Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9611182 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Dynamic response of an Ising system to a pulsed field

Muktish Acharyya, Jayanta K. Bhattacharjee, Bikas K. Chakrabarti

Published Mar 1, 1997• 18 citations
arXivHTMLPDF

Abstract

The dynamical response to a pulsed magnetic field has been studied here both using Monte Carlo simulation and by solving numerically the mean-field dynamical equation of motion for the Ising model. The ratio ${\mathrm{R}}_{\mathrm{p}}$ of the response magnetization half-width to the width of the external field pulse has been observed to diverge and pulse susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ (ratio of the response magnetization peak height and the pulse height) gives a peak near the order-disorder transition temperature ${\mathrm{T}}_{\mathrm{c}}$ (for the unperturbed system). The Monte Carlo results for the Ising system on a square lattice show that ${\mathrm{R}}_{\mathrm{p}}$ diverges at ${\mathrm{T}}_{\mathrm{c}}$, with the exponent \ensuremath{\nu}z\ensuremath{\cong}2.0, while ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ shows a peak at ${\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}$, which is a function of the field pulse width \ensuremath{\delta}t. A finite-size (in time) scaling analysis shows that ${\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}$=${\mathrm{T}}_{\mathrm{c}}$+C(\ensuremath{\delta}t${)}^{\mathrm{\ensuremath{-}}1\mathrm{/}\mathrm{x}}$, with x=\ensuremath{\nu}z\ensuremath{\cong}2.0. The mean-field results show that both the divergence of R and the peak in ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ occur at the mean-field transition temperature, while the peak height in ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$\ensuremath{\sim}(\ensuremath{\delta}t${)}^{\mathrm{y}}$, y\ensuremath{\cong}1 for small values of \ensuremath{\delta}t. These results also compare well with an approximate analytical solution of the mean-field equation of motion.

DOI
10.1103/physreve.55.2392

Related papers

Dynamic magnetization-reversal transition in the Ising model

Arkajyoti Misra, Bikas K. Chakrabarti

We report the results of mean field and the Monte Carlo study of the dynamic magnetization-reversal transition in the Ising model, brought about by the application of an external field pulse applied in opposition to the existing order before the application of the pulse. The parameters controlling the transition are the strength ${h}_{p}$ and the duration $\ensuremath{\Delta}t$ of the pulse. In the mean field case, an approximate analytical expression is obtained for the phase boundary which agrees well with that obtained numerically in the small $\ensuremath{\Delta}t$ and large $T$ limit. The order parameter of the transition has been identified, and is observed to vary continuously near the transition. The order parameter exponent $\ensuremath{\beta}$ was estimated both for the mean field $(\ensuremath{\beta}=1)$ and the Monte Carlo $(\ensuremath{\beta}=0.90\ifmmode\pm\else\textpm\fi{}0.02$ in two dimensions) cases. The transition shows a ``critical slowing down'' type behavior near the phase boundary with diverging relaxation time. The divergence was found to be logarithmic in the mean field case and exponential in the Monte Carlo case. The finite size scaling technique was employed to estimate the correlation length exponent $\ensuremath{\nu}$ $(=1.5\ifmmode\pm\else\textpm\fi{}0.3$ in two dimensions) in the Monte Carlo case.

arXiv:cond-mat/9805289•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1998•18 citations

Response of Ising systems to oscillating and pulsed fields: Hysteresis, ac, and pulse susceptibility

