Dissipative Particle Dynamics

Transcription

1 Dissipative Particle Dynamics A Study of the Methodological Background Ulf Daniel Schiller 9th June 2005 Diploma Thesis Supervised by Prof. Dr. Friederike Schmid Condensed Matter Theory Group Faculty of Physics University of Bielefeld

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3 Thus natural science appears completely to lose from sight the large and general questions; but all the more splendid is the success when, groping in the thicket of special questions, we suddenly find a small opening that allows a hitherto undreamt of outlook on the whole. LUDWIG BOLTZMANN

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5 Contents 1 Introduction 1 2 Theoretical Foundations Classical Mechanics Liouville Formulation of Classical Mechanics Statistical Mechanics and Thermodynamics Ensembles, Ergodicity, Averages, Partition Function Thermodynamics Hydrodynamics Continuum Mechanics Kinetic Theory Computer Simulations Molecular Dynamics Liouville Formulation of the Velocity-Verlet Algorithm Langevin Dynamics Measurements in Computer Simulations Averages and Errors Thermodynamic Properties Structural Properties Correlation Functions Transport Coefficients Local Observables Coarse-Graining Levels of Description Theory of Coarse Graining The GENERIC Structure Dissipative Particle Dynamics A Short History of Dissipative Particle Dynamics Dissipative Particle Dynamics Stochastic Differential Equations and Fokker-Planck Equations for DPD H-Theorems for DPD Hydrodynamics for DPD DPD with Finite Time Step Energy Conserving DPD Microscopic Foundations of DPD Voronoi Fluid Particles Soft Fluid Particles iii

6 Contents 3.9 The Theory of Marsh, Backx and Ernst Boltzmann Theory for Dissipative Particle Dynamics The ESPResSo-Package and Extensions ESPResSo - Extensible Software Package for Research on Soft Matter General Parameters Particles Interactions Force Calculation Integration Constraints Thermostats Periodic Boundary Conditions Verlet Lists and Cell Lists Parallelization Reentering the Integrator How to Set Up a Simulation with ESPResSo Setting Parameters System Setup Warm-Up Integration and Equilibration Integration and Measurements Extensions to ESPResSo Local Observables Local Volume of Global Constraints Case Studies with Dissipative Particle Dynamics Simulation of an Ideal Dissipative Fluid Simulation Setup Simulation Results Discussion of Results Simulation of Stokes Flow around a Sphere Simulation Setup Simulation Results Summary and Discussion of Results Summary and Outlook 142 Acknowledgments 145 A Source Code 147 A.1 Simulation of the Ideal Dissipative Fluid A.2 Simulation of Stokes Flow around a Sphere A.3 Monte-Carlo Integration of Collision Integrals A.4 Pressure Correlation Function A.5 Local Observables in ESPResSo A.6 Local Subvolumes of a Sphere Bibliography 175 iv

7 1 Introduction Computer simulations have beyond doubt become one of the most important research tools in modern physics. The reason for this is the fact that our theoretical description of nature is expressed in mathematical equations, which only in a few exceptional cases can be solved exactly. This holds true even for relatively simple theories, such as Newtonian mechanics, where already the motion of three interacting bodies cannot be predicted in terms of an analytical solution. Hence in the majority of non-trivial cases, we have to resort to approximations in order to obtain predictions from the theoretical models. Such approximations may be either analytical or numerical in nature. While theories are the basis for comprehending nature, experiments provide the observations that are to be comprehended. Therefore, it is essential to compare the theoretical predictions against the experimental results in order to verify the validity of the theoretical models. At this point, approximate theories impose difficulties: The concrete effects of the approximation are often uncontrollable, such that in case of significant deviations between theory and experiment it is difficult to assess whether they are caused by the approximations or not. This means that the verification of approximate theories against the experimental observations is rather unreliable. Computer simulations serve as a bridge between theory and experiment. With the advent of digital computers it became possible to solve complex problems without having to rely on approximations. This is achieved by the powerful calculational capabilities of the computer, which enable us to obtain solutions that are apart from numerical and discretization errors exact results. These exact results can be compared with the predictions of approximate theories and thus serve as a test of theories. The other way round, results of a computer simulation can be compared with experimental measurements and thus serve as a test of models. In this way, the comparison of theory and experiment becomes more conclusive because the origin of discrepancies can now explicitly be assigned to the approximations or to the model itself. Another aspect of simulations, which maybe is more pronounced today, is to view them as computer experiments. A computer simulation provides the connection between the microscopic details of a model and the macroscopic properties of interest, that is, the observables being measured in an according experiment. Sometimes it may be difficult or even impossible to obtain measurements from an experimental setup, for example under extreme conditions like high temperature or high pressure. Then the computer can function as a virtual laboratory, where perfect control of all parameters is possible and accurate measurements can be acquired. It is hence possible to probe the properties of interest for a theoretical model in great detail, which in turn can lead to a better understanding of the theory. Moreover, a computer simulation allows to explore how the microscopic structure is related to the bulk properties. This is possible by switching on or off the different features of the model and study the effects separately. In doing so, one can identify the contributing factors that are responsible for the emergence of collective phenomena. There are many examples where computer simulations have helped in elucidating the physical mechanisms of a complex system, and the construction of new hypothetical models is today inextricably linked 1

