%0 Book Section %1 statphys23_0167 %A Chamati, H. %A Korutcheva, 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 dynamical exponent impurities model point quenched random statphys23 topic-2 %T Finite-size scaling in systems with quenched impurities: Critical dynamics %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=167 %X In the present work we study the critical dynamics of a finite system of size $L$, with periodic boundary conditions, described by the Landau--Ginzburg model in the case of a non conserved order parameter (The Model A): equationeqGLW H=-\frac12 ınt_L^d d^dx łeftt|\psi|^2+\varphi|\psi|^2 + c|\nabla\psi|^2 +u12|\psi|^4\right, equation where $\psi\equiv\psi(x)$ is a $n$--component field with $|\psi|^2=\sum_i=1^n\psi_i^2$ and the parameters $t$, $c$ and $u$ are model constants. The random variable $\varphi\equiv\varphi(x)$, introduced to shift the temperature locally due to the presence of impurities, has a Gaussian distribution with zero mean and variance $\varphi(\mathbfx)\varphi(x')= \Delta\delta^d(x - x')$. The physically interesting case corresponds to the values of the components of the order parameter within the interval $1<n<4$. We extend our investigation to the case of critical dynamics of the model by using the Langevin equation: equation \partial\psi_i\partial\tau =-\partial\mathcalH\partial\psi_i +\zeta_i. equation Here $łambda$ is the Onsager kinetic coefficient and $\zeta_i\equiv\zeta_i(x,\tau)$ is a Gaussian random--white noise having zero mean and variance: \\ $łeft<\zeta_i(x,\tau)\zeta_j(x',\tau')\right>= 2\delta^dłeft(x-x'\right)\delta(\tau-\tau')\delta_ij.$ By using the $\epsilon=(4-d)$--formalism up to one--loop expansion and the analogy with the corresponding quantum system, we derive the explicit form for the linear relaxation time (Figure 1): eqnarrayrelaxa \tau_R(y)&=&L^złambda \frac14\pi\fracn-13\epsilon\nonumber\\ &&\timesłeft1-3n16(n-1) \epsilonłeft(1+y +4\partialyF_4,2(y)\right)\right\nonumber\\ &&g_nłeft(\frac12\mu(y)\right). eqnarray Here $z=2+4-n8(n-1)\epsilon+O(\epsilon^2)$ is the dynamical exponent and eqnarray*eqF F_d,2(x)=ınt^ınfty_0dz\expłeft(-xz(2\pi)^2\right)\times\\ \timesłeftłeft(\sum_l=-ınfty^ınftye^-zl^2 \right)^d-1-łeft(\piz\right)^d/2\right eqnarray* is a universal finite--size scaling function. The function $\mu(y)$ is the inverse gap of the corresponding quantum mechanical Hamiltonian, which has been determined by using variational parameters. The scaling variable is $y=tL^1/\nu$ with $\nu^-1=2-3n8(n+2)\epsilon+O(\epsilon^2)$. The expression for the critical relaxation time in the case of finite systems with short range correlated quenched impurities gives the correct asymptotic behavior characteristics of the bulk limit. Up to the first order in $\epsilon$, the direct effect of the presence of quenched impurities is through the fixed point values of the parameter $u$. Acknowledgments: We thank Prof. N.S. Tonchev and Prof. J.J. Ruiz--Lorenzo for stimulating discussions during all the stages of the work. H.C. is supported by grant No F-1517 of the Bulgarian Fund for Scientific Research. E.K. also acknowledges the financial support from Spanish Grant No.DGI.M.CyT.FIS2005--1729.

