Non-equilibrium molecular dynamics integrators using Maple

In this work we characterize the chaotic properties of atomic fluids subjected to planar mixed flow, which is a linear combination of planar shear and elongational flows, in a constant temperature thermodynamic ensemble. With the use of a recently developed nonequilibrium molecular dynamics algorithm, compatible and reproducible periodic boundary conditions are realized so that Lyapunov spectra analysis can be carried out for the first time. Previous studies on planar shear and elongational flows have shown that Lyapunov spectra organize in different ways, depending on the character of the defining equations of the system. Interestingly, planar mixed flow gives rise to chaotic spectra that, on one hand, contain elements common to those of shear and elongational flows but also show peculiar, unique traits. In particular, the influence of the constituent flows in regards to the conjugate-pairing rule (CPR) is analyzed. CPR is observed in homogeneously thermostated systems whose adiabatic (or unthermostated) equations of motion are symplectic. We show that the component associated with the shear tends to selectively excite some of those degrees, and is responsible for violations in the rule.

It has recently been shown by Evans, Cohen, and Morriss [Phys. Rev. A 42, 5990 (1990)] that bulk thermal-transport coefficients can be calculated from the maximal Lyapunov exponents by using the conjugate-pairing rule. In the present paper we use computer simulation to explore the validity of the rule under conditions which are more general than those used in the derivation by Evans, Cohen, and Morriss. We show that the existence of a IsteadyP state for dissipative dynamics and the satisfaction of the condition known as the adiabatic incompressibility of phase space (i.e., area preserving in the absence of a thermostat) are insufficient for the rule to hold. We also show that the linear field-induced shift of the Lyapunov exponents with respect to the exponent index, is among the nonequilibrium molecular-dynamics algorithms, apparently peculiar to the SLLOD algorithm (so named because of its close relationship to the Dolls tensor algorithm) for shear flow.

Gaussian isokinetic and isoenergetic deterministic thermostats are reviewed in the correct historical context with their later justification using Gauss' principle of least constraint. The Nose-Hoover thermostat for simulating the canonical ensemble is also developed. For some model systems the Lyapunov exponents satisfy the conjugate pairing rule and a Hamiltonian formulation is obtained. We prove the conjugate pairing rule for nonequilibrium systems where the force is derivable from a potential. The generalized symplectic structure and Hamiltonian formulation is discussed. The application of such thermostats to the Lorentz gas is considered in some detail. The periodic orbit expansion methods are used to calculate averages and to categorize the generic transitions in the structure of the attractor. We prove that the conductivity in the nonequilibrium Lorentz gas is non-negative. (c) 1998 American Institute of Physics.

It is known, that i.r. of double point groups can be expressed in terms of Wigner functions. Using the projection method proposed by O. Barut and R. Raczka the appropriate basis of i.r. of double point groups within the harmonic oscillator space have been derived. The appropriate numeric and symbolic algebra programs have been constructed.

