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Surprise ballistic and scaling inverted dynamics of a system coupled to a Hamiltonian thermostat

We study the induced generalized Brownian dynamics of a Hamiltonian thermostat, and specifically the diffusive properties. For a tagged particle regulated by a logarithmic-oscillator thermostat and moving in an external logarithmic potential, we reveal a distinct inversion of the scaling exponents for the mean-squared displacement ⟨Δx 2(t)⟩ ∼ t λ around the exponent λ = 2 − α covering regimes from ballistic diffusion to confinement motion (i.e. 0 ⩽ α ⩽ 2). This behavior contrasts with the expanding behavior associated with superdiffusive processes in a tilted periodic potential. The Hamiltonian thermostat is shown to maintain a fixed kinetic energy and, although this system is nonergodic in the force-free case, leads to a confined system that approaches thermal equilibrium slowly with the same temperature as the thermostat. In addition, we find its displacement has a significant dependence on λ via the amplitude of the external logarithmic potential. The continuous reduction of the scaling exponent is vital in the quantitative evaluation of all diffusive processes.