Time-dependent Order Parameter Research Articles

Domain growth and finite-size-scaling in the kinetic Ising model

This paper describes the application of finite-size scaling concepts to domain growth in systems with a non-conserved order parameter. A finite-size-scaling ansatz for the time-dependent order parameter distribution function is proposed, and tested with extensive Monte-Carlo simulations of domain growth in the 2-D spin-flip kinetic Ising model. The scaling properties of the distribution functions serve to elucidate the configurational self-similarity that underlies the dynamic scaling picture. Moreover, it is demonstrated that the application of finite-size-scaling techniques facilitates the accurate determination of the bulk growth exponent even in the presence of strong finite-size effects, the scale and character of which are graphically exposed by the order parameter distribution function. In addition it is found that one commonly used measure of domain size-the scaled second moment of the magnetisation distribution-belies the full extent of these finite-size effects.

Schwinger pair production in the orbital dynamics of 3He-B

When the B-phase order parameter precesses in an external magnetic field, pairwise excitation of spin 1 2 quasiparticles with opposite momentum will occur, in complete analogy to particle physics, where electron-positron pairs are created from the vacuum by a strong electric field. In superfluid 3He the analog of the electric field vector is represented by the time derivative of the orbital momentum of Cooper pairs. It turns out that the effective electric force acting on quasiparticles subject to a time-dependent order parameter is substantially stronger than in particle physics. As a result the pair creation probability at low temperature is predicted to govern the magnetic relaxation in the nonlinear NMR regime. Under certain conditions the distribution of the emitted quasiparticles appears to be thermal, with the temperature proportional to the precession frequency. This is interpreted as an analog to the Unruh effect in particle physics, where accelerated objects in the vacuum emit thermal radiation with temperature proportional to the acceleration.

Computer simulation of models for orientational glasses

Monte Carlo studies of two- and three-dimensional lattice models where quadrupoles interact with a nearest-neighbor Gaussian coupling are reviewed. None of these models has a thermodynamic glass phase transition at non-zero temperature like the Ising spin glass: rather, phase transitions at zero temperature occur that exhibit a dynamical freeze-in spread out over a wide temperature range and are characterized by a strongly non-exponential relaxation. The time-dependent glass order parameter, q(t), decays with time, t, compatible with a stretched exponential decay q( t) ∼ exp [− ( t/ τ) y ] with a strongly temperature-dependent exponent. While the static glass ‘susceptibility’ for isotropic orientational glasses diverges as x G ∼ T − γ 0 as T → 0, for the three-dimensional three-state Potts glass an exponential divergence is found, 1n x G ∼ T −2, implying that the system is at its lower critical dimension.

Time-dependent Landau theory for order/disorder processes in minerals

AbstractRecent advances in the use of time-dependent order parameter theory to describe the kinetics of order/disorder transitions are reviewed. The time dependence of a macroscopic order parameter,Q, follows, to a good approximation:For systems in which the order parameter has a long correlation length (large ξ) and is not conserved (small ξC), the Ginzburg-Landau equation provides a general kinetic solution:Specific rate laws can be derived from this general solution depending on whether the crystals remain homogeneous with respect to the order parameter,Q. The advantages of the overall approach are, firstly, that it does not depend on the detailed structure of the material being examined; secondly, that the order parameter can be followed experimentally through its relationship with other properties, such as spontaneous strain, excess entropy, intensities of superlattice reflections, etc.; and, finally, that conventional Landau expansions inQmay be used to describe the thermodynamic driving forces.For a simple second-order transition in crystals which remain homogeneous inQthe rate law is:If the free energy of activation varies with the state of order of the crystal, this becomes:Simplifying assumptions can be introduced into the mathematics, or the integrals can be solved numerically. For crystals which remain homogeneous, the simplest solution valid only over small deviations from equilibrium is:For crystals which develop heterogeneities inQ, the rate laws change significantly and we find as an extreme case:where theAcoefficient may be temperature dependent.Experimental data available for a limited number of minerals (omphacite, anorthite, albite, cordierite and nepheline) are used to demonstrate the practical implications of the overall approach. As anticipated from the theory, modulated structures commonly develop during kinetic experiments, the observed rate laws depend on whether the critical point of the ordering is located at the centre or boundary of the Brillouin zone, and the rate laws for ordering and disordering can be quite different. The importance of different length scales, not only in the different techniques for characterizing states of order (IR, NMR, calorimetry, X-ray diffraction, etc.) but also for interpreting observed mechanisms and rate laws, is also outlined.Use of the order parameter in Landau expansions and in Ginzburg-Landau rate laws provides, in principle, a means of predicting the equilibrium and non-equilibrium evolution of minerals in nature.

