Boltzmann density operator systems

Here, Hx0t is the total Hamiltonian of the system. The Boltzmann density operator pTot is... 

Here, (3 has its usual statistical meaning, whereas X. is a scalar, and pB is the Boltzmann density operator describing the system. It was been written [Go//(f)]° in place of [Gofl(t)]° in order to make clear the differences in the fashion so we could calculate the ACF. 

As mentioned in Section 2.1, the usual Boltzmann equation conserves the kinetic energy only. In this sense the Boltzmann equation is referred to as an equation for ideal systems. For nonideal systems we will show that the binary density operator, in the three-particle collision approximation, provides for an energy conservation up to the next-higher order in the density (second virial coefficient). For this reason we consider the time derivative of the mean value of the kinetic energy,12 16 17... 

In Eq. (10), E nt s(u) and Es(in) are the s=x,y,z components of the internal electric field and the field in the dielectric, respectively, and p u is the Boltzmann density matrix for the set of initial states m. The parameter tmn is a measure of the line-width. While small molecules, N<pure solid show well-defined lattice-vibrational spectra, arising from intermolecular vibrations in the crystal, overlap among the vastly larger number of normal modes for large, polymeric systems, produces broad bands, even in the crystalline state. When the polymeric molecule experiences the molecular interactions operative in aqueous solution, a second feature further broadens the vibrational bands, since the line-width parameters, xmn, Eq. (10), reflect the increased molecular collisional effects in solution, as compared to those in the solid. These general considerations are borne out by experiment. The low-frequency Raman spectrum of the amino acid cystine (94) shows a line at 8.7 cm- -, in the crystalline solid, with a half-width of several cm-- -. In contrast, a careful study of the low frequency Raman spectra of lysozyme (92) shows a broad band (half-width 10 cm- -) at 25 cm- -,... 

Z is the partition function, a is the scalar quantity conjugate to A and ks is the Boltzmann constant a could be, e.g., an electric field and A the electric dipolar moment operator of the system. We assume that at the initial time the system is not far from equilibrium and expand the density to first order in a (see Ref. [62]). The expected value of an observable B is expressed as the Mori scalar product that refers to the equilibrium density [62, 63]... 

Equation [18] is valid when the polarizability of the dielectric is proportional to the electrostatic field strength. " The operator V in the Cartesian coordinate system has the form (didx, didy, d/dz). When one deals with a system composed of a macromolecule immersed in an aqueous medium containing a dissolved electrolyte, the partial charges of each atom of the macromolecule can be described as fixed charges charges of the dissolved electrolyte can be described as mobile charges with density determined by a Boltzmann s distribution, and Eq. [18] can be written in the following form, known as the Poisson-Boltzmann equation ... 

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