Some connections between the Classical Calogero-Moser model and the Log Gas

Abstract

In this work we discuss connections between a one-dimensional system of $N$ particles interacting with a repulsive inverse square potential and confined in a harmonic potential (Calogero-Moser model) and the log-gas model which appears in random matrix theory. Both models have the same minimum energy configuration, with the particle positions given by the zeros of the Hermite polynomial. Moreover, the Hessian describing small oscillations around equilibrium are also related for the two models. The Hessian matrix of the Calogero-Moser model is the square of that of the log-gas. We explore this connection further by studying finite temperature equilibrium properties of the two models through Monte-Carlo simulations. In particular, we study the single particle distribution and the marginal distribution of the boundary particle which, for the log-gas, are respectively given by the Wigner semi-circle and the Tracy-Widom distribution. For particles in the bulk, where typical fluctuations are Gaussian, we find that numerical results obtained from small oscillation theory are in very good agreement with the Monte-Carlo simulation results for both the models. For the log-gas, our findings agree with rigorous results from random matrix theory.

arXiv comment23 pages, 11 figures

References in corpus (15)

• Physics and Mathematics of Calogero particles
• Correlation Functions, Cluster Functions and Spacing Distributions for Random Matrices
• Fluctuations of eigenvalues of random normal matrices
• Gaussian fluctuations of eigenvalues in the GUE
• Exact extremal statistics in the classical $1d$ Coulomb gas
• Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
• A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix
• Random matrix ensembles associated with Lax matrices
• Finite N Fluctuation Formulas for Random Matrices
• Universal dynamics of a soliton after an interaction quench
• The classical hydrodynamics of the Calogero-Sutherland model
• Hydrodynamics of local excitations after an interaction quench in 1D cold atomic gases
• Emergence of Calogero family of models in external potentials: Duality, Solitons and Hydrodynamics
• Soliton solutions of Calogero model in harmonic potential
• Poisson Structures of Calogero-Moser and Ruijsenaars-Schneider Models

Cited by in corpus (15)

• Coulomb and Riesz gases: The known and the unknown
• Edge fluctuations and third-order phase transition in harmonically confined long-range systems
• The one-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: limit distributions
• Gap probability and full counting statistics in the one dimensional one-component plasma
• Interacting, running and tumbling: the active Dyson Brownian motion
• Harmonically confined long-ranged interacting gas in the presence of a hard wall
• Dynamically emergent correlations in bosons via quantum resetting
• Spatio-temporal spread of perturbations in power-law models at low temperatures: Exact results for OTOC
• An exact formula for the variance of linear statistics in the one-dimensional jellium mode
• Duality in a hyperbolic interaction model integrable even in a strong confinement: Multi-soliton solutions and field theory
• Fluctuations in the active Dyson Brownian motion and the overdamped Calogero-Moser model
• Full counting statistics of 1d short-range Riesz gases in confinement
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• Crystal to liquid cross-over for active particles with inverse-square power-law interaction
• Truncated linear statistics in the one dimensional one-component plasma