8 1 Introduction to their exploration by means of computer simulation. To summarize the role of computer simulation, I adopt the view of Landau and Binder [2000] here, where theory, experiment and simulation are depicted as three vertices of a triangle surrounding our understanding of nature. There is a vast number of application fields for computer simulations. Generally, they can be used to study the equilibrium and non-equilibrium properties of solids, fluids and gases. More concrete examples, where computer simulations are applied, are phase-transitions and critical phenomena, the properties of liquid crystals and the study of biomolecules like proteins or DNA. Maybe the most challenging application of computer simulations are fluid systems, as the structure and dynamics of complex fluids still comprises many phenomena which are not well understood. This ranges from colloidal systems over fluid dynamics and rheology to biological systems like membranes and finally living cells. Today a variety of simulation methods exists, and strictly speaking, the term computer simulation is ambiguous. The first simulation method came up over 50 years ago, when in March 1952 the Los Alamos computer MANIAC was put into operation. It is the famous Monte Carlo (MC) method, which uses random numbers to sample the phase space of a given system in order to evaluate statistical observables in terms of phase-space integrals. Since in the Monte Carlo method, the momentum part of the phase space has been integrated out, no real dynamics can be simulated. However, MC is very successful in simulating systems at equilibrium and the method is, though well advanced in years, by no means out-dated. A method to simulate the dynamic properties of a system was first used in the late 1950s. The Molecular Dynamics (MD) method solves Newton s equations of motion and obtains the trajectory of the system in discrete time steps. Many refinements to the MD method have been developed over the years, but as with MC, the basic MD algorithm is today still effectively used. Monte Carlo and Molecular Dynamics are only the two most basic simulation techniques. Many sophisticated refinements and extensions have been developed for various contexts. Although computer facilities have been subject to rapid improvement since the pioneering simulations, machine performance still sets limitations in terms of CPU time and memory requirements. This primarily restricts the size of the system that can be simulated, e.g. the number of particles that can be handled with the given resources. In this sense, fluid systems are especially demanding because the solvent has to be represented by a large number of molecules. Furthermore, the interesting phenomena in complex fluids appear on time scales much larger than the motion of the individual solvent particles. Therefore in a conventional MD simulation, a great deal of computing time is used for rather uninteresting behavior. As a consequence, there still are problems for which a simulation turns out to be inefficient or even intractable. Fortunately, there exist strategies to overcome these limitations. They aim at reducing the number of degrees of freedom by representing the system through a set of relevant variables that evolve on a more suitable time scale. The general procedure to remove uninteresting degrees of freedom is called coarse-graining. One such approach is known as Brownian dynamics and is based on the Langevin equations. Like MD, it is a continuum-dynamical method, but the forces have a stochastic part that models a fluctuating environment, e.g. a solvent. The attempt to reduce computational costs also led to the invention of lattice-based algorithms, many of which are based on cellular automata. These methods, such as the lattice gas (LG) or lattice Boltzmann (LB) approach, rely on a spatial discretization of the system. They are quite efficient but also introduce some conceptual difficulties. The stochastic forces of Brownian dynamics are not momentum conserving, and the lattice-based 2

9 methods violate Galilean invariance. Consequently one may argue that these methods are not the best choice for the simulation of fluid dynamics. In this thesis, I study Dissipative Particle Dynamics (DPD), a method invented for carrying out particle based simulations of hydrodynamic behavior. DPD combines ideas from Langevin dynamics and lattice gas algorithms. The according equations are constructed such that Galilean invariance and momentum conservation are fulfilled. Hence the method is expected to produce correct hydrodynamic behavior. The aim of this thesis is to give a survey of the DPD method, to review the theoretical background and to explore its practical applicability. The itinerary of this work is organized as follows: In chapter 2, I provide the theoretical foundations that are used in this work. The material is meant to give a self-contained introduction of the necessary concepts and shall make the reader familiar with the notation. The Dissipative Particle Dynamics method is introduced in chapter 3. I give a review of the basic technique and a survey of important theoretical aspects. Further, I present the extension to energy conserving DPD, and I discuss the microscopic foundations with respect to the theory of coarse-graining. The chapter is closed with a presentation of two different theories for deriving the transport properties of the ideal DPD fluid. Chapter 4 contains the part of this work that deals with programming. I sketch the basic techniques for implementing a computer simulation considering as example the ESPResSo software package. I also explain how a concrete simulation can be carried out with ESPResSo. In addition, I describe the new extensions that have been implemented during this work. In chapter 5, I carry out two case studies with dissipative particle dynamics. These shall demonstrate how the method works and how the results can be interpreted. Besides having tested some theoretical predictions for the ideal dissipative fluid, I have applied the method to a concrete hydrodynamic problem, namely Stokes flow around a sphere. I give a summary of the different aspects of DPD in chapter 6 and sketch some perspectives for future research with the dissipative particle dynamics method. 3

