Short-time dynamics of finite-size mean-field systems
We study the short-time dynamics of a mean-field model with non-conserved order parameter (Curie-Weiss with Glauber dynamics) by solving the associated Fokker-Planck equation. We obtain closed-form expressions for the first moments of the order parameter, near to both the critical and spinodal points, starting from different initial conditions. This allows us to confirm the validity of the short-time dynamical scaling hypothesis in both cases. Although the procedure is illustrated for a particular mean-field model, our results can be straightforwardly extended to generic models with a single order parameter.
pacs:64.60.Ht, 05.10.Gg, 64.60.My, 64.60.an
Universal scaling behavior appears to be an ubiquitous property of critical dynamic systems. While initially believed to hold only in the long time limit, it was realized during the last decade that the dynamical scaling hypothesis can be extended to the short-time limit JaScSc1989 (). This is accomplished by assuming that, close to the critical point, the moment of the order parameter obeys the homogeneity relation
where is time, is the reduced temperature , is the linear system size, is the initial value of the order parameter and is a spatial rescaling parameter. is a universal exponent that describes the short-time behavior, while , , and are the usual critical exponents. When we recover the usual dynamic scaling relation, from which a power law relaxation at the critical point (for instance, in the magnetization ) when and follows. This is the critical slowing down.
On the other hand, the short-time dynamics (STD) scaling properties of the system depend on the initial preparation, i.e., on the scaling field . Setting , from Eq. (1) one obtains for small (but non-null) values of in the large limit
Hence, at the critical point an initial increase of the magnetization is observed. For the second moment the dependency on can be neglected when . Since in the large limit ( is the spatial dimension), one obtains at the critical point
The short-time universal scaling behavior has been verified in a large variety of critical systems both by renormalization group (RG) calculations JaScSc1989 (); PrPrKaTs2008 () and Monte Carlo (MC) numerical simulations Zh1998 (); Zh2006 (); PrPrKrVaPoRy2010 (). The hypothesis also applies when the system starts in the completely ordered state, i.e., . In this case it is assumed that the homogeneity relation
holds even for short (macroscopic) time scales. Hence, in the large limit we have
and taking the derivative of ,
While the scaling hypothesis starting from the disordered state is supported both by numerical simulations and RG, its validity for an initial ordered state relies up to now only on numerical simulations.
Recently, numerical simulations have shown that the short-time scaling hypothesis (1) holds not only close to a critical point, but also close to spinodal points in systems exhibiting a first-order phase transition, both for mean-field and short-range interactions models LoFeCaGr2009 (). This is particularly interesting, because it suggests the existence of some kind of diverging correlation length associated to a spinodal point. Since the proper concept of spinodal in short-range interactions systems is still a matter of debate (see Ref. LoFeCaGr2009 () and references therein), a deeper understanding of the microscopic mechanisms behind the observed short time scaling could shed some light on this problem. One way of achieving this goal is to look for exact solutions of particular models. A first step in that direction is to analyze mean-field (i.e., infinite-range interactions) models, for which the concept of spinodal is well defined LoFeCaGr2009 (). That is the objective of the present work: we analyze the exact STD behavior of far from equilibrium mean-field systems with non-conserved order parameter.
Non-equilibrium phenomena in physics and other fields are commonly studied through Fokker-Planck equations (FPEs). In particular, non-equilibrium dynamical aspects of phase transitions can be analyzed by means of the FPE associated to the master equation describing the microscopic dynamics binder (); hanggi (); munoz (). In fact, this tool was proved to be useful in the description of the relaxation of metastable states binder (), finite-size effects ruffo () or the impact of fluctuations in control parameters politi (), and have been considered for mean-field spin models binder (); mori () and coupled oscillators ruffo (), amongst many others.
As soon as the degrees of freedom of the system can be reduced to a few relevant ones, a low-dimensional FPE can be found. Although this description is suitable for properties that do not depend on the details of the dynamics, or for mean-field kinetics, many conclusions are expected to hold in more general instances.
For a single order parameter , the FPE for its probability is
where the drift and diffusion coefficients are determined by the Hamiltonian and the particular dynamics (e.g., Glauber or Metropolis).
