Carlo Danieli (Dipartimento di Matematica, Universita' degli Studi di Padova): Energy localization in DNA models

My thesis concerns a particular model of a disordered chain of particles originating from
a class of DNA chain models (A. Ponno, draft). The model equations of motion are
˙
ιψm = σm ψm + ε ψm+1 + ψm−1 + |ψm |2 ψm
(1)
where the σm are random coefficients taking values ±1 according to some probability
distribution P, ε is a small parameter and periodic boundary conditions are assumed
(m ∈ ZL ).
Equations (1) are the Hamilton equations associated to the Hamiltonian H = h + εP
where
h=
σm |ψm |2
m∈ZL
and
P =
m∈ZL
1
∗
ψm (ψm+1 + ψm−1 ) + |ψm |4
2
This model comes from the classical Peyard-Bishop model of DNA, and it is obtained
taking into account the inhomogenety due to the distribution of different base pairs
along the molecule. Under certain hypotesys, and making use of standard techniques of
Hamiltonian perturbation theory, we show how such model reduce to a weakly disper-
sive, weakly non linear, discrete Schroedinger equation with ”spin disorder”.

 

We have studied the dynamics of the model above from the point of view of Hamil-
tonian perturbation theory, writing the Hamiltonian H of the model in normal form
with respect to h up to order n. We show that at each order n the ”truncated” (neglect-
ing the remainder Rn+1 ) normal form Hamiltonian
n
H
(n)
εk P k
=h+
k=1
consists in the sum of independent non-interacting Hamiltonians {HDn }i∈In defined each
i
i
on certain disjoint domains Dn := {Dn }i∈In whose complexity increases with n. These
domains are unions of blocks of nearby particles having the same value of the spin σm ,
and are such that at order n two blocks belong to the same domain if they are separated
by at most n − 1 consecutive spins of opposite sign with respect to that of the blocks.
So, each normalized perturbation term Pk describes how and how far the blocks interact
between themselves, and the truncated Hamiltonian H (n) reads
H (n) − Rn+1 =
HD n
i
Dn
To prove the existence of this normal form H (n) of the Hamiltonian H, we have esti-
mated each normal expansion term Pk and the remainder Rn+1 of the expansion series.

We have obtained a Nekhoroshev-like result, determining an optimal truncation order
N ∈ N of the perturbative expansion and an exponential estimate for the time of energy
localization in the domains.
From the explicit expression of the normalized perturbation term Pk , it was possible
to perform a phenomenological analysis of the model. We have extracted a complete
description of the chaotic motion inside each single local domain of the family Dn , ex-
plaining the role of the weak non linearity in the model and understanding that its
presence does not change the optimal truncation order N and exponential time estimate
of the stability of the localization in domain Dn .
The last part we are facing now concerns the characterization of the localization do-
mains in terms of the probability distribution P of the ”spin disorder”, and how its
choise provides existence conditions of the localization phenomena, in particular on the
perturbation coefficient ε.