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In this chapter we review recent advances in numerical simulation of micro and nanoflows. For coarse-grained simulation of microfuidics, we present an overview of Lattice Boltzmann, Brownian dynamics, stochastic rotation dynamics, and smoothed particle hydrodynamics methods and discuss the dissipative particle dynamics method in detail as it shares many features with the other methods. In the area of nanoflows, we review recent advances in non-equilibrium molecular dynamics methods focusing on the development of self-consistent and grand canonical methods for electric-field mediated transport. We present examples showing the significance of quantum effects in nanoflows. Finally, we discuss multiscale modeling focusing on direct coupling of molecular dynamics with Navier-Stokes equations and hierarchical coupling of quantum, molecular dynamics and classical fluid equations.

Coarse-grained molecular dynamics (CGMD) is a computer modeling technique that couples conventional molecular dynamics (MD) in some spatial regions of the simulation to a more coarse-grained description in others. This concurrent multiscale modeling approach allows a more efficient use of computer power as it focuses only on those degrees of freedom that are physically relevant. In the spirit of finite element modeling (FEM), the coarse-grained regions are modeled on a mesh with variable mesh size. CGMD is derived solely from the MD model, however, and has no continuum parameters. As a result, it provides a coupling that is smooth and provides control of errors that arise at the coupling between the atomistic and coarse-grained regions. In this chapter, we review the formulation of CGMD, describing how coarse graining, the systematic removal of irrelevant degrees of freedom, is accomplished for a finite temperature system. We then describe practical implementation of CGMD for large-scale simulations and some tests of validity. We conclude with an outlook on some of the directions future development may take.

In the gaseous state, molecules are to a good approximation isolated entities traveling in space at high speed with sparse and near elastic collisions. At the other extreme, a perfect crystal has a periodic and symmetric intermolecular structure. The structure is dictated by intermolecular forces, and molecules can only perform small oscillations around their equilibrium positions. In between these two extremes, matter has many more ways of aggregation. This chapter deals with proper liquids and the liquid state. Molecular diffusion in liquids occurs approximately on the timescale of nanoseconds, to be compared with the timescale of molecular or lattice vibrations. This chapter also discusses molecular dynamics (MD), the Monte Carlo (MC) method, structural and dynamic descriptors for liquids, physicochemical properties of liquids from MD or MC simulations, simulations of enthalpy, heat capacity and density, crystal and liquid equations of state, polarisability and dielectric constants, free energy simulations, and simulation of water.

Computational modelling of materials behaviour is becoming a reliable tool of scientific investigation, complementary to traditional theory and experimentation. The Multiscale Materials Modelling (MMM) approach reflects the realization that continuum and atomistic analysis methods are complementary. This chapter describes the most popular numerical techniques of each component that make up the MMM paradigm for modelling nano- and micro-systems: Quantum Mechanics (QM), Molecular Dynamics (MD), Monte Carlo (MC), Dislocation Dynamics (DD), Statistical Mechanics (SM), and Continuum Mechanics (CM).

In Chapter 8 five directions of future research are tersely mentioned in order of increasing importance. The investigation of Diophantine relations, such as those reported in Appendix C. The investigation of the behavior in a quantal context of systems described by isochronous Hamiltonians. The investigation of isochronous systems outside the phase space sectors where they behave isochronously. The investigation of the behavior of the isochronous many-body problems treated in Section 5.5, in particular, when the corresponding unmodified system is a realistic molecular dynamics model characterized by a chaotic evolution yielding—in the context of statistical mechanics and thermodynamics—a steadily increasing entropy—which, however, must eventually decrease when the corresponding Ω-modified system returns to its initial state, as entailed by its isochronous character. And, last but by no means least, the investigation of isochronous systems in an applicative context—be it physics, chemistry, biology, medicine, economy, and so on.

