Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nonequilibrium structure factor for conserved spin dynamics: Abrupt temperature increase

James H. Luscombe, Marshall Luban

Published Sep 1, 1996• 2 citations

Abstract

We consider the nonequilibrium, elastic-scattering structure factor S(q,t) (q denotes the wave vector, t the time), for the Kawasaki spin-conserving kinetic Ising model of a one-dimensional system with nearest-neighbor interactions, initially in equilibrium at temperature ${\mathit{T}}_{\mathit{I}}$, that is suddenly placed in contact with a heat bath at temperature ${\mathit{T}}_{\mathit{F}}$, with ${\mathit{T}}_{\mathit{F}}$\ensuremath{\gg}${\mathit{T}}_{\mathit{I}}$. We present detailed results for the case of ${\mathit{T}}_{\mathit{F}}$=\ensuremath{\infty}, for which we have succeeded in calculating the exact form of S(q,t). For finite ${\mathit{T}}_{\mathit{F}}$, we present an approximation scheme for the higher-order nonequilibrium correlation functions that leads to closure of the hierarchy of equations of motion. The merits of this approximation are that (i) S(q,t) is guaranteed to satisfy an exact sum rule over the Brillouin zone (BZ) of wave vectors q, and (ii) S(q,t) evolves to the correct value in the long-time limit. For antiferromagnetic coupling, the structure factor, initially dominated by the Bragg peak associated with ${\mathit{T}}_{\mathit{I}}$ at the edge of the BZ, decays exponentially with time, ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathit{t}/{\mathrm{\ensuremath{\tau}}}_{\mathit{q}}}$ while approximately preserving its shape in q space, since the lifetime ${\mathrm{\ensuremath{\tau}}}_{\mathit{q}}$ is nearly independent of q. Except near the center of the BZ, after the Bragg peak has decayed sufficiently, the dependence of S(q,t) on q can be characterized as though the spins rapidly quasiequilibrate to the equilibrium structure factor associated with ${\mathit{T}}_{\mathit{F}}$, \ensuremath{\chi}(q,${\mathit{T}}_{\mathit{F}}$), in that S(q,t)/\ensuremath{\chi}(q,${\mathit{T}}_{\mathit{F}}$) is independent of q, but is time dependent, slowly approaching unity as ${\mathit{t}}^{\mathrm{\ensuremath{-}}1/2}$ for large t. For q\ensuremath{\approxeq}0 the initial form of S remains in effect until the value of t is of order ${\mathit{q}}^{\mathrm{\ensuremath{-}}2}$. For ferromagnetic coupling, the initial Bragg peak for q\ensuremath{\approxeq}0 does not preserve its shape while decaying exponentially, since the lifetime ${\mathrm{\ensuremath{\tau}}}_{\mathit{q}}$ strongly depends on the wave-vector q, diverging as ${\mathit{q}}^{\mathrm{\ensuremath{-}}2}$ for q\ensuremath{\rightarrow}0, and, in particular, it is as though the spins for q\ensuremath{\approxeq}0 remain ``frozen'' at ${\mathit{T}}_{\mathit{I}}$. Analogous to the behavior for antiferromagnetic interactions, away from the center of the BZ, we find that S(q,t)/\ensuremath{\chi}(q,${\mathit{T}}_{\mathit{F}}$) is independent of q and is a function of t/${\mathit{t}}_{\mathit{w}}$, very slowly approaching unity. The characteristic ``waiting time'' ${\mathit{t}}_{\mathit{w}}$ is anomalously long, proportional to ${\ensuremath{\xi}}^{2}$, where \ensuremath{\xi} is the equilibrium correlation length at temperature ${\mathit{T}}_{\mathit{I}}$. This behavior of ${\mathit{t}}_{\mathit{w}}$ can be related to the random walk of domain boundaries. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physreve.54.2266

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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nonequilibrium structure factor for conserved spin dynamics: Abrupt temperature increase

James H. Luscombe, Marshall Luban

Published Sep 1, 1996• 2 citations

Publisher

Abstract

{\mathit{T}}_{\mathit{I}}

, that is suddenly placed in contact with a heat bath at temperature

{\mathit{T}}_{\mathit{F}}

, with

{\mathit{T}}_{\mathit{F}}

\ensuremath{\gg}

{\mathit{T}}_{\mathit{I}}

. We present detailed results for the case of

{\mathit{T}}_{\mathit{F}}

=\ensuremath{\infty}, for which we have succeeded in calculating the exact form of S(q,t). For finite

