Probability of finding particle

This is the probability of finding particle 1 with coordinate rx and velocity vx (within drx and dVj), particle 2 with coordinate r2 and velocity v2 (within phase space with velocity rather than momentum for convenience since only one type of particle is being considered, this causes no difficulties in Liouville s equation.) The -particle probability distribution function ( < N) is... 

The distribution function p,v(x1.. . xN, t) in the 6AT-dimen-sional phase space represents the probability of finding particle i with phase x ----- rf, p4, particle / with phase x, = r, etc.,.. . at the instant t. This function obeys the Liouville equation... 

For an amorphous mass of particles there is no correlation between particles that are far apart. The joint probability of finding particles at r and ri is simply the product of the individual probabilities. Let us define a configurational pair-correlation function g(ri, r2) as... 

Gives probable position of particle with mass m Probability of finding particle in a region is proportional to V /"... 

The last equality follows from the normalization of p R ) i.e., the probability of finding particle 1 in any place in V is unity. The normalization condition (2.19) can also be obtained directly from (2.13). 

Here, (Si) dS is the average number of particles occupying the element of volume dS. For the present treatment, we limit our discussion to spherical molecules only. As we have already stressed in section 2.1, the quantity defined in (2.88) can be assigned two different meanings. The first follows from the first form on the rhs of (2.88), which is an average quantity in the T, V, Nensemble. The second form on the rhs of (2.88) provides the probability of finding particle 1 in the element of volume dSi. Clearly, this probability is given by N iS dSJN. 

The first factor is simply the ensemble average of the quantity exp(iq n). If the system is homogeneous the particles are distributed randomly so that the probability of finding particle / in the neighborhood d3r is d3rV x where V is the illuminated volume of the sample (not the volume of the scattering cell). This is... 

S Write the expression for the probability of finding particle number 1 with its x coordinate between 0 and 2 for (a) a one-particle, one-dimensional system (b) a one-particle, three-dimensional system (c) a two-particle, three-dimensional system. 

Physically, the semidefinite conditions on the D, Q, and matrices restrict the probabilities of finding particle-particle, hole-hole, and particle-hole pairs to be nonnegative, respectively. Even though the nonnegativity constraints in Eqs. (6-8) are non-redundant, these matrices contain equivalent information as each matrix can be expressed in a one-to-one mapping of another by the fermionic anticommutation relations. The (2,2)-positivity conditions are often denoted as DQG. Contraction of the positive semidefinite D, Q, and matrices generates one-particle D and one-hole 2 matrices that are also positive semidefinite. 

Fig. U. A particle moves along the x axis and is in the state described by the wave funetion Ffx, t). Fig. (a) shows how the probability of finding particle in an infinitesimalh small seetion of the length dx at xq (at time t = tp) is ealculated. Fig. (b) shows how to calculate the probability of finding the partiele at t =

We can now use actual experimental results and Equations 6.4 and 6.5 to estimate the probability of finding particle-free drops in the case of a putative steady state where a random distribution of particles over drops is formed as a result of successive splitting and coalescence events. In the study of polydimethylsiloxane-hydrophobed silica antifoam deactivation given by Denkov et al. [6], drop radii, both before and after deactivation, lie in the range of 1-5 microns. The silica concentration is about 5% by weight and the silica density about 2 g cm" and that of the oil about 1 g cm. Silica particle sizes lie in the range of 0.1-5 microns. Both drops and particles are clearly rather polydisperse. The ratio of antifoam drop to particle size is therefore in the range of 1 < r p /Rp < 100 with an overall particle volume fraction of -0.026. 

We have explored the possibility that deactivation of polydimethylsiloxane-hydrophobed silica antifoams by disproportionation is a consequence of the random distribution of particles across the drops formed when the antifoam is dispersed to achieve a steady state by processes of drop splitting and coalescence. Simple mass balance considerations permit the estimation of the composition of deactivated antifoam dispersions if we assume they are made up of particle-free drops together with particle-rich drops and agglomerates of known silica content From that composition, it is possible to calculate the ratio of the number of particles to the number of drops, NM, in a deactivated antifoam if the distribution of particles across drops is assumed to be random. This analysis reveals that for monodisperse particles and drops where NIM < 1 this necessarily implies a proportion of particle-free drops irrespective of the nature of any distribution If, on the other hand, NIM > 1, then the probability of finding particle-free drops is always vanishingly small. 

We now integrate this function over the space and spin coordinates of particle 2 to find the probability of finding particle 1. The space integration yields unity by the normalization of the space orbital iAioo(2). Integrating the spin coordinates of particle 2 gives zero for the second and third terms in the sum of spin functions by the... 

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