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Robert K. Niven

Publications and source records attributed to Robert K. Niven.

At least 19 recordsLinked to original sources
arXiv2026

Clustering and surface distributions of buoyant particles in open-channel flows

This study investigates the clustering behaviour and surface distributions of buoyant particles at the air-water interface in open-channel turbulent flow, focusing on the interplay between capillary attraction, hydrodynamic drag, and flow-driven lateral transport. Using controlled laboratory flume experiments, we systematically examine clustering dynamics for two particle types differing in size and density. To interpret the observed behaviour, we extend capillary-based clustering frameworks to open-channel flows by introducing a dimensionless clustering Weber number (We_cl) that captures the balance between the flow-induced disruptive force and capillary attraction, providing a compact description of the observed clustering behaviour. In addition, we demonstrate that secondary currents play a central role in surface particle transport, producing systematic lateral accumulation that depends on channel aspect ratio. Together, these findings extend capillary-driven clustering theory to open-channel turbulence and reveal secondary currents as a key mechanism controlling particle surface distributions.

physics.flu-dyn↗
arXiv2024

Dynamical System Identification, Model Selection and Model Uncertainty Quantification by Bayesian Inference

This study presents a Bayesian maximum \textit{a~posteriori} (MAP) framework for dynamical system identification from time-series data. This is shown to be equivalent to a generalized Tikhonov regularization, providing a rational justification for the choice of the residual and regularization terms, respectively, from the negative logarithms of the likelihood and prior distributions. In addition to the estimation of model coefficients, the Bayesian interpretation gives access to the full apparatus for Bayesian inference, including the ranking of models, the quantification of model uncertainties and the estimation of unknown (nuisance) hyperparameters. Two Bayesian algorithms, joint maximum \textit{a~posteriori} (JMAP) and variational Bayesian approximation (VBA), are compared to the {LASSO, ridge regression and SINDy algorithms for sparse} regression, by application to several dynamical systems with added {Gaussian or Laplace} noise. For multivariate Gaussian likelihood and prior distributions, the Bayesian formulation gives Gaussian posterior and evidence distributions, in which the numerator terms can be expressed in terms of the Mahalanobis distance or ``Gaussian norm'' $||\vec{y}-\hat{\vec{y}}||^2_{M^{-1}} = (\vec{y}-\hat{\vec{y}})^\top {M^{-1}} (\vec{y}-\hat{\vec{y}})$, where $\vec{y}$ is a vector variable, $\hat{\vec{y}}$ is its estimator and $M$ is the covariance matrix. The posterior Gaussian norm is shown to provide a robust metric for quantitative model selection {for the different systems and noise models examined.}

stat.ME↗
arXiv2023

Dimensionless Groups by Entropic Similarity

We propose an additional category of dimensionless groups based on the principle of {\it entropic similarity}, defined by ratios of (i) entropy production terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes. Since all processes involving work against friction, dissipation, diffusion, dispersion, mixing, separation, chemical reaction, gain of information or other irreversible changes are driven by (or must overcome) the second law of thermodynamics, it is appropriate to analyse these processes directly in terms of competing entropy-producing and transporting phenomena and the dominant entropic regime, rather than indirectly in terms of forces. In this study, we derive entropic groups for a wide variety of phenomena relevant to fluid mechanics, classified into diffusion and chemical reaction processes, dispersion mechanisms and wave phenomena. It is shown that many dimensionless groups traditionally derived by kinematic or dynamic similarity (including the Reynolds number) can also be recovered by entropic similarity -- albeit with a different entropic interpretation -- while a large number of new dimensionless groups are also identified. The analyses significantly expand the scope of dimensional analysis and similarity arguments for the resolution of new and existing problems in science and engineering.

physics.flu-dyn↗
arXiv2021

New Classes of Conservation Laws Based on Generalized Fluid Densities and Reynolds Transport Theorems

