Boltzmann equation correlation

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. 

In addition to the nearest-neighbor interaction, each ion experiences the electrostatic potential generated by the other ions. In the literature this has generally been equated with the macroscopic potential 0 calculated from the Poisson-Boltzmann equation. This corresponds to a mean-field approximation (vide infra), in which correlations between the ions are neglected. This approximation should be the better the low the concentrations of the ions. 

We then write down the equation of evolution for the distribution function in the limit of long times. This is the generalized Boltzmann equation, which, this time, is instantaneous because, in the limit of long times, the variation of the distribution function during the time interval rc becomes slow. Also, in the long-time limit, we briefly discuss the equation which gives the correlations. 

Hie singlet distribution ff changes when a collision occurs between the particle ot and any one of the remaining 3 = (N — 1) particles (labelled 2 during the collision process). The probability of particles ot and /3 being able to collide depends upon their mutual position in space and their velocity, which is described by the doublet density f"0. Now, in the Boltzmann equation analysis, this leads to a factorisation of the doublet density in terms of the singlet density of both particles. Because any correlation in the position and motion of these particles is lost by this procedure, an alternative approach must be tried to estimate the doublet density. 

The doublet correlation function x 3 is not xTXi, else fjf would be equal to ff and the equations would reduce to the Boltzmann equation. 

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]. 

There are many possible improvements to the Poisson-Boltzmann equation and an extensive discussion of the refinements has been presented by Bell and Levine3 01). The relative permittivity is field dependent and the ions are polarizable. In Eq.(6.2) the correlation between the ions is neglected so are specific chemical effects in the... 

In the limit of an infinite micellar radius, i.e. a charged planar surface, the salt dependence of Ge is solely due to the entropy factor. A difficult question when applying Eq. (6.13) to the salt dependence of the CMC is if Debye-Hiickel correction factors should be included in the monomer activity. When Ge is obtained from a solution of the Poisson-Boltzmann equation in which the correlations between the mobile ions are neglected, it might be that the use of Debye-Hiickel activity factors give an unbalanced treatment. If the correlations between the mobile ions are not considered in the ionic atmosphere of the micelle they should not be included for the free ions in solution. 

It has been long known that the over-simplified Poisson-Boltzmann equation is accurate in predicting the double layer interaction only in a relatively narrow range of electrolyte concentrations. One obvious weakness of the treatment is the prediction that the ions of the same valence produce the same results, regardless of their nature. In contrast, experiment shows marked differences when different kinds of ions are used. The ion-specific effects can be typically ordered in series (the Hofmeister series [36]), and the placement of ions in this series correlates well with the hydration properties of the ions in bulk water. 

In a typical macroscopic assumption of proportionality between polarization and applied electric field, P = e0(c — 1 )E, where e is the dielectric constant, and eq3 reduces to the traditional Poisson—Boltzmann equation (the concentrations cH and c0h being in general much smaller than ce). However, if the correlations between neighboring dipoles are taken into account, the following constitutive equation relating the polarization to the macroscopic electric field is obtained7... 

Eqs. (1), (3) and (4) represent a Modified Poisson-Boltzmann approach, which, when the ions interacts only via the mean field potential ip(z) (e.g. AWa=AWc=0), reduces to the well-known Poisson-Boltzmann equation. The only change in the polarization model is the replacement of the constitutive equation Eq. (1) by Eq. (2), which accounts for the correlation in the orientation of neighboring dipoles. However, since independent functions, four boundary conditions are needed to solve the system composed of Eqs. (2) (3) and (4). For only one surface immersed in an electrolyte, t]/(z-co)-0, m(z-co)=0 whereas for two identical surfaces, the symmetry of ip and m requires that = 0 and m(z=0)=0 where z- 0 represents... 

In all of the discussion above, comparisons have been made between various types of approximations, with the nonlinear Poisson-Boltzmann equation providing the standard with which to judge their validity. However, as already noted, the nonlinear Poisson-Boltzmann equation itself entails numerous approximations. In the language of liquid state theory, the Poisson-Boltzmann equation is a mean-field approximation in which all correlation between point ions in solution is neglected, and indeed the Poisson-Boltzmann results for sphere-sphere [48] and plate-plate [8,49] interactions have been derived as limiting cases of more rigorous approaches. For many years, researchers have examined the accuracy of the Poisson-Boltzmann theory using statistical mechanical methods, and it is... 

A charged surface and the ions, which neutralize the surface, together create an electric double layer. The distribution of the ions can be evaluated from the Poisson-Boltzmann equation where the ions are treated as point particles and the primitive model is used. Further, all correlations between the ions are neglected, which means that the ions are interacting directly only with the colloids and through an external field given by the average distribution of the small ions. The distribution of the particles are assumed to follow Boltzmann s theorem [11]... 

Similar approaches utilising such indicators as FPE to visualise the membrane surface potential ( )s are also routinely employed in our laboratories (17). By correlating the change of the fluorescence and hence surface potential with the addition of net electric charges from the macromolecule that becomes bound, it is possible to quantitate on the basis of the poisson-boltzmann equation above, the number of molecules that become bound. This allows us for example to determine localised molecular interactions on the membrane surface (17, 34). 

In complementary computational studies, Gunner et al. have explored the role of long-range electrostatic interaction on electron transfer processes in the Rhodo-bacter sphaeroides reaction center [38]. The interaction domains were identified by mapping electrostatic potentials, calculated from the Poisson-Boltzmann equation, on to calculated encounter surfaces for each of the components of the reaction center. From qualitative correlation of electron transfer processes with these low-resolution potential maps, it is apparent that long-range interactions profoundly affect the reduction potential of the cofactors in the reaction center. 

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. 

If the exciton-phonon coupling is sufficiently weak, the solution of the equation for the correlation function (B (t)Bm(t)B, (0)Bm (0)) is equivalent to the solution of the Boltzmann equation (11). In this coherent limit, the exciton states of... 

Numerically, it is now a common practice to calculate within the dielectric continuum formulation but employing cavities of realistic molecular shape determined by the van der Waals surface of the solute. The method is based upon finite-difference solution of the Poisson-Boltzmann equation for the electrostatic potential with the appropriate boundary conditions [214, 238, 239]. An important outcome of such studies is that even in complex systems there exists a strong linear correlation between the calculated outer-sphere reorganization energy and the inverse donor-acceptor distance, as anticipated by the Marcus formulation (see Fig. 9.6). More... 

Figure 9.6. Linear correlation between the outer-sphere reorganization energy, calculated by finite-difference solution of the Poisson-Boltzmann equation, and the inverse of the donor-acceptor distance in PCI-Am (X = CHj, R = Am) complex in acetonitrile. (Reproduced from [239] with permission. Copyright (1997) by the American Chemical Society.)...

The simplified Boltzmann equation can be solved using the Lattice Boltzmann Method (LBM) for the distributed function on a regular lattice. LBM considers each lattice structure as a volume element that consists of a collection of particles in the fluid. This simplified Boltzmann equation approximates the collision term, Q f, / ), in Eq. 37 using a relaxation time, t, providing a linear correlation. The most well-known form of the LBM is the BGKLBM, where the relaxation time is a constant. 

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