English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this paper we develop the theory of drift-diffusion on a semi-infiniteCayley tree with stochastic resetting. In the case of a homogeneous tree with aclosed terminal node and no resetting, it is known that the system undergoes aclassical localization-delocalization (LD) transition at a critical meanvelocity $v_c= -(D/L)\ln (z-1)$ where $D$ is the diffusivity, $L$ is the branchlength and $z$ is the coordination number of the tree. If $v <v_c$ then thesteady state concentration at the terminal node is non-zero (drift-dominatedlocalized state), whereas it is zero when $v >v_c $ (diffusion-dominateddelocalized state). This is equivalent to the transition between recurrent andtransient transport on the tree, with the mean first passage time (MFPT) to beabsorbed by an open terminal node switching from a finite value to infinity.Here we show how the LD transition provides a basic framework for understandinganalogous phase transitions in optimal resetting rates. First, we establish theexistence of an optimal resetting rate $r^{**}(z)$ that maximizes thesteady-state solution at a closed terminal node. In addition, we show thatthere is a phase transition at a critical velocity $v_c^{**}(z)$ such that$r^{**}>0$ for $v>v_c^{**}$ and $r^{**}=0$ for $v<v_c^{**}$. We then identify acritical velocity $v^*(z)$ for a phase transition in a second optimal resettingrate $r^*$ that minimizes the MFPT to be absorbed by an open terminal node.Previous results for the semi-infinite line are recovered on setting $z=2$. Thecritical velocity of the LD transition provides an upper bound for the othercritical velocities such that $v_c^*(z)<v_c^{**}(z)<v_c(z)$ for all finite $z$.Only $v_c(z)$ has a simple universal dependence on the coordination number $z$.We end by considering the combined effects of quenched disorder and stochasticresetting.

Drift-diffusion on a Cayley tree with stochastic resetting: the localization-delocalization transition