Boltzmann’s method

Since the time of my first graduation in science, I have been an enthusiastic reader of Boltzmann, whose dynamical vision of physical becoming was for me a model of intuition and penetration. Nonetheless, I could not but notice some unsatisfying aspects. It was clear to me that Boltzmann introduced hypotheses foreign to dynamics under such assumptions, to talk about a dynamical justification of thermodynamics seemed to me an excessive conclusion, to say the least. In my opinion, the identification of entropy with molecular disorder could contain only one part of the truth if, as I persisted in thinking, irreversible processes were endowed with the constructive role that I never cease to attribute to them. For another part, the applications of Boltzmann s methods were restricted to dilute gases, while I was most interested in condensed systems. [Prigogine, 1977/1993, p. 258]... 

Liouville s equation, derivation of Boltzmann s equation from, 41 Littlewood, J. E., 388 Lobachevskies method, 79,85 Local methods of solution of equations, 78... 

The common method of estimation of the interfacial potential in microheteroge-neous systems is to use pH indicators adsorbed at for instance charged micelles. The local interfacial proton activity is related to the bulk activity and to an interfacial potential by Boltzmann s law according to the equation [17,70-72]... 

Other than an effect on backbone solvation, side chains could potentially modulate PPII helix-forming propensities in a number of ways. These include contributions due to side chain conformational entropy and, as discussed previously, side chain-to-backbone hydrogen bonds. Given the extended nature of the PPII conformation, one might expect the side chains to possess significant conformational entropy compared to more compact conformations. The side chain conformational entropy, Y.S ppn (T = 298°K), available to each of the residues simulated in the Ac-Ala-Xaa-Ala-NMe peptides above was estimated using methods outlined in Creamer (2000). In essence, conformational entropy Scan be derived from the distribution of side chain conformations using Boltzmann s equation... 

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... 

Derivation of the Boltzmann distribution function is based on statistical mechanical considerations and requires use of Stirling s approximation and Lagrange s method of undetermined multipliers to arrive at the basic equation, (N,/No) = (g/go)exp[-A Ae/]. The exponential term /3 defines the temperature scale of the Boltzmann function and can be shown to equal t/ksT. In classical mechanics, this distribution is defined by giving values for the coordinates and momenta for each particle in three-coordinate space and the lin-... 

The traditional method of dealing with irreversible processes is, of course, the use of the Boltzmann integro-differential equation and its various extensions. But this method leads to two serious difficulties. The first is that Boltzmann s equation is neither provable nor even meaningful except in the context of molecular encounters, i.e., under the assumption that the intermolecular forces are of such short range in comparison with molecular distances that a molecule spends only a negligible fraction of its time within the influence of others. This drastically restricts the field of applicability, confining the treatment to gases close to the ideal state. But even then the equation can only be established on the basis of an essential assumption of molecular probabilistic independence ( micromolecular chaos ).3... 

Here, R is the radius of the sphere, q is the coefficient of viscosity, kB is Boltzmann s constant, and T is temperature. Equation (4) implies that rR should vary linearly with volume, or mass, in the range where the Stokes-Einstein equation is valid. Figure 5 shows the roughly linear behavior of rR for these compounds, and illustrates why polymeric conjugates of Gd3+ chelates remain a very attractive method of modulating both rR and the intravenous retention time (t1/2) of BPCAs. 

At temperatures where the potential minimum, e, is comparable to kT, equation (18) is not valid because the method of treating the thermal velocity distribution is no longer appropriate. When e becomes comparable to kT, orientation effects are believed to become important because in general the depth of the potential minimum is dependent on the configuration of both molecules. The probability of a particular orientation can be derived from Boltzmann s equation. 

The articles of L. S. Ornstein (1908-1909) are, first, very valuable contributions to the clarification of the foundations of Gibbs s treatment. In addition, they show (1) that operations with canonical ensembles occasionally furnish a more convenient computational scheme for the treatment of complicated problems of equilibrium (e.g., the equilibrium in capillary transition layers) than does Boltzmann s procedure (2) the reason why the two methods give the same result in a large group of equilibrium problems. 

The difficulties in simulating polymer systems stem from the long relaxation times these systems display. Long runs are needed in order to ensure adequate equilibration. We have employed the method of Wall and Mandel (21) as modified for continuum three dimensional polymers by Webman, Ceperley, Kalos and Lebowitz (22). Each chain is considered in order and one end is chosen randomly as a bead. Suppose the initial chain coordi-nates are C = X, .. Xn A new position of that bead, X, is selected such that X = X + Ax where Xn is the initial head position and Ax is a vector randomly chosen via a rejection technique from the probability distribution exp(-BUfl(AX))(3=l/kBT, kfi Boltzmann s constant, T the temperature) and Ujj is iv< n in Eq. 

We give below a simple method to derive an approximate solution to the hnear-ized Poisson-Boltzmann equation (1.9) for the potential distribution i/ (r) around a nearly spherical spheroidal particle immersed in an electrolyte solution [12]. This method is based on Maxwell s method [13] to derive an approximate solution to the Laplace equation for the potential distribution around a nearly spherical particle. 

This chapter deals with a method for obtaining the exact solution to the linearized Poisson-Boltzmann equation on the basis of Schwartz s method [1] without recourse to Derjaguin s approximation [2]. Then we apply this method to derive series expansion representations for the double-layer interaction between spheres [3-13] and those between two parallel cylinders [14, 15]. 

We start with the simplest problem of the plate-plate interaction. Consider two parallel plates 1 and 2 in an electrolyte solution, having constant surface potentials i/ oi and J/o2, separated at a distance H between their surfaces (Fig. 14.1). We take an x-axis perpendicular to the plates with its origin 0 at the surface of one plate so that the region 0solution phase. We derive the potential distribution for the region between the plates (0linearized Poisson-Boltzmann equation in the one-dimensional case is... 

The effusion method originally suggested by Knudsen is essentially a method of pressure measurement which utilizes the fact that pressure is the effect of the bombardment of the walls of the containing vessel by the molecules. If a small part of the wall is replaced by a hole leading to an evacuated space, then the molecular shower will pass through the hole, and the rate at which molecules do this depends only on the mean component of velocity of the gas molecules and the number present, and may be calculated by kinetic theory to be apj 2nmkT) molecules per second, where a is the area of the hole, p is the pressure, m is the mass of a molecule, k is Boltzmann s constant, and T is the absolute temperature 2 19 In the derivation of this formula it is assumed that ... 

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