Abstract
In many experimental situations, a physical system undergoes stochastic evolution which may be described via random maps between two compact spaces. In the current work, we study the applicability of large deviations theory to time-averaged quantities which describe such stochastic maps, in particular time-averaged currents and density functionals. We derive the large deviations principle for these quantities, as well as for global topological currents, and formulate variational, thermodynamic relations to establish large deviation properties of the topological currents. We illustrate the theory with a nontrivial example of a Heisenberg spin-chain with a topological driving of the Wess-Zumino type. The Cramér functional of the topological current is found explicitly in the instanton gas regime for the spin-chain model in the weak-noise limit. In the context of the Morse theory, we discuss a general reduction of continuous stochastic models with weak noise to effective Markov chains describing transitions between stable fixed points.
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Abdalla, E., Abdalla, M.C.B., Rothe, K.D.: Non-perturbative Methods in Two-Dimensional Quantum Field Theory. World Scientific, Singapore (1991)
Arous, G.B., Brunaud, M.: Laplace method: variational study of the fluctuations of mean-field type diffusions. Stoch. Stoch. Rep. 31, 79–144 (1990)
Astumian, R.: Adiabatic operation of a molecular machine. Proc. Natl. Acad. Sci. USA 104, 19,715–19,718 (2007)
Astumian, R.: Design principles for Brownian molecular machines: how to swim in molasses and walk in a hurricane. Phys. Chem. Chem. Phys. 9, 5067–5083 (2007)
Berry, M.V.: Quantum phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984)
Bertini, L., Sole, A.D., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Current fluctuations in stochastic lattice gases. Phys. Rev. Lett. 94, 030601 (2005)
Bertini, L., Sole, A.D., 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 nonequilibrium 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)
Bott, R.: Lectures on Morse theory. Old and new. Bull. Am. Math. Soc. (N.S.) 7, 331–358 (1982)
Caroli, B., Caroli, C., Roulet, B., Gouyet, J.F.: A WKB treatment of diffusion in a multidimensional bistable potential. J. Stat. Phys. 22, 515–536 (1980)
Chan, H.B., Dykman, M.I., Stambaugh, C.: Paths of fluctuation induced switching. Phys. Rev. Lett. 100, 130602 (2008)
Chernyak, V., Chertkov, M., Jarzynski, C.: Dynamical generalization of nonequilibrium work relation. Phys. Rev. E 71, 025102 (2005)
Chernyak, V., Chertkov, M., Jarzynski, C.: Path-integral analysis of fluctuation theorems for general Langevin processes. J. Stat. Mech. P08001 (2006)
Chernyak, V., Sinitsyn, N.: Pumping restriction theorem for stochastic networks. Phys. Rev. Lett. 101, 160601 (2008)
Chernyak, V.Y., Chertkov, M., Malinin, S.V., Teodorescu, R.: Non-equilibrium thermodynamics for functionals of current and density. Preprint (2007)
Conley, C.: Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series in Mathematics, vol. 38. American Mathematical Society, Providence (1978)
Conway, J.B.: Functions of One Complex Variable. Springer, New York (1978)
Crooks, G.: Path-ensemble averages in systems driven far from equilibrium. Phys. Rev. E 61, 2361–2366 (2000)
Dawson, D.A., Gärtner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20, 247–308 (1987)
Derrida, B.: Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. J. Stat. Mech., Theory Exp. P07023 (2007)
Derrida, B., Evans, M.R., Hakim, V., Pasquier, V.: Exact solution of a 1d asymmetric exclusion model using a matrix formulation. J. Phys. A, Math. Gen. 26, 1493–1517 (1993)
Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, I. Commun. Pure Appl. Math. 28, 1–47 (1975)
Dykman, M.I., Luchinsky, D.G., McClintock, P.V.E., Smelyanskiy, V.N.: Corrals and critical behavior of the distribution of fluctuational paths. J. Chem. Phys. 77, 5229–5232 (1996)
Dykman, M.I., Mori, E., Ross, J., Hunt, P.M.: Large fluctuations and optimal paths in chemical kinetics. J. Chem. Phys. 100, 5735–5750 (1994)
Ellis, R.: Large deviations for the empirical measure of a Markov chain with an application to the multivariate empirical measure. Ann. Probab. 16, 1496–1508 (1988)
