On free energy calculations using fluctuation theorems of work [Elektronische Ressource] / Aljoscha M. Hahn. Betreuer: Andreas Engel
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Publié par
Publié le
01 janvier 2011
Nombre de lectures
46
Langue
English
Poids de l'ouvrage
3 Mo
Publié par
Publié le
01 janvier 2011
Nombre de lectures
46
Langue
English
Poids de l'ouvrage
3 Mo
On Free Energy Calculations
using Fluctuation Theorems of Work
Doctoral Thesis by Aljoscha M. Hahn
Von der Fakultät für Mathematik und Naturwissenschaften der
CARL VON OSSIETZKY UNIVERSITÄT OLDENBURG
zur Erlangung des Grades und Titels eines Doctor rerum naturalium (Dr. rer. nat.)
angenommene Dissertation von Herrn Aljoscha Maria Hahn,
geboren am 20. Februar 1977 in Hagios Antonios (Kreta).Gutachter: Prof. Dr. Andreas Engel
Zweitgutachter: Prof. Dr. Holger Stark
Tag der Disputation: 15. Oktober 2010Summary
Free energy determination of thermodynamic systems which are analytical intractable is
an intensively studied problem since at least 80 years. The basic methods are commonly
traced back to the works of John Kirkwood in the 1930s and Robert Zwanzig in the
1950s, who developed the widely known thermodynamic integration and thermodynamic
perturbation theory. Originally aiming analytic calculations of thermodynamic proper-
ties with perturbative methods, the full power of their methods was only revealed in
conjunction with modern computer capabilities and Monte Carlo simulation techniques.
In this alliance they allow for effective, nonperturbative treatment of model systems with
highcomplexity, in specificcalculationsoffreeenergydifferences between thermodynamic
states.
The recently established nonequilibrium work theorems, found by Christopher Jarzyn-
ski and Gavin Crooks in the late 1990s, revived traditional free energy methods in a
quite unexpected form. Whilst formerly relying on computer simulations of microscopic
distributions, in their new robe they are based on measurements of work of nonequi-
librium processes. The nonequilibrium work theorems, i.e. the Jarzynski Equation and
the Crooks Fluctuation Theorem meant a paradigmatic change with respect to theory,
experiment, and simulation in admitting the extraction of equilibrium information from
nonequilibrium trajectories.
The focus of the present thesis lies on three directions: first, on understanding and
characterizing elementary methods for free energy calculations originating from the fluc-
tuation theorem. Second, on analytic transformation of data in order to enhance the
performance of the methods, and third, on the development of criteria which allow for
judging the quality of free energy calculations. Calculation hereby actually means sta-
tistical estimation with data sampled or measured from random distributions. The main
work of the present author is summarized as follows.
Inspired by the work of Charles Bennett on his acceptance ratio method for free en-
ergy calculations and its recent revival in the context of Crooks’ Fluctuation Theorem,
we studied this method in great detail to understand its overall observed superiority over
related methods. The acceptance ratio method utilizes measurements of work in both
directions of a process, and it was finally observed by Shirts and co-workers that it can
1also be understood as a maximum likelihood estimator for a given amount of data, which
greatly explains its exquisite properties from a totally different point of view than that of
Bennett. Yet, a drawback of the maximum likelihood approach to the acceptance ratio
method is the implicit switch to another process of data gathering via Bayes’ Theorem,
which no longer reflects the actual process of measurement. This drawback can be re-
moved, aswehaveshown, byaslightlydifferentansatz, which revealstheacceptanceratio
method to be a constrained maximum likelihood estimator. The great difference between
the two approaches is that the latter permits more efficient estimators, whilst the former
does not. Even more efficient estimators can be provided by some other means, but are
always linked to the specific process and require knowledge on the functional dependence
of the work distributions on the free energy. In contrast, the acceptance ratio method is
always a valid method, and in fact the best method we can use with given measurements
of work when having no further information on the work distributions – which is virtually
always the case.
The performance of the acceptance ratio method depends on the partitioning of the
number of work-measurements with respect to the direction of process. Bennett has
already discussed this question in some detail and derived an equation whose solution
specifies the optimal partitioning of measurements. Albeit, he could not gain relevance
for it and suggested the problem to be untreatable in praxis. We have completed this
issue, in first proving that the mean square error is a convex function of the fraction
of measurements in one direction, which guarantees the existence of a unique optimal
partitioning, and then demonstrating its practical relevance for the purpose of free energy
calculations at maximum efficiency. In addition, the convexity of the mean square error
explainsanalyticallywhytheacceptanceratiomethodisgenericallysuperiortofreeenergy
calculations relying on the Jarzynski Equation.
