Emerging crystallization in one dimension

We analytically uncover the emergence of a universal solidlike phase in one-dimensional hard-sphere systems (Tonks gas). The associated pair distribution function g(r) decays algebraically as (r/ξ)^{-1/2}, revealing quasi-long-range order. Beyond that correlation length ξ∼ℓ/(1-ϕ)^{2}, an exponential decay regime of the form e^{-r/ξ} sets in, where ℓ denotes the mean particle spacing and ϕ the packing fraction. We identify the onset of solidlike behavior near ϕ^{*}≃0.8 at ξ≈ℓ, where the crossover point coincides with the third neighbor shell. Our findings establish a clear theoretical framework for emerging crystallization in one dimension and introduce criteria for detecting incipient solid order in constrained geometries.