Robert K. Niven

Steady state of a dissipative flow-controlled system and the maximum entropy production principle

A theory to predict the steady-state position of a dissipative flow-controlled system, as defined by a control volume, is developed based on the maximum entropy principle of Jaynes, involving minimization of a generalized free-energy-like potential. The analysis provides a theoretical justification of a local, conditional form of the maximum entropy production principle, which successfully predicts the observable properties of many such systems. The analysis reveals a very different manifestation of the second law of thermodynamics in steady-state flow systems, which provides a driving force for the formation of complex systems, including life.

Cluster-based reduced-order modelling of a mixing layer

We propose a novel cluster-based reduced-order modelling (CROM) strategy of unsteady flows. CROM combines the cluster analysis pioneered in Gunzburgers group (Burkardt et al. 2006) and and transition matrix models introduced in fluid dynamics in Eckhardts group (Schneider et al. 2007). CROM constitutes a potential alternative to POD models and generalises the Ulam-Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron-Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Secondly, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

Minimization of a free-energy-like potential for non-equilibrium flow systems at steady state.

This study examines a new formulation of non-equilibrium thermodynamics, which gives a conditional derivation of the ‘maximum entropy production’ (MEP) principle for flow and/or chemical reaction systems at steady state. The analysis uses a dimensionless potential function ϕst for non-equilibrium systems, analogous to the free energy concept of equilibrium thermodynamics. Spontaneous reductions in ϕst arise from increases in the ‘flux entropy’ of the system—a measure of the variability of the fluxes—or in the local entropy production; conditionally, depending on the behaviour of the flux entropy, the formulation reduces to the MEP principle. The inferred steady state is also shown to exhibit high variability in its instantaneous fluxes and rates, consistent with the observed behaviour of turbulent fluid flow, heat convection and biological systems; one consequence is the coexistence of energy producers and consumers in ecological systems. The different paths for attaining steady state are also classified.

Exact Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac statistics

Abstract The exact Maxwell–Boltzmann (MB), Bose–Einstein (BE) and Fermi–Dirac (FD) entropies and probabilistic distributions are derived by the combinatorial method of Boltzmann, without Stirlings approximation. The new entropy measures are explicit functions of the probability and degeneracy of each state, and the total number of entities, N. By analysis of the cost of a “binary decision”, exact BE and FD statistics are shown to have profound consequences for the behaviour of quantum mechanical systems.

Combinatorial entropies and statistics

We examine the combinatorial or probabilistic definition (“Boltzmann’s principle”) of the entropy or cross-entropy function H ∝

Remobilization of Residual Non-Aqueous Phase Liquid in Porous Media by Freeze - Thaw Cycles

{\rm ln}\mathbb{W}

Mixed solid/dispersed phase particles in multiphase fluidised beds, Part I: Free energy of stability due to interfacial tension

or D ∝ -

Simultaneous extrema in the entropy production for steady-state fluid flow in parallel pipes

{\rm ln}\mathbb{P}

Cost of s-fold decisions in exact Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac statistics

, where

Origins of the Combinatorial Basis of Entropy

\mathbb{W}