Point-mass particle state

According to classical mechanics, the state of a point-mass particle is specified by its position and its velocity. If the particle moves in three dimensions we can specify its position by the three Cartesian coordinates x, y, and z. These three coordinates are equivalent to a three-dimensional vector, which we denote by r and call the position vector. This vector is a directed line segment that reaches from the origin of coordinates to the location of the particle. We call x, y, and z the Cartesian components of the position vector. We will denote a vector by a letter in boldface type, but it can also be denoted by a letter with an arrow above it, as in T, by a letter with a wavy underscore, as in r, or by its three Cartesian components listed inside parentheses,... 

If the force on a particle is a known function of position, Eq. (E-1) is an equation of motion, which determines the particle s position and velocity for all values of the time if the position and velocity are known for a single time. Classical mechanics is thus said to be deterministic. The state of a system in classical mechanics is specified by giving the position and velocity of every particle in the system. All mechanical quantities such as kinetic energy and potential energy have values that are determined by the values of these coordinates and velocities, and are mechanical state functions. The kinetic energy of a point-mass particle is a state function that depends on its velocity ... 

Equation (1.3) provides abridge between the observable macroscopic states and the microscopic states of any system. If there were a way to know the microscopic state of the system at different times then all thermodynamic properties could be determined. Assuming a classical system of point-mass particles, Newtonian mechanics provides such a way. We can write Newton s second law for each particle i as follows ... 

We present still another example, Newton s law of motion. This law states that under the action of force a particle will undergo acceleration. The acceleration is inversly proportional to the mass. In the treatment, often the particles are considered as point masses, as in an ideal gas. 

The other state variables are the fugacity of dissolved methane in the bulk of the liquid water phase (fb) and the zero, first and second moment of the particle size distribution (p0, Pi, l )- The initial value for the fugacity, fb° is equal to the three phase equilibrium fugacity feq. The initial number of particles, p , or nuclei initially formed was calculated from a mass balance of the amount of gas consumed at the turbidity point. The explanation of the other variables and parameters as well as the initial conditions are described in detail in the reference. The equations are given to illustrate the nature of this parameter estimation problem with five ODEs, one kinetic parameter (K ) and only one measured state variable. 

Consider a system of N particles with masses m in a volume V = L3. Particle i has position r, and velocity v, and the phase point describing the microscopic state of the system is /e (r, v ) = (ri, r2,..., rN, vi, V2,..., v v). We assume that the particles comprising the system undergo collisions that occur at discrete-time intervals x and free stream between such collisions. If the position of particle i at time t is r, its position at time t + x is... 

In LGCA models, time and space are discrete this means that the model system is defined on a lattice and the state of the automaton is only defined at regular points in time with separation St. The distance between nearest-neighbor sites in the lattice is denoted by 5/. At discrete times, particles with mass m are situated at the lattice sites with b possible velocities ch where i e 1, 2,. .., b. The set c can be chosen in many different ways, although they are restricted by the constraint that... 

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... 

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