Finite-Size Scaling in the Ageing Dynamics of the 1D Glauber–Ising Model ^†

Abstract

Single-time and two-time correlators are computed exactly in the $1D$ Glauber-Ising model after a quench to zero temperature and on a periodic chain of finite length N, using a simple analytical continuation technique. Besides the general confirmation of finite-size scaling in non-equilibrium dynamics, this allows for testing the scaling behaviour of the plateau height $C_{\infty }^{\left(2\right)}$, to which the two-time auto-correlator converges when deep in the finite-size regime.

1. Introduction

An important class of physical phenomena arises in the context of ageing phenomena [1,2] which occur after a many-body system has been quenched from some prescribed initial state, either onto a critical point where at least two physical phases become indistinguishable or into a phase co-existence region where two macroscopic physical phases coexist. In either case, the post-quench dynamics are slow, which may come from the effects of critical-point fluctuations or from the competition between relaxation towards at least two distinct physical states. Microscopically, the system separates into many (correlated or ordered) clusters whose mean size, $\mathsf{\ell }\left(t\right)$, grows with time. Phenomenologically, one observes the three defining properties of physical ageing on a macroscopic scale [3]:

• Slow dynamics (relaxations are slower than might be described by simple exponentials);
• Absence of time-translation invariance;
• Dynamical scaling.

These properties manifest themselves in the typical behaviour of correlation functions, which might be thought of in terms of a coarse-grained order-parameter $\varphi =\varphi (t,\mathit{r})$ which depends on the time t and the space coordinates $\mathit{r}$. For example, notably in situations where the average order parameter $⟨\varphi (t,\mathit{r})⟩=0$, one often considers single-time or two-time correlators (which depend on both the waiting time s and the observation time $t>s$)

$$C(t;\mathit{r})=⟨\varphi (t,\mathit{r})\varphi (t,\mathbf{0})⟩,C(t,s;\mathit{r})=⟨\varphi (t,\mathit{r})\varphi (s,\mathbf{0})⟩$$

(1)

We restrict this work to phase ordering, which occurs for a non-conserved order parameter quenched to $T<T_c$. Then, one generically finds, for large enough times (spatial translation and rotation invariance are implicitly admitted)

$$C(t;r):=C(t;|\mathit{r}\left|\right)=F_C\left(1,{\displaystyle \frac{\left|\mathit{r}\right|}{t^{1/z}}}\right),C(t,s;r):=C(t,s;|\mathit{r}\left|\right)=F_C\left({\displaystyle \frac{t}{s}},{\displaystyle \frac{\left|\mathit{r}\right|}{s^{1/z}}}\right)$$

(2)

which is specified here for systems where the typical domain size $\mathsf{\ell }\left(t\right)$∼$t^{1/z}$ increases algebraically for large times. This defines the dynamical exponent z. For a non-conserved order parameter and short-ranged interactions, to which we shall restrict ourselves throughout this work, one has $z=2$ [4,5,6] (For a conserved order parameter, one speaks of phase separation and z takes different values [4,5,6]. Long-range interactions lead to further modifications [5,7,8,9,10,11,12]). Up to metric scale factors, the form of the scaling function $F_C$ is generically expected to be universal, meaning it is independent of most of the ‘details’ of the underlying microscopic physics; see [4,13,14,15,16] for reviews. Knowing the form of $F_C$ is an important theoretical task and is also of practical importance, as a priori knowledge of $F_C$ allows us to make long-time predictions on the basis of short-time data. These expectations (2) are also confirmed experimentally [17,18].