{\mathit{T}}_{\mathit{F}}

, we present an approximation scheme for the higher-order nonequilibrium correlation functions that leads to closure of the hierarchy of equations of motion. The merits of this approximation are that (i) S(q,t) is guaranteed to satisfy an exact sum rule over the Brillouin zone (BZ) of wave vectors q, and (ii) S(q,t) evolves to the correct value in the long-time limit. For antiferromagnetic coupling, the structure factor, initially dominated by the Bragg peak associated with

{\mathit{T}}_{\mathit{I}}

at the edge of the BZ, decays exponentially with time,

{\mathit{e}}^{\mathrm{\ensuremath{-}}\mathit{t}/{\mathrm{\ensuremath{\tau}}}_{\mathit{q}}}

while approximately preserving its shape in q space, since the lifetime

{\mathrm{\ensuremath{\tau}}}_{\mathit{q}}

is nearly independent of q. Except near the center of the BZ, after the Bragg peak has decayed sufficiently, the dependence of S(q,t) on q can be characterized as though the spins rapidly quasiequilibrate to the equilibrium structure factor associated with

{\mathit{T}}_{\mathit{F}}

, \ensuremath{\chi}(q,

{\mathit{T}}_{\mathit{F}}

), in that S(q,t)/\ensuremath{\chi}(q,

{\mathit{T}}_{\mathit{F}}

) is independent of q, but is time dependent, slowly approaching unity as

{\mathit{t}}^{\mathrm{\ensuremath{-}}1/2}

for large t. For q\ensuremath{\approxeq}0 the initial form of S remains in effect until the value of t is of order

{\mathit{q}}^{\mathrm{\ensuremath{-}}2}

. For ferromagnetic coupling, the initial Bragg peak for q\ensuremath{\approxeq}0 does not preserve its shape while decaying exponentially, since the lifetime

{\mathrm{\ensuremath{\tau}}}_{\mathit{q}}

strongly depends on the wave-vector q, diverging as

{\mathit{q}}^{\mathrm{\ensuremath{-}}2}

for q\ensuremath{\rightarrow}0, and, in particular, it is as though the spins for q\ensuremath{\approxeq}0 remain

frozen'' at ${\mathit{T}}_{\mathit{I}}$. Analogous to the behavior for antiferromagnetic interactions, away from the center of the BZ, we find that S(q,t)/\ensuremath{\chi}(q,${\mathit{T}}_{\mathit{F}}$) is independent of q and is a function of t/${\mathit{t}}_{\mathit{w}}$, very slowly approaching unity. The characteristic

waiting time''

{\mathit{t}}_{\mathit{w}}

is anomalously long, proportional to

{\ensuremath{\xi}}^{2}

, where \ensuremath{\xi} is the equilibrium correlation length at temperature

{\mathit{T}}_{\mathit{I}}

. This behavior of

{\mathit{t}}_{\mathit{w}}

can be related to the random walk of domain boundaries. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physreve.54.2266

Related papers

Exact two-spinon dynamic structure factor of the one-dimensional s=12 Heisenberg-Ising antiferromagnet

