Abstract
Complex systems such as glasses, gels, granular materials, and systems far from equilibrium exhibit violation of the ergodic hypothesis (EH) and of the fluctuation-dissipation theorem (FDT). Recent investigations in systems with memory [1] have established a hierarchical connection between mixing, the EH and the FDT. They have shown that a failure of the mixing condition (MC) will lead to the subsequent failures of the EH and of the FDT. Another important point is that such violations are not limited to complex systems: simple systems may also display this feature. Results from such systems are analytical and obviously easier to understand than those obtained in complex structures, where a large number of competing phenomena are present. In this work, we review some important requirements for the validity of the FDT and its connection with mixing, the EH and anomalous diffusion in onedimensional systems. We show that when the FDT fails, an out-of-equilibrium system relaxes to an effective temperature different from that of the heat reservoir. This effective temperature is a signature of metastability found in many complex systems such as spin-glasses and granular materials.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
I. V. L. Costa, R. Morgado, M. V. B. T. Lima, and F. A. Oliveira. The Fluctuation-Dissipation Theorem fails for fast superdiffusion. Europhys. Lett., 63:173, 2003.
L. Boltzmann. On the Development of the Methods of Theoretical Physics in Recent Times. In Theoretical Physics and Philosophical Problems: Selected Writings. Kluwer Academic Publishers, 1974.
R. Kubo. Fluctuation-Dissipation theorem. Rep. Prog. Phys., 29:255, 1966.
R. Kubo, M. Toda, and N. Hashitsume. Statistical Physics II. Springer, Berlin, 1991.
L. Bellon, L. Buisson, M. Ciccotti, S. Ciliberto, and F. Douarche. Thermal noise properties of two aging materials, cond-mat/0501324, 2005.
G. Parisi. Off-Equilibrium Fluctuation-Dissipation Relation in Fragile Glasses. Phys. Rev. Lett., 79:3660, 1997.
W. Kauzmann. The nature of the glassy state and the behavior of liquids at low temperatures. Chem. Rev., 43:219, 1948.
I. Santamaría-Holek, D. Reguera, and J. M. Rubí. Diffusion in stationary flow from mesoscopic nonequilibrium thermodynamics. Phys. Rev. E, 63:051106, 2001.
F. Ricci-Tersenghi, D. A. Stariolo, and J. J. Arenzon. Two Time Scales and Violation of the Fluctuation-Dissipation Theorem in a Finite Dimensional Model for Structural Glasses. Phys. Rev. Lett., 84:4473, 2000.
R. Exartier and L. Peliti. Measuring effective temperatures in out-of-equilibrium driven systems. Eur. Phys. J. B, 16:119, 2000.
T. S. Grigera and N. E. Israeloff. Observation of Fluctuation-Dissipation-Theorem Violations in a Structural Glass. Phys. Rev. Lett., 83:5038, 1999.
J. D. Bao. Comment on “The Fluctuation-Dissipation Theorem fails for fast superdiffusion” by IVL Costa et al. Europhys. Lett, 67:1050–1051, 2004.
I. V. L. Costa, R. Morgado, M. V. B. T. Lima, and F. A. Oliveira. Comment on “The Fluctuation-Dissipation Theorem fails for fast superdiffusion” – Reply. Europhys. Lett., 67:1052, 2004.
R. Brown. A brief account of microscopical observations made in the months on june, july, and august, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Phil. Mag., 4:161, 1828.
R. Brown. Ann. Phys. Chem. B, 14:294, 1828.
A. Einstein. Investigation on the theory of the Brownian Movement. Dover, New York, 1956.
W. Sutherland. A dynamical theory of diffusion for non-elecrolytes and the molecular mass of albumin. Phil. Mag., 9:781–785, 1905.
A. Pais. Subtle Is the Lord: The Science and the Life of Albert Einstein. Oxford University Press University Press, Oxford, 1983.
