THE TWO-STATE DIFFUSION PROBLEM WITH RESETTING. - DÍAZ-PÉREZAY, J. J.; MULET, R. (Revista Cubana de Física, 2024)
The concept of stochastic resetting emerged in 2011, introducing a simple modification to [15] the one dimensional Random Walk that leads to a stationary distribution and where the first-passage time can be easily optimized. In this context, we will investigate what happens when modeling these phenomena in a framework where a Brownian particle alternates between different states. We will derive the Fokker-Planck equations of the model and analyze the behavior of the probability density of the particle positions for long times and solve the corresponding first-passage time problem. Contrasting with the two-state Brownian motion, we will observe how resetting ensures a stationary behavior in the long run leading to a finite expectation for the first-passage problem. Finally, we will demonstrate how the existence of this expectation guarantees minimum values by varying the resetting or state-change parameters. La idea del reajuste estocástico aparece en 2011 abriendo la posibilidad de interpretar un nuevo grupo de fenómenos, y sobre todo llamando la atenci ón sobre las propiedades favorables de este tipo de modelos. Con este sentido estudiaremos qué sucede al modelar este tipo de fenómenos en un espacio de alternancia de los dos estados en que se mueve una part'ıcula browniana. Deduciremos las ecuaciones de Fokker-Planck del modelo y a partir de ellas trabajaremos el comportamiento de la part'ıcula para tiempos largos y el problema del primer cruce. En comparaci ón con el movimiento browniano clásico en dos estados podremos ver cómo el reajuste nos garantiza un comportamiento estacionario para tiempos largos, y por tanto una esperanza finita para el problema del primer cruce. Por último, veremos como la existencia de la esperanza nos garantiza valores m'ınimos de esta al realizar variaciones sobre los parámetros de reajuste o cambio de estados. PACS: Stochastic resetting (restablecimiento estocástico); double diffusivity (doble difusividad); first-passage time (tiempo de primer paso); Fokker-Planck Equation (ecuación de Fokker-Planck); brownian motion (movimiento browniano). The concept of stochastic resetting emerged in 2011, introducing a simple modification to [15] the one dimensional Random Walk that leads to a stationary distribution and where the first-passage time can be easily optimized. In this context, we will investigate what happens when modeling these phenomena in a framework where a Brownian particle alternates between different states. We will derive the Fokker-Planck equations of the model and analyze the behavior of the probability density of the particle positions for long times and solve the corresponding first-passage time problem. Contrasting with the two-state Brownian motion, we will observe how resetting ensures a stationary behavior in the long run leading to a finite expectation for the first-passage problem. Finally, we will demonstrate how the existence of this expectation guarantees minimum values by varying the resetting or state-change parameters.