10,170 research outputs found
Entropy Production in Random Billiards
We introduce a class of random mechanical systems called random billiards to
study the problem of quantifying the irreversibility of nonequilibrium
macroscopic systems. In a random billiard model, a point particle evolves by
free motion through the interior of a spatial domain, and reflects according to
a reflection operator, specified in the model by a Markov transition kernel,
upon collision with the boundary of the domain. We derive a formula for entropy
production rate that applies to a general class of random billiard systems.
This formula establishes a relation between the purely mathematical concept of
entropy production rate and textbook thermodynamic entropy, recovering in
particular Clausius' formulation of the second law of thermodynamics. We also
study an explicit class of examples whose reflection operator, referred to as
the Maxwell-Smolukowski thermostat, models systems with boundary thermostats
kept at possibly different temperatures. We prove that, under certain mild
regularity conditions, the class of models are uniformly ergodic Markov chains
and derive formulas for the stationary distribution and entropy production rate
in terms of geometric and thermodynamic parameters.Comment: 30 pages, 9 figure
Large deviations of the empirical flow for continuous time Markov chains
We consider a continuous time Markov chain on a countable state space and
prove a joint large deviation principle for the empirical measure and the
empirical flow, which accounts for the total number of jumps between pairs of
states. We give a direct proof using tilting and an indirect one by contraction
from the empirical process.Comment: Minor revision, to appear on Annales de l'Institut Henri Poincare (B)
Probability and Statistic
Network as a computer: ranking paths to find flows
We explore a simple mathematical model of network computation, based on
Markov chains. Similar models apply to a broad range of computational
phenomena, arising in networks of computers, as well as in genetic, and neural
nets, in social networks, and so on. The main problem of interaction with such
spontaneously evolving computational systems is that the data are not uniformly
structured. An interesting approach is to try to extract the semantical content
of the data from their distribution among the nodes. A concept is then
identified by finding the community of nodes that share it. The task of data
structuring is thus reduced to the task of finding the network communities, as
groups of nodes that together perform some non-local data processing. Towards
this goal, we extend the ranking methods from nodes to paths. This allows us to
extract some information about the likely flow biases from the available static
information about the network.Comment: 12 pages, CSR 200
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