Point particle

We first consider tlnee examples as a prelude to the general discussion of basic statistical mechanics. These are (i) non-mteracting spin-i particles in a magnetic field, (ii) non-interacting point particles in a box,... 

Consider a system of non-mteracting point particles in a three-dimensional cubical box of volume V= L... 

The equation of state for a gas consisting of non-interacting point particles has the form... 

For the model of free point particles the Newtonian equations present by far the simplest and most efficient analytical fonnalism. In contrast, for chains of rigid bodies, there are several different, but equally applicable, analytical methods in mechanics, with their spe-... 

Let us consider systems which consist of a mixture of spherical atoms and rigid rotators, i.e., linear N2 molecules and spherical Ar atoms. We denote the position (in D dimensions) and momentum of the (point) particles i with mass m (modeling an Ar atom) by r, and p, and the center-of-mass position and momentum of the linear molecule / with mass M and moment of inertia I (modeling the N2 molecule) by R/ and P/, the normalized director of the linear molecule by n/, and the angular momentum by L/. 

Finally, before constructing 7i [p+ r), p- f)] we can note that we have introduced a field-theoretic approach on a heuristic basis where the fields have a clear physical meaning. For the point particle coulomb gas there is a rigorous transformation of the usual statistical mechanics to a field-theoretic formulation in which, however, the field has no apparent physical meaning (see, e.g., [23,24]). 

The first consistent attempt to unify quantum theory and relativity came after Schrddinger s and Heisenberg s work in 1925 and 1926 produced the rules for the quantum mechanical description of nonrelativistic systems of point particles. Mention should be made of the fact that in these developments de Broglie s hypothesis attributing wave-corpuscular properties to all matter played an important role. Central to this hypothesis are the relations between particle and wave properties E — hv and p = Ilk, which de Broglie advanced on the basis of relativistic dynamics. 

The analogy with the virial expansion of PF for a real gas in powers of 1/F, where the excluded volume occupies an equivalent role, is obvious. If the gas molecules can be regarded as point particles which exert no forces on one another, u = 0, the second and higher virial coefficients (42, Azy etc.) vanish, and the gas behaves ideally. Similarly in the dilute polymer solutions when w = 0, i.e., at 1 = , Eqs. (70), (71), and (72) reduce to vanT Hoff s law... 

In order to apply quantum-mechanical theory to the hydrogen atom, we first need to find the appropriate Hamiltonian operator and Schrodinger equation. As preparation for establishing the Hamiltonian operator, we consider a classical system of two interacting point particles with masses mi and m2 and instantaneous positions ri and V2 as shown in Figure 6.1. In terms of their cartesian components, these position vectors are... 

We begin by considering a reactive system with M finite-sized catalytic spherical particles C, and a total of N = NA + NB A and B point particles in a volume V [17]. The C particles catalyze the interconversion between A and B particles according to the reactions... 

CENTRIFUGAL VERSUS CENTRIPETAL CUT POINT PARTICLE SIZE... 

Figure 1 shows the geometry. Strictly speaking, equation 2 applies only to single cylinders in slow cross flow, for point particles with negligible deposition and with the thickness of boundary layer much less than the radius of the cylinder. 

The Euler Lagrangian approach is very common in the field of dilute dispersed two-phase flow. Already in the mid 1980s, a particle tracking routine was available in the commercial CFD-code FLUENT. In the Euler-Lagrangian approach, the dispersed phase is conceived as a collection of individual particles (solid particles, droplets, bubbles) for which the equations of motion can be solved individually. The particles are conceived as point particles which move... 

The laws of classical dynamics were first formulated by Newton. The first law states that any particle will persist in its state of uniform unaccelerated motion unless it is acted upon by a force. Using the notation xiy y, z, for the cartesian coordinates of the ith point particle, of mass mi Newton s equations for n point particles are... 

The relativistic expression for the energy of a free point particle with rest mass m and momentum p is... 

As shown in Fig 1. the model is a one-dimensional array of equal mass, hard-point particles, the even-numbered particles form a set of... 

In fact, with the help of Krein s trace formula, the quantum field theory calculation is mapped onto a quantum mechanical billiard problem of a point-particle scattered off a finite number of non-overlapping spheres or disks i.e. classically hyperbolic (or even chaotic) scattering systems. 

For a classical system of N point particles enclosed in a volume V,at a temperature T, the canonical partition function can be decomposed in two factors. The first one (Qt) comes from the integration over the space of momenta of the kinetic term of the classical Hamiltonian, which represents the free motion of noninteracting particles. The second one, which introduces the interactions between the particles and involves integration over the positions, is the configuration integral. This way, equation (30)... 

The periodic hard-disk Lorentz gas is a two-dimensional billiard in which a point particle undergoes elastic collisions on hard disks which are fixed in the plane in the form of a spatially periodic lattice. Bunimovich and Sinai have proved that... 

M. L. Chabanol, F. Corson, and Y. Pomeau Statistical mechanics of point particles with a gravitational interaction, Europhys. Lett. 50, 148 (2000). 

The equilibrium distribution of soft generalized coordinates for both stiff and rigid classical mechanical systems of point particles may be written in the generic form... 

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