DOCSLIB.ORG

RN Random Numbers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
RN Random Numbers Random Numb ers RN Copyright C June Computational Science Education Pro ject Remarks Keywords Random pseudorandom linear congruential lagged Fib onacci List of prerequisites some exp osure to sequences and series some abilitytowork in base arithmetic helpful go o d background in FORTRAN or some other pro cedural language List of computational metho ds pseudorandom numb ers random numb ers linear congruential generators LCGs lagged Fib onacci generators LFGs List of architecturescomputers any computer with a FORTRAN compiler and bitwise logical functions List of co des supplied ranlcfor linear congruential generator ranlffor lagged Fib onacci generator randomfor p ortable linear congruential generator after Park and Miller getseedfor to generate initial seeds from the time and date implemented for Unix and IBM PCs Scop e lectures dep ending up on depth of coverage Notation Key a integer multiplier a m c integer constant c m D length of needle in Buons needle exp eriment F fraction of trials needle falls within ruled grid in Buons needle exp eriment k lag in lagged Fib onacci generator k lag in lagged Fib onacci generator k M numb er of binary bits in m m integer mo dulus m N numb er of trials N numb er of parallel pro cessors p P p erio d of generator p a prime integer R random real numb er R n n S spacing b etween grid in Buons needle exp eriment th X n random integer n Subscripts and Sup erscripts denotes initial value or seed th j denotes j value th denotes value th k denotes k value th n denotes n value Monte Carlo Algorithms and Random Numbers In this chapter we discuss the application of Monte Carlo metho ds to scientic and engi neering problems There are problems which can b e solved only by the Monte Carlo metho d and there also are problems which are b est solved bytheMonte Carlo metho d However the metho d is often referred to as the metho d of last resort as it is apt to consume large computing resources As such it is a metho d ideally suitable for inclusion in a textb o ok on computational science as Monte Carlo programs due to their nature of consuming vast computing resources have historically had to b e executed up on the fastest computers avail able at the time and employ the most advanced algorithms implemented with substantial programming acumen Here weprovide a brief history of the Monte Carlo metho d A go o d early work is that of Buons needle problem Ross where in Buon a French mathematician exp erimentally determined a value of by casting a needle on a ruled grid Many trials were required to obtain go o d accuracy The ob jectiveofthework wasavalidation of statistical theory as opp osed to Monte Carlo simulations typically applied to day where answers to new physical problems are sought Lord Rayleigh Rayleigh even delved into this eld near the turn of the century Courant Fredericks and Levy in showed how the metho d could b e used to solve b oundary value problems Ensuing work Hurd illustrated that Remarks Needle Length, D S S > D Obtain π After “Many” Trials S Figure Illustration of Buons Needle Problem if the entire eld were sought Monte Carlo metho ds are far from comp etitive However if the solution in a restricted region of space is sought sayatapoint then Monte Carlo metho ds may b e used to great advantage as they may b e implemented in restricted regions of space andor time Exercise Buons Needle Problem to illustrate geometrically some of the prop erties of random numb ers a Write a program to determine the value of bynumerically casting a needle of length D on a ruled grid with spacing S as shown in Figure You must rst determine a random p osition relative ruled grid as shown in Figure You must rst determine a random p osition relative to a ruled line then determine a random angle and thirdly see whether the needle falls within the ruled lines You may nd it instructiveto compute approximations for versus the numb er of trials eg N and For SDyour answer for the fraction of time F the needle falls within the grid should b e Kalos D F S b Verify the ab ove equation What is the corresp onding formula for SDWarning there is considerable eort involved in this derivation c Using a needle and a piece of pap er up on whichyou have placed a ruled grid p erform aphysical exp eriment to determine using the ab ove equation Perform rst then and then trials Comment on the accuracy of these results relativetothe accuracy of the results from the numerical exp eriment Discuss reasons for any bias apparent in the results Enrico Fermi in the s used Monte Carlo in the calculation of neutron diusion and later designed the Fermiac a Monte Carlo mechanical device used in the calculation of crit icalityinnuclear reactors In the s a formal foundation for the Monte Carlo metho d was develop ed byvon Neumann who established the mathematical basis for probability density functions PDFs inverse cumulative distribution functions CDFs and pseudo random numb er generators The work was done in collab oration with Stanislaw Ulam who realized the imp ortance of the digital computer in the implementation of the approach The collab oration resulted from work on the Manhattan pro ject where the ENIACwas employed in the calculation of yield Ulam et al Ulam and Metrop olis Eckhard Metrop olis Individuals in the IBM corp oration were pioneers in the eld of random numb er gen eration p erhaps b ecause they were rst engaged in it due to their participation in the Manhattan pro ject where Richard Feynman then directed their computing op erations a fascinating exp osition of their approach to p erforming largescale computing involving a parallel approach exists in Richard Feynmans Surely