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Brownian motion in logarithmic potential
Berestneva, Ekaterina ; Ryabov, Artem (advisor) ; Chvosta, Petr (referee)
In this thesis we study first-passage properties of a Brownian particle diffusing under the action of logarithmic potential field U(x, t) = g(t) log(x). The main part of this thesis is de- voted to the case of time-dependent potential strength g(t). To obtain the corresponding survival probability, one may try to solve the Fokker-Planck equation. However, its exact solution for the time-dependent potential is yet unknown. In this work we propose a simple asymptotic theory which yields the long-time behaviour of the survival probability and the moments of the particle position. The survival probability exhibits a rather varied behaviour for different functions g(t). We identify three regimes of asymptotic decay: the regular regime, the marginal regime and the regime of enhanced absorption. We also address the question of how will the derived first-passage properties of Brownian motion change when the absorbing boundary is not exactly at the origin. 1
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Stochastic dynamics and thermodynamics in nonequilibrium steady states
Berestneva, Ekaterina ; Ryabov, Artem (advisor) ; Novotný, Tomáš (referee)
In this thesis we study two stochastic models related to operation of a molecular mo- tor. The first is a Brownian particle moving under the action of the highly unstable potential. It can describe fast processes related to individual steps of the motor. We study statistics of trajectories that by chance avoid the unstable region and do not diverge up to a long time. Conditioning on nondivergence gives rise to an effective force, which keeps the particles in the stable area of the potential. We present two stationary distributions which formally resemble the Gibbs canonical distribution with effective potentials and derive asymptotic behaviors of these potentials. The se- cond is the minimal discrete model of the Feynman-Smoluchowski ratchet coupled to two thermal reservoirs. We investigate stationary values of the average steady state currents, activities, and motor efficiency. For the ratchet we construct the driven processes representing mean quantities conditioned on fluctuations of entropy production and show how the entropy production affects mean probability currents and activity. 1
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Brownian motion in logarithmic potential
Berestneva, Ekaterina ; Ryabov, Artem (advisor) ; Chvosta, Petr (referee)
In this thesis we study first-passage properties of a Brownian particle diffusing under the action of logarithmic potential field U(x, t) = g(t) log(x). The main part of this thesis is de- voted to the case of time-dependent potential strength g(t). To obtain the corresponding survival probability, one may try to solve the Fokker-Planck equation. However, its exact solution for the time-dependent potential is yet unknown. In this work we propose a simple asymptotic theory which yields the long-time behaviour of the survival probability and the moments of the particle position. The survival probability exhibits a rather varied behaviour for different functions g(t). We identify three regimes of asymptotic decay: the regular regime, the marginal regime and the regime of enhanced absorption. We also address the question of how will the derived first-passage properties of Brownian motion change when the absorbing boundary is not exactly at the origin. 1
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