Muktish Acharyya, Bikas K. Chakrabarti

We have studied, using Monte Carlo (MC) simulation for ferromagnetic Ising systems in one to four dimensions and solving numerically the mean-field (MF) equation of motion, the nature of the response magnetization m(t) of an Ising system in the presence of a periodically varying external field [h(t)=${\mathit{h}}_{0}$cos(\ensuremath{\omega}t)]. From these studies, we determine the m-h loop or hysteresis loop area A(=\ensuremath{\oint}mdh) and the dynamic order parameter Q(=\ensuremath{\oint}mdt) and investigate their variations with the frequency (\ensuremath{\omega}) and amplitude (${\mathit{h}}_{0}$) of the applied external magnetic field and the temperature (T) of the system. The variations in A are fitted to a scaling form, assumed to be valid over a wide range of parameter (\ensuremath{\omega},${\mathit{h}}_{0}$,T) values, and the best-fit exponents are obtained in all three dimensions (D=2,3,4). The scaling function is Lorentzian in the MF case and is exponentially decaying, with an initial power law, for the MC cases. The dynamic phase boundary (in the ${\mathit{h}}_{0}$-T plane) is found to be frequency dependent and the transition (from Q\ensuremath{\ne}0 for low T and ${\mathit{h}}_{0}$ to Q=0 for high T and ${\mathit{h}}_{0}$) across the boundary crosses over from a discontinuous to a continuous one at a tricritical point. These boundaries are determined in various cases.We find that the response can be generally expressed as m(t)=P(\ensuremath{\omega}(t-${\mathrm{\ensuremath{\tau}}}_{\mathrm{eff}}$)) where P denotes a periodic function with the same frequency \ensuremath{\omega} of the perturbing field and ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$(${\mathit{h}}_{0}$,\ensuremath{\omega},T) denotes the effective delay. We established that this effective delay ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ of the response is the crucial term and it practically determines all the above observations for A, Q, etc. Investigating the nature of the in-phase (\ensuremath{\chi}\ensuremath{'}) and the out-of-phase (\ensuremath{\chi}\ensuremath{'}\ensuremath{'}) susceptibility, defined as \ensuremath{\chi}\ensuremath{'}=(${\mathit{m}}_{0}$/${\mathit{h}}_{0}$)cos(\ensuremath{\varphi}) and \ensuremath{\chi}\ensuremath{'}\ensuremath{'}=(${\mathit{m}}_{0}$/${\mathit{h}}_{0}$)sin(\ensuremath{\varphi}); \ensuremath{\varphi}=\ensuremath{\omega}${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ [and ${\mathit{m}}_{0}$ is the amplitude of m(t)], we find that the loop area A is directly given by \ensuremath{\chi}\ensuremath{'}\ensuremath{'} and also the temperature variation of \ensuremath{\chi}\ensuremath{'}\ensuremath{'}, at fixed \ensuremath{\omega} and ${\mathit{h}}_{0}$, gives a prominent peak at the dynamic transition point. We have also studied the behavior of the response magnetization by the application of a short-duration (compared with the relaxation time) pulsed magnetic field.Here, we observed (both in the MC and MF cases) that the width ratio (of the half-width and the width of the response magnetization and of the pulsed field, respectively) and the susceptibility (the ratio of excess magnetization over its equilibrium value and the height of the pulsed field) both show sharp peaks at the order-disorder (ferromagnetic-paramagnetic) transition point. We have also studied similar response behavior of Ising systems in the presence of time-varying longitudinal and transverse fields, solving numerically the mean-field equation of motion. We have again studied here the nature of the dynamic phase transition and the behavior of the ac susceptibility (both longitudinal and transverse) across the dynamic phase boundary. For a short-duration pulse of the transverse field, the width ratio and the pulse susceptibility are again seen to diverge at the order-disorder transition point.

Physical review. B, Condensed matter•Sep 1, 1995•232 citations

Competing field pulse induced dynamic transition in Ising Models

Arnab Chatterjee, Bikas K. Chakrabarti

We discuss dynamic magnetization-reversal phenomena in the Ising model under a finite-duration external magnetic field, which competes with the existing order for . The nature of the phase boundary is estimated from the mean-field (MF) equation of motion. The susceptibility and relaxation time diverge at the MF phase boundary. A Monte Carlo study also shows divergence of the relaxation time and order-parameter fluctuations at the phase boundary. The fourth-order cumulant shows two distinct behaviors. For low temperatures and pulse durations, the value of the cumulant at the crossing point for different system sizes is far less than that for the static transition in the same dimension. This suggests a new universality class for the dynamic transition. For higher temperatures and pulse durations, the transition falls in a MF-like weak-singularity universality class.

Phase Transitions•May 1, 2004•14 citations

Fluctuation cumulant behavior for the field-pulse-induced magnetization-reversal transition in Ising models

Arnab Chatterjee, Bikas K. Chakrabarti

The universality class of the dynamic magnetization-reversal transition, induced by a competing field pulse, in an Ising model on a square lattice, below its static ordering temperature, is studied here using Monte Carlo simulations. Fourth-order cumulant of the order parameter distribution is studied for different system sizes around the phase boundary region. The crossing point of the cumulant (for different system sizes) gives the transition point and the value of the cumulant at the transition point indicates the universality class of the transition. The cumulant value at the crossing point for low temperature and pulse width range is observed to be significantly less than that for the static transition in the same two-dimensional Ising model. The finite-size scaling behavior in this range also indicates a higher correlation length exponent value. For higher temperature and pulse width range, the transition seems to fall in a mean-field-like universality class.

arXiv:cond-mat/0210135•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 21, 2003•25 citations

Critical behavior of the three-dimensional random-field Ising model: Two-exponent scaling and discontinuous transition

Heiko Rieger

In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(${\mathit{h}}_{\mathit{r}}$/T${)}^{2}$ being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.

arXiv:cond-mat/9503041•Physical review. B, Condensed matter•Sep 1, 1995•138 citations

Critical dynamics of disordered two-dimensional Ising systems: a Monte Carlo study

Vladimir V. Prudnikov, O. N. Markov

The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.

