The Cloudy photoionization codes have been employed to study a spherically distributed cloud, around an arbitrary planetary nebula, with core temperature 105 K. The ionization factor ${\chi (H)}$ is close to unity, in the inner face of the dusty plasma (DP) cloud, which follows a monotonic declining trend, afterwards. For hydrogen density ${n_H} = 10\;\textrm{c}{\textrm{m}^{ - 3}}$, an exponentially falling trend of temperature could be noticed. A grain charging$\setminus$discharging process is also witnessed, which is very common in a DP environment. For ${n_H} = 10\;\textrm{c}{\textrm{m}^{ - 3}}$, photoionization of grains is more common due to higher photon density; compared with ${n_H} > 10\;\textrm{c}{\textrm{m}^{ - 3}}$, where the grain–electron acquiring probability is maximum, because of significant electron density. Owing to the electrostatic interactions between the charged grain and the electrons, an unusual trend in temperature has been observed.

Two-dimensional particle-in-cell (PIC) simulations explore the collisionless tearing instability developing in a Harris equilibrium configuration in a pair (electron–positron) plasma, with no guide field, for a range of parameters from non-relativistic to relativistic temperatures and drift velocities. Growth rates match the predictions of Zelenyi & Krasnosel'skikh (Astron. Zh., vol. 56, 1979, pp. 819–832) modified for relativistic drifts by Hoshino (Astrophys. J., vol. 900, issue 1, 2020, p. 66) as long as the assumption holds that the thickness $a$ of the current sheet is larger than the Larmor radius $\rho _L$, with the fastest growing mode at $ka \approx 1/\sqrt {3}$. Aside from confirming these predictions, we explore the transitions from thick to thin current sheets and from classical to relativistic temperatures. We show that for thinner current sheets ($a< \rho _L$), the growth rate matches the prediction for the case $a=\rho _L$. We also explore the nonlinear evolution of the modes. While the wavenumber with the fastest growth rate initially matches the prediction of Zelenyi & Krasnosel'skikh (1979), these modes saturate moving the dominant mode to lower wavenumbers (especially for thick current sheets with low growth rates). Furthermore, at a late, nonlinear stage, the growth rate (initially following the growth rate prediction proportional to $(\rho _L/a)^{3/2} < 1$) increases faster than exponentially, reaching a maximum growth rate equivalent to the linear growth rate prediction at $\rho _L/a = 1$, before eventually saturating.