A. Hamid Bougourzi, Michael Karbach, Gerhard H. Muller

The exact two-spinon part of the dynamic spin structure factor ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ for the one-dimensional $s=1/2,$ $\mathrm{XXZ}$ model at $T=0$ in the antiferromagnetically ordered phase is calculated using recent advances in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The two-spinon excitations form a two-parameter continuum consisting of two partly overlapping sheets in $(Q,\ensuremath{\omega})$ space. The spectral threshold has a smooth maximum at the Brillouin zone boundary $(Q=\ensuremath{\pi}/2)$ and a smooth minimum with a gap at the zone center $(Q=0).$ The two-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the two-spinon transition rates, the two regimes $0<~Q<{Q}_{\ensuremath{\kappa}}$ (near the zone center) and ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2$ (near the zone boundary) must be distinguished, where ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}0$ in the Heisenberg limit and ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}\ensuremath{\pi}/2$ in the Ising limit. In the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2,$ the two-spinon transition rates relevant for ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ are finite at the lower boundary of each sheet, decrease monotonically with increasing \ensuremath{\omega}, and approach zero linearly at the upper boundary. The resulting two-spinon part of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime $0<{Q}_{\ensuremath{\kappa}}<~\ensuremath{\pi}/2,$ in contrast, the two-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. In the associated two-spinon line shapes of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega}),$ the linear cusps at the continuum boundaries are replaced by square-root cusps. Existing perturbation studies have been unable to capture the physics of the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2.$ However, their line-shape predictions for the regime $0<~Q<{Q}_{\ensuremath{\kappa}}$ are in good agreement with the exact results if the anisotropy is very strong. For weak anisotropies, the exact line shapes are more asymmetric.The exact two-spinon part of the dynamic spin structure factor ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ for the one-dimensional $s=1/2,$ $\mathrm{XXZ}$ model at $T=0$ in the antiferromagnetically ordered phase is calculated using recent advances in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The two-spinon excitations form a two-parameter continuum consisting of two partly overlapping sheets in $(Q,\ensuremath{\omega})$ space. The spectral threshold has a smooth maximum at the Brillouin zone boundary $(Q=\ensuremath{\pi}/2)$ and a smooth minimum with a gap at the zone center $(Q=0).$ The two-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the two-spinon transition rates, the two regimes $0<~Q<{Q}_{\ensuremath{\kappa}}$ (near the zone center) and ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2$ (near the zone boundary) must be distinguished, where ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}0$ in the Heisenberg limit and ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}\ensuremath{\pi}/2$ in the Ising limit. In the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2,$ the two-spinon transition rates relevant for ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ are finite at the lower boundary of each sheet, decrease monotonically with increasing \ensuremath{\omega}, and approach zero linearly at the upper boundary. The resulting two-spinon part of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime $0<{Q}_{\ensuremath{\kappa}}<~\ensuremath{\pi}/2,$ in contrast, the two-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. In the associated two-spinon line shapes of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega}),$ the linear cusps at the continuum boundaries are replaced by square-root cusps. Existing perturbation studies have been unable to capture the physics of the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2.$ However, their line-shape predictions for the regime $0<~Q<{Q}_{\ensuremath{\kappa}}$ are in good agreement with the exact results if the anisotropy is very strong. For weak anisotropies, the exact line shapes are more asymmetric.

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Nonequilibrium dynamics of random field Ising spin chains: Exact results via real space renormalization group

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Gene F. Mazenko

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Dynamics of three-dimensional Ising spin glasses in thermal equilibrium

Andrew T. Ogielski

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Physical review. B, Condensed matterDec 1, 1985773 citations

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Anna Maraga

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We first consider the perturbative expansion in the interaction strength of the long-wavelength structure factor S(q) and show that it does not satisfy the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q) has an analytical component ∼ const. + O(q^2) for q → 0, compatible with thermalization, but it retains also a nonanalytic term ∼ |q| characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity turns in a power-law decay of the density-density correlations, in contrast with the exponential decay associated to thermalization. We next analyze the low-density case, where one can obtain results valid at any order in interaction but at leading in the density, and find that in the steady state the momentum distribution jump at the Fermi surface is strictly finite, though smaller than at equilibrium. -Chapter 3: Nonadiabatic stationary behavior in a driven Ising chain. 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In this Chapter we study the dynamics of the O(N) model (introduced in the previous Chapter) resulting from a different protocol: a linear ramp of one of its parameters. We find that the presence of a dynamical phase transition, as well as its critical properties, are robust against the change of the protocol. We show that a characterization based on the critical dimensions and exponents would suggest that the dynamical phase transition is analogous to the equilibrium thermal one. However, its nonequilibrium nature becomes evident analyzing the statistics of excitations produced in the ramp process. In particular, the critical properties are encoded in the fluctuations in the number of excitations, which display qualitatively different behaviors depending on the ramp being performed above, at, or below the dynamical critical point. These behaviors bear no dependence on the duration of the protocol.

