404
E. J. Brändas
Dividing (3.8) by t, taking the limit t → 0, one obtains the gain-loss equation
known as the Pauli master equation, PME.
25 The limit is taken with the understanding
that we are dealing with an ensemble of systems approaching equilibrium, i.e. with
a higher entropy than the initial system, i.e. S = k B ln r , where S is the change
in entropy, k B is the Boltzmann constant and r the dimension of the state space.
26
Defining the transition rates
W mn = lim
t→0
S n→m (t)
t
one obtains from (3.8) the PME
˙
P m (t) =
n
W mn P n (t) −
n
W nm P m (t)
(3.10)
and similarly, from (3.7) and (3.9) one gets Eq. (3.2), which as we have repeatedly
stated, holds without the use of the condition W mn = W nm .
The somewhat confusing terminology arises from fundamental classical-quantum
differences. Microscopic reversibility characterizes classical trajectories and their
time reversed paths, while time invariant quantum mechanical reversibility entails
n W mn =
n W nm . A more detailed analysis will be given in succeeding sections,
based on rigorous sub-dynamical master equations [49–51] and their extensions.
Rounding off, the equilibrium condition, ˙
P m (t) = 0, as well as the detailed balance
condition
W mn P n (t) = W nm P m (t)
amounts, if P n (t) = P m (t) = P, to W mn = W nm , which again relates back to
the issue of microscopic reversibility. One might additionally consider the present
question with the concerns related to the proof of Boltzmann’s H-theorem [52]. In
what follows we will return to the key issue [30], i.e. how does time symmetric
physics lead to time-asymmetric chemistry, conservative reversible dynamics lead to
dissipative irreversible evolution and unitary time evolution becomes a contractive
semigroup?
25 The limit is usually interpreted as the course-grained limit with the time being smaller than a
process-defined relaxation time, τ rel , but still large enough to randomize the phases. The Prigogine
sub-dynamics does not contain this assumption.
26 We have naively assumed a finite number of reference states, which in a more refined treatment
must be coupled to a continuum, which technically can be formulated by Löwdin’s partitioning
technique [49] or the Nakajima–Zwanzig equation [50, 51].
E. J. Brändas
Dividing (3.8) by t, taking the limit t → 0, one obtains the gain-loss equation
known as the Pauli master equation, PME.
25 The limit is taken with the understanding
that we are dealing with an ensemble of systems approaching equilibrium, i.e. with
a higher entropy than the initial system, i.e. S = k B ln r , where S is the change
in entropy, k B is the Boltzmann constant and r the dimension of the state space.
26
Defining the transition rates
W mn = lim
t→0
S n→m (t)
t
one obtains from (3.8) the PME
˙
P m (t) =
n
W mn P n (t) −
n
W nm P m (t)
(3.10)
and similarly, from (3.7) and (3.9) one gets Eq. (3.2), which as we have repeatedly
stated, holds without the use of the condition W mn = W nm .
The somewhat confusing terminology arises from fundamental classical-quantum
differences. Microscopic reversibility characterizes classical trajectories and their
time reversed paths, while time invariant quantum mechanical reversibility entails
n W mn =
n W nm . A more detailed analysis will be given in succeeding sections,
based on rigorous sub-dynamical master equations [49–51] and their extensions.
Rounding off, the equilibrium condition, ˙
P m (t) = 0, as well as the detailed balance
condition
W mn P n (t) = W nm P m (t)
amounts, if P n (t) = P m (t) = P, to W mn = W nm , which again relates back to
the issue of microscopic reversibility. One might additionally consider the present
question with the concerns related to the proof of Boltzmann’s H-theorem [52]. In
what follows we will return to the key issue [30], i.e. how does time symmetric
physics lead to time-asymmetric chemistry, conservative reversible dynamics lead to
dissipative irreversible evolution and unitary time evolution becomes a contractive
semigroup?
25 The limit is usually interpreted as the course-grained limit with the time being smaller than a
process-defined relaxation time, τ rel , but still large enough to randomize the phases. The Prigogine
sub-dynamics does not contain this assumption.
26 We have naively assumed a finite number of reference states, which in a more refined treatment
must be coupled to a continuum, which technically can be formulated by Löwdin’s partitioning
technique [49] or the Nakajima–Zwanzig equation [50, 51].