J. C. Dyre and T. B. Schroder. Universality of AC conduction in disordered solids. Rev. Mod. Phys., 72:873, 2000.
F. A. Oliveira, R. Morgado, A. Hansen, and J. M. Rubi. Superdiffusive conduction: Ac conductivity with correlated noise. Physica A, 357:115–121, 2005.
M. von Smoluchowski. Zur kinetischen Theorie der Brownschen Molekularbe-wegung und der Suspensionen. Ann. Phys., 21:756, 1906.
P. Langevin. Sur la theorie du mouvement brownien. Comptes Rendus, 146:530, 1908.
R. Toussaint, G. Helgesen, and E. G. Flekkøy. Dynamic Roughening and Fluctuations of Dipolar Chains. Phys. Rev. Lett., 93:108304, 2004.
F. A. Oliveira and P. L. Taylor. Breaking in polymer-chains. 2. The Lennard-Jones chain. J. Chem. Phys., 101:10118, 1994.
F. A. Oliveira and J. A. Gonzalez. Bond-stability criterion in chain dynamics. Phys. Rev. B, 54:3954, 1996.
F. A. Oliveira. Transition-state analysis for fracture nucleation in polymers: The Lennard-Jones chain. Phys. Rev. B, 57:10576, 1998.
A. M. Maroja, F. A. Oliveira, M. Ciesla, and L. Longa. Polymer fragmentation in extensional flow. Phys. Rev. E, 63:061801, 2001.
C. L. Dias, M. Dube, F. A. Oliveira, and M. Grant. Scaling in force spectroscopy of macromolecules. Phys. Rev. E, 72:011918, 2005.
A. Rahman, K. S. Singwi, and A. Sjölander. Stochastic model of a liquid and cold neutron scattering, ii. Phys. Rev., 126:997, 1962.
R. M. Yulmetyev, A. V. Mokshin, and P. Hänggi. Diffusion time-scale invariance, randomization processes and memory effects in Lennard-Jones liquid. Phys. Rev. E, 68:051201, 2003.
J. M. Sancho, A. M. Lacasta, K. Lindenberg, I. M. Sokolov, and A. H. Romero. Diffusion on a Solid Surface: Anomalous is Normal. Phys. Rev. Lett., 92:250601, 2004.
J. D. Bao and Y. Zhuo. Comment on “Diffusion on a solid surface: Anomalous is normal”. Phys. Rev. Lett., 94:188901, 2005.
J. M. Sancho, A. M. Lacasta, K. Lindenberg, and A. H. Romero. Comment on “Diffusion on a solid surface: Anomalous is normal”– Reply. Phys. Rev. Lett., 94:188902, 2005.
J. D. Bao and Y. Z. Zhuo. Ballistic diffusion induced by a thermal broadband noise. Phys. Rev. Lett., 91:138104, 2003.
J. D. Bao. Transport in a flashing ratchet in the presence of anomalous diffusion. Phys. Lett. A, 314:203, 2003.
M. Ciesla, S. P. Dias, L. Longa, and F. A. Oliveira. Synchronization induced by Langevin dynamics. Phys. Rev. E, 63:065202(R), 2001.
H. B. Callen and T. A. Welton. Irreversibility and generalized noise. Phys. Rev., 83:34, 1951.
V. V. Belyi. Fluctuation-dissipation-dispersion relation and quality factor for slow processes. Phys. Rev. E, 69:017104, 2004.
R. Kubo, M. Yokota, and S. Nakajima. Statistical-Mechanical Theory of Irreversible Processes. 1. General Theory and Simple Applications to Magnetic and Conduction Problems. J. Phys. Soc. Jpn., 12:570–586, 1957.
R. Kubo, M. Yokota, and S. Nakajima. Statistical-Mechanical Theory of Irreversible Processes. 2. Response to Thermal Disturbance. J. Phy. Soc. Jpn., 12:1203–1211, 1957.