Building up on Jarzynski’s observation that traditional free energy perturbation can
be markedly improved by inclusion of analytically defined phase space maps, we have
put forward this new and promising direction and derived a fluctuation theorem for a
generalized notion of work, defined with recourse to phase space maps. The generalized
work fluctuation theorem has the same form as Crooks’ Fluctuation Theorem, and can
include it for specific choices of maps. This analogy allowed us to define the acceptance
ratio method also for generalized work.
2The high potential of the mapping methods can also be seen as its drawback: there
is no general receipt for the construction of suitable maps. So the method seems to
depend primarily on the extend of the user’s insight into the problem at hand. However,
we could demonstrate its applicability to the calculation of the chemical potential of
a high-density Lennard-Jones fluid. Thereby we have constructed maps in two ways, by
simulationandbyananalyticalapproach. Intheanalyticcase, themapwasparametrized
and the parameter numerically optimized. The maps in conjunction with the acceptance
ratio method yielded high-accuracy results which outperformed those from traditional
calculations by far, in particular with respect to the speed of convergence.
Convergence is critical to be achieved within statistical calculations for obtaining reli-
able results, but is in general not easy to verify - if possible at all. Because of their strong
dependence on rarely observed events, free energy calculations with the Jarzynski Equa-
tion and the acceptance ratio method suffer from the tendency to seeming convergence.
This means thata running calculation obeys theproperty to settle down on a stable value
over long times – but without having reached the true value of the free energy. Moreover,
seeming convergence is typically accompanied by a small and decreasing sample variance,
which may harden the belief in that the calculation has converged. This is quite problem-
atic, as then there is no reliance on the results of calculations. To resolve this, we have
proposed a measure of convergence for the acceptance ratio method. The convergence
measure relies on a simple-to-implement test of self-consistency of the calculations which
implicitly monitors thesufficient observation of rareevents. Our analytical and numerical
studies validated its reliability as a measure of convergence.
3Zusammenfassung
DieBestimmung freier Energien analytisch unzug¨anglicher Systeme istein seit wenigstens
80 Jahren intensiv studiertes Problem. Die grundlegenden Methoden werden gemein-
hin auf die Arbeiten von John Kirkwood in den 30er Jahren und Robert Zwanzig in
den 50er Jahren zuru¨ckgefu¨hrt. Obgleich urpsru¨nglich zur sto¨rungstheoretischen Berech-
nung thermodynamischer Eigenschaften entwickelt, entfaltete sich die volle Reichweite
ihrer Methoden erst in Verbindung mit der Leistungsfa¨higkeit moderner Computer und
Monte-Carlo Simulationstechniken. In dieser Vereinigung erlauben sie dieeffektive, nicht-
sto¨rungstheoretische Behandlungkomplexer Modellsysteme, insbesondere dieBerechnung
von Differenzen der freien Energie.
Die in ju¨ngster Zeit begru¨ndeten Fluktuationstheoreme der Arbeit im Nichtgleich-
gewicht, entdeckt von Christopher Jarzynski und Gavin Crooks in den spa¨ten 90ern,
hatten eine Wiederbelebung traditioneller Methoden zur Berechnung der freien Energie
in einer recht unerwarteten Form zur Folge. Urspru¨nglich auf Computersimulationen
mikroskopischer Verteilungen gestu¨tzt, beruhen sie in ihrem neuen Gewand auf Messun-
gen der Arbeit in Nichtgleichgewichtsprozessen. Die Fluktuationstheoreme der Arbeit,
d.h. dieJarzynski Gleichung und dasCrooks’sche Fluktuationstheorem, bedeuteten einen
paradigmatischen Wechsel in Bezug auf Theorie, Experiment und Simulation, indem sie
die Bestimmung von Gleichgewichtseigenschaften aus Nichtgleichgewichtstrajektorien er-
lauben.
DievorliegendeDissertationhatdreiSchwerpunkte: Zumersten, dieCharakterisierung
derjenigen elementaren Methoden zur Berechnung von freien Energien, die auf dem Fluk-
tuationstheorem gru¨nden. Zum zweiten, die analytische Datentransformation mit dem
Ziel, die Gu¨te der Methoden zu verbessern; und drittens, die Entwicklung von Kriterien,
die einen Ru¨ckschluß auf die Qualita¨t der Berechnungen erlauben.