Granular gases are composed of macroscopic bodies kept in motion by an external energy source such as a violent shaking. The behaviour of such systems is quantitatively different from that of ordinary molecular gases: due to the size of the constituents, external fields have a stronger effect on the dynamics and, more importantly, the kinetic energy of the gas is no longer a conserved quantity. The key role of the inelasticity of collisions has been correctly appreciated for about fifteen years, and the ensuing consequences in terms of phase behaviour or transport properties studied in an increasing and now vast body of literature. The purpose of this book is to help the newcomer to the field in acquiring the essential theoretical tools together with some numerical techniques. As emphasized by the authors—who were among the pioneers in the domain— the content could be covered in a one semester course for advanced undergraduates, or it could be incorporated in a more general course dealing with the statistical mechanics of dissipative systems. The book is self-contained, clear, and avoids mathematical complications. In order to elucidate the main physical ideas, heuristic points of views are sometimes preferred to a more rigorous route that would lead to a longer discussion. The 28 chapters are short; they offer exercises and worked examples, solved at the end of the book. Each part is supplemented with a relevant foreword and a useful summary including take-home messages. The editorial work is of good quality, with very few typographical errors. In spite of the title, kinetic theory stricto sensu is not the crux of the matter covered. The authors discuss the consequences of the molecular chaos assumption both at the individual particle level and in terms of collective behaviour. The first part of the book addresses the mechanics of grain collisions. It is emphasized that considering the coefficient of restitution e —a central quantity governing the inelasticity of inter-grain encounters—as velocity independent is inconsistent with the mechanical point of view. An asymptotic expression for the impact velocity dependence of e is therefore derived for visco-elastic spheres. The important inelastic Boltzmann equation is introduced in part II and the associated velocity distribution characterized for a force-free medium (so-called free cooling regime). Transport processes can then be analyzed in part III at the single particle level, and part IV from a more macroscopic viewpoint. The corresponding Chapman–Enskog-like hydrodynamic approach is worked out in detail, in a clear fashion. Finally, the tendency of granular gases to develop instabilities is illustrated in part V where the hydrodynamic picture plays a pivotal role. This book clearly sets the stage. For the sake of simplicity, the authors have discarded some subtle points, such as the open questions underlying the hydrodynamic description (why include the temperature among the hydrodynamic modes, and what about the separation of space and time scales between kinetic and hydrodynamic excitations?). Such omissions are understandable. To a certain extent however, the scope of the book is centered on previous work by the authors, and I have a few regrets. Special emphasis is put on the (variable e) visco-elastic model, which enhances the technical difficulty of the presentation. On the other hand, the important physical effects including scaling laws, hydrodynamic behaviour and structure formation, can be understood in two steps, from the results derived within the much simpler constant e model, allowing subsequently $\varepsilon$ to depend on the granular temperature. The authors justify their choice with the inconsistency of the constant e route. The improvements brought by the visco-elastic model remain to be assessed, since the rotational degrees of freedom, discarded in the book, play an important role and require due consideration of both tangential and normal restitution coefficients, that are again velocity dependent. This seems to be the price of a consistent approach, which does not lend itself to much insight. In addition, the behaviour of driven systems is not addressed, whereas in the realm of granular media, force-free systems are the exception rather than the rule. The differences between constant e and visco-elastic models is presumably less pronounced in the driven case. Study of driven systems also reveals that the rheology of granular gases is intrinsically non-Newtonian, which is a key feature. Finally, the powerful direct simulation Monte Carlo technique is not described, whereas it is an important tool, particularly relevant for the physics of the Boltzmann equation, and straightforward to implement in its simplest version. N Brilliantov and T Poschel concentrate on the (equally relevant) molecular dynamics method instead. In conclusion, the book fills a gap in the field. The companion webpage from where molecular dynamics and symbolic algebra programs can be downloaded is also useful.

Computer simulations were used to investigate shear flows of large numbers of viscoelastic, monosized, spherical particles in unbounded and bounded systems with solid fractions ranging from 0.16 to 0.59. A modified hard-sphere model with inelastic, instantaneous particle interactions was found to replicate some results predicted by kinetic theory in an unbounded shear flow at low and moderate solids fractions. This model was found to predict features such as particle lateral diffusive motion even for systems at solid fractions as high as 0.56. However, for higher solid fractions where phenomena such as jamming could occur, a particle dynamics model accounting for particle contacts of finite duration has been developed, in which the viscoelastic behavior of the particles was represented using a nonlinear Hertzian model. The nonlinear viscoelastic model was found to give more reasonable predictions for cluster formation than previously reported linear models, especially when accounting for surface friction in the model. However, neither frictionless nor frictional particle models could predict particle ordering in unbounded flows. As such, simulations were performed for bounded systems using both the modified hard-sphere model and the nonlinear particle dynamic model. For a bounded shear flow, particle ordering could be predicted by the hard-sphere model even in the absence of both particle friction and gravity, with the local solid fraction and wall separation distance governing the flow stability. For these conditions chain formation was found to be quite likely in the disordered layers for frictional particles. The interesting stick-slip dynamics could be clearly observed in the normal stress signal at the bottom wall. Interpretations were proposed for the complex processes observed, which could lay the foundation for further investigations in sheared dense granular systems.

Comprehending the behaviors of ionic solutions under shear flow is essential for the development of lubrication, electroplating, electrochemical sensing, energy conversions, and many other research fields. In this study, we propose a novel coarse-grained simulation method to investigate the physicochemical properties of ionic solutions under Couette shear flow, in which the ions are modeled as charged beads, the solvent molecules are coarse-grained as dipolar beads using the Stockmayer fluid model, and the SLLOD algorithm is employed to account for the movements of ions and solvent under shear. This method can effectively capture the interplay between ion-ion, ion-dipole, and dipole-dipole electrostatic interactions, as well as their coupling with the imposed flow field. We systematically investigate the impacts of ionic concentrations, dipole moments, and shear rates on the microscopic structure, steady-state viscosity, and ionic diffusivity of ionic solutions. Our results reveal that the formation of ionic clusters via cation-anion electrostatic interactions leads to increased viscosity and suppressed ionic diffusion. Increasing dipolar interactions or applying stronger shear fields can both lead to the disassembly of ionic clusters; however, these two effects show distinct impacts on solution viscosity and ionic diffusion. In particular, enhancing the solvent dipole moment strengthens ion solvation, resulting in increased viscosity and diffusion, whereas elevating the shear rate weakens solvation by displacing bound solvent molecules from the ions, thereby reducing viscosity and enhancing ionic diffusivity. Our study confers insights into the fundamental understanding of the physicochemical properties of ionic solutions and provides guidance for the design and optimization of the functional ion-containing liquid materials.

