dc.contributor.author
Ayaz, Cihan
dc.date.accessioned
2023-07-11T07:15:28Z
dc.date.available
2023-07-11T07:15:28Z
dc.identifier.uri
https://refubium.fu-berlin.de/handle/fub188/39971
dc.identifier.uri
http://dx.doi.org/10.17169/refubium-39693
dc.description.abstract
Starting from the Hamiltonian equations of motion of an arbitrary molecular many-body
system, we first derive non-Markovian models in the form of various generalized Langevin
equations (GLEs) using projection operators. The derived GLEs are integrodifferential
equations for observables that are arbitrary functions of atomistic positions. We construct
the projection operators to include nonlinear potentials and nonlinear memory functions
in the GLEs. The primary motivation to introduce nonlinear GLEs is to move as much
information as possible from the part of the GLE that ends up being modeled by a stochastic
process to the deterministic part of the GLE. In this way, we ensure that one loses less
information through the stochastic modeling of the exact GLE.
Following this chapter, we present numerical methods to determine nonlinear memory
functions from time series data. We demonstrate the numerical extraction method using
a trajectory for the dihedral angle of a butane molecule in water generated by molecular
dynamics simulations. From the trajectory, we calculate all previously derived GLEs using
our method and compare them. For the dihedral angle dynamics of the butane molecule, we
find that a position-dependent mass can lead to nonlinear memory effects. This effect can
be eliminated by adjusting the mass dependence of the potential term.
In the next part, we focus on the so-called approximate GLE, in which nonlinear memory
effects are neglected. We discuss under which assumptions the approximate GLE emerges
from a nonlinear GLE. By analytically computing the Kramers-Moyal coefficients of the
approximate GLE, we show that the Fokker-Planck equation does not describe the dynamics
of a non-Markovian system. We extract the friction kernel of the polypeptide Alanine9
from molecular dynamics simulations to quantify the importance of memory effects in
protein folding. After parameterizing our GLE, we use the Markovian embedding method
to simulate the GLE. Our GLE model very well reproduces the mean first passage times of
both the folding and unfolding dynamics. The Kramers-Moyal coefficients and the mean
square displacement, with pronounced anomalous diffusion, are also very well captured by
the GLE. On the other hand, Markovian models based on Langevin equations with nonlinear
friction cannot reproduce the dynamics in both directions with the same accuracy. From this,
we conclude that consistent modeling of protein folding dynamics must take into account
memory effects.
The last part of the thesis is on the Markovian embedding of nonlinear GLEs. We introduce
three different embedding systems that allow computationally efficient simulations of
nonlinear GLEs. The first embedding system allows the simulation of nonlinear memory
effects for a constant effective mass when the memory function has a nonvanishing component
consisting of a delta function in time. The delta component can is not necessary
for the second embedding system. We derive the second embedding from the nonlinear
Zwanzig model by a perturbation expansion. The third embedding system also allows GLE
simulations in the case that, in addition to a nonlinear memory function, the effective mass
depends on the reaction coordinate. This embedding is not based on an approximation of
the Zwanzig model and, like the first system, assumes a delta component in the memory
function.
en
dc.format.extent
105 Seiten
dc.rights.uri
http://www.fu-berlin.de/sites/refubium/rechtliches/Nutzungsbedingungen
dc.subject
Non-Markovian Processes
en
dc.subject
Molecular Dynamics
en
dc.subject
Projection Operator Methods
en
dc.subject
Generalized Langevin Equation
en
dc.subject.ddc
500 Naturwissenschaften und Mathematik::530 Physik::530 Physik
dc.title
Non-Markovian Modeling of Molecular Many-Body Dynamics
dc.contributor.gender
male
dc.contributor.firstReferee
Netz, Roland R.
dc.contributor.furtherReferee
Noé, Frank
dc.date.accepted
2023-07-05
dc.identifier.urn
urn:nbn:de:kobv:188-refubium-39971-2
refubium.affiliation
Physik
dcterms.accessRights.dnb
free
dcterms.accessRights.openaire
open access