Chaos, molecular

Now let us add the possibility of collisions. Before we proceed, we make the following two assumptions (1) only binary collisions occur, i.e. we rule out situations in which three or more hard-spheres simultaneously come together (which is a physically reasonable assumption provided that the gas is sufficiently dilute), and (2) Boltzman s Stosszahlansatz, or his molecular chaos assumption that the motion of the hard-spheres is effectively pairwise uncorrelated i.e. that the pair-distribution function is the product of individual distribution functions ... 

The molecular chaos assumption is made purely for mathematical convenience. We should be quick to point out that while it is certainly possible that some specially prepared systems might initially possess this property, it completely ignores the (almost) inevitable correlations that will develop in time. ... 

The assumption that the probability of simultaneous occurrence of two particles, of velocities vt and v2 in a differential space volume around r, is equal to the product of the probabilities of their occurrence individually in this volume, is known as the assumption of molecular chaos. In a dense gas, there would be collisions in rapid succession among particles in any small region of the gas the velocity of any one particle would be expected to become closely related to the velocity of its neighboring particles. These effects of correlation are assumed to be absent in the derivation of the Boltzmann equation since mean free paths in a rarefied gas are of the order of 10 5 cm, particles that interact in a collision have come from quite different regions of gas, and would be expected not to interact again with each other over a time involving many collisions. 

Moreover, since the mean free path is of the order of 100 times the molecular diameter, i.e., the range of force for a collision, collisions involving three or more particles are sufficiently rare to be neglected. This binary collision assumption (as well as the molecular chaos assumption) becomes better as the number density of the gas is decreased. Since these assumptions are increasingly valid as the particles spend a larger percentage of time out of the influence of another particle, one may expect that ideal gas behavior may be closely related to the consequences of the Boltzmann equation. This will be seen to be correct in the results of the approximation schemes used to solve the equation. 

If, at the time 3 = 0, the particles were sufficiently far from each other so that their mutual interaction had not affected them, and the gas was sufficiently dilute so that the particles had not interacted previously (lie molecular chaos assumption), then /(2) = /tt)/(1) at s = 0. This may be used on the right side cf Eq. (1-130), and leads to ... 

Arcsine distribution, 105, 111 Assumption of molecular chaos, 17 Asymptotic theory, 384 of relaxation oscillations, 388 Asynchronous excitation, 373 Asynchronous quenching, 373 Autocorrelation function, 146,174 Autocovariance function, 174 Autonomous problems, 340 nonresonance oscillations, 350 resonance oscillations, 350 Autonomous systems, 356 problems of, 323 Autoperiodic oscillation, 372 Averages, 100... 

Mintzer, David, 1 Mitropolsky, Y. A361,362 Mixed groups, 727 Modality of distribution, 123 Models in operations research, 251 Modification, method of, 67 Molecular chaos, assumption of, 17 Miller wave operator, 600 Moment generating function, 269 Moment, 119 nth central, 120... 

MSN.80. F. Henin and 1. Prigogine, Entropy, Dynamics and Molecular Chaos, Proc. Natl. Acad. Set 71, 2618-2622 (1974). 

Gibbs defined entropy as the state of molecular chaos of a system , while Tolman defined it as " the extent to which an energy system has run-down ... 

It was pointed out in Chap. 8, Sect. 2.1 that there are primarily two reasons for the failure of the diffusion equation to describe molecular motion on short times. They are connected with each other. A molecule moving in a solvent does not forget entirely the direction it was travelling prior to a collision [271, 502]. The velocity after the collision is, to some degree, correlated with its velocity before the collision. In essence, the Boltzmann assumption of molecular chaos is unsatisfactory in liquids [453, 490, 511—513]. The second consideration relates to the structure of the solvent (discussed in Chap. 8, Sects. 2.5 and 2.6). Because the solvent molecules interact with each other, despite the motion of solvent molecules, some structure develops and persists over several molecular diameters [451,452a]. Furthermore, as two reactants approach each other, the solvent molecules between them have to be squeezed-out of the way before the reactants can collide [70, 456]. These effects have been considered in a rather heuristic fashion earlier. While the potential of mean force has little overall effect on the rate of reaction, its effect on the probability of recombination or escape is rather more significant (Chap. 8, Sect. 2.6). Hydrodynamic repulsion can lead to a reduction in the rate of reaction by as much as 30-40% under the most favourable circumstances (see Chap. 8, Sect. 2.5 and Chap. 9, Sect. 3) [70, 71]. 

