Nonequilibrium Time Reversibility with Maps and Walks

Time-reversible dynamical simulations of nonequilibrium systems exemplify both Loschmidt’s and Zermélo’s paradoxes. That is, computational time-reversible simulations invariably produce solutions consistent with the irreversible Second Law of Thermodynamics (Loschmidt’s) as well as periodic in the time (Zermélo’s, illustrating Poincaré recurrence). Understanding these paradoxical aspects of time-reversible systems is enhanced here by studying the simplest pair of such model systems. The first is time-reversible, but nevertheless dissipative and periodic, the piecewise-linear compressible Baker Map. The fractal properties of that two-dimensional map are mirrored by an even simpler example, the one-dimensional random walk, confined to the unit interval. As a further puzzle the two models yield ambiguities in determining the fractals’ information dimensions. These puzzles, including the classical paradoxes, are reviewed and explored here.

Information Geometry of Reversible Markov Chains

We analyze the information geometric structure of time reversibility for parametric families of irreducible transition kernels of Markov chains. We define and characterize reversible exponential families of Markov kernels, and show that irreducible and reversible Markov kernels form both a mixture family and, perhaps surprisingly, an exponential family in the set of all stochastic kernels. We propose a parametrization of the entire manifold of reversible kernels, and inspect reversible geodesics. We define information projections onto the reversible manifold, and derive closed-form expressions for the e-projection and m-projection, along with Pythagorean identities with respect to information divergence, leading to some new notion of reversiblization of Markov kernels. We show the family of edge measures pertaining to irreducible and reversible kernels also forms an exponential family among distributions over pairs. We further explore geometric properties of the reversible family, by comparing them with other remarkable families of stochastic matrices. Finally, we show that reversible kernels are, in a sense we define, the minimal exponential family generated by the m-family of symmetric kernels, and the smallest mixture family that comprises the e-family of memoryless kernels.

Adjustment of Force–Gradient Operator in Symplectic Methods

Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian H=T(p)+V(q) with kinetic energy T(p)=p2/2 in the existing references. When a force–gradient operator is appropriately adjusted as a new operator, it is still suitable for a class of Hamiltonian problems H=K(p,q)+V(q) with integrable part K(p,q)=∑i=1n∑j=1naijpipj+∑i=1nbipi, where aij=aij(q) and bi=bi(q) are functions of coordinates q. The newly adjusted operator is not a force–gradient operator but is similar to the momentum-version operator associated to the potential V. The newly extended (or adjusted) algorithms are no longer solvers of the original Hamiltonian, but are solvers of slightly modified Hamiltonians. They are explicit symplectic integrators with symmetry or time reversibility. Numerical tests show that the standard symplectic integrators without the new operator are generally poorer than the corresponding extended methods with the new operator in computational accuracies and efficiencies. The optimized methods have better accuracies than the corresponding non-optimized counterparts. Among the tested symplectic methods, the two extended optimized seven-stage fourth-order methods of Omelyan, Mryglod and Folk exhibit the best numerical performance. As a result, one of the two optimized algorithms is used to study the orbital dynamical features of a modified Hénon–Heiles system and a spring pendulum. These extended integrators allow for integrations in Hamiltonian problems, such as the spiral structure in self-consistent models of rotating galaxies and the spiral arms in galaxies.

Confined vortex surface and irreversibility. 1. Properties of exact solution

We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (Migdal, 2021). These surfaces avoid the Kelvin–Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the sheet only in an exceptional case considered long ago by Burgers and Townsend, where it decays as a Gaussian on both sides of the sheet. In generic asymmetric vortex sheets (Shariff, 2021), vorticity leaks to one side or another, making such sheets inadequate for vortex sheet statistics and anomalous dissipation. We conjecture that the vorticity in a turbulent flow collapses on a special kind of surface (confined vortex surface or CVS), satisfying some equations involving the tangent components of the local strain tensor. The most important qualitative observation is that the inequality needed for this solution’s stability breaks the Euler dynamics’ time reversibility. We interpret this as dynamic irreversibility. We have also represented the enstrophy as a surface integral, conserved in the Navier–Stokes equation in the turbulent limit, with vortex stretching and viscous diffusion terms exactly canceling each other on the CVS surfaces. We have studied the CVS equations for the cylindrical vortex surface for an arbitrary constant background strain with two different eigenvalues. This equation reduces to a particular version of the stationary Birkhoff–Rott equation for the 2D flow with an extra nonanalytic term. We study some general properties of this equation and reduce its solution to a fixed point of a map on a sphere, guaranteed to exist by the Brouwer theorem.

Role of transducer inertia in generation, sensing, and time-reversal process of Lamb waves in thin plates with surface-bonded piezoelectric transducers

An analytical model is presented for the generation, sensing, and time-reversible process of Lamb waves in thin isotropic plates with surface-bonded piezoelectric wafer transducers, incorporating the shear-lag effect of the bonding layer and inertia effects of the system in transducer modeling. A one-dimensional dynamic shear-lag model for the actuator-plate interaction is used to obtain the shear stress distribution at the actuator-plate interface. The Lamb wave solution for the plate under this shear traction excitation is obtained using the two-dimensional (2D) elasticity equations. A consistent sensor-plate interaction model incorporating the shear-lag and inertia effects is developed to determine the induced sensor voltage from the Lamb strain at the plate surface. The model is validated by comparing it with the 2D coupled piezoelasticity-based finite element simulation and experimental data. Detailed parametric studies are conducted to illustrate the effect of inclusion of inertia of actuator, sensor, adhesive, and plate in the transducer modeling on the Lamb wave generation, sensing, time reversibility, and the system’s best reconstruction frequency, and to ascertain how various geometrical and material parameters of the system influence the same. The developed closed-form solution will be immensely useful for the design of Lamb wave based structural health monitoring systems.

