Серия лекций НУЛ МУСС: Карлос Мейа Монастерио

Anomalous diffusion on disordered lattices

Transport in disordered systems does not follow the classical laws of diffusion in ordered media, and the interplay between 

the dynamics and the properties of the disorder yield in most situations anomalous diffusion. In this lecture we will consider a

random walk in a two dimensional disordered lattice. Motivated by  the so-called ice disorder  model, the disorder is  set by  randomly assigning  to each  lattice site  one of  the six possible configurations consisting of two  exits and two entrances. 

We show that in  spite of the lattice being characterised  by a mean zero preferential driving,  the statistics of the  walker’s 

displacement is super-diffusive,  with a  scaling  exponent that  depends  on the  way disorder  is  constructed. A particular constrain

of ice disorder is the so-called random Manhattan model in which infinite lanes with a given direction fill the lattice randomly. 

For the random Manhattan model we derive analytic expressions for the anomalous super diffusive transport of a random walker 

and discuss the influence of stochastic interaction with other walkers.

The thermodynamics of the small

Can we formulate a thermodynamic framework for systems at the nano-scale? The answer to this question is under intense theoretical and experimental investigation. Learning about how thermodynamic processes of small systems, coupled to large reservoirs occur, sheds light on the energetic functioning of living cells, the physics of molecular motors, the optimal design of nano engines, among a host of other problems. Small systems are characterised by fluctuations that can be as large as the observables of interest, thus rendering their treatment different than the thermodynamic systems at the usual scales. In this talk we will explore the behaviour of small systems and learn how their fluctuations are characterised in and out of equilibrium, and rooted on fundamental symmetries of the microscopic dynamics. Small systems are naturally described by stochastic models.  After introducing the Stochastic Thermodynamics framework, we will learn how quantities such as heat, work or entropy production can be define, and used to obtain optimal thermodynamic cycles in terms of their efficiency or to minimise dissipation.