Three non-equilibrium computer simulation algorithms are presented in detail: stochastic molecular dynamics, non-equilibrium Monte Carlo, and Brownian dynamics. Stochastic molecular dynamics is based on the stochastic dissipative equations of motion, which do not suffer the disadvantages of non-Hamiltonian deterministic equations or thermostats. Extensive numerical tests are performed for steady heat flow and for a driven Brownian particle in a solvent. A non-equilibrium Monte Carlo algorithm is based upon the non-equilibrium probability distribution. Umbrella sampling and other methods to improve the efficiency of the algorithm are discussed. Results are compared with the stochastic molecular dynamics and with Nose-Hoover equilibrium molecular dynamics. Brownian dynamics using the simple Langevin equation is outlined. The perturbation theory of the preceding chapter is used for a more advanced algorithm suited for concentrated dispersions and macromolecules. The stochastic calculus is discussed in the context of Brownian dynamics and the generalised Langevin equation

This chapter examines pattern-forming phenomena in thin layers of granular materials subjected to low-frequency periodic vertical vibration above the acceleration of gravity. Compared to driven granular gases discussed in Chapter 4, dense layers of granular materials under sufficiently strong excitation exhibit fluid-like motion. The most spectacular manifestation of the fluid-like behavior of granular layers is the occurrence of surface gravity waves which are quite similar to the corresponding patterns in ordinary fluids. To understand the nature of these collective phenomena, many theoretical and computational approaches have been developed. The most straightforward approach is to use molecular dynamics simulations which are feasible for sufficiently thin layers of grains. On the other hand, since the scale of observed pattern typically is much greater than the size of the individual grain, a variety of continuum approaches, ranging from phenomenological Ginzburg-Landau type theories to granular hydrodynamics, are discussed.

This chapter introduces the main theoretical approaches and models employed in the physics of granular media, such as the kinetic theory of diluted granular gases, various methods of molecular dynamics simulations (event driven, soft particles, contact dynamics), order parameter phase-field models, depth-averaged and two-phase models of dense flows, and a variety of other phenomenological theories.

Building on the complementary advantages of Renormalization Group (RG) and multigrid (MG) methods, Systematic Upscaling (SU) comprises rigorous procedures for deriving suitable variables and corresponding numerical equations (or statistical relations) that describe a given physical system at progressively larger scales, starting at some fine scale where the physical laws are known (in the form of a partial differential system, or a statistical-mechanics Hamiltonian, or Newton laws for moving particles, etc.). Unlike RG, the SU algorithms include repeated coarse-to-fine transitions, which are essential for (1) testing the adequacy of the set of coarse-level variables (thus providing a general tool for constructing that set); (2) accelerating the finer-level simulations; and, most importantly (3) confining those simulations to small representative subdomains. No substantial scale separation is assumed; as in MG, small scale ratio between successive levels is in fact important to ensure slowdown-free simulations at all scales. Detailed examples are given in terms of local-interaction systems at equilibrium, and extensions are briefly discussed to long-range interactions, dynamic systems, low temperatures, and more.

This chapter sketches the basic principles of molecular dynamics computations. It presents the results of high density computations for hard spheres and discs. It provides data for the freezing transition of hard spheres and the transition for hard discs. It discusses phase diagrams obtained by numerical methods for the inverse power law potential, hard spheres with an additional attractive square well and the Lenard–Jones potential.

This chapter begins by emphasizing that the initial electron density for a protein will be significantly affected by errors in the experimental phases and, subsequently, additional errors of interpretation will arise when a model of the structure is built. It describes methods for fitting a protein molecule to its electron density map (both manual and automated) and demonstrates the importance of interactive computer graphics in these processes. It then covers the underlying theory of methods by which the model is adjusted to maximise its agreement with the experimental structure factor or intensity data. The chapter describes the role of stereochemical restraints in macromolecular refinement; recently developed methods of improving the efficiency of refinement which exploit molecular dynamics at high temperature and/or maximum likelihood statistics; the exploitation of non-crystallographic symmetry in refinement; and the process of treating the protein, or parts of it, as rigid groups to improve the radius of convergence or analyse the dynamics of the molecule. Methods for calculating electron density maps which minimise the problem of model bias are described in detail along with criteria by which the success or otherwise of refinement may be judged.

This chapter introduces the classical equations of motion for a system of molecules, and describes their solution by stable, accurate, time-stepping algorithms. Simple atomic systems, rigid molecules, and flexible molecules with and without constraints, are treated, with examples of program code. Quaternions are introduced as useful parameters for solving the rigid-body equations of motion of molecules. A simple example of a multiple timestep algorithm is given, and there is a brief summary of event-driven (hard-particle) dynamics. Examples of constant-temperature molecular dynamics using stochastic and deterministic methods are presented, and the corresponding constant-pressure molecular dynamics methods for fixed and variable box-shape are described. The molecular dynamics method is extended to the treatment of polarizable systems, and dynamical simulation of the grand canonical ensemble is mentioned.

