Topic of Research Seminar: Telegrapher’s Equation with Asymmetric Permeable Barriers

Abstract: The telegrapher’s equation describes transport at finite speed, capturing the crossover from ballistic to diffusive motion in persistent random walks. Real environments are partitioned by permeable interfaces, membranes, or contacts between layers of a porous medium, which transmit part of the probability flux and reflect the rest.

Since the flux in a persistent walk depends on the history of the density gradient rather than its instantaneous value, the corresponding boundary conditions cannot be taken over from diffusion theory; we derive them microscopically, from the redistribution of the directional densities at the interface, and obtain exact time-dependent solutions. For a single barrier, a mirror source with reversed initial bias gives two exponentially attenuated ballistic fronts, carrying the memory of the initial direction.

Allowing the transmission probability to depend on the direction of crossing yields the asymmetric interface, where the coefficients enter only through a surface admittance and the asymmetry appears; the result reduces at short times and, in the diffusive limit, to the known Brownian result. For a finite interval between two permeable walls, the image series no longer resums, and an eigenfunction expansion is used instead: the permeabilities displace the eigenvalues and remove the zero mode, tracking the density from the early fronts reflecting between the walls to the slow decay of the surviving probability.

Subject Field of Topic: Random Walks and Stochastic Movements

Name of Speaker: Ms. Noor Fatima Mirza

Professorial Rank of Speaker: MS Student (Mathematics Dept.), SNS – NUST

Email of Speaker: [email protected]

Affiliation of Speaker: School of Natural Sciences (SNS), National University of Sciences and Technology (NUST)

Date and Venue: (Wednesday) 09 September, 2026, 03:30 pm, CR # 403, Old Building, School of Natural Sciences (SNS), NUST Islamabad