Abstract
For the symmetric simple exclusion process on an infinite line, we calculate exactly the fluctuations of the integrated current Q t during time t through the origin when, in the initial condition, the sites are occupied with density ρ a on the negative axis and with density ρ b on the positive axis. All the cumulants of Q t grow like \(\sqrt{t}\) . In the range where \(Q_{t}\sim \sqrt{t}\) , the decay exp [−Q 3 t /t] of the distribution of Q t is non-Gaussian. Our results are obtained using the Bethe ansatz and several identities derived recently by Tracy and Widom for exclusion processes on the infinite line.
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Alexander, S., Pincus, P.: Diffusion of labeled particles on one-dimensional chains. Phys. Rev. B 18, 2011–2012 (1978)
Antal, T., Krapivsky, P.L., Rákos, A.: Logarithmic current fluctuations in non-equilibrium quantum spin chains. Phys. Rev. E 78, 061115 (2008)
Appert-Rolland, C., Derrida, B., Lecomte, V., Van Wijland, F.: Universal cumulants of the current in diffusive systems on a ring. Phys. Rev. E 78, 021122 (2008)
Arratia, R.: The motion of a tagged particle in the simple symmetric exclusion system on Z. Ann. Probab. 11, 362–373 (1983)
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory for stationary non equilibrium states. J. Stat. Phys. 107, 635–675 (2002)
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Current fluctuations in stochastic lattice gases. Phys. Rev. Lett. 94, 030601 (2005)
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Non equilibrium current fluctuations in stochastic lattice gases. J. Stat. Phys. 123, 237–276 (2006)
Bodineau, T., Derrida, B.: Current fluctuations in non-equilibrium diffusive systems: an additivity principle. Phys. Rev. Lett. 92, 180601 (2004)
Bodineau, T., Derrida, B.: Distribution of current in nonequilibrium diffusive systems and phase transitions. Phys. Rev. E 72, 066110 (2005)
Borodin, A., Ferrari, P.L., Prahofer, M., Sasamoto, T.: Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129, 1055–1080 (2007)
Crooks, G.E.: Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60, 2721–2726 (1999)
De Gier, J., Essler, F.H.L.: Bethe ansatz solution of the asymmetric exclusion process with open boundaries. Phys. Rev. Lett. 95, 240601 (2005)
De Gier, J., Essler, F.H.L.: Exact spectral gaps of the asymmetric exclusion process with open boundaries. J. Stat. Mech. (2006) P12011
Derrida, B., Douçot, B., Roche, P.E.: Current fluctuations in the one dimensional symmetric exclusion process with open boundaries. J. Stat. Phys. 115, 717–748 (2004)
Derrida, B., Lebowitz, J.L.: Exact large deviation function in the asymmetric exclusion process. Phys. Rev. Lett. 80, 209–213 (1998)
Derrida, B., Lebowitz, J.L., Speer, E.R.: Large deviation of the density profile in the steady state of the open symmetric simple exclusion process. J. Stat. Phys. 107, 599–634 (2002)
Derrida, B.: Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. J. Stat. Mech. (2007) P07023
Derrida, B., Lebowitz, J.L., Speer, E.R.: Entropy of open lattice systems. J. Stat. Phys. 126, 1083–1108 (2007)
Dorlas, T.C., Priezzhev, V.B.: Finite-time current probabilities in the asymmetric exclusion process on a ring. J. Stat. Phys. 129, 787–805 (2007)
Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Probability of 2nd law violations in shearing steady-states. Phys. Rev. Lett. 71, 2401–2404 (1993)
Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970 (1995)
Giardina, C., Kurchan, J., Peliti, L.: Direct evaluation of large-deviation functions. Phys. Rev. Lett. 96, 120603 (2006)
Golinelli, O., Mallick, K.: The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics. J. Phys. A 39, 12679–12705 (2006)
Gwa, L.-H., Spohn, H.: Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation. Phys. Rev. A 46, 844–854 (1992)
Harris, R.J., Rákos, A., Schütz, G.M.: Current fluctuations in the zero-range process with open boundaries. J. Stat. Mech. (2005) P08003
