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On Free Energy Calculations Using Fluctuation Theorems of Work 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 2010 Summary 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 high complexity, in specific calculations of free energy differences 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 1 also 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, as we have shown, by a slightly different ansatz, which reveals the acceptance ratio 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 explains analytically why the acceptance ratio method is generically superior to free energy 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. 2 The 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 simulation and by an analytical approach. In the analytic case, the map was parametrized 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 that a running calculation obeys the property 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 the sufficient observation of rare events. Our analytical and numerical studies validated its reliability as a measure of convergence. 3 Zusammenfassung Die Bestimmung freier Energien analytisch unzug¨anglicher Systeme ist ein 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 zur¨uckgef¨uhrt. Obgleich urpsr¨unglich zur st¨orungstheoretischen Berech- nung thermodynamischer Eigenschaften entwickelt, entfaltete sich die volle Reichweite ihrer Methoden erst in Verbindung mit der Leistungsf¨ahigkeit moderner Computer und Monte-Carlo Simulationstechniken. In dieser Vereinigung erlauben sie die effektive, nicht- st¨orungstheoretische Behandlung komplexer Modellsysteme, insbesondere die Berechnung von Differenzen der freien Energie. Die in j¨ungster Zeit begr¨undeten Fluktuationstheoreme der Arbeit im Nichtgleich- gewicht, entdeckt von Christopher Jarzynski und Gavin Crooks in den sp¨aten 90ern, hatten eine Wiederbelebung traditioneller Methoden zur Berechnung der freien Energie in einer recht unerwarteten Form zur Folge. Urspr¨unglich auf Computersimulationen mikroskopischer Verteilungen gest¨utzt, beruhen sie in ihrem neuen Gewand auf Messun- gen der Arbeit in Nichtgleichgewichtsprozessen. Die Fluktuationstheoreme der Arbeit, d.h. die Jarzynski Gleichung und das Crooks’sche Fluktuationstheorem, bedeuteten einen paradigmatischen Wechsel in Bezug auf Theorie, Experiment und Simulation, indem sie die Bestimmung von Gleichgewichtseigenschaften aus Nichtgleichgewichtstrajektorien er- lauben. Die vorliegende Dissertation hat drei Schwerpunkte: Zum ersten, die Charakterisierung derjenigen elementaren Methoden zur Berechnung von freien Energien, die auf dem Fluk- tuationstheorem gr¨unden. Zum zweiten, die analytische Datentransformation mit dem Ziel, die G¨ute der Methoden zu verbessern; und drittens, die Entwicklung von Kriterien, die einen R¨uckschluß auf die Qualit¨at der Berechnungen erlauben. Berechnung bedeutet hier genauer statistische Sch¨atzung, denn die den Rechnungen zugrundeliegenden Daten gehorchen statistischen Verteilungen. Die wesentliche Arbeit des gegenw¨artigen Autors l¨asst sich wie folgt zusammenfassen. Inspiriert
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