graining in time

¡

e ¡

e¡ ¡

e¡

LEVEL 7 brownian dynamics

System Rules

Coarse-grained non-equilibrium pro- Velocities given by force and friction,

cesses; colloidal systems; polymer plus noise; Brownian (di¬usive) dy-

systems namics; Onsager ¬‚ux/force relations

e

Approximation

e

No Go

Reduce description to contin-

e

Details of particles

uous densities of constituent ¡

¡

e ¡

species

e¡ ¡

e¡

LEVEL 8 mesoscopic dynamics

System Rules

As for level 7: self-organizing sys- Density description: mass conser-

tems; reactive non-equilibrium sys- vation plus dynamic ¬‚ux equation,

tems with noise.

e

No Go

Approximation e

e

Spontaneous structure forma-

Average over “in¬nite” number

¡

tion driven by ¬‚uctuations

of particles

¡

e ¡

e¡ ¡

e¡

LEVEL 9 reactive fluid dynamics

System Rules

Non-equilibrium macroscopic mix- Energy, momentum and mass conser-

ture of di¬erent species (as the at- vation; reactive ¬‚uxes

mosphere for weather forecasting

12 Introduction

e

Approximation e

No Go

e

Reduce to one species with Reactive processes; non-

¡

Newtonian viscosity Newtonian behavior

¡

e ¡

e¡ ¡

e¡

LEVEL 10 fluid dynamics

System Rules

Non-equilibrium macroscopic ¬‚uids: Energy, momentum and mass conser-

gases and liquids vation; Navier“Stokes equation

e

Approximation e

No Go e

Low ¬‚uid velocities (low

Turbulence ¡

Reynolds number)

¡

e ¡

e¡ ¡

e¡

LEVEL 11 steady-flow fluid dynamics

System Rules

Non-equilibrium ¬‚uids with laminar Simpli¬ed Navier“Stokes equation

¬‚ow

From level 5 onward, not all atomic details are included in the system

description: one speaks of coarse graining in space. From level 6 onward

dynamic details on a short time scale are disregarded by coarse graining in

time.

In the last stages of this hierarchy (levels 8 to 11), the systems are not

modeled by a set of particles, but rather by properties of a continuum. Equa-

tions describe the time evolution of the continuum properties. Usually such

equations are solved on a spatial grid using ¬nite di¬erence or ¬nite elements

methods for discretizing the continuum equations. A di¬erent approach is

the use of particles to represent the continuum equations, called dissipative

particle dynamics (DPD). The particles are given the proper interactions

representing the correct physical properties that ¬gure as parameters in the

continuum equations.

Note that we have considered dynamical properties at all levels. Not all

questions we endeavor to answer involve dynamic aspects, such as the pre-

diction of static equilibrium properties (e.g., the binding constant of a ligand

to a macromolecule or a solid surface). For such static questions the answers

may be found by sampling methods, such as Monte Carlo simulations, that

generate a representative statistical ensemble of system con¬gurations rather

than a trajectory in time. The ensemble generation makes use of random

1.3 Trajectories and distributions 13

displacements, followed by an acceptance or rejection based on a probabilis-

tic criterion that ensures detailed balance between any pair of con¬gurations:

the ratio of forward and backward transition probabilities is made equal to

the ratio of the required probabilities of the two con¬gurations. In this book

the emphasis will be on dynamic methods; details on Monte Carlo meth-

ods can be found in Allen and Tildesley (1987) or Frenkel and Smit (2002)

for chemically oriented applications and in Binder and Heermann (2002) or

Landau and Binder (2005) for physically oriented applications.

1.3 Trajectories and distributions

Dynamic simulations of many-particle systems contain ¬‚uctuations or stoch-

astic elements, either due to the irrelevant particular choice of initial condi-

tions (as the exact initial positions and velocities of particles in a classical

simulation or the speci¬cation of the initial wave function in a quantum-

dynamical simulation), or due to the “noise” added in the method of solution

(as in Langevin dynamics where a stochastic force is added to replace forces

due to degrees of freedom that are not explicitly represented). Fluctuations

are implicit in the dynamic models up to and including level 8.

The precise details of a particular trajectory of the particles have no rele-

vance for the problem we wish to solve. What we need is always an average

over many trajectories, or at least an average property, such as the average

or the variance of a single observable or a correlation function, over one long

trajectory. In fact, an individual trajectory may even have chaotic proper-

ties: two trajectories with slightly di¬erent initial conditions may deviate

drastically after a su¬ciently long time. However, the average behavior is

deterministic for most physical systems of interest.

Instead of generating distribution functions and correlation functions from

trajectories, we can also try to de¬ne equations, such as the Fokker“Planck

equation, for the distribution functions (probability densities) or correlation

functions themselves. Often the latter is very much more di¬cult than gen-

erating the distribution functions from particular trajectories. An exception

is the generation of equilibrium distributions, for which Monte Carlo meth-

ods are available that circumvent the necessity to solve speci¬c equations

for the distribution functions. Thus the simulation of trajectories is often

the most e¬cient “ if not the only possible “ way to generate the desired

average properties.

While the notion of a trajectory as the time evolution of positions and

velocities of all particles in the system is quite valid and clear in classical

mechanics, there is no such notion in quantum mechanics. The description

14 Introduction