Collision events, uncorrelated

The presence of a high density of solvent molecules leads to recollisions between the potentially reactive pair of molecules. Some examples of such recollision events are shown schematically in Fig. 7.2. In Fig. 1.1a the solute molecules A and B collide elastically, and after collision of A with a solvent molecule S, the A molecule recollides with B and reacts. An event of this type is extremely unlikely in a dilute gas. The description of collision sequences of this type is outside of the scope of a Boltzmann equation, which accounts only for uncorrelated binary collision events. Collision sequences of this kind are expected to play an increasingly important role as the solvent density increases, and as we shall see, they are often the dominant contribution to the dynamics. A similar sequence of reactive events is shown in Fig. 1.2b. 

This equation accounts only for dynamically uncorrelated collisions, and thus misses many of the important classes of correlated collision events discussed earlier. We need a new kinetic equation that does not suffer from this limitation. 

The contributions to the collision operator A (l z) describe the following types of dynamic event the A operators are Enskog collision operators and describe uncorrelated binary collision events describes uncorrelated elastic collisions of A with solvent molecules... 

This is certainly an idealized model for the true collision dynamics it assumes that the velocity is randomized at each collision. All characteristics of the solvent and solute molecule properties, including the effects of the uncorrelated and correlated collision events described earlier, are implicitly contained in the collision frequency parameter. Nevertheless, the virtue of this model lies in its simplicity the full collision operator is rather complex... 

Gordon s model assumes that molecules in a liquid are undergoing collision-interrupted free rotation. A collision is defined as an event which changes the angular momentum of a molecule. It is furthermore assumed that (a) collisions are of zero duration, (b) collisions change the molecule s rotational velocity but do not change its orientation, (c) successive hard-core collisions are uncorrelated that is, the in-... 

Up to the early 1970s a kinetic approach to the time-dependent properties of fluids was synonymous with a framework based on the Boltzmann equation and its extension by Enskog, in which a central role is played by those dynamical events referred to as uncorrelated binary collisions [29]. Because of this feature the Boltzmann equation is in general not applicable to dense fluids, where the collisions are so frequent that they are likely to interfere with each other. The uncorrelation ansatz is clearly equivalent to a loss of memory, or to a Markov approximation. As a result, for dense fluids the traditional kinetic approach should be critically revised to allow for the presence of non-Markovian effects. 

The Andersen thermostat is very simple. After each time step Si, each monomer experiences a random collision with a fictitious heat-bath particle with a collision probability / coll = vSt, where v is the collision frequency. If the collisions are assumed to be uncorrelated events, the collision probability at any time t is Poissonian,pcoll(v, f) = v exp(—vi). In the event of a collision, each component of the velocity of the hit particle is changed according to the Maxwell-Boltzmann distribution p(v,)= exp(—wv /2k T)/ /Inmk T (i = 1,2,3). The width of this Gaussian distribution is determined by the canonical temperature. Each monomer behaves like a Brownian particle under the influence of the forces exerted on it by other particles and external fields. In the limit i —> oo, the phase-space trajectory will have covered the complete accessible phase-space, which is sampled in accordance with Boltzmann statistics. Andersen dynamics resembles Markovian dynamics described in the context of Monte Carlo methods and, in fact, from a statistical mechanics point of view, it reminds us of the Metropolis Monte Carlo method. 

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