Abstract
We consider a system consisting of a planar random walk on a square lattice, subjected to stochastic elementary local deformations. Both numerical and theoretical results are reported. Depending on the deformation transition rates, and specifically on a parameter η which breaks the symmetry between the left and right orientation, the winding distribution of the walk is modified, and the system can be in three different phases: folded, stretched and glassy. An explicit mapping is found, leading to consider the system as a coupling of two exclusion processes: particles of the first one move in a landscape defined by particles of the second one, and vice-versa. This can be viewed as an inhomogeneous exclusion process. For all closed or periodic initial sample paths, a convenient scaling permits to show a convergence in law (or almost surely on a modified probability space) to a continuous curve, the equation of which is given by a system of two non linear stochastic differential equations. The deterministic part of this system is explicitly analyzed via elliptic functions. In a similar way, by using a formal fluid limit approach, the dynamics of the system is shown to be equivalent to a system of two coupled Burgers equations.
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REFERENCES
C. L. Berthelsen, J. Glazier, and M. Skolnick, Global fractal dimension of human DNA sequences treated as pseudorandom walks, Phys. Rev. A 45:8902–8913 (1992).
O. Bénichoux and J. Desbois, Windings of the 2d free rouse chain, J. Phys. A. Math. Gen. 33:6655–6667 (2000).
J. Burgers, A mathematical model illustrating the theory of turbulences, Adv. Appl. Mech. 1:171–199 (1948).
S. Cebrat and M. Dudek, The effect of DNA phase structure on DNA walks, Eur. J. Phys. 3:271–276 (1998).
M. Clincy, B. Derrida, and M. Evans, Phase Transition in the ABC Model, cond/mat 0209674 (2002).
A. Comtet, J. Desbois, and S. Ouvry, Winding of planar brownian curves, J. Phys. A: Math. Gen. 23:3563–3572 (1990).
B. Derrida, M. Evans, V. Hakim, and V. Pasquier, Exact solution for 1d asymmetric exclusion model using a matrix formulation, J. Phys. A: Math. Gen. 26:1493–1517 (1993).
M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1986).
R. Durrett, Ten lectures on particle system, in Lecture Notes in Maths., Vol. 1608 (Springer, 1995), pp. 97–201.
M. Evans, D. P. Foster, C. Godrèche, and D. Mukamel, Spontaneous symmetry breaking in a one dimensional driven diffusive system, Phys. Rev. Lett. 74:208–211 (1995).
P. D. Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, NY, 1979).
O. Kallenberg, Foundations of Modern Probability, 2nd Ed. (Springer, 2001).
M. Kardar, G. Parisi, and Y. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56:889–892 (1986).
J. Komlos, G. Major, and P. Tusnady, An approximation of partial sums of independent rv's and the sample df i,ii., Z. Warsch. verw. Gebiete 32:111–131 (1975, 1976).
B. Øksendal, Stochastic Differential Equations (Springer, 1985).
R. Lahiri, M. Barma, and S. Ramaswamy, Strong phase separation in a model of sedimenting lattices, Phys. Rev. E 61:1648–1658 (2000).
T. M. Liggett, Interacting Particle Systems, Grundlehren der mathematischen Wissenschaften (Springer, 1985).
P. Lévy, Processus stochastiques et mouvement Brownien (Gauthier-Villars, 1948).
J. W. Pitman and M. Yor, Further asymptotic laws of planar brownian motion, Ann. Probab. 17:965–1011 (1989).
P. Rouse, A theory of linear viscoelastic properties of dilute solutions of coiling polymers, J. Chem. Phys. 21:1272–1280 (1953).
F. Spitzer, Some theorems concerning 2-dimensional brownian motion, Trans. Amer. Math. Soc. 87:187–197 (1958).
H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, 1991).
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Fayolle, G., Furtlehner, C. Dynamical Windings of Random Walks and Exclusion Models. Part I: Thermodynamic Limit in ℤ2 . Journal of Statistical Physics 114, 229–260 (2004). https://doi.org/10.1023/B:JOSS.0000003111.88829.9d
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DOI: https://doi.org/10.1023/B:JOSS.0000003111.88829.9d