10 2 Theoretical Foundations This chapter gives a brief introduction into the theoretical concepts that are used in this work. Due to the broad range of aspects that are related to the topic it is not possible to cover all of them exhaustively. However, I try to present an overview of the foundations that is as much selfcontained as possible in the present scope. At first, I review the foundations of classical mechanics, hydrodynamics and thermodynamics. Then I outline the basics of computer simulation techniques, particularly Molecular Dynamics and Langevin Dynamics. I close the chapter with a sketch of the theory of coarse-graining. 2.1 Classical Mechanics The methodology of this work is based on the laws of classical physics, i.e. neither relativistic nor quantum effects will be taken into account. For the systems under consideration this means that the velocity of particles is assumed to be small compared to the speed of light, and the frequency ν is assumed to satisfy hν < k B T. These assumptions are very well justified for many materials [cf. Frenkel and Smit, 1996]. In classical (Hamiltonian) mechanics, the time evolution of a system of particles is given by the Lagrangian equation of motion 1 ( ) d L L = 0, (2.1) dt q i q i where q i and q i are generalized coordinates and velocities, respectively [Goldstein, 1991]. The Lagrangian L is a function of (q, q), given in terms of the kinetic energy K and the potential V : L = K V. (2.2) Using Cartesian coordinates r i and the usual definition of the kinetic energy K = 1 m i ṙ 2 i (2.3) 2 i and forces f i f i = L = ri ri V, (2.4) the Euler-Lagrange equation (2.1) yields Newton s equation of motion m i r i f i = 0. (2.5) 1 There exist several equivalent formalisms in classical mechanics, and it is a matter of taste which one prefers. The Lagrangian formulation is perhaps more suited for extensions to field theories, where a corresponding Euler-Lagrange equation can be formulated in terms of a Lagrangian density L. 4

11 2.1 Classical Mechanics Introducing generalized momenta p i conjugate to the coordinates we can define the Hamiltonian for the system p i = L q i, (2.6) H = i q i p i L. (2.7) If the potential does not depend on the velocities q i and the time t, the Hamiltonian resembles the energy [Goldstein, 1991]. The Hamiltonian equations of motion are For cartesian coordinates we get q i = H p i, ṗ i = H q i. ṙ i = p i m i, ṗ i = f i. (2.8) (2.9) While (2.5) is a system of 3N second-order differential equations, (2.9) is a system of 6N first order differential equations. Both systems are equivalent but they can lead to different discrete algorithms for their solution. The Euler algorithm introduced in section (2.4.1) uses the first order system while the Verlet algorithm uses the second order system Liouville Formulation of Classical Mechanics All positions and momenta together form the phase-space Γ = {r i,p i, which contains all information about the microscopic state of the system. A trajectory in this phase-space is denoted by Γ(t), and the density distribution in phase-space is denoted by ρ(γ; t). For an arbitrary function A(Γ) in phase-space (e.g. an observable), the time evolution can be derived from the equations of motion (2.9) d dt A(Γ) = i ( A ṙ i + A ) ṗ i = ila(γ), (2.10) r i p i where we have introduced the Liouville operator il = ( ) ṙ i + ṗ i. (2.11) r i i p i From the equations of motion follows Liouville s theorem: It implies the continuity equation for the phase-space density Γ Γ = 0. (2.12) ρ ( t + Γ ρ Γ ) = 0. (2.13) 5

12 2 Theoretical Foundations As a consequence, the phase-space density along a trajectory is conserved and application of the chain rule yields Liouville s equation d ρ(γ;t) = 0, (2.14) dt ρ(γ;t) + ilρ(γ;t) = 0 (2.15) t The Liouville formulation can be used to derive symplectic algorithms for Molecular Dynamics. 2.2 Statistical Mechanics and Thermodynamics In complex systems there are usually very many particles. Such systems are better described by their macroscopic properties like temperature, pressure etc., than by every single trajectory of the constituting particles. The description of systems in terms of macroscopic parameters is the field of thermodynamics. Historically, thermodynamics is a phenomenological theory, which is directly based on experimental observations. It is founded on intuitive postulates and many concepts have been developed, for example the equations of state, thermodynamic transformations, etc. [Huang, 1987]. The First Law of Thermodynamics was formulated by ROBERT MAYER in The Second Law of Thermodynamics was stated in 1850 by W. THOMSON, later known as LORD KELVIN. Statistical Mechanics puts thermodynamics on a microscopic basis and explains the macroscopic observations by the molecular motion of the system. In this sense, the macroscopic parameters are statistical averages of microscopic properties. In this section, I will sketch the basics of statistical mechanics and thermodynamics Ensembles, Ergodicity, Averages, Partition Function A system is described by the positions and momenta of all particles in the 6N-dimensional phase space Γ = (r N,p N ). These microscopic degrees of freedom evolve according to the equations of classical mechanics (cf. section 2.1). However, since we are interested in macroscopic parameters only, it does not matter in which microscopic state the system is, as long as it gives rise to the correct macroscopic values. This will usually be a whole set of systems, which is called an ensemble. The distribution of the systems in phase space is given by the phase space density ρ(γ; t). Usually, one makes the (quasi-)ergodic assumption, which states that the system will come arbitrarily close to any point in the accessible phase space, i.e. almost every trajectory will cover almost the entire accessible phase space. If a system is (quasi-)ergodic, time averages can be replaced by ensemble averages: 1 T lim dt A(Γ) = dγ ρ(γ) A(Γ). (2.16) T T 0 Ω To obtain the phase-space densities ρ(γ;t), one often makes another postulate 2 : The postulate of equal a priori probability or maximal ignorance. With the phase-space densities, one can finally 2 Strictly speaking, this postulate is not necessary. However, it simplifies the treatment considerably. 6