Youre Joking Mr Feynman It is interesting to note the extremely primitive computing environments in existence at that time and the challenges this presented to researchers see Hurd Uses of Monte Carlo metho ds have b een manyandvaried since that time However due to computer limitations the metho d has not yet fully lived up to its p otential as discussed by Metrop olis Metrop olis Indeed this is reected in the stages the metho d has undergone in the elds of engineering In the late s and s the metho d was tested in a variety of engineering elds Meuller Ehrlich Polgar and Howell Ha jiSheikh and Chandler et al At that time even simple problems were computeb ound The metho d has since b een extended to more complex problems Howell Mo dest Maltby a and b Maltby and Burns and Cro ckett et al Since then attention was fo cused up on muchneeded convergence enhancement pro cedures Kahn and Marshall Emery and Carson Burghart and Stevens Lanore Shamsunder et al Zinsmeister and Sawyer Larsen and Howell Many complex problems remained intractable through the seventies With the advent of highsp eed sup ercomputers the eld has received increased attention particularly with parallel algorithms whichhavemuch higher execution rates In his PhD dissertation Brown intro duced the concept of the event step Brown enabling ecientvectorization of Monte Carlo algorithms where the particles do not interact This approachwas later successfully exploited by several investigators Martin et al Martin et al rep orted sp eedups of a factor of ve on an IBM with vector units Nearly linear sp eedup was rep orted Sequent Computer Systems on a parallel architecture for photon tracing Bobrowicz et al Bobrowicz et al a and b obtained sp eedup factors of from ve to eight in an algorithm where particles are accumulated in queues until ecient vector lengths are obtained allowing physics algorithms such as the Los Alamos b enchmark GAMTEB to b e eectively vectorized Burns et al Suchadvanced co ding techniques Remarks have enabled much bigger problems to b e attacked with improved accuracyHowever there are still a host of problems which remain intractable even with an eectively vectorized algorithm Moreover some imp ediments to eectivevectorization have b een identied and analyzed Zhong and Kalos Zhong and Kalos analyzed the straggler problem where few particles p ersist for manyevent steps inhibiting p erformance due to the overhead incurred where vectors are short Pryor and Burns Pryor and Burns Burns and Pryor rep orted for one Monte Carlo problem where particles interact sp eedups on the order of half of those observed where the particles do not interact alb eit with a vectorized algorithm of greatly increased complexity The same problem has b een attacked with dierentphysics McDonald and Bagano Dagum in a more ecient algorithm Go o d general references on Monte Carlo ab ound Beckenback Hammersly and Handscomb Schreider Kleinjnen Rub enstein Binder Ha ji Sheikh in the literature Most haveadistinctbent usually either statistics or physics Unfortunately there is a dearth involving largescale engineering applications No tutorial on largescale Monte Carlo simulation can b e complete without a discussion of pseudorandom numb er generators Where billions of random numb ers are required it is essential that the generator b e of long p erio d have go o d statistical prop erties and b e vectorizable On bit machines these criteria are usually satised with a multiplicative generator if the constants are carefully chosen Kalos and Whitlo ck Indeed wehave come a long way from the early days of random numb er generators Knuth For example Crays FORTRAN callable generator ranf has a cycle length of and is very ecientvectorized with lowoverhead We summarize this intro duction with our impression of the present
Load more

Recommended publications
  • The Convergence of Markov Chain Monte Carlo Methods 3
    The Convergence of Markov chain Monte Carlo Methods: From the Metropolis method to Hamiltonian Monte Carlo Michael Betancourt From its inception in the 1950s to the modern frontiers of applied statistics, Markov chain Monte Carlo has been one of the most ubiquitous and successful methods in statistical computing. The development of the method in that time has been fueled by not only increasingly difficult problems but also novel techniques adopted from physics. In this article I will review the history of Markov chain Monte Carlo from its inception with the Metropolis method to the contemporary state-of-the-art in Hamiltonian Monte Carlo. Along the way I will focus on the evolving interplay between the statistical and physical perspectives of the method. This particular conceptual emphasis, not to mention the brevity of the article, requires a necessarily incomplete treatment. A complementary, and entertaining, discussion of the method from the statistical perspective is given in Robert and Casella (2011). Similarly, a more thorough but still very readable review of the mathematics behind Markov chain Monte Carlo and its implementations is given in the excellent survey by Neal (1993). I will begin with a discussion of the mathematical relationship between physical and statistical computation before reviewing the historical introduction of Markov chain Monte Carlo and its first implementations. Then I will continue to the subsequent evolution of the method with increasing more sophisticated implementations, ultimately leading to the advent of Hamiltonian Monte Carlo. 1. FROM PHYSICS TO STATISTICS AND BACK AGAIN At the dawn of the twentieth-century, physics became increasingly focused on under- arXiv:1706.01520v2 [stat.ME] 10 Jan 2018 standing the equilibrium behavior of thermodynamic systems, especially ensembles of par- ticles.
