Complex-Dynamic Fractality and Its Applications in Life Sciences
contain not only the ordinary “expectation” value for a large series of events, but remain
also valid for any single event observation and even before it, providing a priori
probability and its universal dynamic origin. A practically useful probability definition
is given also by the generalised Born rule [9,11,12], derived by dynamic matching and
presenting the wavefunction in a physically transparent form of probability distribution
density (or its amplitude, for the “wave-like” levels of complexity):
αr = ψ (Xr )∣ 2 , (15)
where Xr is the r-th realisation configuration, while the wavefunction can be found
from the universal, causally derived Schrodinger equation [9,11,12].
Dynamic complexity, C, can be universally defined now as any growing function
of system realisation number, or rate of their change, equal to zero for only one system
realisation: C = C (Nw), dC∕dN^ > 0, C (1) = 0. It is the latter case of zero unreduced
complexity that is invariably considered in the canonical, dynamically single-valued, or
unitary, theory, which explains all its old and new difficulties at various levels of world
dynamics [9-13]. The unreduced dynamic complexity is presented by the majority of
actually measured quantities, such as energy, mass, momentum, action, and entropy,
now provided with a universal and essentially nonlinear interpretation in terms of the
underlying interaction processes. Space and time are two universal, physically real
forms of complexity, causally derived as tangible quality of dynamically entangled
structure and immaterial rate (frequency) of realisation change events, respectively.
Complex dynamics is a structure emergence process (dynamically multivalued self-
organisation) and can be described by the universal Hamilton-Jacobi equation for the
generalised action, which is dualistically related to the universal Schrodinger equation
mentioned above through the causal quantization condition (it reflects realisation
change by transition through the intermediate realisation of the wavefunction) [9,11,12].
Note finally that dynamic complexity thus defined represents at the same time universal
measure of genuine and omnipresent chaoticity and (generalised) entropy.
The complex-dynamic, intrinsically probabilistic fractality represents the
inevitable development and internal content of dynamic entanglement (nonseparability),
complexity and chaoticity. It is related to problem nonintegrability as it appears in EP
dependence on the unknown solutions of the auxiliary system of equations, eqs. (10).
After we have revealed dynamic system splitting into chaotically changing realisations
at the first level of nonperturbative dynamics, we should now proceed with further
analysis of the auxiliary system solutions, which introduce additional structure in the
general solution. Due to the unrestricted universality of the generalised EP method, it
can be applied to the truncated system (10), transforming it into a single effective
equation, quite similar to the first-level EP result of eq. (8):
h о (ξ) +Veff (ξ; ηn )"∣ ψn (ξ) = ηnψn (ξ), (16)
where the second-level EP action is analogous to the combined version of eqs. (9):
Vnn' (ξψnn (ξ) ∫ dξψ0ni * (ξ')Vn' n (ξ'ψn (ξ')
Vnff (ξ',ηn ψrn (ξ) = Vnn (ξψn (ξ)+
Ωξ
ηn - ηn0i + εn0 - εn'0
n '≠ n, i
(17)