Variability of complex oscillatory regimes in the stochastic model of cold-flame combustion of a hydrocarbon mixture

Processes of the cold-flame combustion of a mixture of two hydrocarbons are studied on the base of a three-dimensional nonlinear dynamical model. Bifurcation analysis of the deterministic model reveals mono- and bistability parameter zones with equilibrium and oscillatory attractors. For this model, effects of random disturbances in the bistability parameter zone are studied. We show that random forcing causes transitions between coexisting stable equilibria and limit cycles with the formation of complex stochastic mixed-mode oscillations. Properties of these oscillatory regimes are studied by means of statistics of interspike intervals. A phenomenon of anti-coherence resonance is discussed. This article is part of the theme issue ‘Transport phenomena in complex systems (part 2)’.

Rate of Entropy Production in Stochastic Mechanical Systems

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker–Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.

Power Spectral Density Analysis of Nanowire-Anchored Fluctuating Microbead Reveals a Double Lorentzian Distribution

In this work, we investigate the properties of a stochastic model, in which two coupled degrees of freedom are subordinated to viscous, elastic, and also additive random forces. Our model, which builds on previous progress in Brownian motion theory, is designed to describe water-immersed microparticles connected to a cantilever nanowire prepared by polymerization using two-photon direct laser writing (TPP-DLW). The model focuses on insights into nanowires exhibiting viscoelastic behavior, which defines the specific conditions of the microbead. The nanowire bending is described by a three-parameter linear model. The theoretical model is studied from the point of view of the power spectrum density of Brownian fluctuations. Our approach also focuses on the potential energy equipartition, which determines random forcing parametrization. Analytical calculations are provided that result in a double-Lorentzian power density spectrum with two corner frequencies. The proposed model explained our preliminary experimental findings as a result of the use of regression analysis. Furthermore, an a posteriori form of regression efficiency evaluation was designed and applied to three typical spectral regions. The agreement of respective moments obtained by integration of regressed dependences as well as by summing experimental data was confirmed.

Random World and Quantum Mechanics

Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω , to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.

Random forcing with a constant power input for two-dimensional gyrokinetic simulations

A method of random forcing with a constant power input for two-dimensional gyrokinetic turbulence simulations is developed for the study of stationary plasma turbulence. The property that the forcing term injects the energy at a constant rate enables turbulence to be set up in the desired range and energy dissipation channels to be assessed quantitatively in a statistically steady state. Using the developed method, turbulence is demonstrated in the large-scale fluid and small-scale kinetic regimes, where the theoretically predicted scaling laws are reproduced successfully.

Development of stochastically perturbed parameterization scheme for the Noah Land Surface Model with the optimized random forcing parameters using the micro-genetic algorithm

<p>The ensemble data assimilation system is beneficial to represent the initial uncertainties and flow-dependent background error covariance (BEC). In particular, the inevitable model uncertainties can be expressed by ensemble spread, that is the standard deviation of ensemble BEC. However, the ensemble spread generally suffers from under-estimated problems. To alleviate this problem, recent studies employed stochastic perturbation schemes to increases the ensemble spreads by adding the random forcing in the model tendencies (i.e., physical or dynamical tendencies) or parameterization schemes (i.e., PBL, convective scheme, etc.). In this study, we focus on the near-surface uncertainties which are affected by the interactions between the land and atmosphere process. The land surface model (LSM) provides various fluxes as the lower boundary condition to the atmosphere, influencing the accuracy of hourly-to-seasonal scale weather forecasting, but the surface uncertainties were not much addressed yet. In this study, we developed the stochastically perturbed parameterization (SPP) scheme for the Noah LSM. The Weather Research and Forecasting (WRF) ensemble system is used for regional weather forecasting over East Asia, especially over the Korean Peninsula. As a testbed experiment with the newly-developed Noah LSM-SPP system, we first perturbed the soil temperature — a crucial variable for the near-surface forecasts by affecting sensible heat fluxes, land surface skin temperature and surface air temperature, and hence lower-tropospheric temperature. Here, the random forcing used in perturbation is made by the tuning parameters for amplitude, length scale, and time scales: they are commonly determined empirically by trial and error. In order to find optimal tuning parameter values, we applied a global optimization algorithm — the micro-genetic algorithm (micro-GA) — to achieve the smallest root-mean-squared errors. Our results indicate that optimization of the random forcing parameters contributes to an increase in the ensemble spread and a decrease in the ensemble mean errors in the near-surface and lower-troposphere uncertainties. Further experiments will be conducted by including soil moisture in the testbed.</p>

Random World and Quantum Mechanics

Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω, to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.

Estimating the moments of a random forcing field of 2D fluid from image sequences using energy minimisation method

In this paper, we consider a version of energy minimisation technique applied to images of a 2D fluid flow. The two Navier–Stokes equations describe the static flow of a 2D fluid in terms of velocity field, (u, v), pressure field, p and forcing field, f. Apart from these two Navier–Stokes equations, we have the incompressibility condition to evaluate the three parameters. While implementing this system, random noise (usually non-Gaussian) creeps into the random force field $$\underline{f}(x,y)$$ f ̲ ( x , y ) . We denote this random field by $$\delta \underline{f}(x,y)$$ δ f ̲ ( x , y ) having zero mean and non-trivial second and third moments. We assume that these two moments are known except for some unknown parameters $$\underline{\theta }$$ θ ̲ (like mean, variance, co-variance, skewness, etc.) which we wish to estimate. In the proposed technique, we first calculate the approximate shift in the average fluid energy defined as a quadratic function of the velocity field. The energy method then requires that $$\underline{\theta }$$ θ ̲ should be such that this average increases in the energy due to the random forcing component be minimised. We should, however, note that the standard statistical approach to force field estimation is to calculate the velocity field as a function of the force field and then adopt the statistical moment matching technique. Such an approach assumes spatial ergodicity of the velocity field. This approach to force field estimation is more accurate from the statistical moment matching view point but works only if velocity measurements are made. The former technique of energy minimisation does not require any velocity measurements. Both of these techniques are discussed in this paper and MATLAB simulations presented.