Boltzmanns ideas

The Stosszahlansatz gives for each time interval At only the most probable value of the number of collisions. (Correspondingly the //-theorem gives for each At only the most probable value of the change in H.) 

The actual number of collisions (and the actual change in H) fluctuates about this most probable value and can also assume other values with a small, but nonzero probability.159 

The relative probabilities of the various changes in H should at the same time be in agreement with the relative probabilities of the different values of H as they are given by statement (I ) of Section 14. 

However, here again large gaps appear as soon as we replace, in the above statements, the abbreviating term probability by giving the corresponding frequencies.  

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. 

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Boltzmann s derivations depended on the existence of matter being, ultimately, particulate. This is consistent with modern atomic theory. Boltzmanns ideas— including the idea that atoms behave statistically—have been accepted as a correct understanding of matter. 

The idea may be illustrated by considering first a method for increasing the acceptance rate of moves (but at the expense of trying, and discarding, several other possible moves). Having picked an atom to move, calculate the new trial interaction energy for a range of trial positions t = 1.. . k. Pick the actual attempted move from this set, with a probability proportional to the Boltzmann factor. This biases the move selection. 

Calculate the % difference between L found by this method and the modem value of 6.022 X 10. Does this support the idea that the Boltzmann constant is the gas constant per particle ... 

SASA), a concept introduced by Lee and Richards [9], and the electrostatic free energy contribution on the basis of the Poisson-Boltzmann (PB) equation of macroscopic electrostatics, an idea that goes back to Born [10], Debye and Htickel [11], Kirkwood [12], and Onsager [13]. The combination of these two approximations forms the SASA/PB implicit solvent model. In the next section we analyze the microscopic significance of the nonpolar and electrostatic free energy contributions and describe the SASA/PB implicit solvent model. 

This result comes from the idea of a variational rate theory for a diffusive dynamics. If the dynamics of the reactive system is overdamped and the effective friction is spatially isotropic, the time required to pass from the reactant to the product state is expected to be proportional to the integral over the path of the inverse Boltzmann probability. 

The simplest theoretical model proposed to predict the strain response to a complex stress history is the Boltzmann Superposition Principle. Basically this principle proposes that for a linear viscoelastic material, the strain response to a complex loading history is simply the algebraic sum of the strains due to each step in load. Implied in this principle is the idea that the behaviour of a plastic is a function of its entire loading history. There are two situations to consider. 

The expressions in Eq. 1 and Eq. 6 are two different definitions of entropy. The first was established by considerations of the behavior of bulk matter and the second by statistical analysis of molecular behavior. To verify that the two definitions are essentially the same we need to show that the entropy changes predicted by Eq. 6 are the same as those deduced from Eq. 1. To do so, we will show that the Boltzmann formula predicts the correct form of the volume dependence of the entropy of an ideal gas (Eq. 3a). More detailed calculations show that the two definitions are consistent with each other in every respect. In the process of developing these ideas, we shall also deepen our understanding of what we mean by disorder. ... 

Lattice-Boltzmann is an inherently time-dependent approach. Using LB for steady flows, however, and letting the flow develop in time from some starting condition toward a steady-state is not a very good idea, since the LB time steps need to be small (compared to, e.g., FV time steps) in order to meet the incompressibility constraint. 

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. 

Twelve years later, Choh and Uhlenbeck8 published the first explicit generalization of the Boltzmann equation involving triple collisions. Their work rests on Bogolubov s ideas and formal results. Green11 and Rice, Kirkwood, and Harris26 also obtained the triple collision equation by other methods. 

In Section II, we summarize the ideas and the results of Bogolubov,3 Choh and Uhlenbeck,6 and Cohen.8 Bogolubov and Choh and Uhlenbeck solved the hierarchy equations and derived two- and three-body generalized Boltzmann operators Cohen used a cluster expansion method and obtained two-, three-, and four-body explicit results which he was able to extend to arbitrary concentrations. 

Choh and Uhlenbeck6 developed Bogolubov s ideas and extended his formal results. They established a generalized Boltzmann equation which takes account of three-particle collisions. The extension of their results to higher orders in the concentration poses no problem in principle, but it appears difficult, in this formalism, to write a priori the collision term with an arbitrary number of particles. 

The generalization of Boltzmann s solution turned out to be especially difficult. In their 1973 paper, PGHR performed a synthesis of the projector method of C. George and the idea of a transformation of p. The PGHR paper was considered for several years as the bible of Prigogine s group. The... 

Traditional thermodynamics gives a clear definition of entropy but unfortunately does not tell us what it is. An idea of the physical nature of entropy can be gained from statistical thermodynamics. Kelvin and Boltzmann recognised diat there was a relationship between entropy and probability (cf., disorder) of a system with the entropy given by... 

The observed rotational distributions are in agreement with the ideas of Baronavski on the ICN photolysis but disagree with the theoretical calculations. These rotational distributions are not, however, simply Boltzmann distributions. 

Stern combined the ideas of Helmholtz and that of a diffuse layer [64], In Stern theory we take a pragmatic, though somewhat artificial, approach and divide the double layer into two parts an inner part, the Stern layer, and an outer part, the Gouy or diffuse layer. Essentially the Stern layer is a layer of ions which is directly adsorbed to the surface and which is immobile. In contrast, the Gouy-Chapman layer consists of mobile ions, which obey Poisson-Boltzmann statistics. The potential at the point where the bound Stern layer ends and the mobile diffuse layer begins is the zeta potential (C potential). The zeta potential will be discussed in detail in Section 5.4. 

However, like conventional quadrature, this method is of little use for the evaluation of averages such as in Eq. (1.1) because most of the computing is spent at points where the Boltzmann factor is negligible and the integral is therefore zero. Obviously, it would be more preferable to sample many points in regions where the Boltzmann factor is larger than zero and few elsewhere. This is the basic idea behind importance sampling. 

Gorban et al. in their works (Gorban, 2007 Gorban et al., 2001,2006) seems to be more comprehensive for our discussion. The works unfolded the idea of the Ehrenfests (1959) on the isolated system tending toward the Boltzmann equilibrium trajectory as a result of "agitations."... 

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