Macroscopic Variables Research Articles

A Novel Lattice Boltzmann Scheme with Single Extended Force Term for Electromagnetic Wave Propagating in One-Dimensional Plasma Medium

A one-dimensional plasma medium is playing a crucial role in modern sensing device design, which can benefit significantly from numerical electromagnetic wave simulation. In this study, we introduce a novel lattice Boltzmann scheme with a single extended force term for electromagnetic wave propagation in a one-dimensional plasma medium. This method is developed by reconstructing the solution to the macroscopic Maxwell’s equations recovered from the lattice Boltzmann equation. The final formulation of the lattice Boltzmann scheme involves only the equilibrium and one non-equilibrium force term. Among them, the former is calculated from the macroscopic electromagnetic variables, and the latter is evaluated from the dispersive effect. Thus, the proposed lattice Boltzmann scheme directly tracks the evolution of macroscopic electromagnetic variables, which yields lower memory costs and facilitates the implementation of physical boundary conditions. Detailed conduction is carried out based on the Chapman–Enskog expansion technique to prove the mathematical consistency between the proposed lattice Boltzmann scheme and Maxwell’s equations. Based on the proposed method, we present electromagnetic pulse propagating behaviors in nondispersive media and the response of a one-dimensional plasma slab to incident electromagnetic waves that span regions above and below the plasma frequency ωp, and further investigate the optical properties of a one-dimensional plasma photonic crystal with periodic thin layers of plasma with different layer thicknesses to verify the stability, accuracy, and flexibility of the proposed method.

Kinetic and thermodynamic examinations for the unsteady couette flow problem of a plasma using the BGK cylindrical model

We aim to investigate the kinetic and irreversible thermodynamic treatment of the Couette flow of gaseous plasma (GP) confined between two coaxial rotating circular rigid cylinders. The Bhatnagar-Gross-Krook (BGK) model of the Boltzmann kinetic equation (BKE) is solved by applying the perturbation method coupled with the moments' approach with a two-stream Maxwellian distribution function (MDF). Nonlinear partial differential equations are precisely solved. As an actual application, we are considering the laboratory Argon GP. An interesting comparison between the non-equilibrium electron velocity distribution function (EVDF) and the equilibrium EVDF is made carefully with 3-Dimensional graphics in various time values. We found that the system goes to an equilibrium state (ES) with time compatible with Le Chatelier's principles. Moreover, we determined with very high precision the value of the system equilibrium time, which is tequ.≅2.4. That great advantage cannot be reached by solving other macroscopic magnetohydrodynamic models. The relations between the various macroscopic variables of the GP are studied. The irreversible non-equilibrium thermodynamics (NT) properties of the system are presented. We present the extended Gibbs equation (EGE) for the internal energy variation (IEV) in cylindrical geometry for GP for the first time, to the best of our knowledge. We have shown that our model is compatible with the H-theorem of Boltzmann and the Onsager-Casimir relations. The various participations in IEV are introduced. The significance of this search is due to its vast implementations in various fields, such as in GP physics, electrical engineering, medicine, biological systems, and numerous commercial and industrial deployments.

Dynamic micromechanical model for smart composite and reinforced shells

AbstractA dynamic micromechanical model based on the asymptotic homogenization pertaining to smart composite and reinforced shells with piezoelectric and piezomagnetic constituents is developed in this paper. Central to the work is the recovery of the so‐called unit cell problems which allow the calculation of the effective coefficients. In turn, these are substituted into the governing equations to obtain a set of basic macroscopic variables which eventually yield asymptotic expansions of all field variables (mechanical stress, electric and magnetic displacement, heat flux etc.). Highlights of the presented dynamic model include the following: (1) the effective properties of the homogenized structure depend strongly on the curvature of the middle surface of the shell; (2) the effective properties are also temporal functions and not merely spatial ones as predicted by other models; (3) the effective properties reflect the influence of all involved constituent material parameters; and (4) the model captures not only the common product properties (magnetoelectricity, pyroelectricity, pyromagnetism) but also other ones relating current density to mechanical displacement, magnetic field, and temperature. These features essentially mean that the homogenized shell is characterized by inhomogeneity or quasi‐homogeneity after the homogenization process and also exhibits memory effect reminiscent of viscoelastic structures. If electrical conductivity is ignored and the effective properties are averaged over the entire time spectrum the results of the model converge to those of previously published quasi‐static models. If, as well, the electrical and magnetic effects are also suppressed the results of the model converge to those of the classical composite shell model.