13 2.2 Statistical Mechanics and Thermodynamics write down the ensemble average of an observable A(Γ): A = dγ ρ(γ) A(Γ). (2.17) Ω It is convenient to introduce the partition function that contains all necessary information about the system. In the canonical ensemble it is 1 Z K = N! h 3N dγexp ( βh(γ)), (2.18) Ω where H(Γ) is the Hamiltonian of the System, h is a constant to make Z dimensionless, and the inverse temperature β is (k B is the Boltzmann constant) β = 1 k B T. (2.19) With the partition function we can rewrite the average of equation (2.17): A = 1 dγ A(Γ) exp ( βh(γ)). (2.20) Z K Ω Thermodynamics From the point of view of thermodynamics, all information about the system is contained in the thermodynamic potential. The thermodynamic potentials for different ensembles are related to each other by Legendre transformations. The canonical ensemble, for example, is described by the Helmholtz free energy F : F(T,V,N) = 1 β ln Z K(T,V,N). (2.21) It is a function of the temperature T, the volume V and the number of particles N. The differential of the free energy is df = S dt P dv + µ dn (2.22) From this follows that macroscopic quantities can be obtained by taking derivatives of the thermodynamic potential: ( ) ( ) ( ) F F F S = P = µ =, (2.23) T V,N V T,N N T,V where S is the entropy, P is the pressure, and µ is the chemical potential. The internal energy of the system can also be obtained from the free energy E = T 2 (F/T). (2.24) T From these, we can derive the thermodynamic coefficients, for example the coefficient of thermal expansion α = 1 ( ) V, (2.25) V T P,N 7

14 2 Theoretical Foundations the specific heat and the isothermal compressibility c V = T N κ T = 1 V ( ) S, (2.26) T V,N ( ) V. (2.27) P T,N Fluctuations The thermodynamic coefficients can be related to fluctuations in the extensive variables. The specific heat is for example given by the fluctuations in the total energy k B T 2 c V = E 2 E 2 = ( E) 2. (2.28) The fluctuations of a quantity are also related to the response of the quantity to an external field. Consider a quantity A that is coupled to an external field. The susceptibility χ A is then given by χ A = β ( A 2 A 2) = ( A) 2. (2.29) Equation (2.29) is called a linear response theorem. Linear response theory will be useful to derive expressions for transport coefficients in section Hydrodynamics Hydrodynamics or fluid dynamics is the theory of motion of liquids and gases. In the following, I will use the term fluids as a synonym for both liquids and gases. A typical fluid system consists of several atoms or molecules. It is hardly possible to describe the fluid by the solution of Newton s equation for every atom or molecule. Moreover, this would not make much sense because the phenomena of interest in a fluid are collective phenomena that appear on a much larger scale than the size of a single atom or molecule. The theory of hydrodynamics therefore uses different descriptions. In this section, I briefly outline the continuum mechanical description, and I give an introduction to the kinetic theory of fluids Continuum Mechanics Since the trajectory of a single atom or molecule is well below the relevant scale for hydrodynamic phenomena, a fluid can be viewed as a continuum. In this continuum picture, the fluid consists of small volumes, so called fluid elements or fluid particles, which themselves contain still many atoms or molecules. These elements are considered to move as point-like entities through the fluid and determine the state of the fluid. This state is described by the density ρ(r,t), the velocity v(r,t), and the pressure p(r,t) in each point r at time t. For the derivation of the basic equations that describe the evolution of these quantities, I will follow the presentation of Landau and Lifschitz [1966]. 8

15 2.3 Hydrodynamics The equations of hydrodynamics are based on two fundamental assumptions: mass conservation and momentum conservation. From the conservation of mass follows the continuity equation where j = ρv is the mass flux density. ρ t + j = 0, (2.30) The conservation of momentum yields the motion equation for a fluid element subject to the pressure p from the surrounding fluid elements, Euler s equation: v t + (v )v = 1 p. (2.31) ρ This equation does not yet contain any viscous effects and no heat transport. It is therefore only valid for ideal fluids, for which heat transport and viscous effects can be neglected. The pressure p is a function p = p(ǫ,ρ) of the internal energy ǫ and the density ρ and is assumed to satisfy a local equilibrium assumption, i.e. the functional dependence is the same as in equilibrium. The pressure tensor σ takes the form σ ij = p δ ij. (2.32) Using the pressure tensor, equation (2.31) can be written in the form v i t + v i v j = 1 σ ij (2.33) x j j ρ x j j To incorporate friction and heat transport, we have to substitute equation (2.32) with the stress tensor for a viscous fluid. It has the general form with σ ij = p δ ij + σ ij (2.34) ( ) σ ij = η v i + v j 2 x j x i 3 δ v k ij x k k + ζδ ij k v k x k. (2.35) Here, η is the shear viscosity and ζ is the bulk viscosity, both of which are functions of the pressure and the temperature and may vary through the liquid. Introducing this pressure tensor into the right-hand side of Euler s equation we get the general motion equation for a viscous fluid: ρ v i t + j v i v j x j = p x i + j σ ij x j. (2.36) If the variation of the viscosities in the fluid can be neglected, the motion equation can be brought into vectorial form ( ) v ( ρ t + (v )v = p + η v + ζ + η ) grad div v. (2.37) 3 In an incompressible fluid, divv = 0, and the last term of (2.37) vanishes. We finally arrive at the Navier-Stokes equation v t + (v )v = 1 p + ν v, (2.38) ρ 9