P. Hänggi and H. Thomas. Stochastic processes: Time evolution, symmetries and linear response. Phys. Rep, 88:207, 1982.
P. Hänggi and H. Thomas. Time evolution, crelations, and linear response of non-markov processes. Z. Physik B, 26:85, 1977.
P. Hänggi. Stochastic-processes. 2. response theory and fluctuation theorems. Helvetica Physica Acta, 51:202, 1978.
G. N. Bochkov and Yu. E. Kuzovlev. Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics : I. generalized fluctuation-dissipation theorem. Physica A, 106:443–79, 1981.
S. Arrhenius. Über die Reaktionngeschwindigkeit bei des Inversion von Rohrzucker durch Säuren. Zeit. Phys. Chent., 4:226, 1889.
H. A. Kramers. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica, 7:284, 1940.
P. Hänggi and F. Moijtabai. Thermally activated escape rate in presence of long-time memory. Phys. Rev. A, 26:1168, 1982.
R. F. Grote and J. T. Hynes. The stable states picture of chemical-reactions.2. Rate constants for condesed and gas-phase reaction models. J. Chem. Phys., 73:2715, 1980.
E. Pollak, S. C. Tucker, and B. J. Berne. Variational transition state theory for reaction-rates in dissipative systems. Phys. Rev. Lett., 65:1399, 1990.
P. Hänggi, P. Talkner, and M. Borkovec. Reaction-rate theory – 50 years after Kramers. Rev. Mod. Phys., 62:251, 1990.
F. A. Oliveira. Reaction rate theory for non-Markovian systems. Physica A, 257:128, 1998.
J. A. Gonzalez and F. A. Oliveira. Nucleation theory, the escaping processes, and nonlinear stability. Phys. Rev. B, 59:6100, 1999.
H. Mori. Transport, Collective Motion, and Brownian Motion. Prog. Theor. Phys., 33:423, 1965.
R. Zwanzig. Nonequilibrium Statistical Mechanics. Oxford University Press, New York, 2001.
D. J. Evans and G. P. Morris. Statistical Mechanics of Nonequilibrium Liquids. Academic Press, London, 1990.
M. H. Lee. Derivation of the generalized Langevin equation by a method of recurrence relations. J. Math. Phys., 24:2512, 1983.
M. H. Lee. Fick’s Law, Green-Kubo Formula, and Heisenberg’s Equation of Motion. Phys. Rev. Lett., 85:2422, 2000.
M. H. Lee. Ergodic Theory, Infinite Products, and Long Time Behavior in Hermitian Models. Phys. Rev. Lett., 87:250601, 2001.
R. A. Marcus. Theory of oxidation-reduction reactions reactions involving electron transfer. 4. A statistical-mechanical basis for treating contributions from solvent, ligands, and inert salt. Discuss. Faraday Soc., 29:21, 1960.
S. Nakajima. On quantum theory of transport phenomena steady diffusion. Prog. Theor. Phys., 20:948, 1958.
R. Morgado, F. A. Oliveira, G. G. Batrouni, and A. Hansen. Relation between Anomalous and Normal Diffusion in Systems with Memory. Phys. Rev. Lett., 89:100601, 2002.
M. R. Spiegel. Theory and Problems of Laplace Transforms. McGraw-Hill, New York, 1965.
S. Alexander and R. Orbach. Density of states on fractals – fractons. J. Phys. (france) Lett., 43:L625, 1982.
E. Baskin and A. lomin. Superdiffusion on a Comb Structure. Phys. Rev. Lett., 93:120603, 2004.
A. A. Budini and M. Caceres. Functional characterization of generalized Langevin equations. J. Phys. A. Math. Gen., 37:5959–5981, 2004.
J. L. Ferreira, G. O. Ludwig, and A. Montes. Experimental investigations of ion-acoustic double-layers in the electron flow across multidipole magnetic fields. Plasma Phys. Controlled Fusion, 33:297–311, 1991.
