In this thesis, we study a wide class of physical systems, all described by stochastic processes, sharing important common features. Most of the physical systems that we consider here can be modeled via stochastic processes displaying externally driven sudden events. This is a very wide class of multilayered stochastic processes. They are constituted by a main underlying process, and one (or more) extra stochastic mechanism which modifies the evolution of the system by abruptly changing in time one of the state variables. In Chapter 1, we consider as an example stochastic resetting: in this case, the controlling mechanism randomly resets the position of the particle to some prescribed position. Specifically, we analyze how spaceinhomogeneous resetting and topological constraints modify the statistics of the Brownian motion. Namely, we study minimal stochastic models of diffusing particles evaporating or resetting from a finite onedimensional region with either open, periodic and/or absorbing boundary conditions. Eventually, we use our results to model the fluctuating motion of RNA polymerase during transcriptional pauses, allowing us to analytically predict relevant statistical properties which have been measured experimentally. In Chapter 2, we derive new analytical predictions concerning the firstpassage statistics of several models belonging to the class of runandtumble particles. These systems consist of selfpropelled particles moving ballistically (run phase) with random changes of direction (tumble phase). Runandtumble motion has been observed in several species of bacteria and it is a representative model of the class of active matter: the motion of the particles is sustained by energy consumption and is characterized by entropy production, with resulting breaking of timereversal symmetry and fluctuationdissipation theorem. We employ the runandtumble particle as a model of a random searcher. By extending previous works on the Brownian motion, we investigate the effect on the statistics of the firstpassage time of having a (random) location of the initial position of the particle, and of resetting. Finally, we collect all these ingredients to discuss an experimentally relevant protocol that allows one to reproduce resetting by means of optical lasers traps. In Chapter 3, we introduce a multilayered process based on underlying dynamics, identified with the Brownian motion of a particle, and on an external stochastic driving, due to a harmonic potential with equilibrium position coinciding with a timedependent dichotomous stochastic process. Hence, the motion of the particle consists of the alternate relaxation towards the two minima of the stochastic potential. In this sense, the system represents a minimal model for nonMarkovian linear active oscillators, an important subclass of active matter whose main degrees of freedom display oscillatory dynamics. Among the variety of beautiful examples of active oscillators appearing in nature, we apply our finding to describe and fit the spontaneous oscillations of the hair bundles in the ear of the bullfrog, for which experimental measurements are available. Then, we use the values of the fitted parameters in order to predict quantitatively the stationary dissipated power, a quantity that cannot be accessed experimentally. Finally, in Chapter 4, instead of looking at systems that can be described effectively by stochastic processes, we show how these processes may naturally arise from the analysis of nonequilibrium dynamics of quantum systems. Recent works in the field of quantum spin chains have revealed an exact mapping between the quantum evolution of these systems and a set of stochastic differential equations. Following a similar path, we show how this procedure can be extended to the case of a zerodimensional bosonic system. In particular, we consider the quantum anharmonic oscillator with quartic potential, which represents the simplest anharmonic extension of the quantum harmonic oscillator. Then, via a sequence of exact transformations, we parametrize the dynamics of the time evolution operator of the quantum quartic oscillator in terms of stochastic differential equations: for this quantum model, quantum fluctuations can be mapped into classical stochastic ones. Accordingly, the expectation value of observables in this stochastic approach translates into an average over the ensemble of trajectories generated by these stochastic processes. This leads to an intuitive implementation of the numerical analysis of these observables. In general, this approach allows one to use all the analytical and numerical tools of stochastic processes and classical field theories to analyze the dynamics of an anharmonic quantum system. Indeed, we expect that this approach can be extended to higher spatial dimensions, and eventually to manybody bosonic systems with quartic couplings.
Stochastic processes in biological, active and quantum systems / Tucci, Gennaro.  (2022 Feb 07).
Stochastic processes in biological, active and quantum systems
Tucci, Gennaro
20220207
Abstract
In this thesis, we study a wide class of physical systems, all described by stochastic processes, sharing important common features. Most of the physical systems that we consider here can be modeled via stochastic processes displaying externally driven sudden events. This is a very wide class of multilayered stochastic processes. They are constituted by a main underlying process, and one (or more) extra stochastic mechanism which modifies the evolution of the system by abruptly changing in time one of the state variables. In Chapter 1, we consider as an example stochastic resetting: in this case, the controlling mechanism randomly resets the position of the particle to some prescribed position. Specifically, we analyze how spaceinhomogeneous resetting and topological constraints modify the statistics of the Brownian motion. Namely, we study minimal stochastic models of diffusing particles evaporating or resetting from a finite onedimensional region with either open, periodic and/or absorbing boundary conditions. Eventually, we use our results to model the fluctuating motion of RNA polymerase during transcriptional pauses, allowing us to analytically predict relevant statistical properties which have been measured experimentally. In Chapter 2, we derive new analytical predictions concerning the firstpassage statistics of several models belonging to the class of runandtumble particles. These systems consist of selfpropelled particles moving ballistically (run phase) with random changes of direction (tumble phase). Runandtumble motion has been observed in several species of bacteria and it is a representative model of the class of active matter: the motion of the particles is sustained by energy consumption and is characterized by entropy production, with resulting breaking of timereversal symmetry and fluctuationdissipation theorem. We employ the runandtumble particle as a model of a random searcher. By extending previous works on the Brownian motion, we investigate the effect on the statistics of the firstpassage time of having a (random) location of the initial position of the particle, and of resetting. Finally, we collect all these ingredients to discuss an experimentally relevant protocol that allows one to reproduce resetting by means of optical lasers traps. In Chapter 3, we introduce a multilayered process based on underlying dynamics, identified with the Brownian motion of a particle, and on an external stochastic driving, due to a harmonic potential with equilibrium position coinciding with a timedependent dichotomous stochastic process. Hence, the motion of the particle consists of the alternate relaxation towards the two minima of the stochastic potential. In this sense, the system represents a minimal model for nonMarkovian linear active oscillators, an important subclass of active matter whose main degrees of freedom display oscillatory dynamics. Among the variety of beautiful examples of active oscillators appearing in nature, we apply our finding to describe and fit the spontaneous oscillations of the hair bundles in the ear of the bullfrog, for which experimental measurements are available. Then, we use the values of the fitted parameters in order to predict quantitatively the stationary dissipated power, a quantity that cannot be accessed experimentally. Finally, in Chapter 4, instead of looking at systems that can be described effectively by stochastic processes, we show how these processes may naturally arise from the analysis of nonequilibrium dynamics of quantum systems. Recent works in the field of quantum spin chains have revealed an exact mapping between the quantum evolution of these systems and a set of stochastic differential equations. Following a similar path, we show how this procedure can be extended to the case of a zerodimensional bosonic system. In particular, we consider the quantum anharmonic oscillator with quartic potential, which represents the simplest anharmonic extension of the quantum harmonic oscillator. Then, via a sequence of exact transformations, we parametrize the dynamics of the time evolution operator of the quantum quartic oscillator in terms of stochastic differential equations: for this quantum model, quantum fluctuations can be mapped into classical stochastic ones. Accordingly, the expectation value of observables in this stochastic approach translates into an average over the ensemble of trajectories generated by these stochastic processes. This leads to an intuitive implementation of the numerical analysis of these observables. In general, this approach allows one to use all the analytical and numerical tools of stochastic processes and classical field theories to analyze the dynamics of an anharmonic quantum system. Indeed, we expect that this approach can be extended to higher spatial dimensions, and eventually to manybody bosonic systems with quartic couplings.File  Dimensione  Formato  

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