Dynamics, Volume 3, Issue 3 (September 2023) – 13 articles

Computational complexity reduction is at the basis of a new formulation of many-body quantum states according to tensor network ansatz, originally framed in one-dimensional lattices. In order to include long-range entanglement characterizing phase transitions, the multiscale entanglement renormalization ansatz (MERA) defines a sequence of coarse-grained lattices, obtained by targeting the map of a scale-invariant system into an identical coarse-grained one. The quantum circuit associated with this hierarchical structure includes the definition of causal relations and unitary extensions, leading to the definition of ground subspaces as stabilizer codes. The emerging error correcting codes are referred to logical indices located at the highest hierarchical level and to physical indices yielded by redundancy, framed in the AdS-CFT correspondence as holographic quantum codes with bulk and boundary indices, respectively. In a use-case scenario based on errors consisting of spin erasure, the correction is implemented as the reconstruction of a bulk local operator. Full article

We review from a different perspective the approach and solution to the torque-free Euler equations, also called the free asymmetric top equations. We aim to simplify and broaden the study of the asymmetric free rigid body. This is an old but important integrable problem that has two first integrals: the energy and the angular momentum. We reduce this problem by eliminating the time as the independent variable in the three autonomous Euler equations written in cylindrical dimensionless variables, which allows a geometric study of the solution as a function of the cylindrical angle variable $\psi$, by means of continuous deformations dependent on the two independent parameters $\kappa$ and $e_0$. The parameter space is divided into six disjoint regions, whose boundaries are the separatices and degenerated cases. The solutions are given in terms of trigonometric functions of the independent cylindric angle $\psi$. Full article

Boltzmann’s H-theorem is considered a great triumph of science. Though some modifications are necessary to adapt it to modern dynamical theories, it is well established that one of its main tenets remains widely accepted: the introduction of probability is a key element in achieving a transition from time-reversible, deterministic dynamical laws at the microscopic level to irreversible laws describing the approach to equilibrium of isolated macroscopic systems. Thus, it is somehow surprising that we still find instances where this subject is labeled as paradoxical and elusive. More remarkable is the fact that this often happens in texts that succeed in presenting Boltzmann’s ideas with clarity. In order to shed light on how probability allows us to go form microscopic reversibility to macroscopic irreversibility, we use numerical results from a two-dimensional lattice gas composed of distinguishable particles. We discuss the roles played by noise, coarse graining, and probability. The simplicity of our model might help the newcomer to this area in better grasping Boltzmann’s fundamental breakthrough. Full article

This paper investigates the modelling of Korteweg-type fluids and hence the dependence of the stress tensor on gradients of mass density. This topic, originating from the need for describing capillarity effects, is mainly of interest in connection with nanosystems where the mean free path may be comparable with the geometric dimensions of the system. In addition to the Korteweg fluid model, the paper gives a review of the stress tensor function arising in quantum fluid hydrodynamics. Next, thermodynamic consistency is established for a fluid involving first- and second-order density gradients. The modelling investigated is a generalization of the classical Korteweg fluid and allows a better understanding of previous thermodynamic restrictions. The restrictions determined for the general scheme with second-order gradients are applied to the particular cases of the Korteweg fluid and the quantum fluid. Further, to allow for discontinuity wave solutions with finite speed of propagation, a model is established which involves higher-order derivatives and reduces to the Korteweg fluid in stationary conditions. Full article

For a linear ordinary differential equation (ODE in short) of the third order, results are presented that supplement the theory of conjugate points and extremal solutions by W. Leighton, Z. Nehari, M. Hanan. It is especially noted the sensitivity of solutions to the initial data, which makes their numerical study difficult. Similar results were obtained for the third-order nonlinear equations of the Emden-Fowler type. Full article

Two experiments and a dynamic model forhuman limb selection are reported. In Experiment 1, left-handed and right-handed participants (N = 36) repeatedly used one hand for grasping a small cube. After a clear switch in the cube’s location, perseverative limb selection was revealed in both handedness groups. In Experiment 2, the cubes were presented in a clockwise and counter-clockwise sequence to right-handed participants (N = 15). A spatial shift in the switch point between right-hand use and left-hand use was observed. The model simulates the experiments by implementing the nonlinear multiple-timescale dynamics of the action-selection process underlying limb selection. It integrates two mechanisms that were earlier proposed to underlie this selection aspect of manual activity: limb dominance and attentional information. Finally, the model is used to simulate an influential earlier experiment, by establishing a conceptual link between cross-lateral inhibition asymmetry and the direction and strength of handedness. Full article

This manuscript relates to the exploiting of the abstract calculus pattern (ACP) for the (numerical) solution of ordinary differential equation (ODEs) systems, which are ubiquitous mathematical formulations of many physical (dynamical) phenomena. We present FOODIE, a software suite aimed to numerically solve ODE problems by means of a clear, concise, and efficient abstract interface. The results presented prove manifold findings, in particular that our ACP approach enables ease of code development, clearness and robustness, maximization of code re-usability, and conciseness comparable with computer algebra system (CAS) programming (interpreted) but with the computational performance of compiled programming. The proposed programming model is also proven to be agnostic with respect to the parallel paradigm of the computational architecture: the results show that FOODIE applications have good speedup with both shared (OpenMP) and distributed (MPI, CAF) memory architectures. The present paper is the first announcement of the FOODIE project: the current implementation is extensively discussed, and its capabilities are proved by means of tests and examples. Full article