S. Frank, P. Poncharal, Z. L. Wang, and W. A. de Heer. Carbon nanotube quantum resistors. Science, 280:1744, 1998.
V. Bellani, E. Diez, R. Hey, L. Toni, L. Tarricone, G. B. Parravicini, F. Domínguez-Adame, and R. Gómez-Alcalá. Experimental Evidence of Delocalized States in Random Dimer Superlattices. Phys. Rev. Lett., 82:2159, 1999.
V. Bellani, E. Diez, A. Parisini, L. Tarricone, R. Hey, G. B. Parravicini, and F. Domínguez-Adame. Experimental evidence of delocalization in correlated disorder superlattices. Physica E, 7:823, 2000.
P. Poncharal, C. Berger, Y. Yi, Z. L. Wang, and W. A. de Heer. Room temperature ballistic conduction in carbon nanotubes. J. Phys. Chem. B, 106:12104, 2002.
A. F. G. Monte, S. W. da Silva, J. M. R. Cruz, P. C. Morals, and A. S. Chaves. Asymmetric carrier transport in InGaAs quantum wells and wires grown on tilted InP substrates. Physica E, 17:169, 2003.
A. Bakk, J. O. Fossum, G. J. da Silva, H. M. Adland, A. Mikkelsen, and A. Elgsaeter. Viscosity and transient electric birefringence study of clay colloidal aggregation. Phys. Rev. E, 65:021407, 2002.
F. A. Oliveira. Dynamical renormalization of anharmonic lattices at the onset of fracture: Analytical results for scaling, noise, and memory. Phys. Rev. B, 52:1009, 1995.
S. R. A. Salinas. Introduction to Statistical Physics. Springer, 2001.
S. Chandrasekhar. Stochastic Problems in Physics and Astronomy. Rev. Mod. Phys., 15:1–89, 1943.
L. Rayleigh. Scientific papers of Lord Rayleigh. Dover, New York, 1964.
R. Loudon. The Quantum Theory of Light. Oxford University Press, New York, 2000.
A. Scalabrin, A. S. Chaves, D. S. Shin, and S. P. S. Porto. Temperature dependence of A1 and E optical phonons in Batio3. Phys. Status Solidi B, 79:731–742, 1977.
D. L. Rousseau, R. P. Bauman, and S. P. S. Porto. Normal mode determination in crystals. J Raman Spectrosc, 10:253–290, 1981.
A. S. Chaves. A fractional diffusion equation to describe Lévy flights. Phys. Lett. A, 239:13, 1998.
R. Metzler and J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 339:1, 2000.
R. Metzler and J. Klafter. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A. Math. Gen., 37:161, 2004.
P. Lévy. Théorie de l’Addition des Variables Aléatoires. Guthier-Villars, Paris, 1954.
F. A. Oliveira, B. A. Mello, and I. M. Xavier. Scaling transformation of random walk distributions in a lattice. Phys. Rev. E, 61:7200, 2000.
F. A. Oliveira, J. A. Cordeiro, A. S. Chaves, B. A. Mello, and I. M. Xavier. Scaling transformation of random walk and generalized statistics. Physica A, 295:201, 2001.
A. Figueiredo, I. Gleria, R. Matsushita, and S. da Silva. On the origins of truncated Lévy flights. Phys. Lett. A, 315:51, 2003.
A. Figueiredo, I. Gleria, R. Matsushita, and S. da Silva. Autocorrelation as a source of truncated Lévy flights in foreign exchange rates. Physica A, 323:601, 2003.
A. Figueiredo, I. Gleria, R. Matsushita, and S. da Silva. Autocorrelation and the sum of stochastic variables. Phys. Lett. A, 326:166, 2004.
A. Figueiredo, I. Gleria, R. Matsushita, and S. da Silva. Lévy flights, autocorrelation, and slow convergence. Physica A, 337:369, 2004.
