Particle state vector continuous

According to the first postulate, the state of a physical system is completely described by a state function fifiq, /) or ket T1), which depends on spatial coordinates q and the time t. This function is sometimes also called a state vector or a wave function. The coordinate vector q has components q, q2, , so that the state function may also be written as q, q2, , t). For a particle or system that moves in only one dimension (say along the x-axis), the vector q has only one component and the state vector XV is a function of x and t Tfix, /). For a particle or system in three dimensions, the components of q are x, y, z and I1 is a function of the position vector r and t Tfi r, /). The state function is single-valued, a continuous function of each of its variables, and square or quadratically integrable. 

In general we may conclude that the choice of the particle state is determined by the variables needed to specify (i) the rate of change of those of direct interest to the application, and (ii) the birth and death processes. The particle state may generally be characterized by a finite dimensional vector, although in some cases it may not be sufficient. For example, in a diffusive mass transfer process of a solute from a population of liquid droplets to a surrounding continuous phase (e.g., liquid-liquid extraction) one would require a concentration profile in the droplet to calculate the transport rate. In this case, the concentration profile would be an infinite dimensional vector. Although mathematical machinery is conceivable for dealing with infinite dimensional state vectors, it is often possible to use finite dimensional approximations such as a truncated Fourier series expansion. Thus it is adequate for most practical applications to assume that the particle state can be described by a finite dimensional vector. ... 

The generalizations of (2.10.2) and (2.10.3) for the general vector case including continuous phase dependence are identified as follows. Let the rate of change of particle state be given by stochastic differential equations of the... 

Consider the problem in the general setting of the vector particle state space of Section 2.1 in an environment with a continuous phase vector as... 

We shall consider here a population of particles distinguished from one another by a finite dimensional vector x of internal coordinates and distributed uniformly in space. Further, we shall be concerned with the open system of Section 2.8 whose behavior is dictated by the population balance equation (2.8.3). Thus the number density in the feed,/i jn(x), may be assumed to be Nff x) where Nf is the total number density in the feed stream and /(x) is probability density of particle states in it. It will also be assumed that the continuous phase plays no role in the behavior of the system. Relaxing this assumption does not add to any conceptual difficulty, although it may increase the computational burden of the resulting simulation procedure. 

First, the master density function is introduced in Section 7.1. The scalar particle state is discussed in detail in Section 7.1.1 with directions for generalization to the vector case in Section 7.1.2. Coupling with the continuous phase variables is ignored in the foregoing sections, but the necessary modifications for accommodating the environmental effect on the particles are discussed in Section 7.1.3. Thus, from Section 7.1, the basic implements of the stochastic theory of populations along with their probabilistic interpretations become available. These implements are the master density, moment densities that are called product densities, and the resulting mathematical machinery for the calculation of fluctuations. 

Ad a. The set may be discrete, e.g. heads or tails the number of electrons in the conduction band of a semiconductor the number of molecules of a certain component in a reacting mixture. Or the set may be continuous in a given interval one velocity component of a Brownian particle (interval — oo, +00) the kinetic energy of that particle (0, 00) the potential difference between the end points of an electrical resistance (— 00, + 00). Finally the set may be partly discrete, partly continuous, e.g., the energy of an electron in the presence of binding centers. Moreover the set of states may be multidimensional in this case X is often conveniently written as a vector X. Examples X may stand for the three velocity components of a Brownian particle or for the collection of all numbers of molecules of the various components in a reacting mixture or the numbers of electrons trapped in the various species of impurities in a semiconductor. 

Since the formal chemical kinetics operates with large numbers of particles participating in reaction, they could be considered as continuous variables. However, taking into account the atomistic nature of defects, consider hereafter these numbers N as random integer variables. The chemical reaction can be treated now as the birth-death process with individual reaction events accompanied by creation and disappearance of several particles, in a line with the actual reaction scheme [16, 21, 27, 64, 65], Describing the state of a system by a vector N = TV),..., Ns, we can use the Chapmen-Kolmogorov master equation [27] for the distribution function P(N, t)... 