Harris, R.J., Rákos, A., Schütz, G.M.: Breakdown of Gallavotti-Cohen symmetry for stochastic dynamics. Europhys. Lett. 75, 227–233 (2006)
Hurtado, P.I., Garrido, P.L.: Current fluctuations and statistics during a large deviation event in an exactly-solvable transport model. J. Stat. Mech. (2009) P02032
Imamura, T., Sasamoto, T.: Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition. J. Stat. Phys. 128, 799–846 (2007)
Imparato, A., Lecomte, V., van Wijland, F.: Equilibrium-like fluctuations in some boundary-driven open diffusive systems. arXiv:0904.1478
Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997)
Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)
Kardar, M., Parisi, G., Zhang, Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)
Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer, Berlin (1999)
Kipnis, C., Olla, S., Varadhan, S.R.S.: Hydrodynamics and large deviations for simple exclusion processes. Commun. Pure Appl. Math. 42, 115–137 (1989)
Kumar, D.: Diffusion of interacting particles in one dimension. Phys. Rev. E 78, 021133 (2008)
Liggett, T.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Fundamental Principles of Mathematical Sciences, vol. 324. Springer, Berlin (1999)
Povolotsky, A.M., Priezzhev, V.B.: Determinant solution for the totally asymmetric exclusion process with parallel update. J. Stat. Mech. (2006) P07002
Prähofer, M., Spohn, H.: Universal distributions for growth processes in 1+1 dimensions and random matrices. Phys. Rev. Lett. 84, 4882–4885 (2000)
Prähofer, M., Spohn, H.: Current fluctuations for the totally asymmetric simple exclusion process. In: In and Out of Equilibrium: Probability with a Physics Flavor. Progress in Probability, vol. 51, pp 185–204. Bikhäuser, Boston (2002)
Priezzhev, V.B.: Exact nonstationary probabilities in the asymmetric exclusion process on a ring. Phys. Rev. Lett. 91, 050601 (2003)
Prolhac, S., Mallick, K.: Current fluctuations in the exclusion process and Bethe ansatz. J. Phys. A: Math. Theor. 41, 175002 (2008)
Rákos, A., Schütz, G.M.: Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process. J. Stat. Phys. 118, 511–530 (2005)
Richards, P.M.: Theory of one-dimensional hopping conductivity and diffusion. Phys. Rev. B 16, 1393–1409 (1977)
Saada, E.: A limit theorem for the position of a tagged particle in a simple exclusion process. Ann. Probab. 15, 375–381 (1987)
Sasamoto, T.: Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques. J. Stat. Mech. P07007 (2007)
Schütz, G.M.: Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Phys. 88, 427–445 (1997)
Simon, B.: Trace Ideals and Their Applications. Mathematical Surveys and Monographs, vol. 120. AMS, Providence (2005)
Spohn, H.: Long range correlations for stochastic lattice gases in a non-equilibrium steady state. J. Phys. A: Math. Gen. 16, 4275–4291 (1983)
Spohn, H.: Stretched exponential decay in a kinetic Ising model with dynamical constraint. Commun. Math. Phys. 125, 3–12 (1989)
Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)
Tailleur, J., Kurchan, J., Lecomte, V.: Mapping out of equilibrium into equilibrium in one-dimensional transport models. J. Phys. A 41, 505001 (2008)
Tracy, C.A., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279, 815–844 (2008)
Tracy, C.A., Widom, H.: A Fredholm determinant representation in ASEP. J. Stat. Phys. 132, 291–300 (2008)
Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. arXiv:0807.1713 (2008)
Vanbeijeren, H., Kehr, K.W., Kutner, R.: Diffusion in concentrated lattice gases 3. Tracer diffusion on a one-dimensional lattice. Phys. Rev. B 28, 5711–5723 (1983)
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We acknowledge the support of the French Ministry of Education through the ANR BLAN07-2184264 grant.
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Derrida, B., Gerschenfeld, A. Current Fluctuations of the One Dimensional Symmetric Simple Exclusion Process with Step Initial Condition. J Stat Phys 136, 1–15 (2009). https://doi.org/10.1007/s10955-009-9772-7
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DOI: https://doi.org/10.1007/s10955-009-9772-7