B. A. Mello, A. S. Chaves, and F. A. Oliveira. Discrete atomistic model to simulate etching of a crystalline solid. Phys. Rev. E, 63:041113, 2001.
F. D. A. A. Reis. Dynamic transition in etching with poisoning. Phys. Rev. E., 68:041602, 2003.
F. D. A. A. Reis. Universality in two-dimensional kardar-parisi-zhang growth. Phys. Rev. E., 69:021610, 2004.
A. F. G. Monte, S. W. da Silva, J. M. R. Cruz, P. C. Morais, A. S. Chaves, and H. M. Cox. Symmetric and asymmetric fractal diffusion of electron-hole plasmas in semiconductor quantum wells. Phys. Lett. A, 268:430–435, 2000.
A. F. G. Monte, S. W. da Silva, J. M. R. Cruz, P. C. Morais, and A. S. Chaves. Experimental evidence of asymmetric carrier transport in InGaAs quantum wells and wires grown on tilted InP substrates. Appl. Phys. Lett., 81:2460–2462, 2002.
A. O. Caldeira and A. J. Leggett. Quantum tunnelling in a dissipative system. Ann. phys., 149:374–456, 1983.
M. H. Vainstein, I. V. L. Costa, R. Morgado, and F. A. Oliveira. Non-exponential relaxation for anomalous diffusion. to be published., 2006.
M. H. Vainstein, R. Morgado, and F. A. Oliveira. Spatio-temporal conjecture for diffusion. Physica A, 357:109–114, 2005.
J. Loschmidt. Über den zustand des wärmegleichgewichtes eines systems von körpern mit rücksicht auf die schwerkraft. Wien. Ber., 73:139, 1876.
K. Huang. Statistical Mechanics. John Wiley & Sons, New york, 1987.
U. Balucani, M. H. Lee, and V. Tognetti. Dynamical Correlactions. Phys. Rep., 373:409, 2003.
F. A. Oliveira. Time-reversal symmetry in light scattering by excitations in a film. Sol. Stat. Comm., 40:859–861, 1981.
F. A. Oliveira. How to build the Green-function for the elementary excitations in a film once we know those for a single interface. Sol. Stat. Comm., 85:1051, 1993.
F. Moraes, A. M. de M. Carvalho, I. V. L. Costa, F. A. Oliveira, and C. Furtado. Topological interactions in spacetimes with thick line defects. Phys. Rev. D, 68:043512, 2003.
J. A. McLennan. Onsager’s theorem and higher-order hydrodynamic equations. Phys. Rev. A, 10:1272–1276, 1974.
J. W. Dufty and J. M. Rubí. Generalized Onsager symmetry. Phys. Rev. A, 36:222–225, 1987.
A. L. Barabási and H. E. Stanley. Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge, 1995.
M. H. Lee. Can the velocity autocorrelation function decay exponentially? Phys. Rev. Lett., 51:1227–1230, 1983.
X. Xia and P. G. Wolynes. Microscopic Theory of Heterogeneity and Non-exponential Relaxations in Supercooled Liquids. Phys. Rev. Lett., 86:5526, 2001.
M. H. Vainstein, D. A. Stariolo, and J. J. Arenzon. Heterogeneities in systems with quenched disorder. J. Phys. A: Math. Gen., 36:10907–10919, 2003.
F. Benmouna, B. Peng, J. Gapinski, A. Patkowski, J. Ruhe, and D. Johannsmann. Dynamic light scattering from liquid crystal polymer brushes swollen in a nematic solvent. Liq. Cryst., 28:1353, 2001.
M. B. L. Santos, E. A. Oliveira, and A. M. F. Neto. Rayleigh scattering of a new lyotropic nematic liquid crystal system: crossover of propagative and diffusive behavior. Liq. Cryst., 27:1485, 2000.
P. Licinio and M. B. L. Santos. Pretransitional scaling close to a double critical point in a potassium laurate, 1-decanol, and heavy water lyotropic liquid crystal. Phys. Rev. E., 65:031714, 2002.