Time dependent magnetic response in RBa/sub 2/Cu/sub 3/O/sub 7- delta /: effect of magnetic ordering

High-T/sub c/ superconductors show time-dependent magnetic response due mainly to flux flow. In an investigation of the flux flow at temperatures far above any magnetic transition, no difference was found between the time-dependent response of Y1:2:3 and R1:2:3 (R=Gd, Dy), leading to the conclusion that the paramagnetic susceptibility does not contribute to the time dependence in general. On the contrary, near the antiferromagnetic ordering temperature of Gd1:2:3 and in the region of reduced local moment in Dy1:2:3, an anomalous time dependence is observed which can only be explained by assuming a time-dependent paramagnetic susceptibility. The susceptibility can be interpreted by the assumption of a time-dependent magnetic order parameter, which is an unusual feature of magnetic transitions. A possible explanation is spin-glass-like behavior of the local moments.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Monte Carlo study of the Potts glass with nearest-neighbour random Gaussian interaction

The three-state Potts model with nearest-neighbour random Gaussian interaction on the simple cubic lattice is investigated by Monte Carlo simulation. Both static quantities (e.g. glass ordering susceptibility and the correlation function) and dynamic quantities (the analogue of the time-dependent Edwards-Anderson order parameter q(t) for Potts spins) are obtained. As for related models for orientational glasses, it is found that q(t) is consistent with the Kohlrausch law q(t) varies as exp(-(t/ tau )y) for a wide range of temperatures, with a strongly temperature-dependent exponent y, with y becoming very small as the temperature T to 0. The relaxation time tau increases dramatically as T is lowered; thus the system could only be equilibrated for temperatures where the correlation length is rather small. Since the critical region has not been reached, it cannot be distinguished whether the critical temperature Tc is non-zero or at T=0. If Tc=0, the divergences of tau and the ordering susceptibility are probably exponential, i.e. the system is then at its lower critical dimensionality.

Stability of a supercooled liquid to periodic density waves and dynamics of freezing

We present a theoretical analysis of the dynamic stability of a supercooled simple liquid to finite amplitude, periodic, density waves. The analysis is based on a nonlinear diffusion equation (NLDE) which is known to provide an accurate hydrodynamic description of the density fluctuations occurring on a molecular length scale. The NLDE leads to a system of coupled, nonlinear equations for the temporal evolution of the time-dependent order parameters. These equations are then cast in a form appropriate for perturbative evaluation of the generalized relaxation rates. Numerical values of the relaxation rates are obtained for the bcc and the fcc density waves and also for the 2d hcp lattice. A stability limit is reached when the relaxation rates of all the order parameters go to zero simultaneously. The values of the amplitudes of the density waves at the stability limit are found to remain finite at large supercooling, indicating the absence of a classical spinodal point for a simple liquid. The coupling between various density modes at the reciprocal lattice vectors of the solid is found to play an important role in the dynamics of ordering in a supercooled liquid. We find that because of these mode coupling effects, there is a dynamical constraint on the formation of the bcc phase. Such a constraint is absent for simple fcc forming systems. The relation of the present work with nucleation is discussed.

Order parameters of the spin glass mean field theory and initial conditions

The time-dependent order parameter q(t2-t1) is re-examined as a function of the time difference t2-t1 and number of spins N, for the infinite-range spin glass. In the statistical mechanics limit, t2-t1 to infinity before N to infinity , the authors find q(t2-t1)=q= integral q(x)dx, where q(x) is the order parameter function of Parisi (1979, 1980, 1983), provided the system has a canonical (Boltzmann) distribution at the earlier time t1. In the opposite limit, N to infinity before t2-t1 to infinity , they obtain q(t2-t1)=q(x=1). If the initial distribution differs slightly from a canonical one they obtain q(t2, t1)=q(x=0) (i.e. the statistical mechanics limit) and the Sompolinsky solution (1981) appears to be correct under these (non-equilibrium) conditions.

The connection between the measured glass transition temperature and the time dependent order parameter description of the glass transition

The time dependent order parameter concept of the glass transition is used in a description of the relaxation behaviour of the glassforming material. The connection between the macroscopic theory and the measured glass transition temperature has been determined. It has been found that in a case when the Prigogine-Defay ratio is greater than unity, different values of the relaxation functions of entropy and volume obtained by different experiments (e.g., isobaric dilatometry and isothermal compression) should be used in order to obtain the identical Tg. The Prigogine-Defay ratio is found to be a decreasing function of the observation time reaching values between unity in the limit of infinite time and a limit which corresponds to extremely short experiments.