16 2 Theoretical Foundations where we have introduced the kinematic viscosity ν ν = η ρ. (2.39) The stress tensor for an incompressible fluid has the simple form ( vi σ ik = p δ ik + η + v ) k, (2.40) x k x i which contains the shear viscosity η. The solutions of the Navier-Stokes equation (2.38) have to be determined with respect to initial and boundary conditions. The flow of a viscous fluid can be very complex, ranging from laminar flow to turbulent flow. Similarity and Dimension Theory Similarity and dimensional considerations can help in constructing the solutions of the hydrodynamic equations. We consider for example a flow solution of the Navier-Stokes equation with certain boundary conditions. These might be walls with a certain geometry or a geometric body in the fluid. If the flow is stationary, the velocity u of the fluid flowing against the body will be constant. The geometrical boundary conditions have a characteristic length L. Together with the dynamic viscosity ν of the fluid, we have three independent parameters that describe the flow. Any other quantity is a function of these parameters. We can further construct a dimensionless quantity from the three parameters ν, L and u, the so called Reynolds number: Re = L u ν = ρ L u. (2.41) η If we use the dimensionless quantities r/l and v/u, the solution of the Navier-Stokes equation can be written in the form: 3 v ( r ) u = f L,Re, (2.42) where f(, ) denotes a vector-valued function of two dimensionless parameters. This holds for every flow with the same Reynolds number, that is, the functional dependence of the dimensionless flow velocity v/u of the dimensionless position r/l is equal for every flow of this type with equal Reynolds number. Such flows are called similar flows, and the respective solution can be obtained by simple rescaling. Stokes Flow around a Sphere As an example we consider the flow around a sphere of radius R for low Reynolds number. For the stationary flow, the Navier-Stokes equation becomes (v )v = 1 p + ν v. (2.43) ρ 3 This is a consequence of the Pi theorem [cf. Kiselev et al., 1999]. It states that for every physical process described by n independent parameters among which k have independent dimensions, every relation between n + 1 dimensional quantities can be written as a relation between n + 1 k dimensionless quantities. 10

17 2.3 Hydrodynamics For low Reynolds number, the term (v )v can be neglected compared to the term ν v. Hence we get the linear equation p + η v = 0. (2.44) The solution of this equation is v = 3R 4 u +ˆr(u ˆr) r R3 4 u 3ˆr(u ˆr) r 3 + u, (2.45) where u is the velocity of the fluid at infinity, and ˆr is the unit vector in the direction of the position vector. The pressure is given by p = p 3η 2 and the force exerted by the flow on the sphere is This is the well known Stokes formula. u ˆr R, (2.46) r2 F = 6πηRu. (2.47) Diffusion We now turn to a mixture of fluids. The parameter describing the mixture is the concentration c. It is defined as the fraction of the total mass of the fluid in a certain volume. The continuity equation and the Navier-Stokes equation remain valid. There is, however, another form of transport in the medium, namely diffusion. The diffusive flux j D and the concentration c can be related via another continuity equation for the mixture: (ρc) t = (ρcv) j D. (2.48) The diffusive flux j D is driven not only by the concentration gradient, but also by the temperature and the pressure gradients. This is described by the equation ( j D = ρd c + k T T T + k ) p p p. (2.49) D is the diffusion coefficient which relates the diffusive flux to the concentration gradient. The coefficients k T and k p determine the ratio of driving from the temperature and pressure gradients, respectively. 11

18 2 Theoretical Foundations Kinetic Theory Kinetic theory provides the link between the microscopic dynamics of Newton s equations and the description of the macroscopic properties in continuum mechanics [Résibois and De Leener, 1977; Hansen and McDonald, 2000]. The kinetic theory was first developed by Boltzmann for dilute gases. Boltzmann s theory describes the statistical properties of the fluid via distribution functions. The number of particles in a volume element dr around r that have a velocity in the volume dv around v is at time t given by f (1) (r,v;t) dr dv, (2.50) were f (1) (r,v;t) is the one particle distribution function. The macroscopic quantities can be derived from this distribution function as average values: The local number density n(r;t) = dv f (1) (r,v;t), (2.51) the local velocity or in general u(r;t) = 1 n(r; t) A(r;t) = 1 n(r; t) dv v f (1) (r,v;t), (2.52) dv A(v) f (1) (r,v;t). (2.53) The distribution in the whole phase space of the system is given by the N-particle distribution function f (N) (r N,p N ;t), which resembles the phase space density ρ(γ;t). The motion equation for f (N) is the Liouville equation (2.15). The BBGKY Hierarchy If we are only interested in simple observables of the type of equations (2.51) to (2.53), we do not need the full phase-space density. Therefore, we define reduced distribution functions by integrating out a number of degrees of freedom f (n) (r n,v n N! ;t) = dr (N n) dv (N n) f (N) (r N,v N ;t). (2.54) (N n)! For n = 2 we get the pair distribution function f (2) (r 1,r 2,v 1,v 2 ;t). The Liouville equation yields a relation for the reduced distribution function [ ( n t + v i + X i + n j=1 F )] ij f (n) (r n,v n ;t) r i=1 i m v i n F i,n+1 = dr n+1 dv n+1 m f (n+1) (r n+1,v n+1 ;t), (2.55) v i i=1 where X i is the external force on particle i, and F ij is the pair force between particles i and j. For n = 1,2,... this is the BBGKY hierarchy, which is named after Born, Bogolyubov, Green, Kirkwood and Yvon. It relates the one particle distribution function to the pair distribution function, which in turn is related to the three particle distribution function, and so on. 12

