# Séminaire de Mécanique d'Orsay

## Le Vendredi 4 juillet à 14h00 - Salle de conférences du LIMSI

### Maximum entropy analysis in fluid flow systems and networks

## Robert Niven

School of Engineering and Information Technology
University of New South Wales
Camberra, Australia

The concept of a "network" - a set of nodes connected by flow paths - unites many seemingly disparate disciplines, including electrical circuit, communications, water distribution, vehicular transport, chemical reaction and ecological systems, and is now a popular representation for human financial, political and social systems. Historically, the state of a flow network has been analysed by conservation and potential difference (Kirchhoff's) laws or network mappings (e.g. Tellegen's theorem), and more recently by various optimisation methods and dynamical simulation. A relatively unexplored approach, however, is the use of Jaynes' maximum entropy (MaxEnt) method [1]. This method is founded on the generic concept of "entropy", representing the uncertainty associated with a system. When maximised, this gives the most uncertain or most probable state of the system, and so can be used to infer its state.
Recently, the author presented a new formulation of non-equilibrium thermodynamics for the analysis of infinitesimal flow systems, based on a direct application of MaxEnt [2,3]. The analysis invokes an entropy over the set of instantaneous flow and reaction states, giving a potential function (analogous to the Planck potential) which is minimised at steady-state flow. The analysis provides a steady-state analog of the MaxEnt formulation of equilibrium thermodynamics [4]. In this seminar, the formulation of this MaxEnt framework and its implications are presented in detail. The framework is then extended to enable the MaxEnt analysis of generalised networks of any type of flows. The analysis is sufficiently general to allow the inclusion of multiple connections between nodes, sources/sinks at each node, multiple species flows, and frictional and capacity constraints.
This project has received funding from the DAAD (Germany) and Go8 (Australia).
[1] E.T. Jaynes (G.L. Bretthorst, ed.) Probability Theory: The Logic of Science, Cambridge U.P., Cambridge, 2003.
[2] R.K. Niven, Physical Review E 80(2): 021113 (2009).
[3] R.K. Niven, Philosophical Transactions B, 365: 1323-1331 (2010).
[4] H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., John Wiley, NY, 1985.