The number statistics and optimal history of non-equilibrium steady states of mortal diffusing particles

Abstract

Suppose that a point-like steady source at x = 0 injects particles into a half-infinite line. The particles diffuse and die. At long times a non-equilibrium steady state sets in, and we assume that it involves many particles. If the particles are non-interacting, their total number N in the steady state is Poisson-distributed with meanN predicted from a deterministic reaction-diffusion equation. Here we determine the most likely density history of this driven system conditional on observing a given N . We also consider two prototypical examples of interacting diffusing particles: (i) a family of mortal diffusive lattice gases with constant diffusivity (as illustrated by the simple symmetric exclusion process with mortal particles), and (ii) random walkers that can annihilate in pairs. In both examples we calculate the variances of the (non-Poissonian) stationary distributions of N .

Mortal Brownian motion: Three short stories

“…It appears at a nonzero winding angle and results from the singularity of the short-time large-deviation function of the winding angle for immortal Brownian motion [10]. It would be interesting to extend some of our analysis to ensembles of MBPs and, more generally, to ensembles of interacting diffusing particles which can be modelled as lattice gases [3]. The macroscopic fluctuation theory (see Ref.…”

Heterogeneous continuous-time random walks

“…One can also easily treat partially absorbing boundaries [52][53][54][55][56][57][58][59][60] by allowing nonzero leakage probability from the sink site x * . If a particle can disappear or loose its activity during diffusion, FPT problems for such "mortal" walkers [61][62][63][64][65][66][67][68] can be treated by introducing two sink sites, x * 1 and x * 2 , that represent an absorbing boundary and a reactive bulk. Using the exchange time distributions ψ xx * 2 (t) depending on x, one can model space-dependent bulk reaction rates.…”

The escape problem for mortal walkers

“…This can be seen by comparing the asymptotic expansion of Eq. (27) to the integral over x 0 of the Taylor expansion (16).…”

“…finite fluorescence lifetime); the latter mechanism should be taken into account for reliable interpretation of such measurements; (iii) in order to trigger translation, messenger RNA should not be degradated before reaching a ribosome [19]; (iv) in spite of a very high mortality rate, the spermatozoa that search for a small egg in the uterus or in the Fallopian tubes, need to remain alive to complete the fertilization [20][21][22]; (v) molecules should remain active or intact before reaching a reactive site on the surface of a catalyst; (vi) protective materials may trap, bind or deactivate dangerous species via a bulk reaction before they leak through defects in the boundary of the container; (vii) for a safe storage of nuclear wastes, the motion of radioactive nuclei should be slowed down enough to ensure their disintegration or at least to reduce the amount of released nuclei, etc. While some first passage problems have been recently extended to mortal walkers (see [22][23][24][25][26][27] and references therein), the effect of a finite lifetime of a walker onto the escape through the boundary of two-and three-dimensional confining domains has not been investigated. Since the escape is not certain, because of a possible "death" of the walker, the contribution of long trajectories towards the escape region can be greatly reduced, thus strongly affecting the conventional results.…”