Journal of Physics A Mathematical and General•Mar 23, 1995•5 citations
The dynamical response to a pulsed magnetic field has been studied here both using Monte Carlo simulation and by solving numerically the mean-field dynamical equation of motion for the Ising model. The ratio Rp{\mathrm{R}}_{\mathrm{p}}Rp​ of the response magnetization half-width to the width of the external field pulse has been observed to diverge and pulse susceptibility \ensuremathχp{\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}\ensuremathχp​ (ratio of the response magnetization peak height and the pulse height) gives a peak near the order-disorder transition temperature Tc{\mathrm{T}}_{\mathrm{c}}Tc​ (for the unperturbed system). The Monte Carlo results for the Ising system on a square lattice show that Rp{\mathrm{R}}_{\mathrm{p}}Rp​ diverges at Tc{\mathrm{T}}_{\mathrm{c}}Tc​, with the exponent \ensuremath{\nu}z\ensuremath{\cong}2.0, while \ensuremathχp{\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}\ensuremathχp​ shows a peak at Tce{\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}Tce​, which is a function of the field pulse width \ensuremath{\delta}t. A finite-size (in time) scaling analysis shows that Tce{\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}Tce​=Tc{\mathrm{T}}_{\mathrm{c}}Tc​+C(\ensuremath{\delta}t)\ensuremath−1/x{)}^{\mathrm{\ensuremath{-}}1\mathrm{/}\mathrm{x}})\ensuremath−1/x, with x=\ensuremath{\nu}z\ensuremath{\cong}2.0. The mean-field results show that both the divergence of R and the peak in \ensuremathχp{\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}\ensuremathχp​ occur at the mean-field transition temperature, while the peak height in \ensuremathχp{\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}\ensuremathχp​\ensuremath{\sim}(\ensuremath{\delta}t)y{)}^{\mathrm{y}})y, y\ensuremath{\cong}1 for small values of \ensuremath{\delta}t. These results also compare well with an approximate analytical solution of the mean-field equation of motion.
We report the results of mean field and the Monte Carlo study of the dynamic magnetization-reversal transition in the Ising model, brought about by the application of an external field pulse applied in opposition to the existing order before the application of the pulse. The parameters controlling the transition are the strength hp{h}_{p}hp​ and the duration $\ensuremath{\Delta}t$ of the pulse. In the mean field case, an approximate analytical expression is obtained for the phase boundary which agrees well with that obtained numerically in the small $\ensuremath{\Delta}t$ and large TTT limit. The order parameter of the transition has been identified, and is observed to vary continuously near the transition. The order parameter exponent $\ensuremath{\beta}$ was estimated both for the mean field $(\ensuremath{\beta}=1)$ and the Monte Carlo $(\ensuremath{\beta}=0.90\ifmmode\pm\else\textpm\fi{}0.02$ in two dimensions) cases. The transition shows a ``critical slowing down'' type behavior near the phase boundary with diverging relaxation time. The divergence was found to be logarithmic in the mean field case and exponential in the Monte Carlo case. The finite size scaling technique was employed to estimate the correlation length exponent $\ensuremath{\nu}$ $(=1.5\ifmmode\pm\else\textpm\fi{}0.3$ in two dimensions) in the Monte Carlo case.
We have studied, using Monte Carlo (MC) simulation for ferromagnetic Ising systems in one to four dimensions and solving numerically the mean-field (MF) equation of motion, the nature of the response magnetization m(t) of an Ising system in the presence of a periodically varying external field [h(t)=h0{\mathit{h}}_{0}h0​cos(\ensuremath{\omega}t)]. From these studies, we determine the m-h loop or hysteresis loop area A(=\ensuremath{\oint}mdh) and the dynamic order parameter Q(=\ensuremath{\oint}mdt) and investigate their variations with the frequency (\ensuremath{\omega}) and amplitude (h0{\mathit{h}}_{0}h0​) of the applied external magnetic field and the temperature (T) of the system. The variations in A are fitted to a scaling form, assumed to be valid over a wide range of parameter (\ensuremath{\omega},h0{\mathit{h}}_{0}h0​,T) values, and the best-fit exponents are obtained in all three dimensions (D=2,3,4). The scaling function is Lorentzian in the MF case and is exponentially decaying, with an initial power law, for the MC cases. The dynamic phase boundary (in the h0{\mathit{h}}_{0}h0​-T plane) is found to be frequency dependent and the transition (from Q\ensuremath{\ne}0 for low T and h0{\mathit{h}}_{0}h0​ to Q=0 for high T and h0{\mathit{h}}_{0}h0​) across