Collection of N Particles

All results of classical mechanics of one particle in three dimensions presented so far do directly transform to the situation of N particles. For a system of N 

In the case of non-interacting particles, the (conservative) force acting on particle number i i = 1. N) is given as the gradient of the potential energy with respect to the coordinates of the particle. 

It is important to realize that the potential energy U is still a scalar, i.e., realvalued function, but depends on all 3N Cartesian coordinates of the N-particle 

All aspects of Newtonian mechanics can equally well be formulated within the more general Lagrangian framework based on a single scalar function, the Lagrangian. These formal developments are essential prerequisites for the later discussion of relativistic mechanics and relativistic quantum field theories. As a matter of fact the importance of the Lagrangian formalism for contemporary physics cannot be overestimated as it has strongly contributed to the development of every branch of modem theoretical physics. We will thus briefly discuss its most central formal aspects within the framework of classical Newtonian mechanics. 

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Corresponding to any collection of n particles, there exists a tune-dependent function T (q ... 

On the other hand, the macroscopic features are determined by all particles together. Thus one expects the importance of fluctuations to be relatively small when the system is large. In fact, this has been amply illustrated by the examples of linear systems treated so far. They led to the rule of thumb that in a collection of N particles the fluctuations are of order N1/2. Their effect on the macroscopic properties will therefore be of order iV-1/2. Thus it is clear that the size of the system is a parameter that measures the relative importance of the fluctuations. We shall therefore introduce a size parameter Q. The precise definition of Q depends on the system, but we here formulate its general properties. 

Thermodynamic principles arise from a statistical treatment of matter by studying different idealized ensembles of particles that represent different thermodynamic systems. The first ensemble that we study is that of an isolated system a collection of N particles confined to a volume V, with total internal energy E. A system of this sort is referred to as an NVE system or ensemble, as N, V, and E are the three thermodynamic variables that are held constant. N, V, and E are extensive variables. That is, their values are proportional to the size of the system. If we combine NVE subsystems into a larger system, then the total N, V, and E are computed as the sums of N, V, and E of the subsystems. Temperature, pressure, and chemical potential are intensive variables, for which values do not depend on the size of the system. 

Here we derive the theorem for classical systems. The classical laws of motion can be formulated in terms of the Hamiltonian function for the particles in a system, which is defined in terms of the particle positions scalar quantities pt and qi be the entries of the vectors p and q. For a collection of N particles p e 9t3N and q e 9t3N are the collective positions and momenta vectors listing all 3/V entries. The Hamiltonian function is an expression of the total energy of a system ... 

We consider the classical action S [4] for a collection of N particles, defined... 

For a collection of N particles, we can express the total potential energy of the system V as a sum over contributions arising from interacting pairs, triplets, etc. 

The shape of a coordination polyhedron can be defined by its moments of inertia and the shape of the corresponding momental ellipsoid (Figure 4). Thus, consider a rigidly oriented collection of n particles such as a set of n ligands located at the vertices of a coordination polyhedron. The moment of inertia of this collection of particles about any axis passing through its center of mass is defined as... 

The dynamical state of a classical particle / is defined by its coordinate (position) qi and momentum pi. The state of a collection of N particles is defined by specifying the collective coordinate = q, ..., q 4) and collective momentum = p, ...,pn). Alternatively, we can define the state of an A/ -particle system by specifying the phase point = (p, q ) in a 6AAdimensional space called... 

Let us begin with a collection of N atoms of diameter a in a volume V. We shall assume that the atoms are spherical and mutually impenetrable, that the interparticle energy of attraction between any two particles is given by (-j3u r 6) for r > a, with r the center-to-center... 

First, let us consider what is meant by internuclear coordinates and, in particular, how many of these coordinates are needed in order to specify the electronic energy. We consider a collection of N atomic nuclei, which in this context are considered as point particles. In the following, we will for convenience refer to any collection of nuclei and electrons as a molecule . The atomic nuclei and the electrons may form one or more stable molecules but this is of no relevance to the following argument. The internuclear coordinates are defined as coordinates that are invariant to overall translation and rotation. These coordinates can, for example, be chosen as internuclear distances and bond angles. 

The most fundamental characterization of the molecular size of a macromolecule is its radius of gyration, s. The squared radius of gyration of a rigid collection of n + 1 particles indexed by i running from 0 to n, and with particle i weighted as m, is... 

