A Universe in Our Brain: Carnot’s Engine and Maxwell’s Demon

1. 1.

   Note that some obvious misprints in the equation between Eqs. (27, 29) have been corrected in later work [8], see also Eq. (4.16) below.

2. 2.
3. 3.

   Equation (2.2) may contain a dimensional constant, such as \(1/4\pi\), depending on the situation, see next section. However, this will not alter the entropy discussion.

4. 4.

   \({\Gamma }^{(2)}\) should contain a ‘tail’ contribution of unpaired spin-orbitals which is washed away in the thermalization. This is the technical reason why the trace of \({\Gamma }^{(2)}\) in (3.2.4) is \(N/2\) and not \(\left(\genfrac{}{}{0pt}{}{N}{2}\right)\).

5. 5.

   This is superconductivity from the quantum chemists point of view [23], \({\lambda }_{L}\to N/2\) and \({\lambda }_{S}\to 0\) as \(n\to \infty\). However, we will not take the thermodynamic limit just keeping \(n\ge \frac{N}{2}\) finite.

6. 6.

   The analytically continued density matrix obeys complex symmetry. The ‘miracle’, as described above, is that the thermalized \(\varrho\) becomes essentially proportional to \({\varvec{Q}}\) and that \({\varvec{B}}\) occurs again!

7. 7.

   Cf. the change of basis \(|{\varvec{h}}\rangle {\varvec{B}}\) that diagonalizes the matrix \({\varvec{\gamma}}\), which here appears accidental.

8. 8.

   One might indicate a superscript l in the transformation \({\varvec{B}}\) to distinguish the transformation associated with each CDS component. However, since they will here have the same dimension, \(n\), we have omitted this throughout when there is no case for confusion.

9. 9.

   Note that the negative sign, -i, in the off-diagonal elements, is chosen by convention.

10. 10.

    Not to be confused by \(\beta = 1/kt\) in Eq. (3.1.3).

11. 11.

    The notation might be slightly confusing as one should not mix up \(\upsilon = p/m\) and \(\nu = p/c\).

12. 12.

    See footnote 1.

13. 13.

    Communication in a restricted sense, i.e., syntax and possibly pragmatics but not semantics, see also [19, 29], where the authenticity of scientific research is discussed from the viewpoint of telicity in biology and the concept of Communication Simpliciter.

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Authors and Affiliations

1. Department of Chemistry, Uppsala University, Uppsala, Sweden

   Erkki J. Brändas

Corresponding author

Correspondence to Erkki J. Brändas .

Editors and Affiliations

1. Department of Mathematics, Odessa State Environmental University, Odessa, Ukraine

   Alexander V. Glushkov

2. Department of Mathematics, Odessa State Environmental University, Odessa, Ukraine

   Olga Yu. Khetselius

3. LCP-MR, CNRS and Sorbonne-Universités, Paris, France

   Jean Maruani

4. Department of Chemistry, Uppsala University, Uppsala, Sweden

   Erkki Brändas