We study an interacting system of N classical particles on a line at thermal equilibrium. The particles are confined by a harmonic trap and repel each other via pairwise interaction potential that ...behaves as a power law ∝∑i≠jN|xi−xj|−k (with k>−2) of their mutual distance. This is a generalization of the well-known cases of the one-component plasma (k=−1), Dyson's log gas (k→0+), and the Calogero-Moser model (k=2). Because of the competition between harmonic confinement and pairwise repulsion, the particles spread over a finite region of space for all k>−2. We compute exactly the average density profile for large N for all k>−2 and show that while it is independent of temperature for sufficiently low temperature, it has a rich and nontrivial dependence on k with distinct behavior for −2<k<1, k>1 and k=1.
We study the distribution P(ω) of the random variable ω=x1/(x1+x2), where x1 and x2 are the wealths of two individuals selected at random from the same tempered Paretian ensemble characterized by the ...distribution Ψ(x)∼ϕ(x)/x1+α, where α>0 is the Pareto index and ϕ(x) is the cut-off function. We consider two forms of ϕ(x): a bounded function ϕ(x)=1 for L≤x≤H, and zero otherwise, and a smooth exponential function ϕ(x)=exp(−L/x−x/H). In both cases Ψ(x) has moments of arbitrary order. We show that, for α>1, P(ω) always has a unimodal form and is peaked at ω=1/2, so that most probably x1≈x2. For 0<α<1 we observe a more complicated behavior which depends on the value of δ=L/H. In particular, for δ<δc–a certain threshold value–P(ω) has a three-modal (for a bounded ϕ(x)) and a bimodal M-shape (for an exponential ϕ(x)) form which signifies that in such ensembles the wealths x1 and x2 are disproportionately different.
►x1 and x2 are the wealths of two members of the same tempered Paretian ensemble (TPE). ► TPE is described by the distribution ψ(x)=φ(x)/x1+α, where φ(x) is the cut-off function. ► We evaluate the distribution P(ω) of the random variable ω=x1/(x1+x2). ► For α>1, P(ω) is unimodal and peaked at ω=1/2, so that, most probably, x1=x2. ► For 0<α<1 we predict a symmetry-breaking transition to multimodal forms.
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