    [Show full text]
  • Lecture #8: Monte Carlo Method
    Lecture #8: Monte Carlo method ENGG304: Uncertainty, Reliability and Risk Edoardo Patelli Institute for Risk and Uncertainty E: [email protected] W: www.liv.ac.uk/risk-and-uncertainty T: +44 01517944079 A MEMBER OF THE RUSSELL GROUP Edoardo Patelli University of Liverpool 18 March 2019 1 / 81 Lecture Outline 1 Introduction 2 Monte Carlo method Random Number Generator Sampling Methods Buffon’s experiment Monte Carlo integration Probability of failure 3 Summary 4 Computer based class Some useful slides Assignments Edoardo Patelli University of Liverpool 18 March 2019 2 / 81 Introduction Programme ENGG304 1 28/01/2019 Introduction 2 08/02/2019 Human error 3 11/02/2019 Qualitative risk assessment: Safety analysis 4 18/02/2019 Qualitative risk assessment: Event Tree and Fault Tree 5 25/02/2019 Tutorial I 6 04/03/2019 Model of random phenomena 7 11/03/2019 Structural reliability 8 18/03/2019 Monte Carlo simulation I + Hands-on session 9 25/03/2019 Tutorial II 10 01/04/2019 Monte Carlo simulation II + Tutorial III (Hands-on session) Edoardo Patelli University of Liverpool 18 March 2019 3 / 81 Introduction Summary Lecture #7 Safety Margin Fundamental problem Performance function defined as “Capacity” - “Demand” Safety margin: M = C − D = g(x) 2 For normal and independent random variables: M ∼ N(µM ; σM ) q 2 2 µM = µC − µD σM = σC + σD Reliability index β β = µM /σM represents the number of standard deviations by which the mean value of the safety margin M exceeds zero Edoardo Patelli University of Liverpool 18 March 2019 4 / 81 Introduction
    [Show full text]
  • Computer History a Look Back Contents
    Computer History A look back Contents 1 Computer 1 1.1 Etymology ................................................. 1 1.2 History ................................................... 1 1.2.1 Pre-twentieth century ....................................... 1 1.2.2 First general-purpose computing device ............................. 3 1.2.3 Later analog computers ...................................... 3 1.2.4 Digital computer development .................................. 4 1.2.5 Mobile computers become dominant ............................... 7 1.3 Programs ................................................. 7 1.3.1 Stored program architecture ................................... 8 1.3.2 Machine code ........................................... 8 1.3.3 Programming language ...................................... 9 1.3.4 Fourth Generation Languages ................................... 9 1.3.5 Program design .......................................... 9 1.3.6 Bugs ................................................ 9 1.4 Components ................................................ 10 1.4.1 Control unit ............................................ 10 1.4.2 Central processing unit (CPU) .................................. 11 1.4.3 Arithmetic logic unit (ALU) ................................... 11 1.4.4 Memory .............................................. 11 1.4.5 Input/output (I/O) ......................................... 12 1.4.6 Multitasking ............................................ 12 1.4.7 Multiprocessing .........................................