19 2.3 Hydrodynamics The Boltzmann Equation Setting n = 1 in equation (2.55) we get the famous Boltzmann equation: ( ) f (1) t + v r f (1) + F m vf (1) = f (1) t coll, (2.56) where we have used a short hand notation for the right hand side. To obtain a closed form for this equation, Boltzmann made the two fundamental assumptions that only binary collisions between the particles take place and that the collisions are uncorrelated. Both of these assumptions are very well justified for dilute gases, and Boltzmann s description is also very successful for many other cases. Boltzmann s assumptions lead to the Stosszahlansatz or molecular chaos assumption: f (2) (r 1,r 2,v 1,v 2 ;t) = f (1) (r 1,v 1 ;t)f (1) (r 2,v 2 ;t). (2.57) The collision term on the right hand side of equation (2.56), which basically describes scattering collisions, then becomes ( f (1) t ) coll = dv 2 dω σ(θ,v 12 ) v 12 [ ] f (1) (r,v 1,t)f(1) (r,v 2,t) f(1) (r,v 1,t)f (1) (r,v 2,t). (2.58) Here, σ(θ,v 12 ) is the scattering cross section and v 12 = v 1 v 2. With this expression, the Boltzmann equation yields a closed equation for the one particle distribution function. The Boltzmann equation is a nonlinear integro-differential equation which is in general complicated to solve. In most situations, one is only interested in the linear response behavior of the system, and the deviations from equilibrium are assumed to be small. The one-particle distribution function can then be written as f (1) (r,v;t) = f 0 (v) + δf(r,v;t) = f 0 (v)(1 + φ(r,v;t)), (2.59) where φ(r, v; t) describes a small perturbation δf of the equilibrium state φ(r,v;t) = δf(r,v;t). (2.60) f 0 (v) With these definitions, the Boltzmann equation reduces to the linear Boltzmann equation in the absence of external forces F = 0 δf t where the collision operator C is given by Cδf(r,v 1 ;t) = dv 2 Terms of order (δf) 2 have been neglected. + v δf = Cδf, (2.61) dω σ(θ,v 12 ) v 12 f 0 (v 1 )f 0 (v 2 ) [ φ(r,v 1 ;t) + φ(r,v 2 ;t) φ(r,v 1;t) φ(r,v 2 ;t) ]. (2.62) 13

20 2 Theoretical Foundations The collision operator on the right hand side of the linear Boltzmann equation is also linear, and therefore it is much easier to treat than the full nonlinear case. The properties of the solutions can be described to a considerable degree, and it is possible to give explicit expressions for the transport coefficients by analyzing the hydrodynamic modes. With the scalar product g h = dv 1 f 0 (v) g (v)h(v) (2.63) C becomes a Hermitian operator in an abstract Hilbert space. We consider the eigenvalue problem for the linear Boltzmann collision operator. It can be shown that C φ 0 i = λ 0 i φ 0 i (2.64) λ 0 i = φ0 i C φ0 i φ 0 i φ0 i 0, (2.65) and the operator C has five zero eigenvalues that correspond to the collision invariants. The explicit form of the eigenfunctions is φ 0 1(v) = f 0 (v), m φ 0 i (v) = k B T v if 0 (v), ( 2 mv φ 0 2 5(v) = 3 2k B T 3 ) f 0 (v). 2 (2.66) In a spatially uniform system, the solution of the linear Boltzmann equation can be written as 4 δf(v;t) = i c i φ i (v)exp(λ i t) = t 5 c α φ α (v). (2.67) α=1 This equation shows that the new equilibrium state is completely determined by the collision invariants of the system. Since the operator C is isotropic in velocity space, the eigenfunctions can be expanded in spherical harmonics φ 0 j(v) = φ rl (v)y lm (θ v,φ v ). (2.68) However, the explicit form of the functions φ rl is only known for Maxwell molecules, i.e. for an r 4 potential. In this case, the eigenfunctions can be expressed in terms the of Sonine polynomials. We will use this below to approximate the general solution by an expansion in Sonine polynomials. 4 We assume that the spectrum is discrete and that the eigenfunctions form a basis of the Hilbert space. 14

21 2.3 Hydrodynamics Hydrodynamic modes In order to calculate the transport coefficients from the Boltzmann theory, we first identify them phenomenologically as coefficients of the hydrodynamic modes. We transform into Fourier space, where we can substitute iq in the hydrodynamic flow equations. For example, the momentum flow equation takes the form (cf. equation 2.38) ρ t u q (t) = i ( p ρ ) T q ρ q i ( p T ) q T q η q 2 u q ρ ( ζ + η ) q(q v). (2.69) 3 The complete set of flow equations is a system of ordinary differential equations and can be expressed in the form t Ψ q (t) = M q Ψ q (t), (2.70) where Ψ q (t) is a five-vector of the hydrodynamic variables, and M q is a five-dimensional matrix which is non-hermitian. The eigenvalue problem for M q has five independent solutions, and the left-eigenfunctions are biorthonormal to the right-eigenfunctions. Therefore, we can write the solution for the hydrodynamic modes as Ψ q (t) = 5 c 0 α exp(λ q αt)φ q α, (2.71) α=1 where λ q α are the eigenvalues and φ q α the eigenfunctions of M q. The solution of the eigenvalue problem for M q yields explicit expressions for the transport coefficients in terms of the eigenvalues λ q i. In the limit q 0 they are λ q 1,2 = i c s q Γ s q 2, λ q 3,4 = η q2 ρ, λ q 5 = κ ρc p q 2, (2.72) where c s is the speed of sound, Γ s the sound-absorption coefficient, κ the thermal conductivity and C p the specific heat at constant pressure. Transport coefficients To derive microscopic expressions in terms of the one-particle distribution function, we transform the linear Boltzmann equation into Fourier space t f q + iqv x f q = Cf q. (2.73) We have chosen the x-axis in the direction of q, i.e. q = qê x. The hydrodynamic variables are then given by ρ q (t) = m dvf q (v;t), nu q (t) = dvf q (v;t)v, (2.74) ǫ q (t) = dvf q (v;t) mv

