Abstract
This study deals with continuous limits of interacting one-dimensional diffusive systems, arising from stochastic distortions of discrete curves with various kinds of coding representations. These systems are essentially of a reaction-diffusion nature. In the non-reversible case, the invariant measure has in general a non Gibbs form. The corresponding steady-state regime is analyzed in detail, by using a tagged particle together with a state-graph cycle expansion of the probability currents. As a consequence, the constants appearing in Lotka–Volterra equations—which describe the fluid limits of stationary states—can be traced back directly at the discrete level to tagged particle cycles coefficients. Current fluctuations are also studied and the Lagrangian is obtained via an iterative scheme. The related Hamilton–Jacobi equation, which leads to the large deviation functional, is investigated and solved in the reversible case, just for the sake of checking.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. Anantharam and P. Tsoucas, A proof of the markov chain tree theorem. (ISR Technical Report TR 1988–97, SRC, University of Maryland, 1988).
P. Arndt, T. Heinzel and V. Rittenberg, Stochastic models on a ring and quadratic algebras: The three-species diffusion problem. J. Phys. A: Math. Gen. 31:833–843 (1998).
C. Berge, Théorie des Graphes et ses Applications (vol. II of Collection Universitaire des Mathématiques, Dunod, 2ème ed., 1967).
L. Bertini, A. De Sole, D. Gabrielli, G. Jona Lasinio and C. Landim, Current fluctuations in stochastic lattice gases. Phys. Rev. Lett. 94: 030601 (2005).
P. Billingsley, Convergence of Probability Measures, Wiley Series in Probability and Statistics (John Wiley & Sons Inc., second ed., 1999).
J. Burgers, A mathematical model illustrating the theory of turbulences. Adv. Appl. Mech. 1:171–199 (1948).
M. Clincy, B. Derrida and M. R. Evans, Phase transition in the ABC model. Phys. Rev. E 67:6115–6133 (2003).
A. De Masi and E. Presutti, Mathematical Methods for Hydrodynamic Limits (vol. 1501 of Lecture Notes in Mathematics Springer-Verlag, 1991).
B. Derrida, C. Enaud and J. L. Lebowitz, The asymmetric exclusion process and brownian excursions. J. Stat. Phys. 115: 365–382 (2004).
B. Derrida, M. R. 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).
B. Derrida and K. Mallick, Exact diffusion constant for the one-dimensional partially asymmetric exclusion model. J. Phys. A: Math. Gen. 30:1031–1046 (1997).
S. Ethier and T. Kurtz, Markov Processes, Characterization and Convergence (John Wiley & Sons, 1986).
M. R. Evans, D. P. Foster, G. Godrèche and D. Mukamel, Spontaneous symmetry breaking in a one dimensional driven diffusive system. Phys. Rev. Lett. 74:208–211 (1995).
M. R. Evans, Y. Kafri, M. Koduvely and D. Mukamel, Phase separation and coarsening in one-dimensional driven diffusive systems. Phys. Rev. E 58:2764 (1998).
G. Fayolle and C. Furtlehner, Dynamical windings of random walks and exclusion models. Part I: Thermodynamic limit in Z2. J. Statist. Phys. 114:229–260 (2004).
G. Fayolle and C. Furtlehner, Stochastic deformations of sample paths of random walks and exclusion models (in Mathematics and computer science. III, Trends Math., Birkhäuser, Basel, 2004), pp. 415–428.
G. Fayolle and C. Furtlehner, Stochastic dynamics of discrete curves and exclusion processes. part 1: Hydrodynamic limit of the ASEP system (Rapport de Recherche 5793, Inria, 2005).
P. Ferrari and J. Martin, Stationary distribution of multi-type totally asymmetric exclusion processes. math.PR/0501291.
C. Godrèche and J. Luck, Nonequilibrium dynamics of urns models. J. Phys. Cond. Matter 14:1601 (2002).
O. Kallenberg, Foundations of Modern Probability (Springer, second edition ed., 2001).
M. Kardar, G. Parisi and Y. Zhang, Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56:889–892 (1986).
F. P. Kelly, Reversibility and Stochastic Networks (John Wiley & Sons Ltd., 1979. Wiley Series in Probability and Mathematical Statistics).
C. Kipnis and C. Landim, Scaling Limits of Interacting Particles Systems (Springer-Verlag, 1999).
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, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes (vol. 324 of Grundlehren der mathematischen Wissenschaften Springer, 1999).
K. Mallick, Shocks in the asymmetry exclusion model with an impurity. J. Phys. A: Math. Gen. 29:5375–5386 (1996).
K. Mallick, S. Mallick and N. Rajewsky, Exact solution of an exclusion process with three classes of particles and vacancies. J. Phys. A: Math. Gen. 32:8399–8410 (1999).
J. Murray, Mathematical Biology, vol. 19 of Biomathematics (Springer-Verlag, second ed., 1993).
W. Rudin, Functional Analysis, International Series in Pure and Applied Mathematics (McGrawHill, second ed., 1991).
H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, 1991).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fayolle, G., Furtlehner, C. Stochastic Dynamics of Discrete Curves and Multi-Type Exclusion Processes. J Stat Phys 127, 1049–1094 (2007). https://doi.org/10.1007/s10955-007-9286-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-007-9286-0