An efficient discrete velocity method with inner iteration for steady flows in all flow regimes

An efficient improved discrete velocity method (IDVM) with inner iteration is presented to simulate the steady flows in all flow regimes in this work. It is an extension of our previous implicit IDVM to achieve a faster convergence rate. In the previous method, both the discrete velocity Boltzmann equation (DVBE) and the corresponding macroscopic governing equations are solved synchronously, where the computational discrete cost is dominated by the calculation of the DVBE since the number of distribution functions is far larger than that of macroscopic conservative variables. Furthermore, the convergence rate of the calculation of the DVBE is affected by the predicted equilibrium state obtained from the solution of macroscopic governing equations. To provide a more accurate predicted equilibrium state for the fully implicit discretization of the DVBE, an inner iteration is introduced into the solution of macroscopic governing equations, and the flux Jacobian of these equations is evaluated by the difference of numerical fluxes of Navier–Stokes equations rather than the Euler equation-based flux splitting method used in the previous implicit IDVM. This more accurate prediction procedure endows the developed method to accelerate the computation greatly, especially in the continuum flow regime. Numerical results indicate that, in the continuum flow regime, the present method is about one order of magnitude faster than the previous implicit IDVM and one to two orders of magnitude faster than the conventional semi-implicit DVM.

A Poisson map from kinetic theory to hydrodynamics with non-constant entropy

Kinetic theory describes a dilute monatomic gas using a distribution function f(q,p,t), the expected phase-space density of particles at a given position q with a given momentum p. The distribution function evolves according to the collisionless Boltzmann equation in the high Knudsen number limit. Fluid dynamics provides an alternative description of the gas using macroscopic hydrodynamic variables that are functions of position and time only. The mass, momentum and entropy densities of the gas evolve according to the compressible Euler equations in the limit of vanishing viscosity and thermal diffusivity. Both systems can be formulated as noncanonical Hamiltonian systems. Each configuration space is an infinite-dimensional Poisson manifold, and the dynamics is the flow generated by a Hamiltonian functional via a Poisson bracket. We construct a map J1 from the space of distribution functions to the space of hydrodynamic variables that respects the Poisson brackets on the two spaces. This map is therefore a Poisson map. It maps the p-integral of the Boltzmann entropy flogf to the hydrodynamic entropy density. This map belongs to a family of Poisson maps to spaces that include generalised entropy densities as additional hydrodynamic variables. The whole family can be generated from the Taylor expansion of a further Poisson map that depends on a formal parameter. If the kinetic-theory Hamiltonian factors through the Poisson map J1, an exact reduction of kinetic theory to fluid dynamics is possible. However, this is not the case. Nonetheless, by ignoring the contribution to the Hamiltonian from the entropy of the distribution function relative to its local Maxwellian, a distribution function defined by the p-moments ∫dnp(1,p,|p|2)f, we construct an approximate Hamiltonian that factors through the map. The resulting reduced Hamiltonian, which depends on the hydrodynamic variables only, generates the compressible Euler equations. We can thus derive the compressible Euler equations as a Hamiltonian approximation to kinetic theory. We also give an analogous Hamiltonian derivation of the compressible Euler–Poisson equations with non-constant entropy, starting from the Vlasov–Poisson equation.

Weyl–Wigner description of massless Dirac plasmas: ab initio quantum plasmonics for monolayer graphene

We derive, from first principles and using the Weyl–Wigner formalism, a fully quantum kinetic model describing the dynamics in phase space of Dirac electrons in single-layer graphene. In the limit ℏ → 0, we recover the well-known semiclassical Boltzmann equation, widely used in graphene plasmonics. The polarizability function is calculated and, as a benchmark, we retrieve the result based on the random-phase approximation. By keeping all orders in ℏ, we use the newly derived kinetic equation to construct a fluid model for macroscopic variables written in the pseudospin space. As we show, the novel ℏ-dependent terms can be written as corrections to the average current and pressure tensor. Upon linearization of the fluid equations, we obtain a quantum correction to the plasmon dispersion relation, of order ℏ 2, akin to the Bohm term of quantum plasmas. In addition, the average variables provide a way to examine the value of the effective hydrodynamic mass of the carriers. For the latter, we find a relation in which Drude’s mass is multiplied by the square of a velocity-dependent, Lorentz-like factor, with the speed of light replaced by the Fermi velocity, a feature stemming from the quasi-relativistic nature of the Dirac fermions.

