Some Connections Between the Classical Calogero–Moser Model and the Log-Gas

Agarwal, Shweta; Kulkarni, Manas; Dhar, Abhishek

doi:10.1007/s10955-019-02349-6

Cited by 17 publications

(22 citation statements)

References 41 publications

(48 reference statements)

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“…The case β = 2 is a transition point, as will be discussed in Section VI. Note that the Calogero-Sutherland interpretation of the 1D log gas at s = 0 also revels a link with the short range case s = 2 8,56 . The latter is the same model, but classical and at positive temperature.…”

Section: B Long Range Case S < Dmentioning

confidence: 70%

Coulomb and Riesz gases: The known and the unknown

Lewin

2022

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We review what is known, unknown and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in R d interacting with the Riesz potential ±|x| −s (resp. − log |x| for s = 0). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g. in random matrix theory). The main question addressed in the article is how to properly define the associated infinite point process and characterize it using some (renormalized) equilibrium equation. This is largely open in the long range case s < d. For the convenience of the reader we give the detail of what is known in the short range case s > d. In the last part we discuss phase transitions and mention what is expected on physical grounds.

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Section: B Long Range Case S < Dmentioning

confidence: 70%

Coulomb and Riesz gases: The known and the unknown

Lewin

2022

Preprint

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“…For large but finite N , y max fluctuates from sample to sample and we numerically observe that the standard deviation σ ymax = y 2 max − y max 2 describing the typical fluctuation is of order N −η k . It is known that, for inverse temeprature β = O(1), for the Dyson's loggas η 0 = 2/3 [17,26], for the 1dOCP η −1 = 1 [33,34] while for the Calegoro-Moser system η 2 = 5/6 ‡ [58]. We have computed η k numerically for different values of k via Monte-Carlo (MC) simulation using the Metropolis-Hastings algorithm and the result are shown in Fig.…”

Section: Summary Of the Resultsmentioning

confidence: 99%

“…By expanding the energy around the ground state and truncating it at bilinear order, as is done within the Hessian theory, we find that the resulting exponent of the variance fits the numerically obtained exponent remarkably well. Further, we provide ‡ The values of η 0 [17,26] and η −1 [33,34] are analytically established whereas that of η 2 is numerically established [58]. ).…”

Section: Summary Of the Resultsmentioning

confidence: 99%

“…We start by noting that the Riesz gas family for different value of k corresponds to various well known models. Some of the well studied cases include k = 2 which is an interacting integrable system, known as the Calogero-Moser model [13,[55][56][57][58] and k = −1, which corresponds to the one dimensional one component plasma (1dOCP) [33,34,59] discussed above. It is interesting to note that the k → ∞ limit, corresponds to the hard-rod gas [60,61].…”

Section: Some Properties Of Riesz Gas and Important Notationsmentioning

confidence: 99%

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Edge fluctuations and third-order phase transition in harmonically confined long-range systems

Kethepalli,

Kulkarni,

Kundu

et al. 2021

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We study the distribution of the position of the rightmost particle x max in a N -particle Riesz gas in one dimension confined in a harmonic trap. The particles interact via long-range repulsive potential, of the form r −k with −2 < k < ∞ where r is the inter-particle distance. In equilibrium at temperature O(1), the gas settles on a finite length scale L N that depends on N and k. We numerically observe that the typical fluctuation of y max = x max /L N around its mean is of O(N −η k ). Over this length scale, the distribution of the typical fluctuations has a N independent scaling form. We show that the exponent η k obtained from the Hessian theory predicts the scale of typical fluctuations remarkably well. The distribution of atypical fluctuations to the left and right of the mean y max are governed by the left and right large deviation functions, respectively. We compute these large deviation functions explicitly ∀k > −2. We also find that these large deviation functions describe a pulled to pushed type phase transition as observed in Dyson's log-gas (k → 0) and 1d one component plasma (k = −1). Remarkably, we find that the phase transition remains 3 rd order for the entire regime. Our results demonstrate the striking universality of the 3 rd order transition even in models that fall outside the paradigm of Coulomb systems and the random matrix theory. We numerically verify our analytical expressions of the large deviation functions via Monte Carlo simulation using an importance sampling algorithm.

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“…∝ |x i − x j |. A more general long-range model is the harmonically confined Riesz gas [30] where the pairwise repulsion takes the form ∝ |x i − x j | −k with k > −2 [31,32]. The 1dOCP is a special case of the Riesz gas in the limit k → −1.…”

Section: Discussionmentioning

confidence: 99%

Truncated linear statistics in the one dimensional one-component plasma

Flack,

Majumdar,

Schehr

2021

Preprint

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In this paper, we study the probability distribution of the observable s = (1/N )representing the ordered positions of N particles in a 1d one-component plasma, i.e., N harmonically confined charges on a line, with pairwise repulsive 1d Coulomb interaction |x i − x j |. This observable represents an example of a truncated linear statistics -here the center of mass of the N = κ N (with 0 < κ ≤ 1), rightmost particles. It interpolates between the position of the rightmost particle (in the limit κ → 0) and the full center of mass (in the limit κ → 1). We show that, for large N , s fluctuates around its mean s and the typical fluctuations are Gaussian, of width O(N −3/2 ). The atypical large fluctuations of s, for fixed κ, are instead described by a large deviation form P N,κ (s) exp −N 3 φ κ (s) , where the rate function φ κ (s) is computed analytically. We show that φ κ (s) takes different functional forms in five distinct regions in the (κ, s) plane separated by phase boundaries, thus leading to a rich phase diagram in the (κ, s) plane. Across all the phase boundaries the rate function φ(κ, s) undergoes a third-order phase transition. This rate function is also evaluated numerically using a sophisticated importance sampling method, and we find a perfect agreement with our analytical predictions.

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Some Connections Between the Classical Calogero–Moser Model and the Log-Gas

Cited by 17 publications

References 41 publications

Coulomb and Riesz gases: The known and the unknown

Coulomb and Riesz gases: The known and the unknown

Edge fluctuations and third-order phase transition in harmonically confined long-range systems

Truncated linear statistics in the one dimensional one-component plasma

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