M. Peyrard. Glass transition in protein hydration water. Phys. Rev. E, 64:011109, 2001.
F. Colaiori and M. A. Moore. Stretched exponential relaxation in the mode-coupling theory for the Kardar-Parisi-Zhang equation. Phys. Rev. E, 63:057103, 2001.
J. P. Bouchaud, M. Mézard, and J. S. Yedidia. Variational theory for disordered vortex lattices. Phys. Rev. Lett., 67:3840, 1991.
A. Pérez-Madrid. A model for nonexponential behavior and aging in dissipative systems. J. Chem. Phys., 122:214914, 2005.
A. Cavagna, I. Giardina, and T. S. Grigera. Glassy dynamics, metastability limit and crystal growth in a lattice spin model. Europhys. Lett., 61:74, 2003.
V. V. Belyi. Fluctuation-dissipation dispersion relation for a Nonlocal Plasma. Phys. Rev. Lett., 88:255001, 2002.
V. V. Belyi. Fluctuation-dissipation-dispersion relations for a time and space nonlocal Plasma. Int. J. Quantum Chem., 98:183–190, 2004.
J. M. G. Vilar and J. M. Rubí. Thermodynamics “beyond” local equilibrium. Proc. Nat. Acad. Sci, 98:11081–11084, 2001.
H. B. Callen and R. F. Greene. On a Theorem of Irreversible Thermodynamics. Phys. Rev., 86:702, 1952.
R. F. Greene and H. B. Callen. On a Theorem of Irreversible Thermodynamics. II. Phys. Rev., 88:1387, 1952.
S. R. deGroot and P. Mazur. Non-Equilibrium Thermodynamics. Dover, New York, 1984.
J. M. Rubí and D. Bedeaux. Brownian-motion in a fluid in elongational flow. J. Stat. Phys., 53:125, 1988.
J. M. Rubí A. Pérez-Madrid, D. Reguera. Origin of the Violation of the Fluctuation-Dissipation Theorem in Systems with Activated Dynamics. Physica A, 329:357, 2003.
M. Naspedra, D. Reguera, A. Pérez-Madrid, and J. M. Rubí. Glassy dynamics: effective temperatures and intermittencies from a two-state model. Physica A, 351:14–21, 2005.
A. Perez-Madrid, J. M. Rubí, I. Santamaría-Holek. Slow dynamics and local quasi-equilibrium – relaxation in supercooled colloidal systems. J. Phys – Condens. mat., 16:2047, 2004.
B. B. Hu, E. A. de Souza, W. H. Knox, J. E. Cunningham, M. C. Nuss, A. V. Kuznetsov, and S. L. Chuang. Identifying the Distinct Phases of Carrier Transport in Semiconductors with 10 fs Resolution. Phys. Rev. Lett., 74:1689–1692, 1995.
M. Blasone, P. Jizba, and G. Vitiello. Dissipation and quantization. Phys. Lett. A, 287:205–210, 2001.
T. S. Biró, S. G. Matinyan, and B. Müller. Chaotic quantization of classical gauge fields. Found. Phys. Lett., 14:471–485, 2001.
M. Suzuki. New unified formulation of transient phenomena near the instability point on the basis of the Fokker-Planck equation. Physica A, 117:103, 1983.
I. An, S. Chen, and H. Guo. Search for the symmetry of the Fokker-Planck equation. Physica A, 128:520, 1984.
G. Cicogna and D. Vitali. Generalised symmetries of Fokker-Planck-type equations. J. Phys. A: Math. Gen, 22:L453, 1989.
W. M. Shtelen and V. I. Stogny. Symmetry properties of one-dimensional and two-dimensional Fokker-Planck equations. J. Phys. A: Math. Gen, 22:L539, 1989.
