expansions of Correlation Functions and the Ornstein–Zernike Equation

Phase Transitions and Coarse-Graining for a System of Particles in the Continuum

Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

Contact Info

Product

Resources

About