22 2 Theoretical Foundations The solutions can be obtained by solving the eigenvalue problem (C iqv x ) φ q i = λq i φq i. (2.75) We assume that we can expand the eigenvalues in powers of q. In the hydrodynamic limit q 0,t, five of the eigenvalues will tend to zero with q such that the solution of equation (2.73) can be written f q (v;t) = t,q 0 5 c q α(0)exp(λ q αt)φ q α(v). (2.76) α=1 By inserting this expression for f q in the definition of the hydrodynamic variables (2.74) and comparing it with equation (2.71) we can identify the eigenvalues λ q α with the hydrodynamic modes and hence with the transport coefficients (see equation 2.72). 2.4 Computer Simulations Although the equations of classical mechanics are relatively simple, they can be solved for very few systems only. Already for a system of three interacting bodies no analytical solution of Newton s equations can be obtained. Most systems of interest in condensed matter physics or materials science consist of many particles. For such systems, we have to rely on approximations or numerical solutions, the latter of which nowadays are mostly obtained with computer simulations [Allen and Tildesley, 1987, 1993; Frenkel and Smit, 1996]. Computer simulations enable us to explore the solutions of theoretical models which otherwise could only be treated in terms of approximate models. These solutions can then be compared to experimental data in order to verify the theory. Moreover, computer simulations can provide a better understanding of the theoretical models and their parameters because the effects of changes of the model or the parameters can be tested immediately. From another point of view, computer simulations can also serve as a kind of experiment. This is especially useful in areas where not enough experimental data is available. The data generated by the computer simulation can be used as a test for theoretical predictions. The role of computer simulations is thus twofold: on the one hand, they are used to explore theoretical models and their parameters, and on the other hand, they serve as computer experiments for testing a certain theory. This relationship to theory and experiment is nicely depicted by Landau and Binder [2000], which view theory, experiment and simulation as the three vertices of a triangle surrounding our understanding of nature. Mathematically speaking, a computer simulation is used to solve high dimensional integrals like the one in equation (2.20). The two most important techniques to tackle this are Monte Carlo and Molecular Dynamics. The Monte Carlo method performs the integration by stochastic sampling of the phase space, that is, configurations are generated randomly and used as supporting points for a numerical integration. The challenge in developing a Monte Carlo simulation is to generate the random configurations in a clever way. While simple sampling draws the configurations just randomly, importance sampling uses a Markov chain to generate the configurations according to a prescribed distribution. A key condition in importance sampling is the detailed balance condition. The most famous algorithm for importance sampling is the Metropolis algorithm, which is mostly used to generate a Boltzmann distribution. Many other sampling methods have been developed, for example Rosenbluth sampling for polymers and umbrella sampling for estimating free energy differences. For details on Monte Carlo simulation, I refer to the book of Landau and Binder 16

23 2.4 Computer Simulations [2000]. Since Monte Carlo simulations perform a stochastic sampling of the phase space, they do not generate a real trajectory of the system. Therefore, they are unsuitable for evaluating dynamic and transport properties of the system. For this purpose, Molecular Dynamics simulations are more appropriate because they perform a numerical integration of the real trajectory of the system. The problem is to find an exact and efficient integration scheme. In the remainder of this section, I give an introduction to Molecular Dynamics Molecular Dynamics In Molecular Dynamics, the time evolution of a system is simulated by numerically integrating Newton s equations of motion (2.9). An algorithm for this purpose should meet several requirements [Allen and Tildesley, 1987]: it should produce the exact trajectory as close as possible, it should satisfy the conservation laws and the symmetries of the system, it should be computationally efficient (fast execution, low memory requirements), and it should be easy to implement. These points should be discussed a little further [cf. Frenkel and Smit, 1996]. It can be expected that the first two points are in conflict with the latter two. The integration schemes will be derived from a series expansion. The more terms are incorporated, the more exact will the algorithm be, but it will also be more complicated and probably less efficient. Moreover, there are two aspects for the exactness of an algorithm: short-term and long-term stability. Since we deal with complex systems, the real trajectories are likely to be in the regime of Lyapunov instability. That is, two trajectories that are initially close will diverge exponentially. Therefore in a simulation, where numerical deviations always will occur, the simulated trajectory will always diverge exponentially from the real trajectory. However, since we want to predict the average behavior of the system, this is not a problem as long as the second point is fulfilled. An important property of Newton s equations is time reversibility, hence an algorithm for Molecular Dynamics should also be time reversible. It is even more important to avoid an energy drift which typically appears in naive integration methods. One therefore uses symplectic algorithms whose dynamics preserve the volume in phase space. 5 While a non volume-preserving algorithm will expand the volume in phase-space and thus break energy conservation, a symplectic algorithm produces a flow close to the exact Hamiltonian of the system and the energy error is bounded. It can be that an algorithm which has very good short-term stability shows up with a disastrous long-term energy drift. Hence it is preferable to accept moderate short-term errors to gain better long-term energy conservation. Short-term errors will actually always occur due to the finite precision of the computer. The speed of the algorithm is at the second glance not so important as it first seems. It turns out that the most time consuming part of a computer simulation is the calculation of all the forces in the system. Therefore, the efficiency of the integration scheme is of minor importance. It is rather desirable to be able to use the algorithm with a large time step in order to have a minimum of force calculations for a certain simulation time. On the other hand, a large time step increases 5 This is formally expressed by Liouville s theorem, cf. section

