In this appendix, it is shown that all eigenvalues of the ‘master matrix’
are real and negative, except that which corresponds to equilibrium and is equal to 0. The indices i
denote the eigenstates of the (time+independent) Hamiltonian and the Θ matrix appears in equation (5.12
) which can be written as
The transition probabilities γji
are real and positive and satisfy the principle of detailed balance
It is appropriate to introduce the real, symmetric, and therefore Hermitian matrix
Let v i
be an eigenvectorof Θ with the eigenvalue λ:
) one obtains
Thus, any eigenvalue λ of Θ is also an eigenvalue of the matrix Γji − δji αi, which is real, symmetric, and therefore Hermitian. This implies that λ is real.
To show that these eigenvalues are non+positive, it is sufficient to show that the quadratic form
cannot be positive. Indeed, it is easily shown that (D.9
The only non-vanishing contributions are those of coefficients Γij
which are positive equal to zero. Therefore, expression (D.10
) is negative except for xi
= exp(βε i
/2), which corresponds to thermal equilibrium.
The desired property is thus proven.