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Mathematical Aspects of Nonequilibrium Stochastic Processes
- Oggetto:
Mathematical Aspects of Nonequilibrium Stochastic Processes
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Academic year 2017/2018
- Teacher
- Carlos Mejia Monasterio
- Type
- Basic
- Delivery
- Formal authority
- Language
- Italian
- Attendance
- Obligatory
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Sommario del corso
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Course objectives
Strochastic dynamics have been used successfully both for theoretical modelling of complex physical and biological systems and for the analysis of experimental data, where the stochastic aspects represent the uncontrolled interaction with the environment, Thermal fluctuation or hidden internal dynamics
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Results of learning outcomes
In this series of lectures we will provide a systematic mathematical exposition of the conceptual problems of nonequilibrium statistical mechanics with an emphasis to stochastic dynamics. We will concentrate on diffusion processes to model physical and biological systems. We will derive and understand the reaches and limitations of the theory of stochastic trajectories. We will learn the basics of stochastic control theory and use to study the optimal operation of small nano-engines and biological molecular motors. We will establish the relation between these optimal processes and the recent mathematical theory of optimal transportation.
In particular we will identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the Markov jump problem obey a system of differential equations of Hamilton-Jacobi-Bellman-type, used in dynamical programming. We will extend stochastic thermodynamics to Markov-jump processes, allowing us to model processes in quantum systems. As a final application to biological systems, we will learn the basics of first passage time for diffusion processes and learn how such models can be used to model metabolic processes of the cell.- Oggetto:
Course delivery
15 lectures of 2 hours
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Program
Outline of topics
Diffusion processes:
A brief introduction to probability theory
continuous time Markov processes
Markov Jump processes
first passage time processes
A primer in stochastic control theory and dynamical programming
Stochastic Thermodynamics:
Energetic balance of the Langevin Dynamics
Entropy production, irreversibility and Markov Chains
Optimal transportation and minimal dissipation processes
Efficiency and maximum power efficiencyStochastic processes in cell kinetics:
Stochastic processes in the living Cell
First passage time probability and the backward Fokker- Planck equation
Diffusion in confined geometries
Molecular motorsSuggested readings and bibliography
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