Probability distribution function of momenta

Figure 6. Probability distribution functions of momenta for scaled time.

The velocity distribution f(l> is sufficient for calculating most properties of a gas at low density. The distribution function /(1) gives the probability of finding a particular molecule with three coordinates represented by r(1) and three momenta represented by p(l) the locations and velocities of the other N — 1 molecules in the system are not specified. We will not deal with velocity distribution functions of higher order than /(1), and so the superscript will be dropped and implicitly implied from here on (i.e., / = /(1)). We will, however, consider mixtures of gases, and the velocity distribution function for a molecule of type i or type j will be denoted /) (r, p,-, t), /)(r, p , t), and so on. 

The key ingredient in a statistical description of classical, many-particle systems where each particle labeled by i is described by its momentum, pi, and position, 9, is the probability distribution function, P pi,qi ). This distribution is a function of the momenta and positions of each particle and is taken to be normalized to unity — i.e., the sum of the probabilities of finding a particle with all possible momenta and positions is unity ... 

In molecular dynamics simulations, the fundamental dynamical variables are the coordinates and canonically conjugated momenta. The time evolution of the probability distribution function f(7), where T (P, P, -, R 2 sv... 

In order to characterize quantitatively the polydisperse morphology, the shape and the size distribution functions are constructed. The size distribution function gives the probability to find a droplet of a given area (or volume), while the shape distribution function specified the probability to find a droplet of given compactness. The separation of the disconnected objects has to be performed in order to collect the data for such statistics. It is sometimes convenient to use the quantity v1/3 = [Kiropiet/ ]1 3 as a dimensionless measure of the droplet size. Each droplet itself can be further analyzed by calculating the mass center and principal inertia momenta from the scalar field distribution inside the droplet [110]. These data describe the droplet anisotropy. 

There are several technical details in a rigorous definition of the autocorrelation function for velocity. First, one has to remember the vectorial character of velocity, because clearly the direction in which the particle is knocked is important to its subsequent dynamic history. Then, according to the way it is defined, one has to take the product of the velocity at f = 0, Vg, and that at the later chosen time, v,. However, it is not as simple as just multiplying together the two veetors, Vg and v,. One has to allow for the distribution of positions and momenta of the particle in the system at the beginning, that is, at i = 0. To allow for this, one can introduce symbolically a probability distribution coefficient, g. Therefore, the expression for the autocorrelation function will involve the product gVgV,. 

A-z-k) in the lab system, respectively, wave function of TT+TT bound state with the Coulomb potential only squared at the origin with the principal quantum number n and the orbital momentum Z = 0, Pb is the Bohr momentum in 2, dan/dpidp2 is the double inclusive production cross section for tt+tt -pairs from short lived sources without taking into account 7r+7T Coulomb interaction in the hnal state, p and P2 are the 7t+ and 7t momenta in the lab system. The momenta of 7t+ and 7t mesons obey the relation Pi P2 The A2W are produced in states with different principal quantum numbers n and are distributed according to n Wi = 83%, W2 = 10.4%, W3 = 3.1%, W >4 = 3.5%. The probability of A2tt production in K, K, p, p, xp and T mesons decay were calculated in [34]. 

How do we calculate the probability of a fluctuation about an equilibrium state Consider a system characterized by a classical Hamiltonian H r, p ) where p and denote the momenta and positions of all particles. The phase space probability distribution isf (r, p ) = Q exp(—/i22(r, p )), where Q is the canonical partition function. 

However, since the particles in a liquid are moving permanently for a complete treatment, also a discussion of the particle momenta p is required. This can be done using the general pair distribution function F( i, pi, t, f2, p2, t2). F is the probability of finding and a particle at f2 with momentum p2 at time t2 when there was an atom at f at time t which had the momentum pi. A simplification of this general pair distribution function is obtained by integrating over the momenta... 

The A-particle distribution function gives the probability of finding at time t the tV ions in the positions ri,..., rjv regardless of the momenta and positions of... 

The basic goal here is to derive equations for the time-dependence of those macroscopic (observable) variables which can be written as functions of the microscopic coordinates and momenta. This might include quantities such as the total dipole moment (of unit volume) of a fluid, or the various probability distributions for the positions and/or momenta of 1,2,...N molecules. To this end, we define a 6N-dimensional vector JT(t) = (t),... q j(t),... 

Since the joint probability density of the complete set of positions and momenta is Gaussian, the distribution of any subset must also be Gaussian. But the characteristic function corresponding to a Gaussian density takes a simple form, and in the present case is... 

Three components (Q — —1,0,+1) of the multipole moment of rank K = 1 form the cyclic components of the vector. They are proportional to the mean value of the corresponding spherical functions (B.l) for angular momenta distribution in the state of the molecule as described by the probability density p 9,(p). These components of the multipole moments enable us to find the cyclic components of the angular momentum of the molecule ... 

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