Nonlinear relaxation at first-order phase transitions: A Ginzburg-Landau theory including fluctuations

Kinetics of phase transitions are investigated in systems with nonconserved one-component order parameter (i.e., generalized time-dependent Ginzburg-Landau models ind dimensions). The correct static critical behavior as well as fluctuation effects on the kinetics are incorporated by a suitable adaptation of the theory on spinodal decomposition by Langer, Baron and Miller. Both the case of quenches from temperaturesT above to below the critical pointT c and the case of magnetic field changesΔH from positive to negative values are treated, and both time-dependent order parameter 〈m(τ)〉 and structure factorS(q, τ) are obtained numerically ford=2, 3. In the case of quenches atH=0, we find that 〈m(τ)〉≡0 andS(q, τ) —S(q, ∞)∝ exp(−τ1/2/7.2)\(\tilde S(q\tau ^{ - 1/2} )\), withS(q, ∞)∝q−2. In the case of field changes we find that forH not exceeding some critical valueH* the system is trapped in a metastable state with infinite lifetime. In contrast to the meanfield-spinodal, the susceptibility does not seem to diverge atH*. These results are compared with other treatments, in particular the Monte Carlo simulations of kinetic Ising models by Binder and Muller-Krumbhaar. While our theory describes some properties of the metastable states reasonably,H* distinctly exceeds the observed limit of metastability. We argue that the present theory does not take into account nucleation fluctuations, and also fails to describe correctly the domain growth in the late stages of the relaxation. Contrary to Langer et al. we suggest that “universality” holds for nonlinear relaxation and spinodal decomposition nearT c .

Analysis of structural phase transitions by computer simulation

This paper presents the results from molecular-dynamics calculations on a lattice-dynamical system which undergoes a structural phase transition. The characteristics of the model system used for the calculations are described in detail; the model is two dimensional, has an antiferro-distortive structural change, and is in the category of displacive structural transitions. The following results are obtained from the molecular-dynamics calculations. First, some equilibrium properties are shown, including the static correlation function, to establish that the system has a second-order phase transition. Second, the results for the time-dependent order parameter or soft-mode correlation function are given. Near the transition the spectral function for this correlation function exhibits a very narrow and intense central peak, in addition to the soft-mode peak. The temperature dependences of the soft-mode frequency and the central-peak width are given. The results obtained here are very similar to experimental observations, particularly on SrTi${\mathrm{O}}_{3}$, and they are used to argue that only intrinsic anharmonic mechanisms are needed to explain the origin of central peaks. Third, extensive results are given for the temperature and wave-vector dependence of the displacement correlation function, which give the wave-vector dependence of the central-peak characteristics. In a small temperature interval around the transition and in a small region of wave-vector space around but not including the soft-mode wave vector, the central peak is found to split so that the maximum is not exactly at zero frequency. Lastly, results for the energy-density correlation function are given for wave vectors around the soft-mode wave vector and for temperatures above the transition. Near the transition the corresponding spectral function is found to develop a pronounced high-frequency peak.

Critical Fluctuations in Superconductors: A Functional-Integral Approach

Using the functional-integral method applied to the BCS Hamiltonian we have calculated the partition function, off-diagonal correlation function, specific heat, and conductivity of a super-conductor in the transition region. The partition function is obtained in the form $\ensuremath{\int}D\ensuremath{\psi}{e}^{\ensuremath{-}\ensuremath{\beta}F[\ensuremath{\psi}]}$, where $F$ is of the Ginzburg-Landau form but with a time-dependent order parameter. An examination of the coefficient $b$ of the fourth-order term in $F$, however, shows that $b$ is sharply peaked at zero frequency, thus justifying the use of a time-independent order parameter. We then apply a self-consistent mean-field approximation similar to that of Mar\ifmmode \check{c}\else \v{c}\fi{}elja to the fourth-order term. The self-consistency condition determines the mean-square fluctuation in the order parameter and a renormalized temperature shift in terms of which the nature of the phase transition (or lack of it) can be understood. We find that only in three dimensions does a phase transition strictly occur, at a slightly lowered transition temperature with critical indices $\ensuremath{\gamma}=2$, $\ensuremath{\nu}=1$. In two, one, and zero dimensions the mean-square fluctuation in the order parameter is bounded for $T&gt;0$ and the mean value of the order parameter vanishes, indicating that the transition is completely suppressed in samples of reduced dimensionality. This behavior is reflected in the off-diagonal correlation function which manifests off-diagonal long-range order only in three dimensions. The specific heat in three dimensions is found to be finite at ${T}_{c}$, with critical index $\ensuremath{\alpha}=\ensuremath{-}1$. In two, one, and zero dimensions the specific-heat transition is rounded relative to the BCS result. We find an anomalous factor of 2 in the specific-heat calculation deriving from our treatment of the fourth-order term in $F$. As a result we conjecture that the Hartree approximation of Mar\ifmmode \check{c}\else \v{c}\fi{}elja may be better than the Hartree-Fock approximation of Tucker and Halperin. The equations for the conductivity are identical to those of Mar\ifmmode \check{c}\else \v{c}\fi{}elja.

Energy currents in superconductors

Various inadequacies and inconsistencies of previous treatments of heat flow in a nonequilibrium superconductor are pointed out. The energy current associated with a quasistatic time-dependent order parameter is calculated near the transition temperature using standard techniques and a result obeying the equation of continuity is obtained. Further microscopic calculations indicate the existence of local temperature gradients which drive the entropy current. An attempt is made to calculate corresponding additional terms in the time-dependent Ginzburg-Landau equation.