%0 Book Section %1 statphys23_0035 %A Arkhincheev, V.E. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K diffusion einstein flights fluctuation-dissipation generalization levy nonlinear relation statphys23 theorem topic-1 %T Nonlinear Generalization of Fluctuation-Dissipation Theorem for Levy Flights Diffusion %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=35 %X As well known the fluctuation-dissipation theorem (FDT) connects the two different quantities such as stochastic random characteristics of system or motion by acting of stochastic forces (fluctuations) with kinetic characteristics as conductivity or mobility, which characterized motion by regular forces. One of first examples of this FDT is Einstein relation between diffusion coefficient $D$and mobility of particle : $$qD=kT $$ Here $T$ is temperature of system, $k$ is Boltzmann’s constant, $q$is a charge of particle. But this relation was obtained using three assumptions. First, the diffusion has usual character: square root mean displacement is linear in time. Second, the response for external electric field is linear ,that is Ohm’s law holds. Thirdly, the Boltzman’s classical distribution is correct. Because the Boltmann’s distribution doesn’t depend from kinetic phenomena, so we use this distribution in our paper again. But when we study the anomalous Levy flight superdiffusion, the first assumption is not correct and we need check the second assumption about linear response for external field. The main of results of paper is that linear response is absent at anomalous superdiffusion or Ohm’s law is not holds at the systems with anomalous superdiffusion. Instead this the electric current depends on arbitrary weak electric field in a nonlinear way: $$ J E^\nu$$ Nevertheless, we show that the fluctuation-dissipation theorem for a Levy flight diffusion exists in the new nonlinear form as relation between exponent, which describe anomalous superdiffusion $mu$ , and other exponent, which describe of nonlinear response, $\nu$: $$=-1$$

@incollection{statphys23_0035, abstract = {As well known the fluctuation-dissipation theorem (FDT) connects the two different quantities such as stochastic random characteristics of system or motion by acting of stochastic forces (fluctuations) with kinetic characteristics as conductivity or mobility, which characterized motion by regular forces. One of first examples of this FDT is Einstein relation between diffusion coefficient $D$and mobility of particle : $$qD=kT \mu $$ Here $T$ is temperature of system, $k$ is Boltzmann’s constant, $q$is a charge of particle. But this relation was obtained using three assumptions. First, the diffusion has usual character: square root mean displacement is linear in time. Second, the response for external electric field is linear ,that is Ohm’s law holds. Thirdly, the Boltzman’s classical distribution is correct. Because the Boltmann’s distribution doesn’t depend from kinetic phenomena, so we use this distribution in our paper again. But when we study the anomalous Levy flight superdiffusion, the first assumption is not correct and we need check the second assumption about linear response for external field. The main of results of paper is that linear response is absent at anomalous superdiffusion or Ohm’s law is not holds at the systems with anomalous superdiffusion. Instead this the electric current depends on arbitrary weak electric field in a nonlinear way: $$ J \sim E^{\nu}$$ Nevertheless, we show that the fluctuation-dissipation theorem for a Levy flight diffusion exists in the new nonlinear form as relation between exponent, which describe anomalous superdiffusion $mu$ , and other exponent, which describe of nonlinear response, $\nu$: $$\nu =\mu -1$$}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Arkhincheev, V.E.}, biburl = {https://www.bibsonomy.org/bibtex/299869b423406a626dcc7c400ae0b3581/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {4925012ba470033458ea13f6fdb9a0ba}, intrahash = {99869b423406a626dcc7c400ae0b3581}, keywords = {diffusion einstein flights fluctuation-dissipation generalization levy nonlinear relation statphys23 theorem topic-1}, month = {9-13 July}, timestamp = {2007-06-20T10:16:10.000+0200}, title = {Nonlinear Generalization of Fluctuation-Dissipation Theorem for Levy Flights Diffusion}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=35}, year = 2007 }