the boundary crosses over from a discontinuous to a continuous one at a tricritical point. These boundaries are determined in various cases.We find that the response can be generally expressed as m(t)=P(\ensuremath{\omega}(t-${\mathrm{\ensuremath{\tau}}}_{\mathrm{eff}}$)) where P denotes a periodic function with the same frequency \ensuremath{\omega} of the perturbing field and ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$(h0{\mathit{h}}_{0}h0​,\ensuremath{\omega},T) denotes the effective delay. We established that this effective delay ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ of the response is the crucial term and it practically determines all the above observations for A, Q, etc. Investigating the nature of the in-phase (\ensuremath{\chi}\ensuremath{'}) and the out-of-phase (\ensuremath{\chi}\ensuremath{'}\ensuremath{'}) susceptibility, defined as \ensuremath{\chi}\ensuremath{'}=(m0{\mathit{m}}_{0}m0​/h0{\mathit{h}}_{0}h0​)cos(\ensuremath{\varphi}) and \ensuremath{\chi}\ensuremath{'}\ensuremath{'}=(m0{\mathit{m}}_{0}m0​/h0{\mathit{h}}_{0}h0​)sin(\ensuremath{\varphi}); \ensuremath{\varphi}=\ensuremath{\omega}${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ [and m0{\mathit{m}}_{0}m0​ is the amplitude of m(t)], we find that the loop area A is directly given by \ensuremath{\chi}\ensuremath{'}\ensuremath{'} and also the temperature variation of \ensuremath{\chi}\ensuremath{'}\ensuremath{'}, at fixed \ensuremath{\omega} and h0{\mathit{h}}_{0}h0​, gives a prominent peak at the dynamic transition point. We have also studied the behavior of the response magnetization by the application of a short-duration (compared with the relaxation time) pulsed magnetic field.Here, we observed (both in the MC and MF cases) that the width ratio (of the half-width and the width of the response magnetization and of the pulsed field, respectively) and the susceptibility (the ratio of excess magnetization over its equilibrium value and the height of the pulsed field) both show sharp peaks at the order-disorder (ferromagnetic-paramagnetic) transition point. We have also studied similar response behavior of Ising systems in the presence of time-varying longitudinal and transverse fields, solving numerically the mean-field equation of motion. We have again studied here the nature of the dynamic phase transition and the behavior of the ac susceptibility (both longitudinal and transverse) across the dynamic phase boundary. For a short-duration pulse of the transverse field, the width ratio and the pulse susceptibility are again seen to diverge at the order-disorder transition point.
We discuss dynamic magnetization-reversal phenomena in the Ising model under a finite-duration external magnetic field, which competes with the existing order for . The nature of the phase boundary is estimated from the mean-field (MF) equation of motion. The susceptibility and relaxation time diverge at the MF phase boundary. A Monte Carlo study also shows divergence of the relaxation time and order-parameter fluctuations at the phase boundary. The fourth-order cumulant shows two distinct behaviors. For low temperatures and pulse durations, the value of the cumulant at the crossing point for different system sizes is far less than that for the static transition in the same dimension. This suggests a new universality class for the dynamic transition. For higher temperatures and pulse durations, the transition falls in a MF-like weak-singularity universality class.
The universality class of the dynamic magnetization-reversal transition, induced by a competing field pulse, in an Ising model on a square lattice, below its static ordering temperature, is studied here using Monte Carlo simulations. Fourth-order cumulant of the order parameter distribution is studied for different system sizes around the phase boundary region. The crossing point of the cumulant (for different system sizes) gives the transition point and the value of the cumulant at the transition point indicates the universality class of the transition. The cumulant value at the crossing point for low temperature and pulse width range is observed to be significantly less than that for the static transition in the same two-dimensional Ising model. The finite-size scaling behavior in this range also indicates a higher correlation length exponent value. For higher temperature and pulse width range, the transition seems to fall in a mean-field-like universality class.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr{\mathit{h}}_{\mathit{r}}hr​/T)2{)}^{2})2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.