Equation of particle motion

Since the form of the particle equation of motion in electric potential and thermal fields is alike, both kinds of motion can be described by following equations ... 

In Section IV, we turn to our model system, namely a classical or quantal dissipative free particle. We focus the study on the so-called Ohmic dissipation case, in which the noise is white and the particle equation of motion can be given the form of a classical nonretarded Langevin equation in the high-temperature regime (the Brownian particle then undergoes normal diffusive... 

One considers a particle interacting linearly with an environment constituted by an infinite number of independent harmonic oscillators in thermal equilibrium. The particle equation of motion, which can be derived exactly, takes the form of a generalized Langevin equation, in which the memory kernel and the correlation function of the random force are assigned well-defined microscopic expressions in terms of the bath operators. 

In the more refined Langevin description, the free Brownian particle equation of motion contains an inertial term and reads... 

We shall derive such an equation by assuming a bilinear coupling of the form —a between the particle and its environment. In the presence of such a coupling, the particle equation of motion is of the form... 

Fluctuating components of gas velocity are selected from a Gaussian distribution with variance derived from the local value of turbulent kinetic energy. Integration of the particle equation of motion yields the particle position at any given instant in time. 

As a consequence of the form, Eqs. II. 15 and II. 16 of the particle equations of motion, it follows that for some given initial conditions e (0), a(0), there passes only one trajectory at any time t through any point in the particle phase space. This means that for an ensemble of systems with the same initial condition e- (0) and a(0) the probability distribution function /(R- ", p-, t) in the particle phase space obeys a continuity equation, Eq. 1.9, and that the theorem, Eq. 1.11, holds for any dynamic variable a(R f p- O-... 

Fluenf/UNS uses fhese conditions as the starting point for its time integration of the particle equations of motion (the trajectory calculations). The PSD is assumed to obey the Rosin-Rammler-Benett law. 

The angle brackets denote the averaging over the ensemble of the realisation of the random forces in the particle equations of motion (57). We use relation (61) to write an expression for the moment of the normal co-ordinate... 

Assume that Stokes Law gives the particle equation of motion (for Rcp < 0.2). Thus for flow across the thin cylinder R — r ... 

In a classical description of systems of particles, equations of motion are the differential equations whose solution is predictive of the positions and velocities of every particle at every instant of time. Very low-mass particles, such as atoms and electrons, behave according to different mechanical laws, the laws of quantum mechanics. In the quantum mechanical world, particles may display character associated with waves and have... 

In this approach, the particle equation of motion is solved, mostly in a precalculated gas flow field. The particle position and velocity are calculated after successive short time intervals, and in this way the particle is tracked through the cyclone or swirl tube. 

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