This chapter reflects on materiality and specifically active biological matter as a novel historiographical nucleus to conceive of the recent history of the life and chemical sciences, as it reflects on the philosophical question about the ontological status of molecular machinery in light of this book’s history. Furthermore, the history of membrane and protein research is positioned within the historiography of the twentieth-century life sciences, making a case to move away from genetics as shaping the dominant narrative, and highlighting the continuous relevance of physiological and chemical research to these sciences’ present states. A number of questions for further research that this book has opened up will be sketched, such as on the temporality and geography of recent sciences. Not least, the chapter discusses the question of how to characterize the persona of 1970s and 1980s life scientists in between academia, entrepreneurship, mainstream science, and application.

These lecture notes give a concise overview of the problem of describing quantum-mechanically the correlated motion of electrons and nuclei in a molecule. The focus is put on the methodology allowing to study the ultrafast, pure electron dynamics triggered by ionization of a molecule. It is shown that due to the electron correlation the removal of an electron from a molecular orbital can create electronic coherences manifesting in the migration of the positive charge throughout the system on a few-femtosecond time scale; a phenomenon known as correlation-driven charge migration. Some interesting perspectives for designing schemes to influence the chemical reactivity of the molecule by manipulating the charge migration dynamics are also briefly discussed.

This chapter focuses on the coupling of electron dynamics and nuclear motion. In principle, electronic and nuclear degrees of freedom both have to be treated quantum mechanically and on an equal footing. This can be formally achieved using multicomponent time-dependent density-functional theory (TDDFT), but this approach has so far been of limited use. In practice one starts from the Born-Oppenheimer approximation and discusses the nuclear dynamics in terms of potential-energy surfaces. It is shown how TDDFT performs in the calculation of potential-energy surfaces. A particular challenge is the so-called conical intersections. Then, various schemes of ab initio molecular dynamics are discussed: Born-Oppenheimer dynamics, the TDDFT-Ehrenfest approach, and surface hopping schemes. It is shown how TDDFT is used for calculating nonadiabatic couplings between potential-energy surfaces. Finally, the Car-Parrinello approach is briefly discussed.

Accessing the full atomistic details of complex biological processes in near physiological conditions remains challenging at the experimental level. The development of molecular modeling approaches, such as molecular dynamics, together with powerful supercomputers allows for rationalizing and predicting at the molecular level the dynamics of large biomolecular assemblies in various conditions. Although still limited in terms of system size and time scales, molecular dynamics (MD) simulations have proven to be a valuable tool in the arsenal of integrated structural biology techniques. In this chapter, the standard concepts underlying molecular simulations are presented. First the basis of molecular mechanics and the concept of the force field are explained. Then the MD method is introduced together with some details of its implementation commonly employed to study biomolecular systems. Finally, the complementarity of experiments and modeling is illustrated using the example of mitochondrial carriers.

Nanoscale research has deeply penetrated biological fields. A new light has been cast on the worlds of proteins and DNA, thanks to nano’s appreciation and mastery of single molecules and its ability to visualize them. Bionano can notably be understood with reference to the interaction of six parameters, presented here as the nanobiology hexagon. It entails structure/form, binding, function, three-dimensionality, environment, and control. It is in the area of nanobiology that questions of dynamics come to the fore at the molecular level. Biology nano style is crystallized in the mechanisms of recognition. The concept of modularity is similarly central to the field. Through this, nanobio opens new horizons to the study of evolutionary process.

This chapter covers the introduction of quantum mechanics into computer simulation methods. The chapter begins by explaining how electronic degrees of freedom may be handled in an ab initio fashion and how the resulting forces are included in the classical dynamics of the nuclei. The technique for combining the ab initio molecular dynamics of a small region, with classical dynamics or molecular mechanics applied to the surrounding environment, is explained. There is a section on handling quantum degrees of freedom, such as low-mass nuclei, by discretized path integral methods, complete with practical code examples. The problem of calculating quantum time correlation functions is addressed. Ground-state quantum Monte Carlo methods are explained, and the chapter concludes with a forward look to the future development of such techniques particularly to systems that include excited electronic states.

This chapter focuses on the numerical approaches for protocell simulation. It discusses the ab initio methods, semiempirical methods, molecular mechanics or molecular dynamics (MM or MD), quantum mechanics and molecular mechanics, coarse-grained molecular dynamics, the lattice Boltzmann (LB) method, and the Ginzburg-Landau (GL) models, and the multiscale methods.