Geometric and topological entropies of sphere packing.

We present a statistical mechanical description of randomly packed spherical particles, where the average coordination number is treated as a macroscopic thermodynamic variable. The overall packing entropy is shown to have two contributions: geometric, reflecting statistical weights of individual configurations, and topological, which corresponds to the number of topologically distinct states. Both of them are computed in the thermodynamic limit for isostatic and weakly underconstrained packings in 2D and 3D. The theory generalizes concepts of granular and glassy configurational entropies for the case of nonjammed systems. It is directly applicable to sticky colloids and predicts an asymptotic phase behavior of sticky spheres in the limit of strong binding.

A fractional-step lattice Boltzmann method for multiphase flows with complex interfacial behavior and large density contrast

In the present study, a robust fractional-step lattice Boltzmann (FSLB) method is proposed to simulate the mass transfer phenomenon in incompressible multiphase flows with complex interfacial behavior and large density contrast. The previous simplified lattice Boltzmann method recovers the continuity equation in first-order accuracy and reconstructs the corrector step by directly applying the complex central difference scheme on the macroscopic variables. However, the present FSLB method employs the Chapman-Enskog expansion analysis to reconstruct the convection and diffusion terms of the macroscopic governing equations, and uses the equilibrium and non-equilibrium distribution functions to establish the predictor-corrector step. The intermediate variables are predicted by the equilibrium distribution functions without the consideration of the source terms, and then the physical variables are corrected by the non-equilibrium distribution functions and the source terms without the evolution of the distribution functions. The reconstructed governing equations in both the predictor and corrector steps can be recovered into fully second-order accuracy through the C-E expansion analysis and the Taylor series expansion analysis. The present FSLB method inherits the excellent performance of kinetic theory from the conventional LB method and the good numerical stability from the matured fractional-step method, which is validated by several benchmark problems, such as Laplace law, bubble merging, interfacial deformation under a magnetic field, Rayleigh-Taylor instability at Reynolds number of 3000, Kelvin-Helmholtz instability at Reynolds number of 5000, and bubble rising at density ratio of 1000. A good agreement between the present numerical results with the published numerical data verifies the capability and reliability of the present FSLB method to handle the multiphase problems with complex interfacial behavior and large density contrast.

Analyses and reconstruction of the lattice Boltzmann flux solver

The lattice Boltzmann flux solver (LBFS) uses the finite volume method (FVM) to update macroscopic variables while uses the local solution of lattice Boltzmann equation (LBE) to calculate fluxes at the cell interface. It overcomes the intrinsic drawbacks of the lattice Boltzmann method which include the limitation of uniform mesh, coupled time interval with mesh spacing, and extra memory requirement. The recovered macroscopic equations indicate that LBFS is a weakly compressible model. In general, directly solving weakly compressible models with the central difference scheme suffers numerical instability and additional treatments are needed to stabilize computation. The present paper firstly recovers the macroscopic equations of LBFS (MEs-LBFS) with actual numerical dissipative terms by approximating its actual computational process and then analyzes the mechanisms of the good performance of LBFS. Based on these mechanisms, the reconstructed LBFS (RLBFS), which directly uses macroscopic variables rather than LBE to calculate fluxes at the cell interface, is proposed. Numerical tests indicate that RLBFS preserves the mechanisms of restraining pressure oscillation and stabilizing numerical computation in LBFS perfectly. It is capable of simulating both steady and unsteady, 2D and 3D, viscous and inviscid flows as well, and has higher computational efficiency than LBFS.

