Polymers in Confined Geometry.pdf

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38 CHAPTER 4. SIMULATION METHODS polymer with N ≫ 1 monomers. Since in MD simulation the smallest time-scale of the model must be simulated, it cannot be effectively applied to our problem. We therefore concentrate on the Monte-Carlo simulation in the following sections. For a very good introduction also refer to [35]. 4.2 The Monte-Carlo method In statistical physics ensemble averages of fluctuating quantities are of special interest. To obtain these averages, integrals of the following form have to be evaluated (schematically for one dimension) 〈A〉 p = +∞ −∞ dx A(x)p(x) . (4.1) φ(x) This example could be an average of an observable A(x) (“score function”) weighted by the probability distribution p(x), where the integral corresponds—in the language of statistical mechanics—to a ‘sum’ over all states. But the method explained in the following is applicable to every integral presentable in this form. When eq. (4.1) is not analytically solvable then one has to resort to numerical methods. The direct way would be to divide the integration range—which can be of any dimension—into n equidistant intervals and evaluate the sum over {φ(xi)} on the discrete set of points {xi} (Riemann integral). In this way the error of the approximative result in this deterministic algorithm reduces with n −1/d depending on the dimensionality d. Therefore this method becomes very ineffective for higher dimensions d. But in particular high-dimensional integrals is what we are calculating in statistical mechanical problems, therefore we need a smart way to increase the efficiency of the numerical integration. The basic idea of Monte-Carlo 1 evaluation of an integral is to choose the n discretization points/sample points {xi} randomly out of the probability distribution p(x) such that the sum has to be performed only over the observable quantity at the so generated points {A(xi)}. This algorithm has the advantage of a statistical error proportional to n −1/2 , independent of the dimension which outperforms the deterministic calculation for n ≥ 3. But both algorithms have the problem, that they are very inefficient, when the problem under investigation has a large inherent variance, due to the fact that the computational time grows proportional to n and the variance is inversely proportional to √ n. This problem is seen in the following way: If the probability distribution of the random numbers is such, that the important region of the observable is poorly sampled and therefore the variance is large, a huge number of samples may be needed to get a reliable result. It may happen, that the interesting region is 1 The name ought to bring the casinos of Monte-Carlo to mind.
4.3. SAMPLING THE CANONICAL ENSEMBLE 39 barely hit at all during the numerical evaluation procedure. Therefore so called variance reduction techniques are needed. The most prominent idea is that of importance sampling. By replacing the initial probability distribution p(x) for the random numbers by an importance density q(x)—which should be chosen such that the poorly sampled regions are hit more frequently, otherwise the problem can be made worse—one can reduce the need for a large number of samples significantly! When using the importance density an additional step in the evaluation has to be performed: Since the probability distribution of the random samples has been changed to q(x) the score function has to be adjusted to Aq(x) = A(x)f(x)/q(x). It can be shown that this results in the same value for the desired average of the integral but the variance can be reduced by orders of magnitudes. The nice idea of simple Monte-Carlo integration can reduce the error in high dimensional integrals, but without the importance sampling method still a large number of problems would not be solvable in a reasonable amount of computing time! 4.3 Sampling the canonical ensemble In our problem of sampling averages for polymers we are working in a canonical ensemble. Hence we have to deal with the probability distribution in the canonical ensemble for a given configuration/micro state C P (C) = 1 Z(β) exp −βH(C) , (4.2) where H(C) is the energy of the micro state and β−1 = kBT is the temperature. The normalization constant is the canonical partition sum Z(β) = exp −βH(C) , (4.3) C the sum over all micro states weighted by the Boltzmann weight. In the case of continuous states the sum becomes an integral. Typically one deals with a large number of particles in a system, but already a system consisting of N = 100 particles with two possible states per particle would involve a number of 2 100 configurations which is not directly summable in a lifespan of an average physicist on a typical computer system (now imagine a system of N ≈ 10 23 particles. . . ). Even simulating this system by the Monte-Carlo approach of choosing n micro states by equal probability (out of a flat/uniform probability distribution) is not effective since most micro states have very low probability 2 . Therefore we have the problem described above, where the 2 E.g. the energy landscape of a high dimensional problem is mainly flat, but has some narrow deep minima. In this case it is very unlikely that the minima are hit. But since they are governing the behavior of the system, they should be sampled effectively.
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: Chapter 4 Simulation methods This c
Page 49 and 50: 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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