Oct 30, 20150 citations

Coarsening in theq-state Potts model and the Ising model with globally conserved magnetization

Clément Sire, Satya N. Majumdar

We study the nonequilibrium dynamics of the q-state Potts model following a quench from the high-temperature disordered phase to zero temperature. The time-dependent two-point correlation functions of the order parameter field satisfy dynamic scaling with a length scale L(t)\ensuremath{\sim} ${\mathit{t}}^{1/2}$ . In particular, the autocorrelation function decays as [L(t)${]}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}(\mathit{q})}$. We illustrate these properties by solving exactly the kinetic Potts model in d=1. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general q and d. We establish a correspondence between the two-point correlations of the q-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization (2/q-1). The dynamics of this Ising model is solved exactly in the large q limit and in the limit of a large number of components n for the order parameter. For general q and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent \ensuremath{\lambda}(q). These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.We study the nonequilibrium dynamics of the q-state Potts model following a quench from the high-temperature disordered phase to zero temperature. The time-dependent two-point correlation functions of the order parameter field satisfy dynamic scaling with a length scale L(t)\ensuremath{\sim}${\mathit{t}}^{1/2}$. In particular, the autocorrelation function decays as [L(t)${]}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}(\mathit{q})}$. We illustrate these properties by solving exactly the kinetic Potts model in d=1. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general q and d. We establish a correspondence between the two-point correlations of the q-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization (2/q-1). The dynamics of this Ising model is solved exactly in the large q limit and in the limit of a large number of components n for the order parameter. For general q and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent \ensuremath{\lambda}(q). These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.

arXiv:cond-mat/9503163Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJul 1, 199578 citations

{\mathit{T}}_{\mathit{I}}

, that is suddenly placed in contact with a heat bath at temperature

{\mathit{T}}_{\mathit{F}}

, with

{\mathit{T}}_{\mathit{F}}

\ensuremath{\gg}

{\mathit{T}}_{\mathit{I}}

. We present detailed results for the case of

{\mathit{T}}_{\mathit{F}}

=\ensuremath{\infty}, for which we have succeeded in calculating the exact form of S(q,t). For finite

{\mathit{T}}_{\mathit{F}}

{\mathit{T}}_{\mathit{I}}

at the edge of the BZ, decays exponentially with time,

{\mathit{e}}^{\mathrm{\ensuremath{-}}\mathit{t}/{\mathrm{\ensuremath{\tau}}}_{\mathit{q}}}

while approximately preserving its shape in q space, since the lifetime

{\mathrm{\ensuremath{\tau}}}_{\mathit{q}}

{\mathit{T}}_{\mathit{F}}

, \ensuremath{\chi}(q,

{\mathit{T}}_{\mathit{F}}

), in that S(q,t)/\ensuremath{\chi}(q,

{\mathit{T}}_{\mathit{F}}

) is independent of q, but is time dependent, slowly approaching unity as

{\mathit{t}}^{\mathrm{\ensuremath{-}}1/2}

for large t. For q\ensuremath{\approxeq}0 the initial form of S remains in effect until the value of t is of order

{\mathit{q}}^{\mathrm{\ensuremath{-}}2}

. For ferromagnetic coupling, the initial Bragg peak for q\ensuremath{\approxeq}0 does not preserve its shape while decaying exponentially, since the lifetime

{\mathrm{\ensuremath{\tau}}}_{\mathit{q}}

strongly depends on the wave-vector q, diverging as

{\mathit{q}}^{\mathrm{\ensuremath{-}}2}

for q\ensuremath{\rightarrow}0, and, in particular, it is as though the spins for q\ensuremath{\approxeq}0 remain

frozen'' at ${\mathit{T}}_{\mathit{I}}$. Analogous to the behavior for antiferromagnetic interactions, away from the center of the BZ, we find that S(q,t)/\ensuremath{\chi}(q,${\mathit{T}}_{\mathit{F}}$) is independent of q and is a function of t/${\mathit{t}}_{\mathit{w}}$, very slowly approaching unity. The characteristic

waiting time''

{\mathit{t}}_{\mathit{w}}

is anomalously long, proportional to

{\ensuremath{\xi}}^{2}

, where \ensuremath{\xi} is the equilibrium correlation length at temperature

{\mathit{T}}_{\mathit{I}}

. This behavior of

{\mathit{t}}_{\mathit{w}}

can be related to the random walk of domain boundaries. \textcopyright{} 1996 The American Physical Society.

