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Chains of interacting Brownian particles under strain

By Michael J. Allman


We consider the behaviour of a one-dimensional chain of interacting Brownian\ud particles being slowly pulled apart. More precisely, the leftmost particle is fixed,\ud while the rightmost is pulled away at slow speed ε > 0. The interaction between\ud particles is through a pairwise potential U of finite range. If we wait for a long\ud enough time, the distance between a pair of neighbouring particles will exceed the\ud range of U so that these two particles no longer interact. When this happens, we\ud consider the chain broken at this point. Our aim is to investigate how the speed of\ud pulling affects where the chain breaks, in the limit as σ < 0, where σ > 0 is the\ud noise intensity.\ud In Chapter 3, we begin by treating the case that U is cut-off strictly convex.\ud In particular, it does not go smoothly to zero. We find, roughly, that if ε > σ\ud then the chain breaks at the end where it is pulled, while if ε < σ it has an equal\ud probability to break at either end. Then in Chapter 4, we consider the case that U\ud goes smoothly to zero. After approximating the shape of the total energy function,\ud we find, roughly, that the threshold between pulling regimes is given by ε = σ4/3.\ud Our approach is based on a careful analysis of sample path behaviour.\ud Although we mostly consider overdamped dynamics, we also show in Chapter\ud 4 that if the particles have mass εβ with β > 2, then the behaviour of the chain\ud is well-approximated by that in the overdamped case

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