Following this stochastic approach, here we study the scaling of the short-time relaxational dynamics in the vicinity of critical and spinodal points. In first approximation, the drift () is generically linear in the vicinity of a critical point and quadratic in the spinodal, following the quadratic and cubic behavior of the drift potential , respectively. Meanwhile, typically in various models, the noise intensity scales as binder (); ruffo (). Therefore, although we will present the STD for a particular spin model, our results can be straightforwardly extended to more general mean-field ones.
Ii Formal FPE solution and moment expansions
The average of an arbitrary quantity can be derived directly from the FPE, by means of the adjoint Fokker-Plank operator , as follows
Therefore, the first two moments of the order parameter are
where and their derivatives are evaluated in . Notice that if and are not state-dependent, the expansion up to first order is exact.
Alternatively, evolution equations for moments can be obtained by integration of Eq. (7), after multiplying each member of the equation by the quantity to be averaged, that is
For we have
Eqs.(10) lead in general to a hierarchy of coupled equations for the moments. Only for a few special cases ( and polynomials in of degree smaller or equal than one and two respectively) these equations decouple. Otherwise, one has to rely on approximated methods to solve their dynamics.
Iii Paradigmatic mean-field model
Let us exhibit our STD analysis for the paradigmatic system of fully connected Ising spins (Curie-Weiss model), subject to a magnetic field , ruled by the mean-field Hamiltonian
Since the Hamiltonian depends only on the total magnetization , the master equation for this model can be written in closed form for binder (); mori (). In the large limit, when the magnetization per spin can be taken as a continuous variable, an expansion of the master equation up to first order in the perturbative parameter leads for the Glauber dynamics to a FP equation (7) with mori ()
where we have defined , with .
In the next sections we derive asymptotic solutions of the FPE with these coefficients, both close to the critical point ( and ) and to spinodal points for . Analytical results are compared against Monte Carlo simulation ones using Glauber algorithm. Time was adimensionalized with the characteristic time of the transition rate . The unit of time in theoretical expressions corresponds to one MC step in simulations. We also performed several checks using Metropolis algorithm. The outcomes were indistinguishable from the Glauber ones, except for a trivial time rescaling factor 2 close to the critical point, as expected binder ().
Iv STD near the critical point
In the vicinity of the critical point (at and ), the coefficients (13) can be approximated for small (i.e., ) respectively by
where and , with .
Within the domain of validity of these approximations and therefore Concerning , its linear term dominates, that is,
This implies a parabolic approximation of the drift potential , whose shape is plotted in Fig. 1 for different values of , found from the integration of in Eq. (13) and of the linearized expression(15), for comparison. For , one has a confining quadratic potential, while for the parabolic potential is inverted, with an unstable point at .
iv.1 Ornstein-Ulhenbeck approximation
where . This solution applies for (Ornstein-Uhlenbeck (OU) process) as well as for , and is valid as long as the probability distribution remains strongly picked so that the inequality (16) holds for any value of with non-negligible probability.
Performing the average with Eq. (17) gives
Therefore, for , that is , the average magnetization decays (grows) exponentially, with characteristic time . Then, for time scales , it remains . Since in the large limit , then the magnetization scales as . This is consistent with Eq. (2), provided that and , in agreement with the mean-field exponents and . The same exponents are displayed by the Gaussian model JaScSc1989 (). For higher-order moments with even , one has
Then, for short times ,
Hence, , consistently with Eq. (3) (), provided that we choose , the upper critical dimension.
The characteristic time scale for STD behavior is then with
If we have , while for we have .
Fig. 2 displays the comparison between numerical simulations and the approximate OU solutions Eqs. (18)-(19), for , and different values of , such that . The OU approximation gives an excellent agreement for time scales up to ( for the present parameter values). Averages were taken over 1000 independent MC runs. The main differences between the theoretical and numerical results appear for and , where finite-size effects shift the equilibrium value of both the average magnetization and its variance.
