Maximum Entropy Analysis of Non-Equilibrium Flow Systems and Flow Networks

This research theme concerns the concept of entropy, a measure of disorder, and one of the most profound but least understood discoveries of human knowledge. As shown by Boltzmann, the maximisation of entropy subject to the constraints on a system provides the tool to predict the most probable state of that system. Although this idea is widely applied in thermodynamics, it is much broader, being applicable to all probabilistic systems. There is wide scope for new applications throughout all branches of science, engineering, mathematics and the social sciences.

Over the past few years, this project has led to several major new developments. Firstly, the project team (Niven and external collaborators) have formulated a new maximum-entropy framework for the analysis of any flow network, e.g. electrical, hydraulic, communications, transportation, chemical reaction, ecological, economic and human social networks. The method enables the probabilistic prediction of physical parameters (such as flows and potential differences) as well as graphical properties of the network, when there is insufficient information to obtain a closed-form solution. In 2014, during an intensive international collaboration in Australia, France and Germany, the main analytical, semi-analytical and numerical tools for this method were developed, requiring the handling of nonlinear constraints – a major new advance for maximum-entropy optimisation methods. The new methods were applied to a major urban pipe flow network, a distributed power electrical network, and a national road network.

Secondly, we have provided a maximum-entropy framework for the closure of oscillatory and turbulent flow systems, using spectral decomposition methods. Applied to an external flow system (cylinder wake at Reynolds number 100), the analysis gives mean amplitudes and modal energy levels in close agreement with direct Navier-Stokes simulations, at much lower computational cost.

Thirdly, we have derived a new formulation of non-equilibrium thermodynamics, based on Jaynes’ maximum entropy principle. This provides a new variational principle to predict the stationary state of a dissipative system, which reduces under different cases to maximum or minimum entropy production (MaxEP or MinEP) principles. The analysis provides a theoretical justification for MaxEP principles, now widely applied as a heuristic method in many branches of science, e.g. for the analysis of planetary climate systems, crystallography, heat convection, fluid turbulence and chemically degrading ecosystems.



Figure:  National road network (Germany) overlaid with graph structure representation, showing internal flows (black), external flows (blue) and node leakages (red).

The research undertaken was funded through Go8 / DAAD Australia-Germany Collaboration Scheme (2013-2014), Australian Research Council Discovery Project DP140104402 (2014-2016), Region Poitou-Charentes, France (2014) and ANR Chair of Excellence TUCOROM, Institut PPrime, Poitiers, France (2014-2015).