P. Rudra. Symmetry classes of the Fokker-Planck type equations. J. Phys. A: Math. Gen., 22:L539, 1990.
G. Cicogna and D. Vitali. Classification of the extended symmetries of Fokker-Planck equations. J. Phys. A: Math. Gen, 23:L85, 1990.
S. Spichak and V. Stognii. Symmetry classification and exact solution of the one-dimensional Fokker-Planck equations with arbitrary coefficients of drift and diffusion. J. Phys. A: Math. Gen, 32:8341, 1999.
V. Cherkasenko. Galilei invariance of the Fokker-Planck equations with non-linearity. Nonlinear Math. Phys., 2:416, 1995.
J. A. Cardeal, A. E. Santana, and T. M. Rocha. Symmetry and Classes of transport equations. Physica A, 308:292–300, 2002.
M. de Montigny, F. C. Khanna, and A. E. Santana. Gauge symmetry in Fokker-Planck dynamics. Physica A, 323:327, 2003.
C. Duval, G. Burdet, H. P. Künzle, and M. Perrin. Bargmann Structures and Newton-Cartan Theory. Phys. Rev. D, 31:1841–1853, 1985.
Y. Takahashi. Towards the Many-Body theory with the Galilei invariance as a guide I. Fortschr. Phys., 36:63, 1988.
M. de Montigny, F. C. Khanna, and A. E. Santana. On Galilei-Covariant Lagrangian Models of fluids. J. Phys. A: Math. Gen, 34:10921, 2001.
A. S. Chaves, J. M. Figueiredo, and M. C. Nemes. Metric fluctuations, thermodynamics, and classical physics – A proposed connection. Ann. Phys., 231:174–184, 1994.
J. L. Acebal, A. S. Chaves, J. M. Figueiredo, A. L. Mota, and M. C. Menes. Statistical approach for quantum gravity fluctuations in QFT. Phys. Lett. B, 445:94, 1998.
P. W. Anderson. Absence of diffusion in certain random lattices. Phys. Rev., 109:1492–1505, 1958.
S. N. Evangelou and D. E. Katsanos. Super-Diffusion in random chains with correlated disorder. Phys Lett. A., 164:456–464, 1992.
F. A. B. F. de Moura, M. D. Coutinho-Filho, E. P. Raposo, and M. L. Lyra. Delocalization and spin-wave dynamics in ferromagnetic chains with long-range correlated random exchange. Phys. Rev. B, 66:014418, 2002.
F. A. B. F. de Moura, M. D. Coutinho-Filho, E. P. Raposo, and M. L. Lyra. Delocalization in harmonic chains with long-range correlated random masses. Phys. Rev. B., 68:012202, 2003.
F. A. B. F. de Moura and M. L. Lyra. Delocalization in the ID Anderson model with long-range correlated disorder. Phys. Rev. Lett, 81:3735–3738, 1998.
M. H. Vainstein, R. Morgado, F. A. Oliveira, F. A. B. F. de Moura, and M. D. Coutinho-Filho. Stochastic description of the dynamics of the random exchange Heisenberg chain. Phys. Lett. A, 339:33–38, 2003.
L. Longa, E. M. F. Curado, and F. A. Oliveira. Roundoff-induced coalescence of chaotic trajectories. Phys. Rev. E, 54:R2201, 1996.
S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou. The synchronization of chaotic systems. Phys. Rep., 366:1–101, 2002.
R. P. Feynman. The character of physical law. The Random House Publishing Group, New York, 1994.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer
About this chapter
Cite this chapter
Vainstein, M., Costa, I., Oliveira, F. (2006). Mixing, Ergodicity and the Fluctuation-Dissipation Theorem in Complex Systems. In: Miguel, M.C., Rubi, M. (eds) Jamming, Yielding, and Irreversible Deformation in Condensed Matter. Lecture Notes in Physics, vol 688. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-33204-9_10
Download citation
DOI: https://doi.org/10.1007/3-540-33204-9_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30028-1
Online ISBN: 978-3-540-33204-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)