Fig. 2 also shows the performance of Eq. (22), which reproduces the simulation results for longer times than Eq. (18), predicting the transient steady state. The lower saturation level observed in simulation outcomes for is due to the presence of fluctuations that drive some trajectories to the equilibrium state with negative magnetization, while the deterministic equation rules the stabilization at the level of the local minimum. Also notice that this discrepancy decreases as departs from the critical value because of the consequent increase of the potential barrier height, which makes such events less probable. For , the system evolves quickly towards the vicinity of the equilibrium state and the saturation level of the second moment is very close to the value given by the (bimodal) steady state distribution . In any case, finite-size higher order corrections can be neglected as far as the STD behavior is concerned.
iv.2 Quartic approximation of the drift potential
In Appendix A we additionally show that finite-size corrections do not change the STD scaling of . For the second moment we obtain
where . Notice that up to a typical time scale , the approximation holds. For , a crossover to a second linear (hence normal diffusive) regime but with a different diffusion constant is predicted, namely , although it typically falls beyond the STD region.
iv.3 Other initial conditions
To investigate the scaling behavior for other initial conditions, we analyzed the STD behavior when . As can be seen in the inset of Fig. 1, the cubic approximation still holds close to . Hence, the thermodynamic-limit expression (22) is expected to apply too, as verified in Fig. 3a. In comparison with the initial condition of Fig. 2, here trajectories get more trapped around the positive minimum, hence the agreement with deterministic Eq. (22) is still better. For finite systems, the intensity of the fluctuations is state dependent following Eq. (13). Therefore, the finite-size corrections derived by assuming do not hold. However, for very short times one still expects , according to Eq. (9), as in fact verified in numerical simulations illustrated in Fig. 3. From Eq. (22) we have that for , in agreement with Eq. (5). The excellent accord between Eq. (22) and numerical simulation outcomes displayed in Fig. 3 when confirms our previous assumptions. Numerical simulations for other values of also verify the above scaling. For the equilibrium (final steady state) values of both mean and variance are quickly approached as in Fig.2. However, when , we see from Fig.3b that all the curves lie below the critical curve, at variance with the behavior observed when (compare with Fig.2b). This is because when almost all the trajectories get trapped in the positive minimum. Thus, the variance stabilizes in a value corresponding to the fluctuations in a single potential minimum. At long enough times, both minima in a finite size system get equally populated and therefore the equilibrium value of will be higher. However, the time scales needed to observe this effect fall outside the STD regime. On the contrary, when , a relatively large number of trajectories cross the barrier between minima and approaches the equilibrium value (which is larger than the steady one), even at very short times, as can be verified by comparing the numerical plateaux in Fig.2b with the equilibrium value
V STD near the spinodal
When the model has a line of first-order transitions at and metastable stationary solutions for a range of values of . Without loss of generality we will restrict hereafter to the metastable solutions with positive magnetization, that is, those analytic continuations of the equilibrium magnetization from positive to negative values of . Defining , the metastable state exists as long as , where the spinodal field is given by
where is the magnetization at the spinodal point LoFeCaGr2009 ().
Suppose now that we start the system evolution from the completely ordered state with and and let us define and . Considering as an order parameter, numerical simulations using Metropolis dynamics LoFeCaGr2009 () showed that close enough to the spinodal point () its moments obey the scaling form (4) with . For temperatures far enough from the spinodal magnetization is close to one and we can expand and in powers of and . Moreover, close to the spinodal we can neglect mori () the finite-size correction of . Then, from Eqs. (13) one has at first order in and second order in :
The moments of can be calculated by means of Eq. (8), namely
where we have defined .
where and . For (hence ), Eq. (29) becomes
Alternatively, Eqs. (29)-(30) can be obtained by integrating Eq. (23), and are in good agreement with numerical simulations, as illustrated in Fig. (5). One observes the following asymptotic behaviors:
(i) For (), a constant level is reached. In fact, since the potential presents a local minimum, the plateau occurs at a level associated to that minimum. This is in accord with numerical simulations (Fig. 5), notice that the local minimum of the potential is at , then , in agreement with the observed level.
(ii) For : (), Eq. (30) yields a rapid decay towards zero attained at finite . This is because the potential is tilted towards the absolute minimum (without local minimum).
