Polymers in Confined Geometry.pdf

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42 CHAPTER 4. SIMULATION METHODS This means in terms of energy: If the energy is lowered the system evolves to the preferred lower energy state. If the energy is increased, the system moves only with a Boltzmann factor into that less preferred region. But one has to allow for this movement “up the energy-hill” since the system is fluctuating around the equilibrium configuration. In an computer algorithmic form this consists of either accepting the move right away since ∆E is lowered. Or a random number of a uniform deviate has to be drawn which must lie below the transition probability exp(−β∆E). So the new state is: Cr accepted with probability: p Ck+1 = . (4.9) Ck rejected with probability: 1 − p Instead of the usual non-differential acceptance probability, eq. (4.8), other schemes are possible—like the Glauber dynamics—but we will not go into the details of this but stick we the equations derived above. The condition of detailed balance is a sufficient but not necessary condition for the Monte-Carlo scheme to work properly. It can be shown that e.g. sequential updating of the particles—after a particle is moved, in the next trials this particle is skipped, so detailed balance is obviously not satisfied—gives a working MCscheme as well (see [13]). 4.3.3 Correlations and error calculation When using the Metropolis algorithm one has to assure that the simulation is exploring the configuration space. There are several problems which have to be taken into account. Normally two configurations which are only separated by some small number of MC steps are “quite close” together in configuration space (in particular when being close to a steep energy minimum, a large number of MC steps are rejected, so that one has to be careful to sample the fluctuation around the minimum efficiently). When taking a sample configuration too frequently (in the worst case every step), one has an ensemble of samples which are correlated. This asks for careful treatment of error estimation and for a smart way of choosing trial steps. Choosing a trial Monte-Carlo step must be done in the following way: • Make as large steps as possible to explore all configuration space without having too many steps rejected, since large changes in energy ∆E are not very probable to be accepted. • On the other hand the steps should be small enough to have an overall high enough acceptance rate. But choosing the step size to small—which makes the acceptance rate large—will not drive the simulation far through configuration space.
4.3. SAMPLING THE CANONICAL ENSEMBLE 43 Therefore for each simulation the acceptance rate must be adjusted before the sampling is started. Acceptance rates in the range 25–50% are usually giving good results. Some additional points have to considered. When starting a simulation from an arbitrary initial configuration, one has to let the system equilibrate for “some time” (the number of MC-steps needed for equilibration, has to determined independently) before the sampling is started. Otherwise non-equilibrium configurations are sampled for an equilibrium average. Furthermore ergodicity has to be ensured: The trial steps have to be chosen such that the whole configuration space is covered, if the simulation would be running long enough. In this context one has to be particularly sure, that the system does not get trapped in local minima, showing quasi ergodicity. Statistical error Having a set of uncorrelated data points {x1, . . . , xn}—by experiment or simulation—a measure for the error of the average over the samples 〈x〉 = 1 n n xi, (4.10) is calculated through the statistical variance, being the mean-square deviation of the data points from the mean σ(x) 2 = 1 n − 1 i=1 n xi − 〈x〉 2 . (4.11) This actually represents the width of the distribution of data points. Since we expect the average to converge to the exact value xs in the limit n → ∞ the estimated error of the mean is given by i=1 ∆x = σ(x) √ n . (4.12) But eq. (4.11) only holds as long as the samples taken are uncorrelated, since otherwise the error calculated is to small compared to the real one—the samples look ‘similar’ as long as they are correlated. The aim is to have a measure after how many MC-steps samples must be taken to have them uncorrelated or to extract a real error estimate from correlated data. Three possible solutions are presented in the following. Correlation function to make independent samples Up to now there was no real time evolution in Monte-Carlo since steps are generated mechanically in fixed intervals which have a priori nothing to do with the real time evolution
Page 1 and 2: Diploma Thesis Polymers in Confined
Page 3: TMTOWTDI—There’s More Than One
Page 7 and 8: Contents 1 Introduction 1 2 Polymer
Page 9 and 10: Chapter 1 Introduction Polymers are
Page 11 and 12: (a) λ-DNA (b) T2-DNA Figure 1.2: A
Page 13 and 14: Chapter 2 Polymer models This chapt
Page 15 and 16: 2.2. IDEAL PHANTOM CHAIN 7 configur
Page 17 and 18: 2.3. SEMI-FLEXIBLE POLYMERS 9 In th
Page 19 and 20: 2.3. SEMI-FLEXIBLE POLYMERS 11 to i
Page 21 and 22: 2.3. SEMI-FLEXIBLE POLYMERS 13 for
Page 23 and 24: 2.5. REAL CHAINS 15 no overhangs al
Page 25 and 26: 2.5. REAL CHAINS 17 Figure 2.5: Sna
Page 27 and 28: 2.5. REAL CHAINS 19 ln R cases intr
Page 29 and 30: Chapter 3 Polymers in confined geom
Page 31 and 32: 3.2. SCALING RELATIONS 23 using eq.
Page 33 and 34: 3.2. SCALING RELATIONS 25 b a d R |
Page 35 and 36: 3.3. ANALYTICAL RESULTS 27 The mean
Page 37 and 38: 3.3. ANALYTICAL RESULTS 29 From thi
Page 39 and 40: 3.3. ANALYTICAL RESULTS 31 a consta
Page 41 and 42: 3.3. ANALYTICAL RESULTS 33 the proj
Page 43 and 44: 3.3. ANALYTICAL RESULTS 35 x 2 (s/
Page 45 and 46: Chapter 4 Simulation methods This c
Page 47 and 48: 4.3. SAMPLING THE CANONICAL ENSEMBL
Page 49: 4.3. SAMPLING THE CANONICAL ENSEMBL
Page 53 and 54: 4.3. SAMPLING THE CANONICAL ENSEMBL
Page 55 and 56: 4.4. SIMULATION OF UNCONFINED WORM-
Page 57 and 58: 4.4. SIMULATION OF UNCONFINED WORM-
Page 59 and 60: 4.5. SIMULATION OF CONFINED WORM-LI
Page 61 and 62: 4.6. SAMPLED QUANTITIES AND THEIR R
Page 63 and 64: 4.6. SAMPLED QUANTITIES AND THEIR R
Page 65 and 66: Chapter 5 Simulation results In thi
Page 67 and 68: 5.2. THE HARMONIC POTENTIAL 59 ˆt
Page 69 and 70: 5.2. THE HARMONIC POTENTIAL 61 5.2.
Page 71 and 72: 5.2. THE HARMONIC POTENTIAL 63 beha
Page 73 and 74: 5.2. THE HARMONIC POTENTIAL 65 R 2
Page 75 and 76: 5.2. THE HARMONIC POTENTIAL 67 R 2
Page 77 and 78: 5.2. THE HARMONIC POTENTIAL 69 R
Page 79 and 80: 5.2. THE HARMONIC POTENTIAL 71 p(R)
Page 81 and 82: 5.3. THE HARD WALLS 73 〈Ra〉 1 0
Page 83 and 84: Chapter 6 Conclusion Emerging from
Page 85 and 86: Appendix A Discrete worm-like chain
Page 87 and 88: which, after taking all the derivat
Page 89 and 90: Appendix B Calculation for the conf
Page 91 and 92: B.2. END-TO-END DISTANCE CORRELATIO
Page 93 and 94: B.3. PARALLEL END-TO-END DISTANCE C
Page 95 and 96: Glossary L filament length lp persi
Page 97 and 98: Bibliography [1] M. Abramowitz and
Page 99 and 100: BIBLIOGRAPHY 91 [29] P. L’Ecuyer.
Page 101:
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