coherent-dissipative structures or a spatio-temporal breathing edifice, cf. the flagellar movement in cytochemistry or gamma waves in the brain, commensurate
with estimates resulting from the present configurations. Since the non-material
spatio-temporal structure to be presented in more detail below, will remind of the
steps regulated by enzymatic pathways as the response to changes in the cell’s
environment or signals from other cells, the semiotic character of the cell above
motivates the name STEM or Spatio-TEmporal Mnemonic entity.
Thus defining a “cell basis” of the STEM units, Eq. (9.1), we will build a higher
order Liouville super operator structure based on the propagator/generator P. Note
that we can form in analogy with h; g; f of Sects. 6 and 7, the corresponding cell
basis H; G; F subject to the same fundamental transformation (where n maybe
different to the one in (6.4), yet might be related, and x ¼ e
ip=n , see more below)
G
j i ¼ B H
j i
ð9:3Þ
The probability argument, cf. the discussion leading up to Eqs. (6.6–6.9), can
now be augmented, appropriately modified with I ¼
P n
k¼1
F k
j i F k
h j to define the
propagator P, see e.g. Refs. [40, 41] for details
P ¼ x 0 s À i
ð
Þ IþJ
ð 9:4Þ
with x 0 the thermal frequency, s ¼ s rel the average lifetime of the cell and s corr the
short timescale, here essentially equal to s lim of the thermal molecular motion. Note
the fundamental difference in the higher level description, Eq. (9.4) and the thermalization derived on the lower one from Eq. (7.4), with each thermally excited
nuclear oscillation, coupled to the environment (of the other degrees of freedom of
the system), yielding the threshold zero energy and the appropriate width. The
Liouville configuration imparts the thermal frequency x 0 and correlation time
s corr ¼ h=kT from the interaction with the environment consisting of other cells.
Hence the build-up of (9.4) is straightforward, including the appearance of the nonconventional operator J , see e.g. the analogy with Eq. (8.10).
J ¼
X nÀ1
k¼1
F k
j i F kþ1
h
j
ð9:5Þ
The n-dimensional Jordan block J appears in analogy with our previous discussion, resulting in prolonged timescales of multiples of s. Hence with the
dimension n of J in resonance (or equal for simplicity) with the dimension
appearing in Eqs. (6.6–6.9) and (8.5–8.10), important information and associated
communication will be transcended from the molecular- to the cellular level. The
scaling-up from the intra-cell degrees of freedom to the inter-cell ones provides the
verification of J above. Consequently, there must be a resonating relation between
272
E.J. Brändas
with estimates resulting from the present configurations. Since the non-material
spatio-temporal structure to be presented in more detail below, will remind of the
steps regulated by enzymatic pathways as the response to changes in the cell’s
environment or signals from other cells, the semiotic character of the cell above
motivates the name STEM or Spatio-TEmporal Mnemonic entity.
Thus defining a “cell basis” of the STEM units, Eq. (9.1), we will build a higher
order Liouville super operator structure based on the propagator/generator P. Note
that we can form in analogy with h; g; f of Sects. 6 and 7, the corresponding cell
basis H; G; F subject to the same fundamental transformation (where n maybe
different to the one in (6.4), yet might be related, and x ¼ e
ip=n , see more below)
G
j i ¼ B H
j i
ð9:3Þ
The probability argument, cf. the discussion leading up to Eqs. (6.6–6.9), can
now be augmented, appropriately modified with I ¼
P n
k¼1
F k
j i F k
h j to define the
propagator P, see e.g. Refs. [40, 41] for details
P ¼ x 0 s À i
ð
Þ IþJ
ð 9:4Þ
with x 0 the thermal frequency, s ¼ s rel the average lifetime of the cell and s corr the
short timescale, here essentially equal to s lim of the thermal molecular motion. Note
the fundamental difference in the higher level description, Eq. (9.4) and the thermalization derived on the lower one from Eq. (7.4), with each thermally excited
nuclear oscillation, coupled to the environment (of the other degrees of freedom of
the system), yielding the threshold zero energy and the appropriate width. The
Liouville configuration imparts the thermal frequency x 0 and correlation time
s corr ¼ h=kT from the interaction with the environment consisting of other cells.
Hence the build-up of (9.4) is straightforward, including the appearance of the nonconventional operator J , see e.g. the analogy with Eq. (8.10).
J ¼
X nÀ1
k¼1
F k
j i F kþ1
h
j
ð9:5Þ
The n-dimensional Jordan block J appears in analogy with our previous discussion, resulting in prolonged timescales of multiples of s. Hence with the
dimension n of J in resonance (or equal for simplicity) with the dimension
appearing in Eqs. (6.6–6.9) and (8.5–8.10), important information and associated
communication will be transcended from the molecular- to the cellular level. The
scaling-up from the intra-cell degrees of freedom to the inter-cell ones provides the
verification of J above. Consequently, there must be a resonating relation between
272
E.J. Brändas