First passage time in multistep stochastic processes with applications to dust charging

An approach was developed to describe the first passage time (FPT) in multistep stochastic processes with discrete states governed by a master equation (ME). The approach is an extension of the totally absorbing boundary approach given for calculation of FPT in one-step processes [N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier Science Publishers, North Holland, Amsterdam, 2007)] to include multistep processes where jumps are not restricted to adjacent sites. In addition, a Fokker-Planck equation (FPE) was derived from the multistep ME, assuming the continuity of the state variable. The developed approach and an FPE based approach [C. W. Gardiner, Handbook of Stochastic Methods, 3rd ed. (Springer-Verlag, New York, 2004)] were used to find the mean first passage time (MFPT) of the transition between the negative and positive stable macrostates of dust grain charge when the charging process was bistable. The dust was in a plasma and charged by collecting ions and electrons, and emitting secondary electrons. The MFPTs for the transitioning of grain charge from one macrostate to the other were calculated by the two approaches for a range of grain sizes. Both approaches produced very similar results for the same grain except for when it was very small. The difference between MFPTs of two approaches for very small grains was attributed to the failure of the charge continuity assumption in the FPE description. For a given grain, the MFPT for a transition from the negative macrostate to the positive one was substantially larger than that for a transition in a reverse order. The normalized MFPT for a transition from the positive to the negative macrostate showed little sensitivity to the grain radius. For a reverse transition, with the increase of the grain radius, it dropped first and then increased. The probability density function of FPT was substantially wider for a transition from the positive to the negative macrostate, as compared to a reverse transition.

Figure 1

Dimensionless growth (solid line) and dissipation (dashed line) times vs dimensionless deviation from the mean when the PDF is governed by a linear FPE.

Figure 2

Stationary macrostates (roots of the macroscopic equation at a stationary state) against $E_M/4k{T}_e$ near a triple-root situation for $E_M/4k{T}_s=30$ (solid line), 32 (dashed line), 35 (dot-dashed line), and 40 (dotted line); ${\delta }_M=15$.

Figure 3

Probability density function of the normalized grain charge ($z=Z/\mathrm{\Omega }$) at a bistable state for grain radius of 100, 30, and 10 nm shown by solid, dashed, and dotted-dashed lines, respectively, through the Fokker-Planck equation, and shown by $\circ$, $\times$, and $+$, respectively, through the master equation; ${\delta }_M=15$, $E_M/4k{T}_e=45$, $E_M/4k{T}_s=32$.

Figure 4

MFPT normalized by ${\tau }_c$, versus grain radius for a transition from the negative macrostate to the positive macrostate through master equation ($+$) and Fokker-Planck equation ($◯$) and for a transition from the positive macrostate to the negative macrostate through master equation ($\times$) and Fokker-Planck equation ($\square$); see caption of Fig. 3 for parameter values.

Figure 5

Probability density function of FPT normalized by ${\tau }_c$ for grain radius of 100 (solid line), 30 (dashed line), and 10 nm (dotted-dashed line) for grain charge transitioning (a) from the positive macrostate to the negative macrostate; and (b) from the negative macrostate to the positive macrostate. See the caption of Fig. 3 for parameter values.