Non-equilibrium Statistical Physics with Application to Disordered Systems by Manuel Osvaldo Cáceres

Author:Manuel Osvaldo Cáceres
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(6.33)

Excursus. Note that the properties (6.31) and (6.32) are not sufficient to ensure that there exists a matrix U such that U −1 ⋅ H ⋅ U is diagonal [something similar happens with the Markov matrix T 1, see Sect. 6.1.1]. Only the principle of detailed balance21 allows to symmetrize H under a certain scalar product.22 This means that in general we can only ensure a similarity transformation to the Jordan form . We propose exercises to clarify this further.

Exercise. Interpret the differences between the Markov matrix T 1 (Markov chains with discrete time) and the matrix H (for continuous-time Markov processes ); in particular the normalizations: and respectively. Hint: assuming a characteristic time scale write and expand the exponential operator.

Excursus. If the matrix H is not reducible (see Appendix B), properties (6.31) and (6.32) are sufficient to ensure that there is a single stationary state , and thus the relaxation to it from any initial condition . That is: .

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