In practical situations, it may be difficult to achieve spatially totally homogeneous samples without any kind of interfaces and/or granular effects. It is therefore of interest to study situations of physical ageing in geometries of finite extent, such as a hyper-cubic form with $N^d$ sites. A simple example illustrates a typical kind of finite-size effect (see Figure 1). If one considers the auto-correlation function (please distinguish carefully between the two-time auto-correlator $C(t,s)$, Equation (3a) below, and the time-space correlator $C(t;r)$, Equation (2)). $C(t,s):=C(t,s;\mathbf{0})$ of a phase-ordering (or phase-separating) system, one finds for a spatially infinite system and for sufficiently large times, where $t,s\gg {\tau }_{\mathrm{micro}}$ and $t-s\gg {\tau }_{\mathrm{micro}}$, that (i) a data collapse occurs and (ii) the characteristic power-law behaviour

$$\begin{array}{cc}\hfill C(t,s)& =F_C\left({\displaystyle \frac{t}{s}},\left|\mathbf{0}\right|\right)=f_C\left({\displaystyle \frac{t}{s}}\right),f_C\left(y\right)\stackrel{y\gg 1}{\sim }{y}^{-\lambda /z}\hfill \end{array}$$

(3a)

for large time ratios $y=t/s>1$. Here, $\lambda$ is the auto-correlation exponent and is independent of the equilibrium critical exponents. A recent list of estimates of z and $\lambda$ for phase-ordering systems can be found in [19]. On the other hand, in a fully finite system, even if the auto-correlator is still close to that of the spatially infinite system for not-too-large y, deviations from (3a) will arise (see again Figure 1). Generically, in a finite system, the auto-correlator should first decrease more rapidly as a function of y than it would in the infinite system. For even larger values of y, the auto-correlator saturates at a plateau of height

$$\begin{array}{c}\hfill \underset{y\to \infty }{lim}C(ys,s;0;N)=C_{\infty }^{\left(2\right)}(s,N)\end{array}$$

(3b)

which, in principle, should depend on the waiting time s and the system size N. Qualitative discussions on this go back a long time (see [20,21]). The systematic study of finite-size effects—by which we mean the consequences of the system being in a restricted spatial volume of linear size N, and the associated finite-size scaling—has a long history indeed, both for equilibrium phase transitions [22,23,24,25] and equilibrium dynamics [26]. Early examples of finite-size studies in non-equilibrium systems include [27,28,29]. Finite-size effects in glassy dynamics are studied theoretically in [30,31,32] and experimentally in [20,21,33,34]. They are also one possible way to artificially create spurious sub-ageing effects [35]). Another early observation of this saturation effect occurs in the Kuramoto model of self-synchronisation [36].

Figure 1. Qualitative dependence of the scaled two-time auto-correlator $C(t,s)$ on the time ratio $y=t/s$ for (i) a spatially infinite system (dashed line) with the power-law behaviour ∼$y^{-\lambda /z}$ and (ii) in a fully finite system (full line) which converges to a characteristic plateau $C_{\infty }^{\left(2\right)}$.

The expected limit behaviour of (3b) can be understood heuristically [19,37]. We recall the argument for quenches to $T<T_c$. For large times, one expects that the auto-correlator in terms of domain sizes $\mathsf{\ell }\left(t\right)$ and $\mathsf{\ell }\left(s\right)$ reads $C(t,s)$∼${\left({\displaystyle \frac{\mathsf{\ell }\left(t\right)}{\mathsf{\ell }\left(s\right)}}\right)}^{-\lambda }$ (see Equation (3a)). If the observation time t becomes so large that the domain size $\mathsf{\ell }\left(t\right)$∼N crosses over into the saturation regime, while the waiting time s is still small enough that the infinite-system rule $\mathsf{\ell }\left(s\right)$∼$s^{1/z}$ applies (hence $\mathsf{\ell }\left(s\right)\ll N$), one would find

$$C_{\infty }^{\left(2\right)}\sim N^{-\lambda }\mathrm{if}s\mathrm{is}\mathrm{fixed},C_{\infty }^{\left(2\right)}\sim s^{\lambda /z}\mathrm{if}N\mathrm{is}\mathrm{fixed}$$

(4)

More formally, (4) is one of the several consequences of the hypothesis of generalised time-translation invariance [38]. Furthermore, one can write generalisations for quenches to all temperatures $T\le T_c$, for both conserved and non-conserved order parameters, and so on. One interesting aspect of (4) is that it offers a new way to numerically estimate the exponents $\lambda$ and $\lambda /z$, respectively.