@incollection{statphys23_0167, abstract = {In the present work we study the critical dynamics of a finite system of size $L$, with periodic boundary conditions, described by the Landau--Ginzburg model in the case of a non conserved order parameter (The Model A): \begin{equation}\label{eqGLW} \mathcal{H}=-\frac12 \int_{L^{d}} d^dx \left[t|\psi|^2+\varphi|\psi|^2 + c|\nabla\psi|^{2} +\frac{u}{12}|\psi|^{4}\right], \end{equation} where $\psi\equiv\psi(\mathbf{x})$ is a $n$--component field with $|\psi|^2=\sum_{i=1}^n\psi_i^2$ and the parameters $t$, $c$ and $u$ are model constants. The random variable $\varphi\equiv\varphi(\mathbf{x})$, introduced to shift the temperature locally due to the presence of impurities, has a Gaussian distribution with zero mean and variance $\overline{\varphi(\mathbf{x})\varphi(\mathbf{x}')}= \Delta\delta^{d}(\mathbf{x} - \mathbf{x}')$. The physically interesting case corresponds to the values of the components of the order parameter within the interval $1<n<4$. We extend our investigation to the case of critical dynamics of the model by using the Langevin equation: \begin{equation} \frac{\partial\psi_i}{\partial\tau} =-\lambda \frac{\partial\mathcal{H}}{\partial\psi_i} +\zeta_i. \end{equation} Here $\lambda$ is the Onsager kinetic coefficient and $\zeta_i\equiv\zeta_i(\mathbf{x},\tau)$ is a Gaussian random--white noise having zero mean and variance: \\ $\left<\zeta_i(\mathbf{x},\tau)\zeta_j(\mathbf{x}',\tau')\right>= 2\lambda \delta^d\left(\mathbf{x}-\mathbf{x}'\right)\delta(\tau-\tau')\delta_{ij}.$ By using the $\epsilon=(4-d)$--formalism up to one--loop expansion and the analogy with the corresponding quantum system, we derive the explicit form for the linear relaxation time (Figure 1): \begin{eqnarray}\label{relaxa} \tau_R(y)&=&\frac{L^z}\lambda \frac1{4\pi}\sqrt{\frac{n-1}{3\epsilon}}\nonumber\\ &&\times\left[1-\frac{3n}{16(n-1)} \epsilon\left(1+\ln y +4\frac{\partial}{\partial y}F_{4,2}(y)\right)\right]\nonumber\\ &&\times g_n\left(\frac12\mu(y)\right). \end{eqnarray} Here $z=2+\frac{4-n}{8(n-1)}\epsilon+O(\epsilon^2)$ is the dynamical exponent and \begin{eqnarray*}\label{eqF} F_{d,2}(x)=\int^{\infty}_{0}dz\exp\left(-\frac{xz}{(2\pi)^{2}}\right)\times\\ \times\left[\left(\sum_{l=-\infty}^{\infty}e^{-zl^{2}} \right)^{d}-1-\left(\frac{\pi}{z}\right)^{d/2}\right] \end{eqnarray*} is a universal finite--size scaling function. The function $\mu(y)$ is the inverse gap of the corresponding quantum mechanical Hamiltonian, which has been determined by using variational parameters. The scaling variable is $y=tL^{1/\nu}$ with $\nu^{-1}=2-\frac{3n}{8(n+2)}\epsilon+O(\epsilon^2)$. The expression for the critical relaxation time in the case of finite systems with short range correlated quenched impurities gives the correct asymptotic behavior characteristics of the bulk limit. Up to the first order in $\epsilon$, the direct effect of the presence of quenched impurities is through the fixed point values of the parameter $u$. Acknowledgments: We thank Prof. N.S. Tonchev and Prof. J.J. Ruiz--Lorenzo for stimulating discussions during all the stages of the work. H.C. is supported by grant No F-1517 of the Bulgarian Fund for Scientific Research. E.K. also acknowledges the financial support from Spanish Grant No.DGI.M.CyT.FIS2005--1729.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Chamati, H. and Korutcheva, E.}, biburl = {https://www.bibsonomy.org/bibtex/22c9018af9fe77dc4c45800281e47ca74/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {3785a7bbdf57394e9c2bd24fee3360c2}, intrahash = {2c9018af9fe77dc4c45800281e47ca74}, keywords = {dynamical exponent impurities model point quenched random statphys23 topic-2}, month = {9-13 July}, timestamp = {2007-06-20T10:16:13.000+0200}, title = {Finite-size scaling in systems with quenched impurities: Critical dynamics}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=167}, year = 2007 }