Parsimonious neural networks learn interpretable physical laws

Machine learning is playing an increasing role in the physical sciences and significant progress has been made towards embedding domain knowledge into models. Less explored is its use to discover interpretable physical laws from data. We propose parsimonious neural networks (PNNs) that combine neural networks with evolutionary optimization to find models that balance accuracy with parsimony. The power and versatility of the approach is demonstrated by developing models for classical mechanics and to predict the melting temperature of materials from fundamental properties. In the first example, the resulting PNNs are easily interpretable as Newton’s second law, expressed as a non-trivial time integrator that exhibits time-reversibility and conserves energy, where the parsimony is critical to extract underlying symmetries from the data. In the second case, the PNNs not only find the celebrated Lindemann melting law, but also new relationships that outperform it in the pareto sense of parsimony vs. accuracy.

Fabrication of Reproducible and Selective Ammonia Vapor Sensor-Pellet of Polypyrrole/Cerium Oxide Nanocomposite for Prompt Detection at Room Temperature

Polypyrrole (PPy) and polypyrrole/cerium oxide nanocomposite (PPy/CeO2) were prepared by the chemical oxidative method in an aqueous medium using anhydrous ferric chloride (FeCl3) as an oxidant. The successful formulation of materials was confirmed by Fourier transform infrared spectroscopy (FT-IR), X-ray diffraction (XRD), thermogravimetric analysis (TGA), scanning electron microscopy (SEM), and transmittance electron microscopy (TEM). A four-in-line probe device was used for studying DC electrical conductivity and ammonia vapor sensing properties of PPy and PPy/CeO2. The significant improvement in both the conductivity and sensing parameters of PPy/CeO2 compared to pristine PPy reveals some synergistic/electronic interaction between PPy and cerium oxide nanoparticles (CeO2 NPs) working at molecular levels. The initial conductivity (i.e., conductivity at room temperature) was found to be 0.152 Scm−1 and 1.295 Scm−1 for PPy and PPy/CeO2, respectively. Also, PPy/CeO2 showed much better conductivity retention than pristine PPy under both the isothermal and cyclic ageing conditions. Ammonia vapor sensing was carried out at different concentration (0.01, 0.03, 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5 vol %). The sensing response of PPy/CeO2 varied with varying concentrations. At 0.5 vol % ammonia concentration, the % sensing response of PPy and PPy/CeO2 sensor was found to be 39.1% and 93.4%, respectively. The sensing efficiency of the PPy/CeO2 sensor was also evaluated at 0.4. 0.3, 0.2, 0.1, 0.05, 0.03, and 0.01 vol % ammonia concentration in terms of % sensing response, response/recovery time, reversibility, selectivity as well as stability at room temperature.

A Numerical Study on Computational Time Reversal for Structural Health Monitoring

Structural health monitoring problems are studied numerically with the time reversal method (TR). The dynamic output of the structure is applied, time reversed, as an external loading and its propagation within the deformable medium is followed backwards in time. Unknown loading sources or damages can be discovered by means of this method, focused by the reversed signal. The method is theoretically justified by the time-reversibility of the wave equation. Damage identification problems relevant to structural health monitoring for truss and frame structures are studied here. Beam structures are used for the demonstration of the concept, by means of numerical experiments. The influence of the signal-to-noise ratio (SNR) on the results was investigated, since this quantity influences the applicability of the method in real-life cases. The method is promising, in view of the increasing availability of distributed intelligent sensors and actuators.

Inferring the number and position of changes in selective regime in a non-equilibrium mutation-selection framework

Background Recovering the historical patterns of selection acting on a protein coding sequence is a major goal of evolutionary biology. Mutation-selection models address this problem by explicitly modelling fixation rates as a function of site-specific amino acid fitness values.However, they are restricted in their utility for investigating directional evolution because they require prior knowledge of the locations of fitness changes in the lineages of a phylogeny. Results We apply a modified mutation-selection methodology that relaxes assumptions of equlibrium and time-reversibility. Our implementation allows us to identify branches where adaptive or compensatory shifts in the fitness landscape have taken place, signalled by a change in amino acid fitness profiles. Through simulation and analysis of an empirical data set of $$\beta $$ β -lactamase genes, we test our ability to recover the position of adaptive events within the tree and successfully reconstruct initial codon frequencies and fitness profile parameters generated under the non-stationary model. Conclusion We demonstrate successful detection of selective shifts and identification of the affected branch on partitions of 300 codons or more. We successfully reconstruct fitness parameters and initial codon frequencies in simulated data and demonstrate that failing to account for non-equilibrium evolution can increase the error in fitness profile estimation. We also demonstrate reconstruction of plausible shifts in amino acid fitnesses in the bacterial $$\beta $$ β -lactamase family and discuss some caveats for interpretation.