Analytical SN solutions in heterogeneous slabs using symbolic algebra computer programs

The problem of analyzing the parameters of a chaotic system of fractional order is considered. As an example, a chaotic system based on a Chua circuit of an integer order was chosen. To obtain a Chua circuit of fractional order, it is proposed to replace the capacitive elements in the original circuit with elements with fractal impedance. The purpose of the work is to create a mathematical model of a fractional order Chua circuit with real values of the circuit elements, on the basis of which to evaluate existing software products in the Matlab environment suitable for modeling chaotic fractional order systems, as well as to search for software products suitable for calculating the spectrum of Lyapunov exponents. An analysis of ready-made solutions in the Matlab environment, from the literature known to us, showed that the most suitable option for modeling a system of differential equations of a fractional order is the «fde_pi12_pc» program. The selected program allows you to set all the system parameters of interest and has a relatively high accuracy due to the use of the predictor-corrector algorithm. Based on this program, a domain for determining fractional indicators of the Chua system was constructed in which the presence of a characteristic attractor is visually observed. Existing software products in the Matlab environment are analyzed, allowing the calculation of Lyapunov exponents for chaotic systems described by fractional order differential equations. Of the solutions considered, the best, in our opinion, is the «FO_NC_Lyapunov» program. Using the selected program, a spatial graph of the spectrum of Lyapunov exponents was constructed when changing the fractional exponents of the Chua system. Based on the spatial graph of the spectrum of Lyapunov exponents and the criteria for the chaotic behavior of the system based on the values of Lyapunov exponents, a definition area was constructed with visualization of zones of chaotic behavior of the Chua system. The domains of definition obtained by visual analysis of the presence of an attractor and calculating the spectrum of Lyapunov exponents for the same mathematical model did not coincide; therefore, there are currently no software products for reliably assessing the chaos of fractional order systems by calculating the spectrum of Lyapunov exponents.

Marine risers may experience a complex vortex-induced vibration (VIV) owing to the enhanced nonlinear effects from structural curved shapes versus sheared flows. Many studies have focused on VIV of straight risers in sheared flows or curved risers in uniform flows. In this study, a numerical prediction model is advanced to investigate the multidirectional VIV of a catenary riser subjected to linearly sheared flow. The prediction model is described by equations of three-dimensional responses of the catenary riser coupled with the spatially distributed nonlinear wake oscillators calibrated against experimentally observed VIV responses of curved cylinders. Hydrodynamic lift and drag forces are mathematically modelled based on fluid-structural variables accounting for effects of space-varying riser local inclinations, vortex excitation frequencies and relative flow-cylinder velocities following the Independence Principle. The coupled VIV system is discretized in space by a second order finite difference scheme and integrated in time. To validate the model, simulations are performed based on a flexible curved cylinder under the concave sheared flow and compared with experimental results in the literature. By varying the relative directions between the riser equilibrium and the incoming flow in planar and non-planar configurations, VIV responses of a long flexible catenary riser in a uniform free stream flow versus a linearly sheared flow are compared and analysed in terms of amplitudes, resonant frequencies, lock-in conditions, and energy transfer mechanisms under perpendicular, convex and concave flows. Typical VIV features are captured alongside travelling wave responses, modulated amplitudes and frequency switching caused by sheared flows. Overall results highlight the influence of sheared flows on multidirectional VIV features.

We report on the computation of full Lyapunov spectra of the boundary-driven Chernov–Lebowitz model for stationary planar shear flow. The Lyapunov exponents are calculated with a recently developed formalism for systems with elastic hard collisions. Although the Chernov–Lebowitz model is strictly energy conserving, any phase-space volume is subjected to a contraction due to the reflection rules of the hard disks colliding with the walls. Consequently, the sum of Lyapunov exponents is negative. As expected for an inhomogeneously driven system, the Lyapunov spectra do not obey the conjugate pairing rule. The external driving makes the system less chaotic, which is reflected in a decrease of the Kolmogorov–Sinai entropy if the driving is increased.