The Reynolds transport theorem occupies a central place in fluid dynamics, providing a generalized integral conservation equation for the transport of any conserved quantity within a fluid, and connected to its corresponding differential equation. Recently, a new generalized framework was presented for this theorem, enabling parametric transformations between positions on a manifold or in a generalized coordinate space, exploiting the underlying multivariate Lie symmetries associated with a conserved quantity. We examine the implications of this framework for fluid flow systems, within an Eulerian position-velocity (phase space) description. The analysis invokes a hierarchy of five probability density functions, which by convolution are used to define five fluid densities and generalized densities appropriate for different spaces. We obtain 11 formulations of the generalized Reynolds transport theorem for different choices of the coordinate space, parameter space and density, only the first of which is known. These are used to generate 11 tables of integral and differential conservation laws applicable to these systems, for eight important conserved quantities (fluid mass, species mass, linear momentum, angular momentum, energy, charge, entropy and probability). These substantially expand the set of conservation laws for the analysis of fluid flow and dynamical systems.

physics.flu-dyn↗
arXiv2020

Rethinking the Reynolds Transport Theorem, Liouville Equation, and Perron-Frobenius and Koopman Operators

The Reynolds transport theorem provides a generalized conservation law for the transport of a conserved quantity by fluid flow through a continuous connected control volume. It is close connected to the Liouville equation for the conservation of a local probability density function, which in turn leads to the Perron-Frobenius and Koopman evolution operators. All of these tools can be interpreted as continuous temporal maps between fluid elements or domains, connected by the integral curves (pathlines) described by a velocity vector field. We here review these theorems and operators, to present a unified framework for their extension to maps in different spaces. These include (a) spatial maps between different positions in a time-independent flow, connected by a velocity gradient tensor field, and (b) parametric maps between different positions in a manifold, connected by a generalized tensor field. The general formulation invokes a multivariate extension of exterior calculus, and the concept of a probability differential form. The analyses reveal the existence of multivariate continuous (Lie) symmetries induced by a vector or tensor field associated with a conserved quantity, which are manifested as integral conservation laws in different spaces. The findings are used to derive generalized conservation laws, Liouville equations and operators for a number of fluid mechanical and dynamical systems, including spatial (time-independent) and spatiotemporal fluid flows, flow systems with pairwise or $n$-wise spatial correlations, phase space systems, Lagrangian flows, spectral flows, and systems with coupled chemical reaction and flow processes.

physics.flu-dyn↗
arXiv2018

Learning Optimal Control of Synchronization in Networks of Coupled Oscillators using Genetic Programming-based Symbolic Regression

Networks of coupled dynamical systems provide a powerful way to model systems with enormously complex dynamics, such as the human brain. Control of synchronization in such networked systems has far reaching applications in many domains, including engineering and medicine. In this paper, we formulate the synchronization control in dynamical systems as an optimization problem and present a multi-objective genetic programming-based approach to infer optimal control functions that drive the system from a synchronized to a non-synchronized state and vice-versa. The genetic programming-based controller allows learning optimal control functions in an interpretable symbolic form. The effectiveness of the proposed approach is demonstrated in controlling synchronization in coupled oscillator systems linked in networks of increasing order complexity, ranging from a simple coupled oscillator system to a hierarchical network of coupled oscillators. The results show that the proposed method can learn highly-effective and interpretable control functions for such systems.

nlin.AO↗
arXiv2016

Prediction of Dynamical Systems by Symbolic Regression

We study the modeling and prediction of dynamical systems based on conventional models derived from measurements. Such algorithms are highly desirable in situations where the underlying dynamics are hard to model from physical principles or simplified models need to be found. We focus on symbolic regression methods as a part of machine learning. These algorithms are capable of learning an analytically tractable model from data, a highly valuable property. Symbolic regression methods can be considered as generalized regression methods. We investigate two particular algorithms, the so-called fast function extraction which is a generalized linear regression algorithm, and genetic programming which is a very general method. Both are able to combine functions in a certain way such that a good model for the prediction of the temporal evolution of a dynamical system can be identified. We illustrate the algorithms by finding a prediction for the evolution of a harmonic oscillator based on measurements, by detecting an arriving front in an excitable system, and as a real-world application, the prediction of solar power production based on energy production observations at a given site together with the weather forecast.

physics.data-an↗
arXiv2014

Control Volume Analysis, Entropy Balance and the Entropy Production in Flow Systems

