Equilibrium distribution density matrix

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

The density matrix ps(0) is an operator in the Hilbert space Hs of S and represents an arbitrary initial condition. The matrix pB(0) operates in HB as in the preceding section we take for it an equilibrium distribution... 

With symmetric boundary conditions at the chosen time t = 0, the microscopic formulation conforms to time reversible laws as expected. The same conclusion follows from an analogous examination of the Liouville equation. In this setting, the initial data at time, t = 0, is a statistical density distribution or density matrix. Although there are celebrated discussions on the problem of the approach to equilibrium, we nevertheless observe that without course graining or any other simplifying approximations the exact subdynamics would submit to the same physical laws as above, i.e., time reversibility and therefore constant entropy. 

One can ask what is the advantage to use the more complex two-time Green functions instead of density matrices There are several reasons. First of all, NGF give, as we shall see below, a clear description of both density of states and distribution of particles over this states. Then, the equations of motion including interactions and the influence of environment can be obtained with the help of a diagrammatic technique, and (very important) all diagrammatic results of equilibrium theory can be easily incorporated. Retardation effects are conveniently taken into account by two-time Green functions. And,. .. finally, one can always go back to the density matrix when necessary. 

Here, the integration over X was performed in Eq. (63) to define W%a (X, ) which is the integrated value of the combination of the spectral density function with the time independent operator. This spectral density function contains the quantum equilibrium structure of the system. (X, t) is the time evolved matrix element of the number operator for the product state B. Thus, to calculate the rate, one samples initial configurations from the quantum equilibrium distribution, and then computes the evolution of the number operator for product state B. The QCL evolution of the species operator is accomplished using one of the algorithms discussed in Sec. 3.2. Alternative approaches to the dynamics may also be used such as the further approximations to the QCLE discussed in Sec. 4. 

In thermodynamic equilibrium the olf-diagonal density-matrix elements are zero. The diagonal elements are determined by the Boltzmann distribution in the eigenbasis of the Hamilton operator Hz,... 

The thermal equilibrium density matrix is calculated from the definition of the density matrix and the Boltzmann equilibrium spin distribution.39 Ignoring all constants, equilibrium density matrix, is simply equal to Iz, as expected. 

To explain the latter now we consider a quasi-equilibrium distribution, described by the following density matrix ... 

Such a form of quasi-equilibrium distribution takes place due to the fact of the availability of two invariants of motion. In Equation 25 parameters a and p linked to the operators Hz and Hss are thermodynamically conjugative parameters for the Zeeman energy and the energy of spin-spin interactions respectively. We can expand the exponent in Equation 25 in jxjwers of xT-Lz and f Hss and keep only the linear terms. As we shall see later such a linearization corresponds to the high temperature approximation. In the linear approximation in x Hz and Hss, the density matrix is reduced to... 

The magnitude of elastic vibrations is controlled by a parameter of units A. While the original formulation of this method allowed for different parameter values for each atom type (or even every atom), traditionally a single size is used for all atoms in the polymer matrix. The fluctuations impose an additional equilibrium probability density distribution... 

This simple form for the density matrix implies that, in thermal equilibrium, the populations obey to the Boltzmann distribution, whereas the coherences are identically zero. 

The microscopic origin of the nonlinear response is the distortion induced in the molecular charge distribution due to the electrical field. The presence of a microscopic dipole produces a macroscopic polarization in the unit volume P = N r) where N is the number density of polarizable units and (er) the expectation value of the dipole moment induced in each unit. In order to evaluate (sr) we will use the density matrix formalism, because it is the easiest way to relate microscopic properties to macroscopic ones and to cope with macroscopic coherence effects. In the absence of fields, the medium is supposed to be described by an unperturbed Hamiltonian Hq and to be at equilibrium. When the fields are applied, the field-matter interaction contributes a time-dependent term V(t) =-E(t)P(t) to the global energy. The evolution of the system under this perturbation can be described through the equation of motion of the density operator ... 

Now, we would like to investigate adsorption of another fluid of species / in the pore filled by the matrix. The fluid/ outside the pore has the chemical potential at equilibrium the adsorbed fluid / reaches the density distribution pf z). The pair distribution of / particles is characterized by the inhomogeneous correlation function /z (l,2). The matrix and fluid species are denoted by 0 and 1. We assume the simplest form of the interactions between particles and between particles and pore walls, choosing both species as hard spheres of unit diameter... 

In this paper we have examined commercial equilibrium catalysts (Beat) to look for correlations between zeolite and matrix area stability of various catalyst families as a function of sodium. We have performed density separation of Feat samples to measure the sodium distribution as a function of catalyst age. Finally, we have revisited the issue of whether fresh catalyst sodium is equivalent to feedstock sodium in its influence on zeolite stability, catalytic activity and selectivity. 

One of the important issues addressed in our simulations is the character of clusters under study. Are these clusters solid or liquid rmder experimental conditions If they arc liquid, then the distribution wc observe in the pick-up and consequently in the photodissociation simulations corresponds to a statistical distribution at a. given temperature. If, however, the cluster is solid then both in the simulations and in the cxj)eriment we observe a quasi-stationary state with a very long lifetime rather than an equilibrium thermodynamical state. This question can be resolved by means of the instantaneous normal modes (INM) density of states (DOS) spectrum. To calculate INM DOS wc construct the Hessian matrix in a mass-weighted atomic Cartesian coordinate basis of N atoms with /r=. r, y, z. The 3N eigenvectors in the form Ci -.Cjj,Cj-,C2, C2/,C2-,.c.vj.,ca/j,c.v de-... 

As an alternative, consider the Backward Euler method q + = q + hp +i, p +i = p — hq +i. Then it is easily shown that both eigenvalues of the matrix lie in the interior of the unit disk, and hence all solutions of the recurrence relation tend to the origin with increasing n (the origin is an attractive equilibrium point). In this case, all densities evolve toward the Dirac distribution centered at the origin (5[ ](5 p]. The only (distributional) solution of Cjp = 0 is again 3[( ]5[p], which in this case is attractive. 

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