In the limit , from Eq. (29) it follows
Hence, at the spinodal point one has for , consistently with Eq. (5) with and , in agreement with previous numerical results LoFeCaGr2009 (). This behavior corresponds to the relaxation towards the saddle point . While in an infinite system such point is an stationary state, finite-size fluctuations destabilize it, with the subsequent exponential relaxation towards the equilibrium value at longer times, as depicted in Fig. 5. Finite-size corrections to Eq. (31), that we compute for , can be obtained by including the diffusion term in Eq. (28). When , following Eq. (27), we have , with , , and . In Appendix B we obtain Eq. (44), furnishing corrected at first order in , that for leads to
Hence, finite-size effects will become relevant only when , with
in agreement with the scaling proposed in Ref. LoFeCaGr2009 (): , with and .
in accord with Eq. (9).
Notice that in this regime, the prefactor of given by Eq. (36) is generically different from that obtained in the very short-time regime following Eq. (34). Fig. (6) illustrates this cross-over for different values of and fixed temperature. The prefactor at small times, varies with (panel a), while at intermediate times the prefactor becomes independently of , which is evident in the linear scale (panel b).
In any case the behavior up to is consistent with the STD scaling hypothesis for the set of mean-field exponents , , and in agreement with numerical outcomes LoFeCaGr2009 ().
Vi Final comments
We studied the short-time dynamical behavior of finite-size mean-field models (infinite-range interactions) with non conserved order parameter dynamics. By solving the associated Fokker-Plank equation we obtained closed expressions for the first moments of the order parameter, in the vicinity of both the critical and spinodal points. This allowed us to confirm the STD scaling hypothesis in both situations, as well as to determine the dynamical ranges of its validity. In particular, we confirmed analytically its validity when the system starts from an ordered state. Moreover, we found that a diffusion-like scaling behavior of the second moment appears for any initial value of the order parameter, but the associated diffusion coefficient presents a crossover between two different values, for short and intermediate times within the STD regime.
We found in general that the scaling behavior of the first moment is mainly determined by the shape of the potential and therefore by the equilibrium generalized free energy , which has the same extrema structure as binder () . The scaling behavior of higher moments, on the other hand, has its origin on the Gaussian nature of finite-size fluctuations close to the singular points. Although our results were obtained for a particular model, it is worth to stress that the above facts are characteristic of mean-field systems, since they depend only on the shape of and on the proportionality . This makes the analysis quite general and independent of the particular mean-field model.
Acknowledgments: The authors would like to thank T. S. Grigera and E. S. Loscar for sharing with us their simulation codes for Metropolis dynamics, as well as for useful discussions. This work was supported by CNPq and Faperj (Brazil), CONICET, Universidad Nacional de Córdoba, and ANPCyT/FONCyT (Argentina).
Appendix A Quartic potential approximation near the critical point
In the limit , we can neglect in a first approximation the diffusion term and compute
By iterating the operator times and identifying the general form of the coefficients of , with the aid of symbolic manipulation programs, we obtain
that coincides with the exact deterministic solution (22).
Fluctuations can be neglected as long as . However, while remains of order one (except for extreme temperatures), typically . Then, a finite-size correction can be included by keeping only the terms of order and in each coefficient of in Eq. (37). This procedure yields the correction term,
where . Then, it results
Notice that the first two terms in the right-hand side come from the expansion of the deterministic Eq. (38) up to first order in .
In particular, exactly at the critical point we have and (hence ). Therefore, as in the case of the OU approximation, one concludes that the magnetization remains up to a characteristic time .
Similarly, for , one obtains the correction
Since in the deterministic limit , then the first two terms in the right-hand side come from the expansion of the squared Eq. (38) up to first order in .
Appendix B Moment calculation near the spinodal
If , from Eq. (8) using , and , the average magnetization is given by
where . Completing squares and making the change of variables with we obtain
Considering the generating function for tangent, with the change of variable , one has tangent ()
To include finite-size effects we have to consider the complete expression
When , from Eq. (27), we have , with , , and . The contributions of order associated to each coefficient of the quadratic approximation of are
where . Summing the -corrections together with the deterministic one, given by Eq. (31), yields
Likewise, we calculate
In this case, the contributions of order are
Summing up the corrections , together with the deterministic term (given by the squared Eq. (31)), yields
Finally, the second moment is obtained through . The purely deterministic terms cancel out and at first order in it remains
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