Polymers in Confined Geometry.pdf

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40 CHAPTER 4. SIMULATION METHODS variance will be large, since we barely hit contributing configurations. Introducing the importance sampling method again and choosing random states directly out of the distribution eq. (4.2) would be smart, but than we are trapped in a vicious circle since the partition sum Z, which would be needed for that, was what we had to calculate in first place. But we were not able to do that. The solution to that problem is found by using Markov chains. 4.3.1 Markov chains The method of Markov chains in connection with Monte-Carlo simulations was introduced by N. Metropolis in [34] together with an algorithm, introduced in the next section 4.3.2. The concept is to generate a sequence of successive configuration Ci which sample the configuration space in a representative manner. Each step in this chain only depends on the present configuration. This already is the Markov property. A specific chain of n configurations {C0, . . . , Cn} out of the whole configuration space occurs with the joint probability P (C0, . . . , Cn). Writing this down as a sequence of events gives a product of conditional probabilities gives P (C0, . . . , Cn) = P (Cn|Cn−1, . . . , C0) · P (Cn−1|Cn−2, . . . , C0) · · · P (C2|C1, C0) · P (C1|C0) · P (C0) n = P (C0) P (Ci|Ci−1), (4.4) i=1 where in the last line the Markovian property has already been introduced (cf. [45]). Markovian property means that the next step only depends on the current state of the system, the past has no influence on future steps 3 . For the conditional probabilities this results in a dependence only on the previous configuration P (Ck|Ck−1, . . . , C0) = P (Ck|Ck−1). (4.5) The advantage of this property is that we can introduce (for a time homogeneous Markov chain) a fixed set of transition rates Wij between every two states Cj → Ci of the configuration space irrespective of all the other states. This rates form a transition matrix with n × n elements. The probabilities of the configurations than follow a discrete time master equation, which should have a equilibrium probability distribution {πi = π(Ci)} (where Ci is some configuration of the allowed space) for our static problem. By demanding detailed balance Wijπj = Wjiπi, (4.6) 3 A famous example is the Markovian Scopa player: This player plays his cards only on basis of the last one played. He does not remember what has been played before. . . (Scopa is a famous Italian card game)
4.3. SAMPLING THE CANONICAL ENSEMBLE 41 it is ensured that the system evolves towards equilibrium. So there is a direct balance between every two states. There are other ways to ensure the evolution— e.g. circular balance between all the states—into an equilibrium distribution, but detailed balance is the easiest. The most important point is, that there is nothing like an absorbing state. In the following sections we will only deal with the detailed balance condition. Knowing this detailed balance condition, it is easy to construct the transition probabilities for the canonical ensemble, since we know the equilibrium distribution, eq. (4.2), Wij Wji = πi πj = exp (−β∆E) , (4.7) where ∆E = H(Ci) − H(Cj) is the energy difference between the configurations. The normalizing factor (the partition sum) Z drops out! We only need ratios of equilibrium probabilities which we know to be the Boltzmann weights. This finally renders the desired evaluations possible. The question which remains is how to construct an algorithm which ‘moves’ us through configuration space with this transition probabilities, since storing the transition matrix W as a whole is not possible for large configurations spaces. 4.3.2 Metropolis algorithm The Metropolis algorithm is a recipe how to generate a Markov chain for an initial configuration C0 → C1 → · · · → Ck−1 → Ck → · · · (cf. [34]). If we choose—as will be described in more detail later—a worm-like chain as an example, then the algorithm would consist of three main steps: 1. Choose one of the N discrete chain segments at random. 2. Change this segment into a new one by an arbitrary rotation. Be careful to have the back-rotation possible as well, to ensure detailed balance! 3. Check if this move is accepted or rejected. These three steps are iterated over for as long as the desired accuracy for the value to calculate is achieved. In this context one has to bear in mind, that successive steps may be correlated so that one should wait a specific time before using a configuration for sampling. This problems will be addressed later. The last step is where the above transition probability comes into play. When starting off a configuration Ck a test configuration Ct is generated, for which we can calculate the energy difference ∆E(Ct) = H(Ct) − H(Ck). With this the transition is accepted with the probability—using eq. (4.7): p = min 1, exp(−β∆E) . (4.8)
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: 4.3. SAMPLING THE CANONICAL ENSEMBL
Page 51 and 52: 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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