Harmonically confined long-ranged interacting gas in the presence of a hard wall
Résumé
In this paper, we compute exactly the average density of a harmonically confined Riesz gas of N particles for large N in the presence of a hard wall. In this Riesz gas, the particles repel each other via a pairwise interaction that behaves as |x i − x j |−k for k > −2, with x i denoting the position of the ith particle. This density can be classified into three different regimes of k. For k ⩾ 1, where the interactions are effectively short-ranged, the appropriately scaled density has a finite support over [−l k (w), w] where w is the scaled position of the wall. While the density vanishes at the left edge of the support, it approaches a nonzero constant at the right edge w. For −1 < k < 1, where the interactions are weakly long-ranged, we find that the scaled density is again supported over [−l k (w), w]. While it still vanishes at the left edge of the support, it diverges at the right edge w algebraically with an exponent (k − 1)/2. For −2 < k < −1, the interactions are strongly long-ranged that leads to a rather exotic density profile with an extended bulk part and a delta-peak at the wall, separated by a hole in between. Exactly at k = −1 the hole disappears. For −2 < k < −1, we find an interesting first-order phase transition when the scaled position of the wall decreases through a critical value w = w*(k). For w < w*(k), the density is a pure delta-peak located at the wall. The amplitude of the delta-peak plays the role of an order parameter which jumps to the value 1 as w is decreased through w*(k). Our analytical results are in very good agreement with our Monte-Carlo simulations.
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