The present work strives to obtain a test of (4) in the context of an exactly solvable model. Since there already exists an exact confirmation of (4) in the spherical model for dimensions $2<d<4$ and $T<T_c$ [19], we consider here the case of the $1D$ Glauber–Ising model (for a short summary of the rôle of the Ising model in equilibrium phase transitions, discovered by Cagniard de la Tour about 200 years ago, see, e.g., [39] and refs. therein), quenched to temperature $T=0$ from a fully disordered initial state (initial correlations are irrelevant at large times in the $1D$ Glauber–Ising model [40]) and whose single- and two-time correlators obey the scaling form (2). Tests of (4) in the $2D$ Glauber–Ising model quenched to $T<T_c$ will be presented elsewhere [37]. We shall be interested in deriving the full size-dependent single-time and two-time spin–spin correlators which will ultimately allow us to perform an explicit test of (4).

This work is organised as follows: In Section 2, we shall introduce the analytic continuation technique and confirm that it reproduces the known exact results. In Section 3, we shall use this technique to compute the finite-size effects in the ageing dynamics and finally confirm (4). Section 4 gives our conclusions. Technical details of the exact solution are given in Appendix A, Appendix B, Appendix C, Appendix D and Appendix E.

2. Critical Relaxations in Infinite-Size Systems

2.1. The $1D$ Glauber–Ising Model

The nearest-neighbour Ising model on a chain $\Lambda \subset \mathbb{Z}$ is defined through the hamiltonian

$$\mathcal{H}=-\sum _{n\in \Lambda }{\sigma }_n{\sigma }_{n+1}$$

(5)

for the Ising spins ${\sigma }_n=\pm 1$ and the exchange coupling is normalised to unity. In a heat-bath formulation, at temperature T, at each time step $\Delta t$, a randomly chosen site $n\in \Lambda$ is updated according to Glauber dynamics [41] with the rates [42]

$${\sigma }_n\left(t\right)\mapsto \pm 1\mathrm{with}\mathrm{the}\mathrm{probability}{\displaystyle \frac{1}{2}}\left(1\pm tanh{\displaystyle \frac{{\sigma }_{n-1}\left(t\right)+{\sigma }_{n+1}\left(t\right)}{T}}\right)$$

(6)

On a discrete chain, the single-time correlator is $C_n\left(t\right):=⟨{\sigma }_n\left(t\right){\sigma }_0\left(t\right)⟩$, where the average is taken over the thermal histories defined by Equation (6). The correlator obeys the equation of motion [41,42,43,44]

$${\partial }_t{C}_n\left(t\right)=-2C_n\left(t\right)+\gamma \left(C_{n-1}\left(t\right)+C_{n+1}\left(t\right)\right)\mathrm{when}n\ne 0,C_0\left(t\right)=1$$

(7)

with the abbreviation $\gamma =tanh(2/T)$ (such that $0\le \gamma \le 1$), and where the microscopic rate constant is normalised to unity. An initial condition must still be given; for an initially fully disordered system, one has $C_n\left(0\right)={\delta }_{n,0}$ [41,42]. Throughout this work, spatial translation invariance is implicitly admitted. We shall give the periodicity conditions for systems on a lattice of finite size N later. Similarly, with the time difference $\tau =t-s$, the two-time correlator is defined as $C_n(\tau ,s):=⟨{\sigma }_n\left(t\right){\sigma }_0\left(s\right)⟩=⟨{\sigma }_n(\tau +s){\sigma }_0\left(s\right)⟩$ and obeys the equation of motion [41,42,43,44]

$${\partial }_{\tau }{C}_n(\tau ,s)=-C_n(\tau ,s)+{\displaystyle \frac{\gamma }{2}}\left(C_{n-1}(\tau ,s)+C_{n+1}(\tau ,s)\right),C_n(0,s)=C_n\left(s\right)$$

(8)

Herein, the single-time correlator $C_n\left(s\right)$ serves as initial value. Solving Equations (7) and (8) constitutes the mathematical problem for the determination of the single-time and two-time correlators.