    [Show full text]
  • Markov Chain Monte Carlo Methods
    Introduction to Machine Learning CMU-10701 Markov Chain Monte Carlo Methods Barnabás Póczos Contents Markov Chain Monte Carlo Methods Sampling • Rejection • Importance • Hastings-Metropolis • Gibbs Markov Chains • Properties Particle Filtering • Condensation 2 Monte Carlo Methods 3 Markov Chain Monte Carlo History A recent survey places the Metropolis algorithm among the 10 algorithms that have had the greatest influence on the development and practice of science and engineering in the 20th century (Beichl&Sullivan, 2000). The Metropolis algorithm is an instance of a large class of sampling algorithms, known as Markov chain Monte Carlo (MCMC) MCMC plays significant role in statistics, econometrics, physics and computing science. There are several high-dimensional problems, such as computing the volume of a convex body in d dimensions and other high-dim integrals, for which MCMC simulation is the only known general approach for providing a solution within a reasonable time. 4 Markov Chain Monte Carlo History While convalescing from an illness in 1946, Stanislaw Ulam was playing solitaire. It, then, occurred to him to try to compute the chances that a particular solitaire laid out with 52 cards would come out successfully (Eckhard, 1987). Exhaustive combinatorial calculations are very difficult. New, more practical approach: laying out several solitaires at random and then observing and counting the number of successful plays. This idea of selecting a statistical sample to approximate a hard combinatorial problem by a much simpler problem is at the heart of modern Monte Carlo simulation. 5 FERMIAC FERMIAC: The beginning of Monte Carlo Methods Developed in the Early 1930’s The Monte Carlo trolley, or FERMIAC, was an analog computer invented by physicist Enrico Fermi to aid in his studies of neutron transport.
    [Show full text]
  • THE BEGINNING of the MONTE CARLO METHOD by N
    THE BEGINNING of the MONTE CARLO METHOD by N. Metropolis he year was 1945. Two earth- Some Background first electronic computer—the ENIAC— shaking events took place: the at the University of Pennsylvania in Phil- successful test at Alamogordo Most of us have grown so blase about adelphia. Their mentors were Physicist and the building of the first elec- computer developments and capabilities First Class John Mauchly and Brilliant tronicT computer. Their combined impact -even some that are spectacular—that Engineer Presper Eckert. Mauchly, fa- was to modify qualitatively the nature of it is difficult to believe or imagine there miliar with Geiger counters in physics global interactions between Russia and was a time when we suffered the noisy, laboratories, had realized that if electronic the West. No less perturbative were the painstakingly slow, electromechanical de- circuits could count, then they could do changes wrought in all of academic re- vices that chomped away on punched arithmetic and hence solve, inter alia, dif- search and in applied science. On a less cards. Their saving grace was that they ference equations—at almost incredible grand scale these events brought about a continued working around the clock, ex- speeds! When he’d seen a seemingly renascence of a mathematical technique cept for maintenance and occasional re- limitless array of women cranking out known to the old guard as statistical sam- pair (such as removing a dust particle firing tables with desk calculators, he’d pling; in its new surroundings and owing from a relay gap). But these machines been inspired to propose to the Ballistics to its nature, there was no denying its new helped enormously with the routine, rela- Research Laboratory at Aberdeen that an name of the Monte Carlo method.
    [Show full text]
  • Benchmarking OLTARIS for Deep Space Dose Analyses Using MCNP6
    Benchmarking OLTARIS For Deep Space Dose Analyses Using MCNP6 By John Daniel Baunach Thesis Submitted to the Faculty of the Graduate School of Vanderbilt University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Physics May, 2015 Nashville, Tennessee Approved: Michael G. Stabin, Ph.D. Todd E. Peterson, Ph.D. Copyright © 2015 by John Daniel Baunach All Rights Reserved ii ACKNOWLEDGEMENTS I much first acknowledge the Nuclear Regulatory Commission, for without their Nuclear Education Grant Program I would not have had the chance to follow this research wherever it led. This opportunity for graduate-level research is something I never expected to have, so on behalf of all students who are supported by these funds -- thank you. An immeasurable debt of gratitude is owed to my advisor, Dr. Michael Stabin. His endless patience in teaching the most stubborn and frustrating of students is an inspiration as a person, as a teacher, and as an aspiring health physicist. The lessons I have learned under his tutelage far exceed any mere memorization of facts, and have taught me to expect a high standard in my own research and professional relationships. For this and much more -- thank you. Much debt is also owed to Dr. Robert Singleterry of NASA's Langley Research Center. I could not ask for a more knowledgeable guide for the content of this thesis, and for impressing on me the true challenges of any potential interplanetary spaceflight. For allowing this student to even be in the room with professionals I have looked up to my entire life -- thank you.
