We consider one-dimensional infinite chains of harmonic oscillators with stochastic perturbations and long-range interactions which have polynomial decay rate x −θ , x → ∞, θ > 1, where x ∈ Z is the interaction range. We prove that if 2 < θ ≤ 3, then the time evolution of the macroscopic thermal energy distribution is superdiffusive and governed by a fractional diffusion equation with exponent 3 7−θ , while if θ > 3, then the exponent is 3 4 . The threshold is θ = 3 because the derivative of the dispersion relation diverges as k → 0 when θ ≤ 3.

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We consider a one-dimensional infinite chain of coupled charged harmonic oscillators in a magnetic field with a small stochastic perturbation of order ǫ. We prove that for a space-time scale of order ǫ −1 the density of energy distribution (Wigner distribution) evolves according to a linear phonon Boltzmann equation. We also prove that an appropriately scaled limit of solutions of the linear phonon Boltzmann equation is a solution of the fractional diffusion equation with exponent 5 6.

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We consider one-dimensional infinite chains of harmonic oscillators with random exchanges of momenta and long-range interaction potentials which have polynomial decay rate |x|−θ , x → ∞, θ > 1 where x ∈ Z is the interaction range. The dynamics conserve total momentum, total length and total energy. We prove that the systems evolve macroscopically on superballistic space-time scale ( y ε − 1 , t ε − θ − 1 2 ) when 1 < θ < 3, ( y ε − 1 , t ε − 1 − log ( ε − 1 ) − 1 ) when θ = 3, and hyperbolic space-time scale (yɛ −1, tɛ −1) when θ > 3. Combining our results and the results in (Suda 2021 Ann. Inst. Henri Poincare B 57 2268–2314), we show the existence of two different space-time scales on which the systems evolve. In addition, we prove the fluctuations of the normal modes of the superballistic wave equation, which are analogues of the Riemann invariants and capture fluctuations along the characteristics. For the normal modes, the space-time scale is superdiffusive when 2 < θ ⩽ 4 and diffusive when θ > 4.