Correspondence between momentum-dependent relaxation time and field redefinition of relativistic hydrodynamic theory

In this article a correspondence has been established between the out of equilibrium system dissipation and the thermodynamic field redefinition of the macroscopic variables through the momentum dependent relaxation time approximation (MDRTA) solution of relativistic transport equation. Here, it has been shown that the out of equilibrium thermodynamic fields are not uniquely defined and are subjected to include dissipative effects from the medium. A second order relativistic hydrodynamic theory has been developed including such dissipative effects. The necessary conditions for developing a hydrodynamic theory has been fulfilled, (i) the thermodynamic identities incorporating such redefined fields have been shown to conserve the energy-momentum tensor perfectly under MDRTA, (ii) the non-negativity of entropy production remains unaffected by the inclusion of such dissipative contributions in hydro fields as long as the independent transport coefficients remain positive.

High-Order Unified Gas-Kinetic Scheme

In this paper, we present a high-order unified gas-kinetic scheme (UGKS) using the weighted essentially non-oscillatory with adaptive-order (WENO-AO) method for spatial reconstruction and the two-stage fourth-order scheme for time evolution. Since the UGKS updates both the macroscopic flow variables and microscopic distribution function, and provides an adaptive flux function by combining the equilibrium and non-equilibrium parts, it is possible to take separate treatment of the equilibrium and non-equilibrium calculation in the UGKS for the development of high-order scheme. Considering the fact that high-order techniques are commonly required for continuum flow with complex structures, and the rarefied flow structure are relatively simple and smooth in the physical space, we apply the high-order techniques in the equilibrium part of the UGKS for the capturing of macroscopic flow evolution, and retain the calculation of distribution function as a second-order method, so that a balance of computational cost and numerical accuracy could be well achieved. The high-order UGKS has been validated by several numerical test cases, including sine-wave accuracy test, sod-shock tube, Couette, oscillating Couette, lid-driven cavity and oscillating cavity flow. It is shown that the current method preserves the multiscale property of the original UGKS and obtains more accurate solutions in several cases.

Macroscopic dynamics of gene regulatory networks revealed by individual entropy

Complex systems are usually high-dimensional with intricate interactions among internal components, and may display complicated dynamics under different conditions. While it is difficult to measure detailed dynamics of each component, proper macroscopic description of a complex system is crucial for quantitative studies. In biological systems, each cell is a complex system containing a huge number of molecular components that are interconnected with each other through intricate molecular interaction networks. Here, we consider gene regulatory networks in a cell, and introduce individual entropy as a macroscopic variable to quantify the transcriptional dynamics in response to changes in random perturbations and/or network structures. The proposed individual entropy measures the information entropy of a system at each instant with respect to a basal reference state, and may provide temporal dynamics to characterize switches of system states. Individual entropy provides a method to quantify the stationary macroscopic dynamics of a gene set that is dependent on the gene regulation connections, and can be served as an indicator for the evolution of network structure variation. Moreover, the individual entropy with reference to a preceding state enables us to characterize different dynamic patterns generated from varying network structures. Our results show that the proposed individual entropy can be a valuable macroscopic variable of complex systems in characterizing the transition processes from order to disorder dynamics, and to identify the critical events during the transition process.

Quantum Hydrodynamics of Spinning Particles in Electromagnetic and Torsion Fields

We develop a many-particle quantum-hydrodynamical model of fermion matter interacting with the external classical electromagnetic and gravitational/inertial and torsion fields. The consistent hydrodynamical formulation is constructed for the many-particle quantum system of Dirac fermions on the basis of the nonrelativistic Pauli-like equation obtained via the Foldy–Wouthuysen transformation. With the help of the Madelung decomposition approach, the explicit relations between the microscopic and macroscopic fluid variables are derived. The closed system of equations of quantum hydrodynamics encompasses the continuity equation, and the dynamical equations of the momentum balance and the spin density evolution. The possible experimental manifestations of the torsion in the dynamics of spin waves is discussed.

