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By: Robert Niven (University of New South Wales and Institut Pprime, Poitiers)

# Maximum Entropy Analysis of Flow Systems and Flow Networks

The maximum entropy (MaxEnt) principle of Jaynes (1957, 2003) provides a powerful technique for the analysis of complex dynamical systems. Although imbued with several philosophical interpretations, the most useful is undoubtedly the original viewpoint of Boltzmann (1877) that, by a constrained maximisation of entropy, we seek the most probable state of the system. This is identified with the stationary state of the system. This viewpoint unites the analysis of (a) physical and chemical thermodynamic systems with fixed contents, in which we seek the equilibrium state; and (b) non-equilibrium flow systems with fixed forcings, for which we seek the steady state; and (c) non-equilibrium chemical reaction systems with fixed inputs/outputs, for which we seek an evolving stationary state. The MaxEnt method can provide a tremendous degree of simplication, e.g. ~23 orders of magnitude in thermodynamic systems.

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 (Niven, 2009, 2010). 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 is analogous in every aspect to the MaxEnt formulation of equilibrium thermodynamics (e.g. Callen, 1985). This common framework and its application to flow systems are first presented in detail. The framework is then extended to the MaxEnt analysis of a flow network (Niven et al., 2013; Waldrip et al., 2013), a representation which crosses many disciplines, including electrical circuit, communications, water distribution, vehicular transport, chemical reaction, ecological and human social systems. The method is illustrated by the analysis of under-constrained pipe flow and traffic flow networks, as specified by flow rates and potentials. The MaxEnt method is sufficiently general that it is readily extended to incorporate other uncertainties in the system specification, even in the network structure itself.

This project has received funding from ARC and Go8 (Australia) and DAAD (Germany).

Boltzmann, L. (1877), Über die Beziehung zwischen dem zweiten Hauptsätze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respective den Sätzen über das Wärmegleichgewicht, Wien. Ber., 76: 373-435.

Callen, H.B. (1985) Thermodynamics and an Introduction to Thermostatistics, 2nd ed., John Wiley, NY.

Jaynes, E.T. (1957), Information theory and statistical mechanics, Physical Review, 106: 620-630.

Jaynes, E.T. (2003) (Bretthorst, G.L., ed.) Probability Theory: The Logic of Science, Cambridge U.P., Cambridge.

Niven R.K. (2009), Physical Review E 80(2): 021113.

Niven, R.K. (2010), Philosophical Transactions B, 365: 1323-1331.

Niven, R.K., Abel, M., Schlegel, M., Noack, B.R., Waldrip, S.H., Abbass, H.A., Shafi, K. (2013), Maximum entropy analysis of flow networks, MaxEnt 2013, Canberra, 15-20 December 2013.

Waldrip, S.H., Niven, R.K., Abel, M., Schlegel, M., Noack, B.R., Abbass, H.A., Shafi, K. (2013), Maximum entropy analysis of hydraulic pipe networks, MaxEnt 2013, Canberra, 15-20 December 2013.

By: Robert Niven (University of New South Wales and Institut Pprime, Poitiers)

# Maximum Entropy Analysis of Flow Systems and Flow Networks

The maximum entropy (MaxEnt) principle of Jaynes (1957, 2003) provides a powerful technique for the analysis of complex dynamical systems. Although imbued with several philosophical interpretations, the most useful is undoubtedly the original viewpoint of Boltzmann (1877) that, by a constrained maximisation of entropy, we seek the most probable state of the system. This is identified with the stationary state of the system. This viewpoint unites the analysis of (a) physical and chemical thermodynamic systems with fixed contents, in which we seek the equilibrium state; and (b) non-equilibrium flow systems with fixed forcings, for which we seek the steady state; and (c) non-equilibrium chemical reaction systems with fixed inputs/outputs, for which we seek an evolving stationary state. The MaxEnt method can provide a tremendous degree of simplication, e.g. ~23 orders of magnitude in thermodynamic systems.

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 (Niven, 2009, 2010). 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 is analogous in every aspect to the MaxEnt formulation of equilibrium thermodynamics (e.g. Callen, 1985). This common framework and its application to flow systems are first presented in detail. The framework is then extended to the MaxEnt analysis of a flow network (Niven et al., 2013; Waldrip et al., 2013), a representation which crosses many disciplines, including electrical circuit, communications, water distribution, vehicular transport, chemical reaction, ecological and human social systems. The method is illustrated by the analysis of under-constrained pipe flow and traffic flow networks, as specified by flow rates and potentials. The MaxEnt method is sufficiently general that it is readily extended to incorporate other uncertainties in the system specification, even in the network structure itself.

This project has received funding from ARC and Go8 (Australia) and DAAD (Germany).

Boltzmann, L. (1877), Über die Beziehung zwischen dem zweiten Hauptsätze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respective den Sätzen über das Wärmegleichgewicht, Wien. Ber., 76: 373-435.

Callen, H.B. (1985) Thermodynamics and an Introduction to Thermostatistics, 2nd ed., John Wiley, NY.

Jaynes, E.T. (1957), Information theory and statistical mechanics, Physical Review, 106: 620-630.

Jaynes, E.T. (2003) (Bretthorst, G.L., ed.) Probability Theory: The Logic of Science, Cambridge U.P., Cambridge.

Niven R.K. (2009), Physical Review E 80(2): 021113.

Niven, R.K. (2010), Philosophical Transactions B, 365: 1323-1331.

Niven, R.K., Abel, M., Schlegel, M., Noack, B.R., Waldrip, S.H., Abbass, H.A., Shafi, K. (2013), Maximum entropy analysis of flow networks, MaxEnt 2013, Canberra, 15-20 December 2013.

Waldrip, S.H., Niven, R.K., Abel, M., Schlegel, M., Noack, B.R., Abbass, H.A., Shafi, K. (2013), Maximum entropy analysis of hydraulic pipe networks, MaxEnt 2013, Canberra, 15-20 December 2013.