First collision time

A rationalization for 6C = 1-5 was based on an effective decay constant determined from numerical simulations of the probability distribution of first collision time, fnCt) for lattices containing L sites and n excitons. A collision was defined as either two excitons arriving at the same site or neighboring sites at the same time. The function NKn(t) was constructed by repeated simulations for a fixed number of excitons. Fig. 8 illustrates the case where n = 3. In this case a collision occurred after only a single hopping time unit. In terms of the simulated distribution function and in the absence of unimolecular decay, the occupation distribution is given by... 

The first component in expression (2.53) corresponds to the long-time behaviour of K( t) described by Markovian perturbation theory, while the second term introduces a correction for times less than zj. Within this time interval (before the first collision occurs) the system should display the dynamic properties of free rotation ( inertial effects ). 

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 collision ending the first diffusional step could have occurred with equal probability at any time between 0 and t. Averaging e(0) e(t) (2) and e (0) e (0 (2) over time of the first collision yields... 

Theories based on the Enskog collision time (84) or other solid-like approaches do not have a strongly temperature-dependent frequency correlation time. But they do have a temperature-dependent factor resulting from the need to create the solvent fluctuations in the first place. Thus, all fast-modulation theories predict that the dephasing rate will go to zero at 0 K. 

Because of the nature of InChIKey (and every hash in general), collisions are possible. This fact comes directly from the limited number of possibilities a 25-char-acters-long string can contain. Even though collisions of InChIKeys are inevitable in the future, it is not possible to say when the first collision will occur. The official InChl documentation (documentation published with the InChl source code, version 1.02-Beta http //www.iupac.org/inchi/download/index.html) states that the probability of a collision in a set of 1 billion InChIKeys is 2.0 x 10 20%. However because the second part of the InChIKey is based on InChl layers that do not exist (are empty) for many structures (such as isotopic layer, stereochemistry layer, etc.), a more realistic estimate must be based on collisions in the first part of the InChIKey alone. In this case the same source states that the probability of a collision in a set of 1 billion structures increases to 2.7 x 10-9 %. However, even this means that unless we are extremely unlucky, InChIKey should remain unique for quite a long time. It... 

The exchange of energy between an oscillator and a simple molecule was first analyzed from a classical viewpoint by I andau and Teller, who showed that, for a very slow collision, the net inelastic transfer is zero. This can be seen intuitively by considering the behavior of an infinitesimal and nearly constant force applied to one atom of a vibrating molecule. On one half cycle when the force and motion are in phase there will be an increase in momentum and kinetic energy of this atom which will be almost precisely compensated in the next half cycle by a decrease in momentum and kinetic energy. Closer analysis shows that the net effect of such a force over a cycle is to slowly accelerate the entire oscillator but not to excite it. The probability of inelastic transfer increases with the hard-ness of the collision. This latter is measured by the ratio of the time of a vibration to the collision time, rtr/rcoii = Vnl Tva, where intermolecular forces/ v is the oscillator frequency, and Vr is the relative collision velocity. 

There are a number of cases in which these extended collision times can have kinetic consequences. One of them concerns the primary dissociation of a molecule into reactive fragments, while the second concerns the recombination of active radicals to form an inactive molecule. If we consider first the hypothetical species A-B decomposing in solution to form active species (e.g., radicals) A + B, we can formulate the kinetic scheme by... 

Fig. 19. Bond distances vs. time for the N2 + O2 reaction in a cluster of 125 Ne atoms at an impact velocity of 12 km/s. Two curves are the two old bonds, N-N and 0-0. The other two curves are N to O distetnces of one oxygen atom from each one of the two nitrogen atoms. Note that dissociation of the O2 molecule that begins after the first collision of the reactants, and before N-O bond formation takes place. We show the O atom distance to the two N atoms so as to emphasize that a transient hot N2O molecule was formed. Reaction occurs by the dissociation of this caged molecule.

Also apparent in Figure 9 are two other major features. The first has an onset near 110-120 eV and arrives at the detector essentially simultaneously with the collision time. This signal corresponds to electrons ejected promptly upon collision. At ISO eV this signel becomes comparable in intensity to that of C(o >... 

To close this Section we comment on two papers that do not fit under any neat heading. The first of these is by Xiao et al,261 who study the final stages of the collapse of an unstable bubble or cavity using MD simulations of an equilibrated Lennard-Jones fluid from which a sphere of molecules has been removed. They find that the temperature inside this bubble can reach up to an equivalent of 6000 K for water. It is at these temperatures that sonolumines-cence is observed experimentally. The mechanism of bubble collapse is found to be oscillatory in time, in agreement with classical hydrodynamics predictions and experimental observation. The second paper, by Lue,262 studies the collision statistics of hard hypersphere fluids by MD in 3, 4 and 5 dimensions. Equations of state, self-diffusion coefficients, shear viscosities and thermal conductivities are determined as functions of density. Exact expressions for the mean-free path in terms of the average collision time and the compressibility factor in terms of collision rate are also derived. Work such as this, abstract as it may appear, may be valuable in the development of microscopic theories of fluid transport as well as provide insight into transport processes in general. 

Because the collision time t is arbitrary and can lead to stiff equations, it is best to solve Eq. (8.96) using operator splitting, i.e. solve first... 

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 second notable feature of these evolution curves is the pronounced shoulder effect seen on short time scales, particularly for the case where the flow is initiated from a site farthest removed from the reaction center. The appearance of shoulders is related to the fact that, for a particle initiating its motion at a specific site somewhere in the lattice, there is a minimum time required for the coreactant to reach the reaction center this time is proportional to the length of the shortest path, and hence the reactive event cannot occur until (at least) that interval of time has expired. This effect is analogous to the one observed in computer simulations of Boltzmann s H function calculated for two-dimensional hard disks [27]. Starting with disks on lattice sites with an isotropic velocity distribution, there is a time lag (a horizontal shoulder) in the evolution of the system owing to the time required for the first collision between two hard particles to occur. 

At low pressures the horizontal parts of the curve become so long that final dissociation occurs at t " every molecule which reaches Eg decomposes before possible deactivation by subsequent collisions. Therefore at low pressures the mean first passage time of molecules from = 0 to = E o is a measure of the dissociation rate (see refs. 84, 85). 

---