This chapter concerns "control volume analysis", the standard engineering tool for the analysis of flow systems, and its application to entropy balance calculations. Firstly, the principles of control volume analysis are enunciated and applied to flows of conserved quantities (e.g. mass, momentum, energy) through a control volume, giving integral (Reynolds transport theorem) and differential forms of the conservation equations. Several definitions of steady state are discussed. The concept of "entropy" is then established using Jaynes' maximum entropy method, both in general and in equilibrium thermodynamics. The thermodynamic entropy then gives the "entropy production" concept. Equations for the entropy production are then derived for simple, integral and infinitesimal flow systems. Some technical aspects are examined, including discrete and continuum representations of volume elements, the effect of radiation, and the analysis of systems subdivided into compartments. A Reynolds decomposition of the entropy production equation then reveals an "entropy production closure problem" in fluctuating dissipative systems: even at steady state, the entropy production based on mean flow rates and gradients is not necessarily in balance with the outward entropy fluxes based on mean quantities. Finally, a direct analysis of an infinitesimal element by Jaynes' maximum entropy method yields a theoretical framework with which to predict the steady state of a flow system. This is cast in terms of a "minimum flux potential" principle, which reduces, in different circumstances, to maximum or minimum entropy production (MaxEP or MinEP) principles. It is hoped that this chapter inspires others to attain a deeper understanding and higher technical rigour in the calculation and extremisation of the entropy production in flow systems of all types.

physics.flu-dyn↗
arXiv2014

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 Gunzburger's group (Burkardt et al. 2006) and and transition matrix models introduced in fluid dynamics in Eckhardt's 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.

physics.flu-dyn↗
arXiv2011

Maximum-Entropy Weighting of Multi-Component Earth Climate Models

A maximum entropy-based framework is presented for the synthesis of projections from multiple Earth climate models. This identifies the most representative (most probable) model from a set of climate models -- as defined by specified constraints -- eliminating the need to calculate the entire set. Two approaches are developed, based on individual climate models or ensembles of models, subject to a single cost (energy) constraint or competing cost-benefit constraints. A finite-time limit on the minimum cost of modifying a model synthesis framework, at finite rates of change, is also reported.

physics.geo-ph↗
arXiv2011

Jaynes' Maximum Entropy Principle, Riemannian Metrics and Generalised Least Action Bound

The set of solutions inferred by the generic maximum entropy (MaxEnt) or maximum relative entropy (MaxREnt) principles of Jaynes - considered as a function of the moment constraints or their conjugate Lagrangian multipliers - is endowed with a Riemannian geometric description, based on the second differential tensor of the entropy or its Legendre transform (negative Massieu function). The analysis provides a generalised {\it least action bound} applicable to all Jaynesian systems, which provides a lower bound to the cost (in generic entropy units) of a transition between inferred positions along a specified path, at specified rates of change of the control parameters. The analysis therefore extends the concepts of "finite time thermodynamics" to the generic Jaynes domain, providing a link between purely static (stationary) inferred positions of a system, and dynamic transitions between these positions (as a function of time or some other coordinate). If the path is unspecified, the analysis gives an absolute lower bound for the cost of the transition, corresponding to the geodesic of the Riemannian hypersurface. The analysis is applied to (i) an equilibrium thermodynamic system subject to mean internal energy and volume constraints, and (ii) a flow system at steady state, subject to constraints on the mean heat, mass and momentum fluxes and chemical reaction rates. The first example recovers the {\it minimum entropy cost} of a transition between equilibrium positions, a widely used result of finite-time thermodynamics. The second example leads to a new {\it minimum entropy production principle}, for the cost of a transition between steady state positions of a flow system.

cond-mat.stat-mech↗
arXiv2010

Simultaneous Extrema in the Entropy Production for Steady-State Fluid Flow in Parallel Pipes

