Scaling laws for harmonically trapped two-species mixtures at thermal equilibrium

Abstract

We discuss the scaling of the interaction energy with particle numbers for a harmonically trapped two-species mixture at thermal equilibrium experiencing interactions of arbitrary strength and range. In the limit of long-range interactions and weak coupling, we recover known results for the integrable Caldeira-Leggett model in the classical limit. In the case of short-range interactions and for a balanced mixture, numerical simulations show scaling laws with exponents that depend on the interaction strength, its attractive or repulsive nature, and the dimensionality of the system. Simple analytic considerations based on equilibrium statistical mechanics and small interspecies coupling quantitatively recover the numerical results. The dependence of the scaling on interaction strength helps to identify a threshold between two distinct regimes. Our thermalization model covers both local and extended interactions allowing for interpolation between different systems such as fully ionized gases and neutral atoms, as well as parameters describing integrable and chaotic dynamics.

Relationship between nonlinearities and thermalization in classical open systems: The role of the interaction range

“…1 can be numerically integrated to machine precision. A plot of the time-averaged interaction energy, at the end of the times considered, versus number of particles, for N A = N B = N , shows power law scaling as detailed in [17].…”

“…Based on our former studies we only focus here on repulsive interspecies interactions, as they show more markedly anomalous scaling (see for example Figs. 3 and 5 in [17]). In particular, for a completely unbalanced mixture (for instance FIG.…”

“…In a more recent contribution, we further explored the dependence of this anomalous scaling with the interparticle interaction strength [17], showing that the anomalous behavior only appears for an intermediate range of interaction strength values. While for simplicity we focused on one-dimensional systems, the phenomenon persists in higher dimensionality, differing only in the specific value of the anomalous scaling exponent.…”

“…In this paper we extend our analysis of the model by FIG. 1: Scaling exponent α versus λ, evaluated, as described in [17], by fitting the interaction energy at long times to power-law in the number of harmonic oscillators, NA = NB = 50, 100, 200, 400. Both short and long-range interactions do not result in anomalous scaling for the exponent α, while at intermediate values the scaling exponents are anomalous.…”