The exact two-spinon part of the dynamic spin structure factor ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ for the one-dimensional

s=1/2,

\mathrm{XXZ}

model at

T=0

in the antiferromagnetically ordered phase is calculated using recent advances in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The two-spinon excitations form a two-parameter continuum consisting of two partly overlapping sheets in $(Q,\ensuremath{\omega})$ space. The spectral threshold has a smooth maximum at the Brillouin zone boundary $(Q=\ensuremath{\pi}/2)$ and a smooth minimum with a gap at the zone center

(Q=0).

The two-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the two-spinon transition rates, the two regimes $0<~Q<{Q}_{\ensuremath{\kappa}}$ (near the zone center) and ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2$ (near the zone boundary) must be distinguished, where ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}0$ in the Heisenberg limit and ${Q}_{\ensuremath{\kappa}}\ensuremath{\rightarrow}\ensuremath{\pi}/2$ in the Ising limit. In the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2,$ the two-spinon transition rates relevant for ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ are finite at the lower boundary of each sheet, decrease monotonically with increasing \ensuremath{\omega}, and approach zero linearly at the upper boundary. The resulting two-spinon part of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega})$ is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime $0<{Q}_{\ensuremath{\kappa}}<~\ensuremath{\pi}/2,$ in contrast, the two-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. In the associated two-spinon line shapes of ${S}_{\mathrm{xx}}(Q,\ensuremath{\omega}),$ the linear cusps at the continuum boundaries are replaced by square-root cusps. Existing perturbation studies have been unable to capture the physics of the regime ${Q}_{\ensuremath{\kappa}}<Q<~\ensuremath{\pi}/2.$ However, their line-shape predictions for the regime $0<~Q<{Q}_{\ensuremath{\kappa}}$ are in good agreement with the exact results if the anisotropy is very strong. For weak anisotropies, the exact line shapes are more asymmetric.

I present an analysis of the dynamic behavior of short-range Ising spin glasses observed in stochastic simulations. The time dependence of the order parameter q(t)=〈

{S}_{x}

(0)

{S}_{x}

(t)〉\ifmmode\bar\else\textasciimacron\fi{}---which is the same as that of the structure factor---and the time dependence of the related dynamic correlation functions have been recorded with good statistics and very long observation times. The spin-glass model with a symmetric distribution of discrete nearest-neighbor \ifmmode\pm\else\textpm\fi{}J interactions on a simple-cubic lattice was used. Simulations were performed with a special fast computer, allowing for the first-time investigation of the equilibrium dynamics for a wide range of temperatures (0.7\ensuremath{\le}kT/J\ensuremath{\le}5.0) and lattice sizes (

{8}^{3}

{16}^{3}

{32}^{3}

, and

{64}^{3}

). I have found that the empirical formula q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$exp(-\ensuremath{\omega}${t}^{\ensuremath{\beta}}$) with temperature-dependent exponents x(T) and \ensuremath{\beta}(T) describes the decay very well at all temperatures above the spin-glass transition. In the spin-glass phase, only the algebraic decay q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$ could be observed, with different temperature dependences of the exponent x(T). The dynamic scaling hypothesis and finite-size scaling explain well the observed temperature and size dependence of the data, and the functional form of the correlation functions is com- patible with the scaling form if corrections to scaling are taken into account. The scaling behavior and the dynamic and static critical exponents found in my simulations are in reasonable agreement with recent experiments performed on insulating spin glasses, showing that despite its simplicity the discrete model of spin glasses analyzed in this work displays behavior similar to that seen in nature.

{\mathit{t}}^{1/2}

. In particular, the autocorrelation function decays as [L(t)${]}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}(\mathit{q})}$. We illustrate these properties by solving exactly the kinetic Potts model in d=1. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general q and d. We establish a correspondence between the two-point correlations of the q-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization (2/q-1). The dynamics of this Ising model is solved exactly in the large q limit and in the limit of a large number of components n for the order parameter. For general q and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent \ensuremath{\lambda}(q). These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.