Article Dans Une Revue Physical Review E Année : 2025

Diffusion with stochastic resetting on a lattice

Résumé

We provide an exact formula for the mean first-passage time (MFPT) to a target at the origin for a single particle diffusing on a -dimensional hypercubic {\em lattice} starting from a fixed initial position and resetting to with a rate . Previously known results in the continuous space are recovered in the scaling limit , with the product fixed. However, our formula is valid for any and any that enables us to explore a much wider region of the parameter space that is inaccessible in the continuum limit. For example, we have shown that the MFPT, as a function of for fixed , diverges in the two opposite limits and with a unique minimum in between, provided the starting point is not a nearest neighbour of the target. In this case, the MFPT diverges as a power law as , but very interestingly with an exponent that depends on the starting point where is the lattice spacing and 's are integers. If, on the other hand, the starting point happens to be a nearest neighbour of the target, then the MFPT decreases monotonically with increasing , approaching a universal limiting value as , indicating that the optimal resetting rate in this case is infinity. We provide a simple physical reason and a simple Markov-chain explanation behind this somewhat unexpected universal result. Our analytical predictions are verified in numerical simulations on lattices up to dimensions. Finally, in the absence of a target, we also compute exactly the position distribution of the walker in the nonequlibrium stationary state that also displays interesting lattice effects not captured by the continuum theory.

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Dates et versions

hal-05273070 , version 1 (22-09-2025)

Identifiants

  • HAL Id : hal-05273070 , version 1

Citer

Satya Majumdar, Alexander K Hartmann. Diffusion with stochastic resetting on a lattice. Physical Review E , 2025, 112, pp.034102. ⟨hal-05273070⟩
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