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):
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.
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