Hulburt and Katz (HI7) developed a framework for the analysis of particulate systems with the population balance equation for a multivariate particle number density. This number density is defined over phase space which is characterized by a vector of the least number of independent coordinates attached to a particle distribution that allow complete description of the properties of the distribution. Phase space is composed of three external particle coordinates x and m internal particle coordinates Xj. The former (Xei, x 2, A es) refer to the spatial distribution of particles. The latter coordinate properties Ocu,Xa,. . , Xt ) give a quantitative description of the state of an individual particle, such as its mass, concentration, temperature, age, etc. In the case of a homogeneous dispersion such as in a well-mixed vessel the external coordinates are unnecessary whereas for a nonideal stirred vessel or tubular configuration they may be needed. Thus (x t)d represents the number of particles per unit volume of dispersion at time t in the incremental range x, x -I- d, where x represents both coordinate sets. The number density continuity equation in particle phase space is shown to be (HI 8, R6)... 

So far we have considered a single mesoscopic equation for the particle density and a corresponding random walk model, a Markov process with continuous states in discrete time. It is natural to extend this analysis to a system of mesoscopic equations for the densities of particles Pi(x,n), i = 1,2,..., m. To describe the microscopic movement of particles we need a vector process (X , S ), where X is the position of the particle at time n and S its state at time n. S is a sequence of random variables taking one of m possible values at time n. One can introduce the probability density Pj(jc, n) = 9P(X < x,S = i)/dx and an imbedded Markov chain with the m x m transition matrix H = (/i ), so that the matrix entry corresponds to the conditional probability of a transition from state i to state j. 

Recall that the continuous phase variables were described by the vector field Y(r, t). In general, the components of this vector field should encompass all continuous phase quantities that affect the behavior of single particles. These could include all dynamic quantities connected with the motion of the continuous phase, the local thermodynamic state variables such as pressure and temperature, concentrations of various chemical constituents, and so on. Clearly, this general setting is too enormously complex for fruitful applications so that it is necessary to suitably constrain our domain of interest. In this connection, the reader may recall our exclusion of the fluid mechanics of dispersions, so that we shall not be interested in the equation... 

In the Lagrangian-picture of a flnid, the history of the systan is encoded in the state variables q ia, t) (indices i,j, k,... = 1,2,3), i.e., the positions of all the distinct fluid elements at time t, each particle being distinguished by a continuously variable vector label a. The particle label may be chosen to be the initial position or more abstractly as a point in some continuum, e.g., it may be a color in a continuous spectrum. In order to have the freedom to choose physical position as a label. 

The LBM is similar to the LGA in that one performs simulations for populations of computational particles on a lattice. It differs from the LGA in that one computes the time evolution of particle distribution functions. These particle distribution functions are a discretized version of the particle distribution function that is used in Boltzmann s kinetic theory of dilute gases. There are, however, several important differences. First, the Boltzmann distribution function is a function of three continuous spatial coordinates, three continuous velocity components, and time. In the LBM, the velocity space is truncated to a finite number of directions. One popular lattice uses 15 lattice velocities, including the rest state. The dimensionless velocity vectors are shown in Fig. 66. The length of the lattice vectors is chosen so that, in one time step, the population of particles having that velocity will propagate to the nearest lattice point along the direction of the lattice vector. If one denotes the distribution function for direction i by fi x,t), the fluid density, p, and fluid velocity, u, are given by... 

Thus we arrive at the important conclusion that a finite chain having N +1 particles has N normal modes of longitudinal vibration. The same arguments used above can be generalized to three dimensions and it can be shown that two transverse modes of vibration are also possible. Therefore, a chain of N +1 atoms can have 3N discrete modes of vibration. Thus the fc-vector, which was continuous for an infinite chain, is now quantized into 3N discrete states. The energy of each of these states is assigned to be hoj. Because of the close resemblance of these quantized vibrational states to photons in a cavity, these quantized waves are called phonons. 

The wavefunctions of the free particle are said to form a continuum of energy states because a continuous range of positive energies (or wave vectors) is possible. Unlike the situation for the harmonic oscillator or the particle in the box, there is no restriction on what energies are allowed. This is a typical feature of a particle that is unbound, meaning the particle s kinetic energy is sufficient to surmount any potential energy barrier. 

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