Mayer and Virial Series at Low Temperature

Jansen, Sabine

doi:10.1007/s10955-012-0490-1

Cited by 17 publications

(11 citation statements)

References 19 publications

(54 reference statements)

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“…When p = p β → 0 slower than any exponential, it is still true that g(β) → e 0 0 . When βp β = exp(−βν) with ν > 0, one can show with [26,27] At pressures vanishing faster than exp(−β|e 0 0 |), the most likely configurations have very large spacings (dilute gas phase, (β) → ∞) and the previous results no longer apply. For lim inf 1 β log(βp β ) > e 0 0 , we expect that large deviations principles with rate functions E 0 bulk and E 0 surf − min E 0 surf still hold (in fact our proofs still show weak large deviations principles).…”

Section: Small Positive Temperaturementioning

confidence: 93%

Surface Energy and Boundary Layers for a Chain of Atoms at Low Temperature

Jansen

König

Schmidt

et al. 2020

Arch Rational Mech Anal

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We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard–Jones type. The pressure (stress) is assumed to be small but positive and bounded away from zero, while the temperature $$\beta ^{-1}$$ β - 1 goes to zero. Our main results are: (1) As $$\beta \rightarrow \infty $$ β → ∞ at fixed positive pressure $$p>0$$ p > 0 , the Gibbs measures $$\mu _\beta $$ μ β and $$\nu _\beta $$ ν β for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals $$\overline{{\mathcal {E}}}_{\mathrm {bulk}}$$ E ¯ bulk and $$\overline{{\mathcal {E}}}_\mathrm {surf}$$ E ¯ surf . The minimizer of the surface functional corresponds to zero temperature boundary layers; (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of $$\overline{{\mathcal {E}}}_\mathrm {surf}$$ E ¯ surf ; (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts; (4) Bounds on the decay of correlations are provided, some of them uniform in $$\beta $$ β .

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Section: Small Positive Temperaturementioning

confidence: 93%

Surface Energy and Boundary Layers for a Chain of Atoms at Low Temperature

Jansen

König

Schmidt

et al. 2020

Arch Rational Mech Anal

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“…Recent works [59,131,133] investigate the exponentially small region of convergence of the Mayer series close to T = ρ = 0, where things can be proved in any dimension. Namely, they considered the limit T → 0 and ρ → 0 with the constraint that T log ρ → ν (recall that ρ ∼ z = e µ/T at small activity, so this is similar to fixing µ).…”

Section: Extensionsmentioning

confidence: 99%

The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

134

191

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In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking thereby the natural translation-invariance of the system. This famous problem is still largely open. Mathematically, it amounts to studying the minima of a real-valued function defined on R 3N where N is the number of particles, which tends to infinity. We review the existing literature and mention several related open problems, of which many have not been thoroughly studied.

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“…It is the smoothing effect of this term that gets lost in that approximation, and possibly a lot of new kinks appear in this way. We know that these additional kinks correspond to cross-overs inside the gas phase, but not to sharp phase transitions (see [8] for a discussion of this). Hence, the full ideal mixture captures the behaviour of the physical system much better than the function studied in [9].…”

Section: Discussionmentioning

confidence: 99%

Ideal Mixture Approximation of Cluster Size Distributions at Low Density

Jansen

König

2012

J Stat Phys

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We consider an interacting particle system in continuous configuration space. The pair interaction has an attractive part. We show that, at low density, the system behaves approximately like an ideal mixture of clusters (droplets): we prove rigorous bounds (a) for the constrained free energy associated with a given cluster size distribution, considered as an order parameter, (b) for the free energy, obtained by minimising over the order parameter, and (c) for the minimising cluster size distributions. It is known that, under suitable assumptions, the ideal mixture has a transition from a gas phase to a condensed phase as the density is varied; our bounds hold both in the gas phase and in the coexistence region of the ideal mixture. The present paper improves our earlier results by taking into account the mixing entropy.

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Mayer and Virial Series at Low Temperature

Cited by 17 publications

References 19 publications

Surface Energy and Boundary Layers for a Chain of Atoms at Low Temperature

Surface Energy and Boundary Layers for a Chain of Atoms at Low Temperature

The crystallization conjecture: a review

Ideal Mixture Approximation of Cluster Size Distributions at Low Density

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