Particles confined in arbitrary potentials with a class of finite-ranged interactions

Abstract

In this paper, we develop a large-$N$ field theory for a system of $N$ classical particles in one dimension at thermal equilibrium. The particles are confined by an arbitrary external potential, $V_{\text{ex}}(x)$, and repel each other via a class of pairwise interaction potentials $V_{\text{int}}(r)$ (where $r$ is distance between a pair of particles) such that $V_{\text{int}}\sim |r{|}^{-k}$ when $r\to 0$. We consider the case where every particle is interacting with $d$ (finite range parameter) number of particles to its left and right. Due to the intricate interplay between external confinement, pairwise repulsion and entropy, the density exhibits markedly distinct behavior in three regimes $k>0$, $k\to 0$ and $k<0$. From this field theory, we compute analytically the average density profile for large $N$ in these regimes. We show that the contribution from interaction dominates the collective behaviour for $k>0$ and the entropy contribution dominates for $k<0$, and both contributes equivalently in the $k\to 0$ limit (finite range log-gas). Given the fact that these family of systems are of broad relevance, our analytical findings are of paramount importance. These results are in excellent agreement with brute-force Monte-Carlo simulations.

arXiv comment8 pages, 4 figures

References in corpus (34)

• Bose-Einstein condensation of chromium
• Observation of a Many-Body Dynamical Phase Transition with a 53-Qubit Quantum Simulator
• A Strongly Dipolar Bose-Einstein Condensate of Dysprosium
• Bose-Einstein condensation of atoms in a uniform potential
• Dipolar collisions of polar molecules in the quantum regime
• Langevin Equation for the Density of a System of Interacting Langevin Processes
• Nonlinear fluctuating hydrodynamics for anharmonic chains
• Critical Dynamics of Spontaneous Symmetry Breaking in a Homogeneous Bose gas
• Thermalization near integrability in a dipolar quantum Newton's cradle
• Physics and Mathematics of Calogero particles
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Introduction to Random Matrices - Theory and Practice
• Comparing contact and dipolar interaction in a Bose-Einstein condensate
• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• Generalized Langevin equations for a driven tracer in dense soft colloids: construction and applications
• Strong Phase Separation in a Model of Sedimenting Lattices
• Quantum Joule-Thomson Effect in a Saturated Homogeneous Bose Gas
• Exact extremal statistics in the classical $1d$ Coulomb gas
• Hydrodynamics of cold atomic gases in the limit of weak nonlinearity, dispersion and dissipation
• Extreme statistics and index distribution in the classical $1d$ Coulomb gas
• Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law
• Universality of the third-order phase transition in the constrained Coulomb gas
• Harmonically confined particles with long-range repulsive interactions
• From single-particle excitations to sound waves in a box-trapped atomic Bose-Einstein condensate
• Universality of the weak pushed-to-pulled transition in systems with repulsive interactions
• Exact correlations in a single file system with a driven tracer
• Emergence of Calogero family of models in external potentials: Duality, Solitons and Hydrodynamics
• Soliton solutions of Calogero model in harmonic potential
• New Integrable Systems from Unitary Matrix Models
• Finite-Range Coulomb Gas Models II: Applications to Quantum Kicked Rotors and Banded Random Matrices
• Finite-Range Coulomb Gas Models of Banded Random Matrices and Quantum Kicked Rotors
• Finite-Range Coulomb Gas Models I: Some Analytical Results
• Duality in a hyperbolic interaction model integrable even in a strong confinement: Multi-soliton solutions and field theory

Cited by in corpus (12)

• Edge fluctuations and third-order phase transition in harmonically confined long-range systems
• Gap probability and full counting statistics in the one dimensional one-component plasma
• 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
• Finite temperature equilibrium density profiles of integrable systems in confining potentials
• Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices
• Full counting statistics of 1d short-range Riesz gases in confinement
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• Crossover in densities of confined particles with finite range of interaction
• Crystal to liquid cross-over for active particles with inverse-square power-law interaction
• Extreme dynamics and relaxation of quantum gases: A hydrodynamic approach