Finite collision time

The behaviour of orientational correlation functions near t = 0 carries information on both free rotation and interparticle interaction during collisions. In the impact approximation this information is lost. As far as collisions are considered as instantaneous, impact Eq. (2.48) holds, and all derivatives of exponential Kj(t) have a break at t = 0. However, 

This information is sufficient to define in a general way the short-time behaviour of 

Beginning with the fourth moment, all of them are sensitive to the strength of the interaction during collisions. 

To illustrate the accuracy of the perturbation theory these results are worth comparing with the well-known values of h and I4 for t = 1 rigorously found from first principles in [8]. It turns out that the second moment in Eq. (2.65a) is exact. The evaluation of I4, however, is inaccurate its first component is half as large as the true one. The cause of this discrepancy is easily revealed. Since M = / and (/) = J/xj, the second component in Ux) is linear in e. Hence, it is as exact in this order as perturbation theory itself. In contrast, the first component in IqXj is quadratic in A and its value in the lowest order of perturbation theory is not guaranteed. Generally speaking 

In the next section we will show how perturbation theory must be developed to provide an exact value of up. 

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Impact processes with finite collision time 27... 

The latter is negligible in the centre of the spectrum (at < 1) which looks like a pure Lorentzian, as in the impact approximation with HWHH 1/t. Only far wings are affected at finite collision time tc. When, however, k > 1 /4 (fc2 = ), the situation changes drastically. To describe it let us... 

Burshtein A. I., McConnell J. Spectral estimation of finite collision times in liquid solutions, Physica A157, 933-54 (1989). 

With two-body collisions only (dilute gases), without chemical reaction for the time being and neglecting finite dimensions of molecules and finite collision times, one may obtain from the kinetic theory the Boltzmann integro-differential equation for/< >. 

Collision time is the duration of collision. In classical physics, collisions are considered instantaneous. However, there is a finite collision time during wave scattering. The collision time is defined as the ratio of wavelength of the carrier and the propagation speed of the carrier. The typical values of collision time (t ) are as follows ... 

Twenty years ago, Bogolubov3 developed a method of generalizing the Boltzmann equation for moderately dense gases. His idea was that if one starts with a gas in a given initial state, its evolution is at first determined by the initial conditions. After a lapse of time—of the order of several collision times—the system reaches a state of quasi-equilibrium which does not depend on the initial conditions and in which the w-particle distribution functions (n > 2) depend on the time only through the one-particle distribution function. With these simple statements Bogolubov derived a Boltzmann equation taking into account delocalization effects due to the finite radius of the particles, and he also established the formal relations that the n-particle distribution function has to obey. 

The second term of Eq. (40) gives the contribution from collisions. These are non-instantaneous processes since the variation of p 0> at the time t depends on the value of this function at the earlier instant t. The evolution is non-Markovian and the system remembers its earlier history. However, this memory extends only over a finite period, as one can see from the expression (44) for the kernel G, (t). This results from supposing that the poles z( are not infinitesimally close to the real axis and thus that the collision time tc is finite (see Eq. (39)). 

Here we apply the finite-volume scheme to simulate two different examples of inhomogeneous kinetic equations. The first example is a non-equilibrium Riemann shock problem with different values of the collision time t. The second example is two ID crossing jets with different collision times. In reality, the collision time is controlled by the number density Moo, which we normalize with respect to unity in these examples. Thus, the reader can interpret the different values of t as different values of the unnormalized number density. As noted above, for the multi-Gaussian quadrature we compute the spatial fiuxes using Ml = 14 and Mo = 4 with a CFL number of unity. 

The list of collision times could not very well contain an entry for each j and V, inasmuch as it would not only be large for large N, but would also consist mostly of values that are infinite. It is more to the point to maintain only the finite values that are less than some preset value r . A manageable table is obtained by letting... 

The Hubbard relation is indifferent not only to the model of collision but to molecular reorientation mechanism as well. In particular, it holds for a jump mechanism of reorientation as shown in Fig. 1.22, provided that rotation over the barrier proceeds within a finite time t°. To be convinced of this, let us take the rate of jump reorientation as it was given in [11], namely... 

The simple fitting procedure is especially useful in the case of sophisticated nonlinear spectroscopy such as time domain CARS [238]. The very rough though popular strong collision model is often used in an attempt to reproduce the shape of pulse response in CARS [239]. Even if it is successful, information obtained in this way is not useful. When the fitting law is used instead, both the finite strength of collisions and their adiabaticity are properly taken into account. A comparison of... 

Ion-pair formation lowers the concentrations of free ions in solution, and hence the conductivity of the solution. It must be pointed out that ion-pair formation is not equivalent to the formation of undissociated molecules or complexes from the ions. In contrast to such species, ions in an ion pair are linked only by electrostatic and not by chemical forces. During ion-pair formation a common solvation sheath is set up, but between the ions thin solvation interlayers are preserved. The ion pair will break up during strong collisions with other particles (i.e., not in all collisions). Therefore, ion pairs have a finite lifetime, which is longer than the mean time between individual collisions. 

In order to have a finite probability that termolecular collisions can occur, we must relax our definition of a collision. We will assume that the approach of rigid spheres to within a distance of one another constitutes a termolecular collision that can lead to reaction if appropriate energy and geometry requirements are met. This approach is often attributed to Tolman (41). The number of ternary collisions per unit volume per unit time between molecules A, B, and C such that A and C are both within a distance of B is given by ZABC. 

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