    [Show full text]
  • Stanislaw M. Ulam Papers 1916-1984 Mss.Ms.Coll.54
    Stanislaw M. Ulam Papers 1916-1984 Mss.Ms.Coll.54 American Philosophical Society 2002 105 South Fifth Street Philadelphia, PA, 19106 215-440-3400 [email protected] Stanislaw M. Ulam Papers 1916-1984 Mss.Ms.Coll.54 Table of Contents Summary Information ................................................................................................................................. 3 Background note ......................................................................................................................................... 4 Scope & content ..........................................................................................................................................7 Administrative Information .........................................................................................................................9 Related Materials ........................................................................................................................................ 9 Indexing Terms ........................................................................................................................................... 9 Bibliography ..............................................................................................................................................11 Collection Inventory ..................................................................................................................................12 Series I. Professional correspondence, 1932-1986................................................................................12
    [Show full text]
  • History of Monte Carlo
    The History of the Monte Carlo Methods Hamed Jafari Research Division of Computer Graphics Institute of Visual Computing & Human-Centered Technology TU Wien, Austria The Monte Carlo Integration The History of Monte Carlo Methods 2 1930s: Enrico Fermi Italian Physicist (1901-1954) „Architect of nuclear age“ Creator of world‘s 1st reactor Developed the idea of statistical sampling Left Italy and became a scientist on the Manhattan Project at Los Alamos in New Mexico The History of the Monte Carlo Methods 3 Early 1940s: The idea of the first digital Computer by John Mauchly and Presper Eckert At University of Pennsylvania They Proposed the idea of electronic computer to Ballistic Research Lab in Aberdeen Presper Eckert (center) Lead to “Project PX” The History of the Monte Carlo Methods 4 The ENIAC The first programmable, electronic, general-purpose digital computer Electronic Numerical Integrator and Computer 27 tons, 167 square meters, 18000 vacuum tubes Completed in 1946 – operational from 1947 until 1955 First review tests were firing table problems for Aberdeen vacuum tubes function tables The History of the Monte Carlo Methods 5 John von Neumann Hungarian-American mathematician, computer scientist and physicist. 1903-1957 1940s: Consultant to both Los Alamos and Aberdeen That connection lead to his involvement with ENIAC The History of the Monte Carlo Methods 6 Nicholas Metropolis Greek-American physicist (1915-1999) Was also working at Los Alamos Known for Metropolis-Hasting algorithm and as one of the creators of MANIAC The History
    [Show full text]
  • ENIAC– the First Electronic Computer, University of Pennsylvania
    Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA The Monte Carlo Method and MCNP – A Brief Review of Our 40 Year History Presentation to the International Topical Meeting on Industrial Radiation and Radioisotope Measurement Applications Conference Avneet Sood, PhD XCP-3 Group Leader 10 July 2017 Operated by Los Alamos National Security, LLC for the U.S. Department of Energy's NNSA LA-UR-17-25633 Outline • Origins of the Monte Carlo method – Development of electronic computers and Monte Carlo method occur simultaneously – Ulam, Von Neumann, Richtmeyer, Metropolis, Fermi • Growth and usage of Monte Carlo codes – 1950’s, 1960’s, and 1970’s • Monte Carlo becomes mainstream; nuclear criticality and reactor • Emergence of MCNP • MCNP’s history • MCNP’s upcoming future *nearly all references can be found at: https://laws.lanl.gov/vhosts/mcnp.lanl.gov/references.shtml Los Alamos National Laboratory 7/14/2017 | 3 Abstract The Monte Carlo method for radiation particle transport has its origins at LANL dating back to the 1940’s. The creators of these methods were Drs. Stanislaw Ulam, John von Neumann, Robert Richtmyer, and Nicholas Metropolis. Monte Carlo methods for particle transport have been driving computational developments since the beginning of modern computers; this continues today. In the 1950’s and 1960’s, these new methods were organized into a series of special-purpose Monte Carlo codes, including MCS, MCN, MCP, and MCG. These codes were able to transport neutrons and photons for specialized LANL applications. In 1977, these separate codes were combined to create the first generalized Monte Carlo radiation particle transport code, MCNP.
