Exchange energy, 200 Hilbert space

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

To stay in Hilbert space would imply the measuring/measured quantum systems to remain in an entangled state unknown to people at the Fence. But all possible changes are there anyway. To disclose them, energy must be exchanged, and consequently, entropy must vary. One is coming close to thermodynamics as soon as the description of phenomena forces to take the systems away Hilbert space arena. In other words, Hilbert space alone is not adequate to handle this type of physics because it is an abstract formalism only. This implies that actual emergence of a particular outcome cannot be accounted for by a quantum theory the space-time occurrence of one click is not predictable by the theory. 

Topological atoms are defined solely by using the gradient of the electron density, g. As a result they are independent of how g is obtained. For example, they can be derived from Gaussian, Slater or plane wave basis fimctions. Since they are not formulated in Hilbert space, the inclusion of electron correlation does not lead to conceptual difficulties. We adhere to this philosophy and extend it to the use of (spinless) reduced density matrices in the formulation of atomic interaction energy. So far our work has only focused on the Coulomb part but the exchange-correlation part can also be treated in a similar but generalised framework, as I show now. 

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