According to the van der Waals picture, attractive and repulsive forces play distinct roles in the structure of simple fluids. Here, we examine their roles in dynamics; specifically, in the degree of deterministic chaos using the Kolmogorov-Sinai (KS) entropy rate and the spectra of Lyapunov exponents. With computer simulations of three-dimensional Lennard-Jones and Weeks-Chandler-Andersen fluids, we find repulsive forces dictate these dynamical properties, with attractive forces reducing the KS entropy at a given thermodynamic state. Regardless of interparticle forces, the maximal Lyapunov exponent is intensive for systems ranging from 200 to 2000 particles. Our finite-size scaling analysis also shows that the KS entropy is both extensive (a linear function of system-size) and additive. Both temperature and density control the "dynamical chemical potential," the rate of linear growth of the KS entropy with system size. At fixed system-size, both the KS entropy and the largest exponent exhibit a maximum as a function of density. We attribute the maxima to the competition between two effects: as particles are forced to be in closer proximity, there is an enhancement from the sharp curvature of the repulsive potential and a suppression from the diminishing free volume and particle mobility. The extensivity and additivity of the KS entropy and the intensivity of the largest Lyapunov exponent, however, hold over a range of temperatures and densities across the liquid and liquid-vapor coexistence regimes.

The collisionally induced random particle velocities in a rapid granular shear flow drive the diffusion of particles in manner directly analogous to the thermal diffusion of molecules or the eddy-induced diffusion in a turbulent flow. This paper reports measurements, via computer simulation, of the anisotropic diffusion tensor for a granular shear flow. The components are determined both by particle tracking and through velocity correlations, which are found to agree with reasonable accuracy. As might be expected from symmetry arguments, there are four non-zero components generated in a simple shear flow: the three diagonal components and one off-diagonal component.

Summary:Microphase separation transition in block copolymer melts and solutions in equilibrium and under shear flow is reviewed. The non‐equilibrium molecular dynamics (NEMD) computer simulation methodology is presented in detail including the derivation of the SLLOD equations of motion, Gaussian thermostat, and operator‐splitting symplectic integrators. Results of our recent NEMD computer simulation studies of diblock copolymers in a selective solvent under shear flow are presented. Shear‐dependent structural, rheological, and microscopical properties are described. New phase transitions are discovered. The parallel‐perpendicular orientational transition in a weak‐strong flow is revealed. Theoretical approaches are reviewed including the Edwards Hamiltonian, Landau‐Ginzburg model, self‐consistent mean field theory, field‐theoretic simulation, as well as the time‐dependent Landau‐Ginzburg framework and its application to the studies of complex fluids.magnified image

This study investigates the impact of lateral boundary conditions on the propagation and dispersion of locally excited Rossby waves in a zonally periodic, barotropic, quasigeostrophic channel model on the β-plane. We use basic flows with either a linear meridional shear or a jet-like profile. On the southern boundary of the channel we impose either a rigid wall or a radiation condition, whereas the northern sidewall is permeable for Rossby waves. We compare the numerical solutions found for a reflecting southern boundary in a weakly dissipative flow to the solutions obtained from a WKB-analysis for the corresponding unforced nondissipative situation. Furthermore, we compare the generalized Eliassen-Palm flux vectors to the ray paths of Rossby wave packets, obtained from WKB ray tracing. In particular, we focus our investigation on the two-dimensional structure of trapped modal waves and wave trains in a simple linear numerical model. Summarizing our results, we find that along the reflective wall, trapped modal wave structures as well as reflected wave trains occur with characteristics (e.g., wave numbers, turning latitudes) similar to the ones computed using asymptotic methods. In a linear sheared flow wave packets are trapped for all zonal wave numbers in contrast to a jet-like mean flow which has a selective effect on the waves; i.e., a turning latitudes phenomenon between the coast and the flow maximum occurs for short waves, while long waves can propagate freely across the zonal mean flow. This comes out clearly when studying the stream lines of the Eliassen-Palm flux vectors of the numerical model simulations. Furthermore, due to the reflected wave activity, the dispersion of Rossby waves is influenced by the southern boundary condition not only in the vicinity of the border but also in regions away from the boundary. These results appear to be important on the one hand for the existence of trapped Rossby waves in large-scale oceanic shear flows along a zonally oriented coast. And, on the other hand for large-scale boundary waves in conceptional atmospheric channel models which can lead to unwanted resonance effects.