Size-independent strain gradient effective models based on homogenization methods: Applications to 3D composite materials, pantograph and thin walled lattices

The notion of strain gradients provides a reliable model for capturing size effects and localization phenomena. The difficulty in identifying corresponding constitutive parameters, on the other hand, restricts the practical applicability of such theory. In this work, we aim at developing homogenization based strain-gradient continuum models to compute the effective Cauchy and strain gradient moduli for a wide class of 3D architectured materials and composites prone to strain gradient effects. The present homogenization method is based on a variational principle in linear elasticity articulated with Hill's lemma, which is extended by considering the generalized kinematics within the framework of periodic homogenization. The microscopic displacement field is decomposed into a polynomial function of the introduced macroscopic kinematic variables and the other is periodic fluctuation. The fluctuating displacement is expressed versus the macroscopic kinematic variables solving sequential classical and higher order unit cell boundary value problems. The computed effective elastic moduli in the current computing framework do not depend on the unit cell size, and are therefore intrinsic to the microstructure within the structure. 3D applications for inclusion based composites, pantographic lattices and thin-walled lattice structures showing pronounced strain gradient effects are carried out in order to exemplify the proposed homogenization strategy.

Tryptophan Production Maximization in a Fed-Batch Bioreactor with Modified E. coli Cells, by Optimizing Its Operating Policy Based on an Extended Structured Cell Kinetic Model.

Hybrid kinetic models, linking structured cell metabolic processes to the dynamics of macroscopic variables of the bioreactor, are more and more used in engineering evaluations to derive more precise predictions of the process dynamics under variable operating conditions. Depending on the cell model complexity, such a math tool can be used to evaluate the metabolic fluxes in relation to the bioreactor operating conditions, thus suggesting ways to genetically modify the microorganism for certain purposes. Even if development of such an extended dynamic model requires more experimental and computational efforts, its use is advantageous. The approached probative example refers to a model simulating the dynamics of nanoscale variables from several pathways of the central carbon metabolism (CCM) of Escherichia coli cells, linked to the macroscopic state variables of a fed-batch bioreactor (FBR) used for the tryptophan (TRP) production. The used E. coli strain was modified to replace the PTS system for glucose (GLC) uptake with a more efficient one. The study presents multiple elements of novelty: (i) the experimentally validated modular model itself, and (ii) its efficiency in computationally deriving an optimal operation policy of the FBR.

Role of phase-dependent influence function in the Winfree model of coupled oscillators.

We consider a globally coupled Winfree model comprised of a phase-dependent influence function and sensitive function, and unravel the impact of offset and integer parameters, characterizing the shape of the influence function, on the phase diagram of the Winfree model. The decreasing value of the offset parameter decreases the degree of positive phase shift among the oscillators by promoting the negative phase shift, which indeed favors the onset of multistability among the synchronous oscillatory state and asynchronous stable steady states in a large region of the phase diagram. Further, large integer parameters lead to brief pulses of the influence function, which again enhances the effect of the offset parameter. There is an explosive transition to a synchronous oscillatory state from an asynchronous steady state via a Hopf bifurcation. Dynamical transitions and multistability emerge through saddle-node, pitchfork, and homoclinic bifurcations in the phase diagram. We deduce two ordinary differential equations corresponding to the two macroscopic variables from the population of globally coupled Winfree oscillators using the Ott-Antonsen ansatz. We also deduce various bifurcation curves analytically from the reduced low-dimensional macroscopic variables for the exactly solvable case. The analytical curves exactly match the simulation boundaries in the phase diagram.

Improved gas-kinetic unified algorithm for high rarefied to continuum flows by computable modeling of the Boltzmann equation

In this paper, we present an improved gas-kinetic unified algorithm (IGKUA) for high rarefied transition to continuum flows by computable modeling of Boltzmann equation. Compared with the original algorithm, the new method utilizes less needed discrete velocity ordinate points to obtain accurate results and removes the dependency of flow regime on computational time step, which can speed up the convergence in continuum flows. One of the novel strategies adopted in IGKUA is to develop a type of gas-kinetic quadrature rule that can exactly preserve conservation constraint of the model by adjusting the integral weights, increasing efficiency, and reducing nonphysical sources. Another key innovation is to introduce the analytical solutions of colliding-relaxation equation by considering the evolutions of associated macroscopic flow variables first, leading to no limit on the permissible time step. Numerical explicit and implicit schemes for unsteady flows are constructed to solve the physical convective equation, and Fourier spectral method is applied for the molecular-velocity convective movement equation analytically when the flows are under external-force fields. The IGKUA is tested using some numerical examples, including the shock-tube problems, Rayleigh flow, Couette flow, lid-driven cavity, external force-driven Poiseuille flow, and hypersonic flow past an infinite flat plate. Simulation results are in high resolution of the flow fields and match well with the results of the analytical, direct simulation Monte Carlo, Navier–Stokes solvers, and other reference methods. In addition, the new algorithm is better than the original one in the aspects of computational amount and time, which are more obvious when simulating the continuum flows.