Ocean Thermal Energy Conversion (OTEC) is a process that can produce electricity by utilizing the temperature difference between deep cold water and surface warm water. The cold-water pipe (CWP) is a key component of OTEC systems, which transports deep cold water to the floating platform. The CWP is subjected to various environmental and operational loads, such as waves, currents, internal flow, and platform motion, which can affect its dynamic response and stability. In this paper, we establish a computational model of the mechanical performance of the CWP based on the Euler–Bernoulli beam theory and the Morrison equation, considering the effects of internal flow, sea current, and wave excitation. We use the differential quadrature method (DQM) to obtain a semi-analytical solution of the lateral displacement and bending moment of the CWP. We verify the correctness and validity of our model by comparing it with the finite element simulation results using OrcaFlex software. We also analyze the effects of operating conditions—such as wave intensity, clump weight at the bottom, and internal flow velocity—on the dynamic response of the CWP using numerical simulation and the orthogonal experimental method. The results show that changing the wave strength and internal flow velocity has little effect on the lateral displacement of the CWP but increasing the current velocity can significantly increase the lateral displacement of the CWP, which can lead to instability. The effects of waves, clump weight, internal flow, and sea current on the maximum bending moment of the CWP are similar; all of them increase sharply at first and then decrease gradually until they level off. The differences in the effects are mainly reflected in the different locations of the pipe sections. This paper suggests some design guidance for CWP in terms of dynamic responses depending on the operating conditions. This paper contributes to the journal’s scope by providing a novel and efficient method for analyzing the mechanical performance of CWP for OTEC systems, which is an important ocean energy resource. Full article

Many physical processes can be described via nonlinear second-order ordinary differential equations and so, exact solutions to these equations are of interest as, aside from their accuracy, they may reveal beforehand key properties of the system’s response. This work presents a method for computing exact solutions of second-order nonlinear autonomous undamped ordinary differential equations. The solutions are divided into nine cases, each depending on the initial conditions and the system’s first integral. The exact solutions are constructed via a suitable parametrization of the unknown function into a class of functions capable of representing its behavior. The solution is shown to exist and be well-defined in all cases for a general nonlinear form of the differential equation. Practical properties of the solution, such as its period, time to reach an extreme value or long-term behavior, are obtained without the need of computing the solution in advance. Illustrative examples considering different types of nonlinearity present in classical physical systems are used to further validate the obtained exact solutions. Full article

To further explore the origins of Life, we consider three self-replicating chemical models. In general, models of the origin of Life include molecular components that can self-replicate and achieve exponential growth. Therefore, chemical self-replication is an essential chemical property of any model. The simplest self-replication mechanisms use the molecular product as a template for its synthesis. This mechanism is the so-called First-Order self-replication. Its regulatory limitations make it challenging to develop chemical networks, which are essential in the models of the origins of Life. In Second-Order self-replication, the molecular product forms a catalytic dimer capable of synthesis of the principal molecular product. In contrast with a simple template, the dimers show more flexibility in forming complex chemical networks since the chemical activity of the dimers can be activated or inhibited by the molecular components of the network. Here, we consider three minimal models: the First-Order Model (FOM), the Second-Order Model (SOM), and an Extended Second-Order Model (ESOM). We construct and analyze the mechanistic dimensionless ordinary differential equations (ODEs) associated with the models. The numerical integration of the set of ODEs gives us a visualization of these systems’ oscillatory behavior and compares their capacities for sustained autocatalytic behavior. The FOM model displays more complex oscillatory behavior than the ESOM model. Full article

The phenomena of electrical conductivity and electromigration in metallic systems are related, since in both cases the basic physical process is the scattering of conduction electrons by metal ions. Numerous searches have been made for equations connecting the conductivity with electromigration. In the case of a liquid metal, when using the Drude–Sommerfeld (DS) conductivity equation, it was not possible to obtain a quantitative relationship between these phenomena, which would be correct. Attempts to find such a relationship when taking into account the N. Mott correction (g-factor) in the DS equation were unsuccessful. This article proposes a different correction (b-factor) to the DS equation, which takes into account the possibility of varying the momentum transferred by the conduction electron to a metal ion during the scattering. This correction allows to establish a quantitative relationship between conductivity and electromigration as well as between electromigration in various binary systems with common components, in agreement with the experiment. The proposed theory describes well, in particular, two- and multi-component metal systems of any concentration (the consistency rule for triangles A–B, B–C, C–A). The value of the b-factor smoothly changes depending on the heat of vaporization of the metal, per unit volume. Full article

This study attempts to shed new light on the dynamics of a turning ship using the principles of free body diagrams (FBDs). Unexpectedly, the literature gap is defined by incomplete and flawed FBDs. The method behind this new approach involves the FBD of a turning ship, with all the essential forces included, namely propulsive force, sideward thruster force (producing the initial turning moment), drag force, lift force, centrifugal force, inertial force, and hydrodynamic force couple. From these forces, the force and moment equations are derived. The accelerations are calculated from the force and moment equilibria to simulate the dynamics from input parameters such as mass m, length L, draught D, and fluid density ρ. The turning dynamics are explained in terms of velocities, accelerations, forces, and moments, based on two conditions: flat and steep angles of attack (AoA) and long and short turning radii R. A critical result is the proportionality of lift and centrifugal forces, leading to the proposal of a pleometric index (m·L^–2·D^–1·ρ^–1), which is nonlinearly proportional to the product of AoA and R/L, characterising the dynamics of a turning ship. The FBD approach of this study also identified missing databases required for accurate simulation of turning dynamics, such as drag and lift coefficients of different hull geometries. Full article