2.2. The Discrete Case

There are many well-known ways [40,41,42,43,44,45,46,47,48,49,50,51,52,53] to solve Equations (7) and (8), using, for example, generating functions [41], free fermions [48], a scaling ansatz [45,50], Laplace transforms [42], grassmannian variables [46], or systems of equations of motion [40,43,44,51,52,53]. Here, we shall adopt a method which easily generalises to finite systems as well.

We begin with the single-time correlator (For a disordered initial condition $⟨{\sigma }_n\left(0\right)⟩=0$, the site-dependent magnetisation $⟨{\sigma }_n\left(t\right)⟩=0$ for all times). In principle, one wishes to decouple the equations of motion (7) by a Fourier transform, but because of the boundary condition $C_0\left(t\right)=1$, this is not straightforward. Rather, we observe first that the single-time correlator $C_n\left(t\right)=⟨{\sigma }_n\left(t\right){\sigma }_0\left(t\right)⟩=⟨{\sigma }_0\left(t\right){\sigma }_n\left(t\right)⟩=⟨{\sigma }_{\left|n\right|}\left(t\right){\sigma }_0\left(t\right)⟩$ is even in n and does not depend on the sign of n. It is enough to restrict the physical interpretation to positive values of $n>0$. In particular, the equation of motion (7) is needed for $n>0$ only. Therefore, we consider that $C_{-n}\left(t\right)$ can be thought of as an entity devoid of physical significance. It can therefore be used for purely calculational, mathematical purposes. We shall define $C_{-n}\left(t\right)$ in such a way that the equation of motion (7) holds for all values of $n\in \mathbb{Z}$. To do so, we write down the ansatz

$$C_{-n}\left(t\right)={\alpha }_n-C_n\left(t\right);n\ge 0$$

(9)

and try to choose ${\alpha }_n$ such that the boundary condition $C_0\left(t\right)=1$ and the equation of motion (7) become valid for all $n\in \mathbb{Z}$. This kind of analytical continuation will be used several times below (it had already been used in the exact treatment of $1D$ coagulation-diffusion processes [54,55,56] to find the single-time and two-time correlation and response functions). It is easy to check that this is indeed possible if the ${\alpha }_n$ obey the (time-independent) recursion relation

$${\alpha }_{n+1}={\displaystyle \frac{2}{\gamma }}{\alpha }_n-{\alpha }_{n-1},{\alpha }_0=2,{\alpha }_1={\displaystyle \frac{2}{\gamma }}$$

(10)

Lemma 1.

The solution of the recursion relation (10) is for all $n\ge 0$

$${\alpha }_n={\left({\displaystyle \frac{\gamma }{1-\sqrt{1-{\gamma }^2}}}\right)}^n+{\left({\displaystyle \frac{\gamma }{1+\sqrt{1-{\gamma }^2}}}\right)}^n$$

(11)

Proof. 

This is easily checked by induction. For $n=0$ and $n=1$, the initial values in (10) are immediately reproduced. Then, it is straightforward to check that (11) obeys indeed the recurrence relation (10). □

It then follows that one has for all $n\in \mathbb{Z}$ the linear equation of motion

$${\partial }_t{C}_n\left(t\right)=-2C_n\left(t\right)+\gamma \left(C_{n-1}\left(t\right)+C_{n+1}\left(t\right)\right)$$