Stochastic approximations such as random walk or molecular chaos, which treat the motion as a succession of simple one- or two-body events, neglecting the correlations between these events implied by the overall deterministic dynamics. The analytical theory of gases, for example, is based on the molecular chaos assumption, i.e. the neglect of correlations betweeen consecutive collision partners of the same molecule. Another example is the random walk theory of diffusion in solids, which neglects the dynamical correlations between consecutive jumps of a diffusing lattice vacancy or interstitial. 

The BE was found intuitively. Here, the hypothesis of molecular chaos, that is, the assumption that any pair of particles enters the collision process uncorrelated (statistical independence) is the most important one. Only this assumption allows the formulation of a closed equation for the single-particle distribution function. 

Further statistical development of the Stosszahlansatz Hypothesis of molecular chaos... 

In this sense Jeans160 has made statement (1) above more precise. Statement (2), on the other hand, which, in our eyes, represents what Boltzmann actually meant by the hypothesis of molecular chaos, 161 is still awaiting a corresponding formulation. The following considerations are based mainly on the work of Jeans and attempt to establish a connection with the criticisms which Bur-bury162 has repeatedly made of the Stosszahlansatz. 

Although the formulation of the hypothesis of molecular chaos still contains many gaps, it certainly shows clearly that the Umkehreinwand and the Wiederkehrein-wand affect only the original formulation of the Stosszahl-amatz in fact, they prove its untenability. The improved statistical version of the Stosszahlamaiz, however, takes into account all those requirements arising from these objections.168... 

This last assumption about equal frequencies is often referred to as the hypothesis of molecular chaos. However, the same phrase is also applied to another, considerably deeper statement, which will be developed in Section 18c. The confusion of the two meanings has an important role in our discussion of the /-theorem. Hence it seemed mandatory to reserve the expression the hypothesis of molecular chaos in our discussion exclusively for the concept introduced in Section 18c. Cf. note 161. 

M. Planck, Acht Vorleaungen Uber theoret. Phyaik. (Leipzig, 1909), 3rd lecture. The "special physical hypothesis introduced by Planck to exclude the spontaneous occurrence of observable decreases in entropy (he calls it the hypothesis of "elementary disorder") consists of the following statement The number of collisions which take place in a real gas never deviates appreciably from the Stoaazahlanaatz (cf. Section 18). The hypothesis denoted in Section 18c as the "hypothesis of molecular chaos" would, on the other hand, permit such deviations. 

For a large enough system, for which we can assume molecular chaos, the configurations generated by a Born-Oppenheimer Molecular Dynamics should then be representative of the microcanonical distribution ... 

We assume molecular chaos. This means that in binary collisions both sets of molecules are randomly distributed so that the molecular velocity is uncorrelated with their position. 

This assumption is difficult to justify because it introduces statistical arguments into a problem that is in principle purely mechanical [85]. Criticism against the Boltzmann equation was raised in the past related to this problem. Nowadays it is apparently accepted that the molecular chaos assumption is needed only for the molecules that are going to collide. After the collision the scattered particles are of course strongly correlated, but this is considered irrelevant for the calculation since the colliding molecules come from different regions of space and have met in their past history other particles and are therefore entirely uncorrelated. 

For completeness it is stated that these relations rely on the molecular chaos assumption (i.e., valid for dilute gases only ), anticipating that the col-lisional pair distribution function can be expressed by (ri, ci, r2, C2, t) =... 

Finally, we impose the assumption of molecular chaos. That is, the pair distribution function can be expressed as [38] [61] ... 

Since the molecular diameters for the different gases are of the same order, the mean free path in any gas at the given temperature and pressure is of the order 10 (cm). In this case it follows that the mean free path is about hundred times the diameter of the molecule, thus the gas is dilute. However, it is noted that at higher gas pressure, say 10132500 (Pa), the mean free path is reduced and comparable with the dimensions of a molecule. In this case the assumption of molecular chaos may not be valid so the gas cannot be considered dilute [12]. 

For a dilute gas molecular chaos is assumed so the velocities of the two colliding particles are uncorrelated. The pair distribution function can then be expressed as the product of two single particle distribution functions = /2 The number of binary collisions per unit time per unit volume N12 can then be expressed as follows ... 

Enskog [20] made two modifications to (4.9) in order to describe the frequency of collisions in a dense gas assumed to consist of rigid spheres, but the assumption of molecular chaos still prevails. Firstly, due to the finite size of... 

We may wish, for example, to know the probability of finding a gas molecule at a definite spot in the box within which we suppose the gas to have been enclosed. If no external forces act on the molecules, we shall be unable to give any reason why a particle of gas should be at one place in the box rather than at another. Similarly, in this case there is no assignable reason why a particle of the gas should move in one direction rather than in another. We therefore introduce the following hypothesis, the principle of molecular chaos For the molecules of gas in a closed box, in the absence of external forces, all positions in the box and all directions of velocity are equally probable. 

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