Thermal density

Temperature differences (thermal density) between the inside and the outside of the building... 

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

The method of latent heat storage based on liquid-solid phase transition is available to make smaller the volume of heat storage tank, because of its higher thermal density than that of sensible heat storage. Therefore, a substance which has a large amount of latent heat of fusion is more profitable as a heat storage material. 

Now that the entanglement of the XY Hamiltonian with impurities has been calculated at Y = 0, we can consider the case where the system is at thermal equilibrium at temperature T. The density matrix for the XY model at thermal equilibrium is given by the canonical ensemble p = jZ, where = l/k T, and Z = Tr is the partition function. The thermal density matrix is diag-... 

These circumstances may explain why it took ten years for the phenomenon to be experimentally observed after the prediction. In 1973, prior to this finding, Lon Hocker, George Benedek, and Tanaka realized that a gel scattered light, and the light intensity fluctuated with time [5]. They established that the scattering is due to the thermal density fluctuations of the polymer network and derived a theory that explained the fluctuation. These fluctuations are similar to... 

Using the theoretical results of Stoof et al.[19] we calculate b(t) by averaging the field and temperature dependent relaxation rates over a thermal density distribution. The contribution due to an individual process may be written as... 

One view of this trace operation is that the usual phase space integral may be obtained by representing the thermal density matrix e in plane-wave momentum states, and performing the trace in that state space (Landau et al, 1980, Section 33. Expansion in powers of h ). Particle distinguishabihty restrictions are essential physical requirements for that calculation. In this book we will confine ourselves to the Boltzmann-Gibbs case so that e = Q n, V, T)/n, since the... 

The mixture of the molecules pure states is given by the thermal density operator... 

A mixture of eigenstates and L., each with 50% probability. This would correspond to a decomposition of the thermal density operator into the pure states as = jD+ + D. ... 

In Fig. 5 some of these different decompositions are visualized with the aid of the Bloch sphere. The thermal density operator Dp= from... 

FIGURE 5 A thermal density operator can be decomposed into pure states in infinitely many different ways. Mbdng the vectors corresponding to eigenstates or chiral states or alternative chiral states with 50% probability always leads to the zero vector, i.e., the center of the Bloch sphere, corresponding to the density operator Dp =... 

In summary, one has infinitely many different ways of decomposing a thermal density operator into pure states. Different decompositions can refer to entirely different physical or chemical points of view viz. ... 

It is, of course, not compulsory to decompose a thermal density operator into only two or finitely many pure states. One could equally well try to decompose a thermal density operator into a continuum of pure states, or even into all the pure states of the system in question. This possibility is illustrated here by use of a two-level system and the relevant Bloch sphere. 

In Fig. 6 again different decompositions of some thermal state D into pure states are shown This time the density operator D is not chosen to be equal to jl. Since every thermal density operator D can be written as a mixture of the eigenstates of the underlying Hamiltonian, the corresponding vector b ) must be some mixture of the north and south pole vectors corresponding to the eigenstates and hence must be on the z-axis, as indicated in Fig. 6. 

A given thermal density operator can be decomposed into stationary or nonstationary pure states. Depending on this choice, two different points of view are adopted, which for simplicity will be referred to as the spectro-scopist s and the chemist s point of view, respectively. 

This dichotomy reflects itself in the different possible choices for decompositions of the thermal density operator Dp into pure states, viz. (i) A decomposition of the thermal density operator Dp into (symmetry-adapted) eigenstates of the Hamiltonian. If superpositions of these eigenstates are not considered, one obtains a classical energy obsen able. (ii) A decomposition of the thermal density operator Dp into pure handed states. If superpositions of these handed states are not considered, one gets a classical chirality observable, (iii) A decomposition fi of the thermal density operator Dp into pure states such that the average dispersion... 

If superpositions of these minimal-dispersion states are not considered, one obtains an approximately classical nuclear structure, which is the best possible nuclear structure compatible with the thermal density operator D. ... 

The main reason for this bewildering observation is that the thermal density operators describe a strictly stationary situation. To make this point clear, consider a thermal state describing an ensemble of left-handed molecules of some given species. Since handed molecules racemize—albeit slowly—one may expect that evolves into -f... 

We therefore end up again with the problem of finding a canonical decomposition of the overall thermal density operator of some molecular species into pure states. Based on such a canonical decomposition of (and dynamical arguments see subsequent text), one can introduce effective thermal states in some ad hoc manner. 

All statistical results in terms of thermal density operators are preserved. Consideration of the canonical decomposition allows us to discuss molecular structure or other fuzzy classical concepts, which cannot be done by considering thermal density operators alone. 

System Main crystalline IJending Deformation Thermal Density Use... 

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