(12)

but now in the absence of any boundary condition. This is the goal we wanted to achieve. The solution of Equation (12) is described in Appendix A. It follows that the physical correlator is

$$C_n\left(t\right)=⟨{\sigma }_{\left|n\right|}\left(t\right){\sigma }_0\left(t\right)⟩={\eta }_{-}^{\left|n\right|}+\sum _{m=1}^{\infty }\left(C_m\left(0\right)-{\eta }_{-}^m\right)e^{-2t}\left[I_{\left|n\right|-m}\left(2\gamma t\right)-I_{\left|n\right|+m}\left(2\gamma t\right)\right]$$

(13)

where $I_n\left(t\right)$ is a modified Bessel function [57] and

$$C_{n,\mathrm{eq}}={\eta }_{-}^{\left|n\right|}={\left({\displaystyle \frac{1-\sqrt{1-{\gamma }^2}}{\gamma }}\right)}^{\left|n\right|}$$

(14)

is the equilibrium correlator $C_{n,\mathrm{eq}}={⟨{\sigma }_{\left|n\right|}{\sigma }_0⟩}_{\mathrm{eq}}$. This is indeed nothing else than Glauber’s time-honoured result [41] (Equation (63)). For any $0\le \gamma <1$, it shows the rapid relaxation towards the equilibrium correlator (14) on a finite time-scale ${\tau }_{\mathrm{rel}}^{-1}=2(1-\gamma )$. Formally, ${\tau }_{\mathrm{rel}}$ diverges if $\gamma \to 1$ (or the temperature $T\to 0$), and ageing is only possible in this latter case (see Section 1).

This whole calculation only has the purpose of showing that the analytical continuation technique used here does reproduce the well-known results in the literature [40,41,42,44,51]. Having verified this, we can now proceed towards the derivation of new results, notably on critical dynamics, with $\gamma =1$.

For the two-time correlator, it is straightforward to solve (8) by a Fourier transform and then insert the single-time correlator (13). Since this leads to lengthy expressions [51], which are not needed below, we do not carry this out explicitly.

2.3. The Continuum Limit

1. Having seen that ageing can only occur for $\gamma =1$, we restrict ourselves to this case from now on. The calculations become shorter in the continuum limit (This is valid in the scaling limit of (7) where $t\to \infty$, $n\to \infty$ but $n^2/t$ is kept fixed. This will always be taken in what follows, where t and x are re-scaled time and space coordinates), where, for the single-time correlator $C(t;x)=⟨\sigma (t,x)\sigma (t,0)⟩$, we have from (7) the equation of motion

$${\partial }_{t}C(t;x)={\partial }_x^{2}C(t;x),C(t;0)=1$$

(15)

where the diffusion constant was scaled to unity. Since the physical correlator $C(t;x)$ is even in x, we can restrict the problem (15) to the half-line $x\ge 0$ and use the function $C(t;-x)$ for computational purposes. The analytic continuation (9) then simplifies to

$$C(t;-x)=2-C(t;x)$$

(16)

for $x\ge 0$. In Appendix B, we show that (with the complementary error function $\mathrm{erfc}\left(x\right)$ [57])

$$C(t;x)=\mathrm{erfc}\left({\displaystyle \frac{\left|x\right|}{2t^{1/2}}}\right)+{\displaystyle \frac{e^{-x^2/4t}}{\sqrt{\pi t}}}{\int }_0^{\infty }\mathrm{d}{x}^{\prime }C(0;x^{\prime })e^{-x^{\prime 2}/4t}sinh\left({\displaystyle \frac{\left|x\right|x^{\prime }}{2t}}\right)$$

(17)

remains for any initial correlations $C(0;x)$. For an initially fully disordered lattice, $C(0;x)=0$ for $x>0$ and the second term in (17) vanishes, as stated countless times in the literature (see e.g., [4,41,42,43,44,45,50] and references therein). In order to further clarify its importance, consider as an example $C(0;x)={\left(1+x^2\right)}^{-\aleph /2}$ with a positive exponent $\aleph >0$. For large times, $C(t;x)={\mathfrak{T}}_1+{\mathfrak{T}}_2$, where the contribution of the second term ${\mathfrak{T}}_2$ will be of the order