Steady-state flow of an incompressible fluid in parallel pipes can simultaneously satisfy two contradictory extremum principles in the entropy production, depending on the flow conditions. For a constant total flow rate, the flow can satisfy (i) a pipe network minimum entropy production (MinEP) principle with respect to the flow rates, and (ii) the maximum entropy production (MaxEP) principle of Ziegler and Paltridge with respect to the choice of flow regime. The first principle - different to but allied to that of Prigogine - arises from the stability of the steady state compared to non-steady-state flows; it is proven for isothermal laminar and turbulent flows in parallel pipes with a constant power law exponent, but is otherwise invalid. The second principle appears to be more fundamental, driving the formation of turbulent flow in single and parallel pipes at higher Reynolds numbers. For constant head conditions, the flow can satisfy (i) a modified maximum entropy production (MaxEPMod) principle of Županović and co-workers with respect to the flow rates, and (ii) an inversion of the Ziegler-Paltridge MaxEP principle with respect to the flow regime. The interplay between these principles is demonstrated by examples.

physics.flu-dyn↗
arXiv2009

Minimisation of a Free-Energy-Like Potential for Non-Equilibrium 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 $ϕ_{st}$ for non-equilibrium systems, analogous to the free energy concept of equilibrium thermodynamics. Spontaneous reductions in $ϕ_{st}$ 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.

q-bio.PE↗
arXiv2009

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 (MaxEnt) principle of Jaynes, involving minimisation of a generalised free energy-like potential. The analysis provides a theoretical justification of a local, conditional form of the Maximum Entropy Production (MEP) 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.

physics.class-ph↗
arXiv2009

Jaynes' MaxEnt, Steady State Flow Systems and the Maximum Entropy Production Principle

Jaynes' maximum entropy (MaxEnt) principle was recently used to give a conditional, local derivation of the ``maximum entropy production'' (MEP) principle, which states that a flow system with fixed flow(s) or gradient(s) will converge to a steady state of maximum production of thermodynamic entropy (R.K. Niven, Phys. Rev. E, in press). The analysis provides a steady state analog of the MaxEnt formulation of equilibrium thermodynamics, applicable to many complex flow systems at steady state. The present study examines the classification of physical systems, with emphasis on the choice of constraints in MaxEnt. The discussion clarifies the distinction between equilibrium, fluid flow, source/sink, flow/reactive and other systems, leading into an appraisal of the application of MaxEnt to steady state flow and reactive systems.

physics.flu-dyn↗
arXiv2009

Combinatorial Entropies and Statistics

We examine the {combinatorial} or {probabilistic} definition ("Boltzmann's principle") of the entropy or cross-entropy function $H \propto \ln \mathbb{W}$ or $D \propto - \ln \mathbb{P}$, where $\mathbb{W}$ is the statistical weight and $\mathbb{P}$ the probability of a given realization of a system. Extremisation of $H$ or $D$, subject to any constraints, thus selects the "most probable" (MaxProb) realization. If the system is multinomial, $D$ converges asymptotically (for number of entities $N \back \to \back \infty$) to the Kullback-Leibler cross-entropy $D_{KL}$; for equiprobable categories in a system, $H$ converges to the Shannon entropy $H_{Sh}$. However, in many cases $\mathbb{W}$ or $\mathbb{P}$ is not multinomial and/or does not satisfy an asymptotic limit. Such systems cannot meaningfully be analysed with $D_{KL}$ or $H_{Sh}$, but can be analysed directly by MaxProb. This study reviews several examples, including (a) non-asymptotic systems; (b) systems with indistinguishable entities (quantum statistics); (c) systems with indistinguishable categories; (d) systems represented by urn models, such as "neither independent nor identically distributed" (ninid) sampling; and (e) systems representable in graphical form, such as decision trees and networks. Boltzmann's combinatorial definition of entropy is shown to be of greater importance for {"probabilistic inference"} than the axiomatic definition used in information theory.

cond-mat.stat-mech↗
arXiv2009

Non-Asymptotic Thermodynamic Ensembles

Boltzmann's principle is used to select the "most probable" realization (macrostate) of an isolated or closed thermodynamic system, containing a small number of particles ($N \llsp \infty$), for both classical and quantum statistics. The inferred probability distributions provide the means to define intensive variables and construct thermodynamic relationships for small microcanonical systems, which do not satisfy the thermodynamic limit. This is of critical importance to nanoscience and quantum technology.

cond-mat.stat-mech↗