Jump to ContentJump to Main Navigation
Molecular Nanomagnets$

Dante Gatteschi, Roberta Sessoli, and Jacques Villain

Print publication date: 2006

Print ISBN-13: 9780198567530

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198567530.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 25 February 2017

(p.329) APPENDIX D BASIC PROPERTIES OF THE MASTER EQUATION

(p.329) APPENDIX D BASIC PROPERTIES OF THE MASTER EQUATION

Source:
Molecular Nanomagnets
Publisher:
Oxford University Press

In this appendix, it is shown that all eigenvalues of the ‘master matrix’

(D.1)
Θ j i = γ j i δ j i i γ i
are real and negative, except that which corresponds to equilibrium and is equal to 0. The indices i, j denote the eigenstates of the (time+independent) Hamiltonian and the Θ matrix appears in equation (5.12) which can be written as
(D.2)
d d t p ( t ) = Θ p ( t ) .
The transition probabilities γji are real and positive and satisfy the principle of detailed balance
(D.3)
γ j i = γ i j exp [ β ( ε j ε i ) ] .

It is appropriate to introduce the real, symmetric, and therefore Hermitian matrix

(D.4)
Γ ij = γ ij exp [ β ( ε j ε i ) / 2 ] = Γ ij .
Let v i be an eigenvectorof Θ with the eigenvalue λ:
(D.5)
j i γ j i v j v i i γ i = λ v i .
Substituting (D.4) one obtains
(D.6)
j i Γ j i u j α i u i = λ u i
where
(D.7)
u i = v i exp [ βε i / 2 ]
and
(D.8)
α i = i γ i = i γ i exp [ β ( ε i ε ) / 2 ] .

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. (p.330) To show that these eigenvalues are non+positive, it is sufficient to show that the quadratic form

(D.9)
Φ = Σ ij Γ ji x i x j Σ i α i x i 2
cannot be positive. Indeed, it is easily shown that (D.9) reads
(D.10)
Φ = ( 1 / 2 ) Σ ij Γ ji { x i exp [ β ( ε i ε j ) / 4 ] x j exp [ ( ε j ε i ) / 4 ] } . 2
The only non-vanishing contributions are those of coefficients Γij with ij 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.