Modeling the convective thermal heat transfer of nanofluids with carbon nanotubes in cylindrical minichannel

This paper is devoted to the development of an algorithm for numerical modeling convective thermal heat transfer of nanofluids with carbon nanotubes. The algorithm is based on a one-liquid description of a nanofluid with common macroscopic variables. The properties of the nanofluid are determined only by the concentration of carbon tubes, and it is assumed that their distribution is uniform and does not change during the flow. A nanofluid can have both Newtonian and non-Newtonian rheology. The fundamental point of this algorithm is the need to use real thermophysical data in solving specific problems, which depend on the concentration of carbon nanotubes naturally. The transport equations are solved using finite volume method. The algorithm was tested by comparing the simulation data with the experimental. The problem of convective thermal exchange of nanofluid with single-walled nanotubes is solved. The corresponding experimental data were previously obtained by the authors of this work. It is shown that the algorithm simulates the considered flow with high accuracy. In addition, its important advantage is the possibility of modeling the flow characteristics, which cannot be measured experimentally. As such example the data on the velocity and temperature profiles of the fluid in the channel are presented.

Rotation-driven transition into coexistent Josephson modes in an atomtronic dc superconducting quantum interference device

By means of a two-mode model, we show that transitions to different arrays of coexistent regimes in the phase space can be attained by rotating a double-well system, which consists of a toroidal condensate with two diametrically placed barriers. Such a configuration corresponds to the atomtronic counterpart of the well-known direct-current superconducting quantum interference device. Due to the phase gradient experimented by the on-site localized functions when the system is subject to rotation, a phase difference appears on each junction in order to satisfy the quantization of the velocity field around the torus. We demonstrate that such a phase can produce a significant change on the relative values of different types of hopping parameters. In particular, we show that within a determined rotation frequency interval, a hopping parameter, usually disregarded in nonrotating systems, turns out to rule the dynamics. At the limits of such a frequency interval, bifurcations of the stationary points occur, which substantially change the phase space portrait that describes the orbits of the macroscopic canonical conjugate variables. We analyze the emerging dynamics that combines the $0$ and $\pi$ Josephson modes, and evaluate the small-oscillation time-periods of such orbits at the frequency range where each mode survives. All the findings predicted by the model are confirmed by Gross-Pitaevskii simulations.

A Free-Energy Landscape Analysis of Calmodulin Obtained from an NMR Data-Utilized Multi-Scale Divide-and-Conquer Molecular Dynamics Simulation.

Calmodulin (CaM) is a multifunctional calcium-binding protein, which regulates a variety of biochemical processes. CaM acts through its conformational changes and complex formation with its target enzymes. CaM consists of two globular domains (N-lobe and C-lobe) linked by an extended linker region. Upon calcium binding, the N-lobe and C-lobe undergo local conformational changes, followed by a major conformational change of the entire CaM to wrap the target enzyme. However, the regulation mechanisms, such as allosteric interactions, which regulate the large structural changes, are still unclear. In order to investigate the series of structural changes, the free-energy landscape of CaM was obtained by multi-scale divide-and-conquer molecular dynamics (MSDC-MD). The resultant free-energy landscape (FEL) shows that the Ca2+ bound CaM (holo-CaM) would take an experimentally famous elongated structure, which can be formed in the early stage of structural change, by breaking the inter-domain interactions. The FEL also shows that important interactions complete the structural change from the elongated structure to the ring-like structure. In addition, the FEL might give a guiding principle to predict mutational sites in CaM. In this study, it was demonstrated that the movement process of macroscopic variables on the FEL may be diffusive to some extent, and then, the MSDC-MD is suitable to the parallel computation.