For a collection of N identical particles the state of the system may be specified by giving the coordinates of each particle so that the system as a whole is represented by a cloud of N points in phase space. Alternatively, the state of the system as a whole constituting the gas phase can be completely specified by a single point in phase space. The phase for the system as a whole is called a y-space or sometimes a F-space, and the phase space for any individual kind of particles (molecules) contained in the system is called a /u-space for that particle [97] [28]. These notations may be linked to the theory of Gibbs [33] and Boltzmann [6], respectively. 

To establish the analogy between a polymer chain and a particle trajectory, de Gennes started with a collection of n beads located at points ri, r2,. .., r the length I between the beads being fixed. Consider the link between the TVth and ( -f-1 )th bead. The beads are subject to external constraints (e.g. thermal motion) so that the average elongation of this link is u... 

Catalyst granules may be of a variety of shapes and forms. It will turn out in later sections that the catalytic properties of almost all shapes except perfect spheres are difficult to describe by mathematical equations. Thus for simplicity it is desirable to consider a particle of arbitrary shape as a collection of N more or less independent pores all of the same mean radius r and of the same length L. The length we should assign to our pores is not immediately obvious, since the average distance... 

The partition function q is for a single particle the corresponding quantity Q for a collection of N non-interacting particles (ideal gas) is given in eq. (13.13). 

It is convenient to define a state of a system in quantum mechanics. All (physical) information that can be known about a quantum mechanical system is contained in a quantum mechanical state function, which is also called a wave function mainly for historical reasons. In order to be able to distinguish different states of a system we introduce the subscript n to label these different states. The term quantum mechanical system will denote an elementary particle or a collection of elementary particles. In chemistry, it is a collection of electrons and atomic nuclei constituting an atom, a molecule or an assembly of atoms and molecules. 

A GCMC simulation for a one-component system is performed as follows. The simulation cell has a fixed volume F, and is placed under periodic boundary conditions. The inverse temperature, = lksT and the chemical potential, /x, are specified as input parameters to the simulation. Histogram reweighting requires collection of data for the probability j N,E) of occurrence of N particles in the simulation cell with total configurational energy in the vicinity of E. This probability distribution function follows the relationship... 

An ensemble is a collection of systems each of which contains N particles, occupies a volume V, and possess energy E. Each system represents one of the possible microscopic states and each is represented by a distribution of points in phase space. A phase space is defined by 3n coordinates and 3n momenta for a dynamic system consisting of n particles. There are three types of ensembles ... 

In a collection of N gas particles in some volume V, each gas particle has its own particular kinetic energy (because we haven t constrained the kinetic energy at all so far, the kinetic energy of any gas particle could be anything). Therefore, there are N equation 19.2s that when added together give the total kinetic energy of the gas. 

It is useful to treat the spin functions as having variables s though these can take only discrete values and from now on to label the Cartesian space variables as r so that x can be used to denote the totality of variables. To construct a many-particle function to describe space and spin for a collection of N identical particles it is apparently simply necessary to form the products ... 

In classical statistical mechanics, the Hamiltonian (p, q) = K(p) + (r) of a system of N particles in a fixed volume V is a sum of the kinetic energy K(p) and the potential energy E(r) of the particles here p and r represent the collective momenta p and positions q of the particles, respectively. The fact that the potential energy is taken to be a function of coordinates only is not always true as happens if we have charged particles in a magnetic field. These cases will not be considered here. The dimensionless total canonical PF Zr(T, V) of the system (we revert back to exhibiting the dependence on V in this section) can be written as a product of two independent integrals... 

Sum of forces acting on the mixture in the control volume (N) Wall lift force acting on a collection of dispersed particles per unit mixture volume (N/m )... 

The statistical mechanics of distinguishable quantum particles is called Boltzmann statistics. For a collection of N such particles, the path integral expression for the partition function is given by equation (4), with r = ri, T2,. .. ryv - Consider as an example the case of N noninteracting free particles all having the same mass m. The Hamiltonian is H = which consists only of a nondiagonal part Hi = H. Applying equation (5) to this case yields... 

Around the turn of the centuiy researchers in the area of Brownian motion theoiy were preoccupied with the irregular motion exhibited by colloidal-sized particles immersed in a fluid. Since then the mathematical apparatus of Brownian theoiy has crept into a number of disciplines and has been used to treat a range of problems involving systems from the size of atoms to systems of stellar dimensions. For the sake of clarity, we shall not consider such a range of problems, conflning our attention to the more traditional problem of describing a collection of N identical classical particles executing motion in a thermal environment. 

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