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Dynamics of a collection of active particles on a two-dimensional periodic undulated surface

We study the dynamics of circular active particles (AP) on a two dimensional periodic undulated surface. Each particle has an internal energy mechanism which is modeled by an active friction force and it is controlled by an activity parameter $v_0$. It acts as negative friction if the speed of the particle is smaller than $v_0$ and normal friction otherwise. Surface undulation is modeled by the periodic undulation of fixed amplitude and wavelength and is measured in terms of a dimensionless ratio of amplitude and wavelength, $\bar{h}$. The dynamics of the particle is studied for different activities, $v_0$ and surface undulations (SU), $\bar{h}$. Three types of particle dynamics are observed on varying activity and SU. For small $v_0 \lesssim 0.1$ and $\bar{h}$ $\gtrsim 0.8$, particles remain confined in a surface minimum, for moderate $v_0 \lesssim \bar{h}$, dynamics of particle shows an intermediate subdiffusion to late time diffusion and for large $v_0 \gtrsim \bar{h}$, it shows initial superdiffusion to late time diffusion. For all $v_0$'s and $\bar{h} \lesssim 0.2$, the dynamics of particle, satisfies the Green-Kubo relation between the effective diffusivity and velocity auto-correlation function. Systematic deviation is found on increasing $\bar{h}$. Hence, an effective equilibrium can be established for a range of system parameters in this nonequilibirum system.