Linearization, of the Boltzmann equation

A Vital Step in the Debye-Huckel Theory of the Charge Distribution around Ions Linearization of the Boltzmann Equation... 

This equation determines (in linear noise approximation) the fluctuations about the solution )) of the Boltzmann equation. 

Taking the natural logarithm of (A3.1.54), we see that In/j + In has to be conserved for an equilibrium solution of the Boltzmann equation. Therefore, In/j can generally be expressed as a linear combination with constant coefficients... 

Perturbation theory in reactor physics is usually formulated and used in the integrodifferential formulation of transport theory (or in the diffusion approximation to this formulation). This formulation is convenient for use because (1) the perturbations in the operators of the Boltzmann equation are linear with the physical perturbations in the system, and (2) most computer codes in use solve the integrodifferential (or the diffusion) equations. 

From these remarks we see that there are three main ingredients in a hydrodynamic description of fluid flow, which we would like to examine with the aid of the Boltzmann equation for a dilute gas. These are (a) conservation laws for number (or mass), momentum, and energy, (b) linear laws relating fluxes in particles, momentum, and energy to gradients of n, o, and T, and (c) an equilibrium-like relation, connecting the hydrostatic pressure p(r, t) to n(r, t) and T r, t).t... 

As the density of the fluid is increased above values for which a linear term in the expansion of equation (5.1) is adequate (crudely above values for which a third virial coefficient is adequate to describe the compression factor of a gas), the basis of even the formal kinetic theory is in doubt. In essence, the difficulty arises because it becomes necessary, at higher densities, to consider the distribution function, in configuration and momentum space, of pairs, triplets etc. of molecules in order to formulate an equation for the evolution of the single-particle distribution function. Such an equation would be the generalization to higher densities of the Boltzmann equation, discussed in Chapter 4 (Ferziger Kaper 1972 Dorfman van Beijeren 1977). 

Hydrodynamic flow also can be incorporated via a discretized version of the Boltzmann equation. This approach is known as the lattice Boltzmann (LB) method. The idea is to solve the linearized Boltzmann equation on an underlying lattice to propagate molecular populations ( fictitious solvent particles), which define the density and velocity on each lattice site. Similar to SRD, the monomers are treated as hydrodynamic point sources in the continuous space. The flow velocity at the monomer positions follows from a linear interpolation between neighboring lattice sites. 

Here, b is the distance between the nearest unit charges along the cylinder (b = 0.34nm for the ssDNA and b = 0.17nm for the dsDNA), (+) and (—) are related to cations and anions, respectively, and a = rss for the ssDNA and a rds for the dsDNA. The expressions (5) and (6) have been obtained using the equations for the electrostatic potential derived in [64, 65], where a linearization of the Poisson-Boltzmann equation near the Donnan potential in the hexagonal DNA cell was implemented. 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

The search for the form of W of vulcanized rubbers was initiated by polymer physicists. In 1934, Guth and Mark2 and Kuhn3) considered an idealized single chain which consists of a number of links jointed linearly and freely, and derived the probability P that the end-to-end distance of the chain assumes a given value. The resulting probability function of Gaussian type was then substituted into the Boltzmann equation for entropy s, which reads,... 

To obtain a useful approximate solution of the PB equation (S8.6-4), we consider the dilute limit in which the electric potential O is weak compared with the ambient thermal energy kT. In this limit, the Boltzmann exponential can be linearized by retaining only the leading term in the power series expansion... 

The first one is that this particular form of H can also be used to prove the approach to equilibrium in the case of Boltzmann s kinetic equation for dilute gases. The Boltzmann equation is nonlinear and a different technique is needed to prove that all solutions tend to equilibrium. This technique is based on (5.6) other convex functions cannot be used. Incidentally, the Boltzmann equation is not a master equation for a probability density, but an evolution equation for the particle density in the six-dimensional one-particle phase space ( /i-space ). The linearized Boltzmann equation, however, has the same structure as a master equation (compare XIV.5). 

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. 

Alternatively, there has been a revival of Debye-Hiickel (DH) theory [196-199] which provides an expression for the free energy of the RPM based on macroscopic electrostatics. Ions j are assumed to be distributed around a central ion i according to the Boltzmann factor exp(—/ , - y.(r)), where y(r) is the mean local electrostatic potential at ion j. By linearization of the resulting Poisson-Boltzmann (PB) equation, one finds the Coulombic interaction to be screened by the well-known DH screening factor exp(—r0r). The ion-ion contribution to the excess free energy then reads... 

For the sake of simplicity, in what follows it will be considered that the double layer potential is sufficiently small to allow the linearization of the Poisson—Boltzmann equation (the Debye—Hiickel approximation). The extension to the nonlinear cases is (relatively) straightforward however, it will turn out that the differences from the DLVO theory are particularly important at high electrolyte concentrations, when the potentials are small. In this approximation, the distribution of charge inside the double layer is given by... 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

In the literature sometimes the statement is made that the Poisson-Boltzmann equation is only compatible with electrostatics if linearized, which is not correct. The argument refers to the superposition principle which relies on the presupposed linearity of Poisson s equation. Note, however, that Poisson s equation is not linear if the charge density depends on f itself in a non-linear way as it is the case here. 

Now, a linear charge density-potential relation is consistent with the law of superposition of potentials, which states that the electrostatic potential at a point due to an assembly of charges is the sum of the potentials due to the individual charges. Thus, when one uses an unlinearized P-B equation, one is assuming the validity of the law of superposition of potentials in the Poisson equation and its invalidity in the Boltzmann equation. This is a basic logical inconsistency which must reveal itself in the predictions that emage from the so-called rigorous solutions. This is indeed the case, as will be shown below. 

ALTERNATIVE METHOD OF LINEARIZATION OF THE POISSON-BOLTZMANN EQUATION... 

Further, we assume that ij/ix) is small enough to obey the following linearized form of Poisson-Boltzmann equations ... 

Using the theory developed by Chapman-Enskog (see Ref. 14), a hierarchy of continuum fluid mechanics formulations may be derived from the Boltzmann equation as perturbations to the Maxwellian velocity distribution function. The first three equation sets are well known (1) the Euler equations, in which the velocity distribution is exactly the Maxwellian form (2) the Navier-Stokes equations, which represent a small deviation from Maxwellian and rely on linear expressions for viscosity and thermal conductivity and (3) the Burnett equations, which include second order derivatives for viscosity and thermal conductivity. 

A common source of non-linearity is the Boltzmann term in the PB equation (15). In this case Eqn. (18) is invalid, and the integration in Eqn. (17) become tedious since the electrostatic equations have to be re-solved many times. Variational expressions for the total electrostatic energy for the non-linear PB model are available (Reiner and Radke 1990 Sharp and Honig 1990a Zhou 1994)... 

Appl3ung the Enskog perturbation method we intend to describe the prop>-erties of gases which are only slightly different from equilibrium. Only under these conditions will the flux vectors be about linear in the derivatives so that the formal deflnitions of the transport coefficients apply. In this limit the distribution function is still nearly Maxwellian, and the Boltzmann equation can be solved by a perturbation method. The resulting solutions are then used to obtain expressions for the heat and momentum fluxes and for the corresponding transport coefficients. 

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