Boltzmann stosszahlansatz

In other words, in approximate accordance with the original paper by Boltzmann [6], we assume that in a given volume element the expected number of collisions between molecules that belong to different velocity ranges can be computed statistically. This assumption is referred to as the Boltzmann Stosszahlansatz (German for Collision number assumption). A result of the Boltzmann H-theorem analysis is that the latter statistical assumption makes Boltzmann s equation irreversible in time (e.g., [28], sect. 4.2). 

If we wish to know the number of (VpV)-collisions that actually take place in this small time interval, we need to know exactly where each particle is located and then follow the motion of all the particles from time tto time t+ bt. In fact, this is what is done in computer simulated molecular dynamics. We wish to avoid this exact specification of the particle trajectories, and instead carry out a plausible argument for the computation of r To do this, Boltzmann made the following assumption, called the Stosszahlansatz, which we encountered already in the calculation of the mean free path ... 

A major preoccupation of nonequilibrium statistical mechanics is to justify the existence of the hydrodynamic modes from the microscopic Hamiltonian dynamics. Boltzmann equation is based on approximations valid for dilute fluids such as the Stosszahlansatz. In the context of Boltzmann s theory, the concept of hydrodynamic modes has a limited validity because of this approximation. We may wonder if they can be justified directly from the microscopic dynamics without any approximation. If this were the case, this would be great progress... 

Let q represent an observable quantity of a macroscopic system such as the circuit in Figure 1. Assuming that there are no other macroscopic observables, one can derive Eq. (15) from the equation of motion of all particles at the expense of a regrettable, but indispensable, repeated randomness assumption, similar to Boltzmann s Stosszahlansatz. 6 It then also follows that, provided q is an even variable, W has a symmetry property called detailed balancing. 6,7... 

Here a is the differential cross-section, and depends only on Pi Pi = l/>3 Pa and on (/U - p2) p2 Pa)-The precise number of molecules in the cell fluctuates around the value given by the Boltzmann equation, because the collisions occur at random, and only their probability is given by the Stosszahlansatz. Our aim is to compute these fluctuations. If / differs little from the equilibrium distribution one may replace the Boltzmann equation by its linearized version. It is then possible to include the fluctuations by adding a Langevin term, whose strength is determined by means of the fluctuation-dissipation theorem.510 As demonstrated in IX.4, however, the Langevin approach is unreliable outside the linear domain. We shall therefore start from the master equation and use the -expansion. The whole procedure consists of four steps. 

In our approach [1, 2] termed the dynamic method the complex susceptibility x = x — ix" is determined by a law of undamped motion of a dipole in a given potential well and by dissipation mechanism often described as stosszahlansatz in the underlying kinetic or Boltzmann equation. In this review we shall refer to this (dynamic) method as the ACF method, since it is actually based on calculation of the spectrum of the dipolar autocorrelation function (ACF). Actually we use a one-particle approximation, in which the form of an employed potential well (being in many cases rectangular or close to it) is taken a priori. Correlation of the particles coordinates is characterized implicitly by the Kirkwood correlation factor g, its value being taken from the experimental data. The ACF method is simple and effective, because we do not employ the stochastic equations of motions. This feature distinguishes our method from other well-known approaches—for example, from those described in books [13, 14]. 

Criticism of the Stosszahlansatz and its corollaries arose as soon as it was recognized as paradoxical that the completely reversible gas model of the kinetic theory was apparently able to explain irreversible processes, i.e., phenomena whose development shows a definite direction in time. These nonstationary,51 irreversible processes were brought into the center of interest by the //-theorem of Boltzmann. In order to show that every non-Max-wellian distribution always approaches the Maxwell distribution in time, this theorem synthesizes all the special irreversible processes (like heat conduction and... 

The monotonic decrease of the //-function (and as a consequence the irreversible approach of any distribution to the Maxwell-Boltzmann distribution) follows from the fact that in the calculations of the //-theorem the Stosszahlansate is used for every successive At, without exception. Therefore the Umkehreinwand and the Wieder-kehreinwand are mainly directed to this application of the Stosszahlansatz.61... 

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. 

When H has reached its minimum value this is the well known Maxwell-Boltzmann distribution for a gas in thermal equilibrium with a uniform motion u. So, argues Boltzmann, solutions of his equation for an isolated system approach an equilibrium state, just as real gases seem to do. Up to a negative factor -k, in fact), differences in H are the same as differences in the thermodynamic entropy between initial and final equilibrium states. Boltzmann thought that his //-theorem gave a foundation of the increase in entropy as a result of the collision integral, whose derivation was based on the Stosszahlansatz. 

Several attempts were made to derive the Boltzmann equation from the Liouville equation, but it was not until the work of Bogoliubov in 1947 and the later work of Cohen and of Green that a satisfactory derivation was given. The approach to the Boltzmann equation from the Liouville equation has proved to be important for two reasons. First, it makes it possible to replace the Stosszahlansatz by a more fundamental assumption about the statistical ensemble that is sampled when an experiment is performed. Second, it allows one to generalize the Boltzmann equation to dense gases. 

Consider a dense gas of hard spheres, all with mass m and diameter a. Since the collisions of hard-sphere molecules are instantaneous, the probability is zero that any particle will collide with more than one particle at a time. Hence we still suppose that the dynamical events taking place in the gas are made up of binary collisions, and that to derive an equation for the single-particle distribution function /(r, v, /) we need only take binary collisions into account. However, the Stosszahlansatz used in deriving the Boltzmann equation for a dilute gas should be modified to take into account any spatial and velocity correlations that may exist between the colliding spheres. The Enskog theory continues to ignore the possibility of correlations in the velocities before collision, but attempts to take into account the spatial correlations. In addition, the Enskog theory takes into account the variation of the distribution function over distances of the order of the molecular diameter, which also leads to corrections to the Boltzmann equation. 

We see that if particles 1 and 2 have suffered earlier collisions with each other, as illustrated in Fig. 22b, or with the same third particle, as in Figs. 22c and 22d, then there will be both spatial and velocity correlations between particles 1 and 2 before the onset of their collision at time t. These dynamical correlations are not taken into account in the Boltzmann equation, since the Stosszahlansatz assumes they do not exist, nor are they taken into account by the Enskog theory for hard spheres, which ignores all velocity correlations. Here we see that dynamical correlations do exist in the gas and that they are accounted for in the generalized Boltzmann equation since it takes the concerted action of three or more particles to produce such correlations. 

This reminds of the classical problem the fundamental equations are invariant for time reversal. An argument like Boltzmann s Stosszahlansatz could possibly be used to resolve it. Of course, there is no quantum mechanical equivalent of Boltzmann s assumption possible. 

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