    [Show full text]
  • Le Prime Macchine Da Calcolo Italiane E I Loro Inventori1
    Le prime macchine da calcolo italiane e i loro inventori1 Silvio Hénin Introduzione Se si tien conto di tutti i tipi di strumenti matematici inventati, fin dalle tavole aritmetiche sumere (ca. 2500 a.C.) e dagli abachi cinesi e greci (V – IV secolo a.C.) si può dedurre che dispositivi per alleviare la noiosa fatica del calcolo numerico apparvero molto presto nella storia delle civiltà [Campbell-Kelly 2003, Needham 1959, Schärlig 2001]. Le tavole delle moltiplicazioni e gli abachi erano più che sufficienti per il limitato fabbisogno di calcolo di mercanti, contabili e amministratori, almeno fino al XVII secolo, quando la rivoluzione scientifica e la crescita delle burocrazie governative richiesero calcoli più complessi e una maggior precisione, tanto da indurre il filosofo e matematico Gottfried Leibniz a esclamare “è indegno che uomini eccellenti debbano sprecare il loro tempo nel calcolo, quando un qualunque contadino potrebbe farlo altrettanto bene con l’aiuto di una macchina” e a inventare un calcolatore meccanico nel 1672, il terzo della storia dopo quelli di Schickard (1623) e di Pascal (1645). Nei successivi due secoli, numerosi scienziati e semplici artigiani si dedicarono all’invenzione e alla costruzione di macchine per il calcolo aritmetico, trasformando lentamente questi congegni da ‘inutili giocattoli […] destinati al fallimento’ [Williams 1990] ‒ seppure ingegnosi ed esteticamente attraenti ‒ in oggetti utili, producibili in massa e venduti in gran numero. Fu l’avvento del capitalismo industriale alla fine del XIX secolo, che causò una ‘crisi del controllo’ [Beninger 1986] che a sua volta generò un mercato ricettivo per macchine calcolatrici veloci e precise, necessarie alla elaborazione di dati numerici e statistici.
    [Show full text]
  • Monte Carlo Methods: Early History and the Basics
    MCMs: Early History and The Basics Monte Carlo Methods: Early History and The Basics Prof. Michael Mascagni Department of Computer Science Department of Mathematics Department of Scientific Computing Florida State University, Tallahassee, FL 32306 USA E-mail: [email protected] or [email protected] URL: http://www.cs.fsu.edu/∼mascagni MCMs: Early History and The Basics Outline of the Talk Early History of Probability Theory and Monte Carlo Methods Early History of Probability Theory The Stars Align at Los Alamos The Problems The People The Technology Monte Carlo Methods The Birth General Concepts of the Monte Carlo Method References MCMs: Early History and The Basics Early History of Probability Theory and Monte Carlo Methods Early History of Probability Theory Early History of Probability Theory I Probability was first used to understand games of chance 1. Antoine Gombaud, chevalier de Méré, a French nobleman called on Blaise Pascal and Pierre de Fermat were called on to resolve a dispute 2. Correspondence between Pascal and Fermat led to Huygens writing a text on “Probability" 3. Jacob Bernoulli, Abraham de Moivre, and Pierre-Simon, marquis de Laplace, led development of modern “Probability" 4. 1812: Laplace, Théorie Analytique des Probabilités MCMs: Early History and The Basics Early History of Probability Theory and Monte Carlo Methods Early History of Probability Theory Early History of Monte Carlo: Before Los Alamos I Buffon Needle Problem: Early Monte Carlo (experimental mathematics) 1. Problem was first stated in 1777 by Georges-Louis Leclerc, comte de Buffon 2. Involves dropping a needle on a lined surface and can be used to estimate 3.
    [Show full text]
  • Computer Meets Theoretical Physics the New Frontier of Molecular Simulation
    THE FRONTIERS COLLECTION Giovanni Battimelli Giovanni Ciccotti Pietro Greco COM PU T ER MEETS THEORETICAL PHYSICS The New Frontier of Molecular Simulation THE FRONTIERS COLLECTION Series Editors Avshalom C. Elitzur, Iyar, Israel Institute of Advanced Research, Rehovot, Israel Zeeya Merali, Foundational Questions Institute, Decatur, GA, USA Thanu Padmanabhan, Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India Maximilian Schlosshauer, Department of Physics, University of Portland, Portland, OR, USA Mark P. Silverman, Department of Physics, Trinity College, Hartford, CT, USA Jack A. Tuszynski, Department of Physics, University of Alberta, Edmonton, AB, Canada Rüdiger Vaas, Redaktion Astronomie, Physik, bild der wissenschaft, Leinfelden-Echterdingen, Germany The books in this collection are devoted to challenging and open problems at the forefront of modern science and scholarship, including related philosophical debates. In contrast to typical research monographs, however, they strive to present their topics in a manner accessible also to scientifically literate non-specialists wishing to gain insight into the deeper implications and fascinating questions involved. Taken as a whole, the series reflects the need for a fundamental and interdisciplinary approach to modern science and research. Furthermore, it is intended to encourage active academics in all fields to ponder over important and perhaps controversial issues beyond their own speciality. Extending from quantum physics and relativity to entropy, consciousness,
    [Show full text]