24 2 Theoretical Foundations the errors introduced by the discretization of the equations of motion. Moreover, there are algorithms that produce the correct equilibrium distribution only in the limit of infinitely small time step. In practice, the time step therefore represents a trade off between speed and stability of the simulation. 6 The last point in the above list is of a more meta-level character. The ease of implementation has no direct consequence on the simulation performance and does not directly influence the results. However, sophisticated algorithms that are complicated to implement bear the risk of introducing programming errors. Such errors can lead to spurious effects in the results. Often these effects are quite subtle and the errors leading to them are hard to find. The history of computer simulations is certainly full of examples (as a matter of fact, most programming errors will never be known by many people since they do not get published, of course). The difficulties with implementing correct algorithms can in some degree be reduced by the use of software engineering techniques. The development of readily available software libraries and tools for computer simulations will also help to relocate future efforts from implementation of algorithms to investigation of results. While the first might be interesting for a software engineer, the latter is by far more interesting for the physicist. Molecular Dynamics algorithms basically fall into two classes: Verlet-like algorithms and Gear predictor-corrector algorithms [Allen and Tildesley, 1987]. While in the predictor-corrector algorithms also higher moments are calculated, Verlet-like algorithms only use the positions r, the velocities v and the forces f to determine their respective new values. Here, we will focus on Verlet-like algorithms. A discretization of the equations of motion is given in terms of the Taylor expansion of (2.9): r i (t + t) = r i (t) + t v i (t) + t2 f i (t) + t3... r i (t) + O( t 4 ) (2.77) 2m i 3! v i (t + t) = v i (t) + t f i (t) + t2 m i 2 v i(t) + t3 3!... v i (t) + O( t 4 ). (2.78) Euler Algorithm Perhaps the most simple integration scheme based on equations (2.77) and (2.78) is realized by the Euler algorithm. The trajectory is calculated according to r i (t + t) = r i (t) + t v i (t) + t2 2m i f i (t) + O( t 3 ) v i (t + t) = v i (t) + t m i f i (t) + O( t 2 ) (2.79) The short-term stability of the Euler algorithm is of the order t. It is neither time reversible nor phase-space preserving, hence long-term stability cannot be guaranteed [cf. Frenkel and Smit, 1996]. The Euler algorithm is therefore rather unfavorable. 6 The size of the time step is usually also dependent on the interaction potentials that are present. While hard potentials require a small time step, it can be larger for soft potentials. Moreover the time step has to be chosen with respect to the average speed of the particles in the system. 18

25 2.4 Computer Simulations Verlet Algorithm An integration scheme that is both simple and accurate is used in the Verlet algorithm. It solves the second order system (2.5) based on the current positions r i (t) and forces f i (t) and the previous positions r i (t t). For the derivation we consider the Taylor expansion for r i (t t) similar to (2.77): r i (t t) = r i (t) t v i (t) + t2 f i (t) t3... r i (t) + O( t 4 ). (2.80) 2m i 3! The updating equation for the positions is obtained by adding (2.77) and (2.80), and for the velocities by subtracting them, respectively: r i (t + t) = 2r i (t) r(t t) + t2 f i (t) + O( t 4 ), m i v i (t) = r (2.81) i(t + t) r i (t t) + O( t 2 ). 2 t The velocities are actually not needed to compute the trajectories, but they are useful for calculating observables like the kinetic energy. However, in the Verlet scheme the velocities v(t) are only available once r(t + t) has been calculated, i.e. one time step later. Moreover, the updating of positions according to (2.81) gives rise to numerical imprecision because a small term of order t 2 is added to a difference of O(1)-terms. Leap-frog Algorithm It is possible to modify the Verlet algorithm in order to circumvent the deficiencies mentioned above. One approach is the leap-frog algorithm. The updating equations are: v i (t + t 2 ) = v i(t t 2 ) + t f i (t), m i r i (t + t) = r i (t) + t v i (t + t (2.82) 2 ). The velocities are updated first. Since they are evaluated at half time steps, they leap ahead the positions. The current velocities can be obtained from v i (t) = v i(t t 2 ) + v i( t + t 2 ). (2.83) 2 Numerical imprecision is minimized in the leap-frog scheme. However, the velocities are still not accessible in an ad-hoc manner. Velocity-Verlet Algorithm An algorithm that yields the positions, velocities and forces at the same time is given by the Velocity-Verlet scheme. The positions and velocities are updated according to r i (t + t) = r i (t) + t v i (t) + t2 m i f i (t) + O( t 3 ), v i (t + t) = v(t) + t 2m i (f i (t) + f i (t + t)) + O( t 3 ). (2.84) 19