Evans, D.J., Cohen, E.G.D., Morris, G.P.: Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71, 2401–2404 (1993)
Fateev, V.A., Frolov, I.V., Schwarz, A.S.: Quantum fluctuations of instantons in the nonlinear σ model. Nucl. Phys. B 154, 1–20 (1979)
Gallavotti, G., Cohen, E.D.G.: Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931–970 (1995)
Gärtner, J.: On large deviations from the invariant measure. Theory Probab. Appl. 22, 24–39 (1977)
Giardina, C., Kurchan, J., Peliti, L.: Direct evaluation of large-deviation functions. Phys. Rev. Lett. 96, 120603 (2006)
Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley-Interscience, New York (1994)
Harris, R.J., Schütz, G.M.: Fluctuation theorems for stochastic dynamics. J. Stat. Mech., Theory Exp. P07020 (2007)
Hirsch, M.W.: Differential Topology. Graduate Texts in Mathematics, vol. 33. Springer, Berlin (1997)
Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997)
Jülicher, F., Ajdari, A., Prost, J.: Modeling molecular motors. Rev. Mod. Phys. 69, 1269–1281 (2002)
van Kampen, N.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1992)
Kurchan, J.: Fluctuation theorem for stochastic dynamics. J. Phys. A, Math. Gen. 31, 3719–3729 (1998)
Landauer, R., Swanson, J.A.: Frequency factors in the thermally activated process. Phys. Rev. 121, 1668–1674 (1961)
Lebowitz, J.L., Spohn, H.: A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333–365 (1999)
Maes, C., Netocny, K.: Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states. Europhys. Lett. 82, 30003 (2008)
Maes, C., Netocny, K., Wynants, B.: Steady state statistics of driven diffusions. J. Stat. Phys. 387, 2675–2689 (2008)
Maier, R.S., Stein, D.L.: Escape problem for irreversible systems. Phys. Rev. E 48, 931–938 (1993)
Manin, Y.I.: Gauge Field Theory and Complex Geometry. Springer, Berlin (1997)
Chertkov, M., Kolokolov, I., Lebedev, V., Turitsyn, K.: Polymer statistics in a random flow with mean shear. J. Fluid. Mech. 531, 251–260 (2005)
Noji, H., Yasuda, R., Yoshida, M., Kinosita, K.: Direct observation of the rotation of F-1-ATPase. Nature 386, 299–302 (1997)
Novikov, S.P.: The Hamiltonian formalism and a many-valued analogue of Morse theory. Russ. Math. Surv. 37(5), 1–56 (1982)
Onsager, L., Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91, 1505–1512 (1953)
Polyakov, A., Wiegmann, P.B.: Theory of non-Abelian Goldstone bosons in two dimensions. Phys. Lett. B 131, 121–126 (1983)
Polyakov, A., Wiegmann, P.B.: Goldstone fields in two dimensions with multivalued actions. Phys. Lett. B 141, 223–228 (1984)
Pra, P.D., den Hollander, F.: McKean-Vlasov limit for interacting random processes in random media. J. Stat. Phys. 84, 735–772 (1996)
Puglisi, A., Rondoni, L., Vulpiani, A.: Relevance of initial and final conditions for the fluctuation relation in Markov processes. J. Stat. Mech., Theory Exp. P08010 (2006)
Rahav, S., Horowitz, J., Jarzynski, C.: Directed flow in nonadiabatic stochastic pumps. Phys. Rev. Lett. 101, 140602 (2008)
Ralpha, D., Stiles, M.: Spin transfer torques. J. Magn. Magn. Mater. 320, 1190–1216 (2008)
Seifert, U.: Entropy production along a stochastic trajectory and an integral fluctuation theorem. Phys. Rev. Lett. 95, 040602 (2005)
Spanier, E.H.: Algebraic Topology. McGraw-Hill, New York (1995)
Tanase-Nicola, S., Kurchan, J.: Metastable states transitions, basins and borders at finite temperatures. J. Stat. Phys. 116, 1201–1245 (2004)
Tserkovnyak, Y., Brataas, A., Bauer, G.E.W., Halperin, B.I.: Nonlocal magnetization dynamics in ferromagnetic heterostructures. Rev. Mod. Phys. 77, 1375–1421 (2005)
Turitsyn, K., Chertkov, M., Chernyak, V.Y., Puliafito, A.: Statistics of entropy production in linearized stochastic systems. Phys. Rev. Lett. 98, 180603 (2007)
van Zon, R., Cohen, E.G.D.: Extension of the fluctuation theorem. Phys. Rev. Lett. 91, 110601 (2003)
Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17, 661–692 (1982)
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Chernyak, V.Y., Chertkov, M., Malinin, S.V. et al. Non-Equilibrium Thermodynamics and Topology of Currents. J Stat Phys 137, 109–147 (2009). https://doi.org/10.1007/s10955-009-9832-z
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DOI: https://doi.org/10.1007/s10955-009-9832-z