**Authors:** Akito Takahashi

This paper gives the follow-up study on the basics for our previous papers, Study on
4D/Tetrahedral Symmetric Condensate Condensation Motion by Non-Linear Langevin Equation,
Symposium 998-Low Energy Nuclear Reactions Source Book, ACS, published on August 2008
from Oxford University Press.
Pertaining to the quantum mechanics, the basics of new approach using the stochastic
differential equation (Langevin equation) is written for quantifying dynamic motion of
known molecules as D_{2}^{+}, D_{2} and D_{3}^{+}
as well as D-atom state. The role of the Platonic
symmetry in these known molecules are discussed for deducing simple one-dimensional
(R_{dd} dependent; here R_{dd} is distance between nearest d-d pair) Langevin equation and
making quantum-mechanical ensemble averaging to obtain equation for expectation value.
The methodology is applied for more complicated D-clusters as 4D/TSC and 6D/OSC which
would keep the Platonic symmetry, by introducing the force fluctuation deviating from
the ideal Platonic symmetry. Time-dependent TSC and OSC trapping potentials which take
balance to getting back to the Platonic symmetry from the distorted states were defined
and used for numerical solution of Langevin equation. Finally, time-dependent fusion
rate formula for simultaneous 4D interaction was obtained based on the Fermi's golden
rule and one-pion exchange potential of strong interaction. The 4D fusion is regarded to
cause radiation-less excess heat and ^{4}He ash in metal-deuterium systems under dynamic conditions

**Comments:** recovered from sciprint.org

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[v1] 14 Feb 2009

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