$${\mathfrak{T}}_2\sim e^{-x^2/4t}{\int }_0^{\infty }\mathrm{d}{x}^{\prime }{t}^{-\aleph /2}{e}^{-x^{\prime 2}/4}sinh\left({\displaystyle \frac{\left|x\right|x^{\prime }}{2t^{1/2}}}\right)\sim t^{-\aleph /2}\to 0$$

when $t\to \infty$ but where $x^2/t$ is kept fixed. This is much smaller than the scale-invariant and finite first term ${\mathfrak{T}}_1=\mathrm{erfc}\left(\left|x\right|/2t^{1/2}\right)$. We conclude that, in the $1D$ Glauber–Ising model at $T=0$, initially decaying correlations are always irrelevant for $t\to \infty$. This merely confirms long-standing results for the two-time correlator in this model [40]. Consequently, only the universal first term in (17) is important in the long-time scaling limit and the initial-condition-dependent second term can be discarded. The leading term in (17) might have been found more readily by inserting the scaling ansatz (2) into (15) [45].

Using the notation of (2), and for an initially fully disordered lattice, the scaled correlator $C(t;x)=F_C(1,|x|/2\sqrt{t})$ is shown in the right panel of Figure 2 as the full black curve, labelled $y=1.0$, as a function of the scaling variable ${\xi }^2=x^2/2t$. For small arguments, one observes a sharp peak at $\xi =0$. This indicates that the interfaces between domains are sharp, as expected for kinetic Ising models and in agreement with Porod’s law [4,14].

Figure 2. Properties of the scaled two-time correlator (2) in the $1D$ Glauber–Ising model. Left panel: Universal decay of the correlator $F_C(y,\xi )$ for large y, with $\xi =[0.0,0.2,0.5,1.0,2.0]$ from top to bottom. The inset shows the expected universal power-law decay (24) for large values of y. Right panel: Decay of the correlator $F_C(y,\xi )$ as a function of $\xi$ for for $y=[1.0,1.5,3.0,4.5]$ from bottom to top on the right of the figure. The inset highlights the expected Gaussian decay for large $\xi$ and the dashed lines indicate the leading decay behaviour (25).

2. In Appendix B, we also compute the characteristic length scale $\mathsf{\ell }\left(t\right)\sim t^{1/z}$ from the second moment of $C(t;x)$ and find

$${\mathsf{\ell }}^2\left(t\right)={\displaystyle \frac{{\int }_0^{\infty }\mathrm{d}x{x}^{2}C(t;x)}{{\int }_0^{\infty }\mathrm{d}xC(t;x)}}={\displaystyle \frac{4}{3}}t$$

(18)

which is exact for a fully disordered initial state. Equation (18) confirms the expected [5] dynamical exponent $z=2$. One might use it in (17) to achieve a data collapse $C(t;x)=\mathrm{erfc}\left(\left|x\right|/\left(\sqrt{3}\mathsf{\ell }\left(t\right)\right)\right)$, up to irrelevant terms.

3. We now turn to the two-time correlator

$$C(\tau ,s;x):=⟨\sigma (\tau +s,x)\sigma (s,0)⟩$$

(19)

For the zero-temperature case $\gamma =1$, we have from (8) the equation of motion

$${\partial }_{\tau }C(\tau ,s;x)={\displaystyle \frac{1}{2}}{\partial }_x^{2}C(\tau ,s;x),C(0,s;x)=C(s;x)=\mathrm{erfc}\left({\displaystyle \frac{\left|x\right|}{2s^{1/2}}}\right)$$

(20)

where, in the initial condition, we merely retain from (17) the most relevant term for large times or use a totally disordered initial state. In what follows, we shall use the scaling variables

$$y={\displaystyle \frac{t}{s}}>1,{\xi }^2={\displaystyle \frac{x^2}{2s}}$$

(21)

for the ratio y of the two times and the re-scaled length $\xi$. Then, the time difference $\tau =t-s=s(y-1)$. From now on, we shall always work in the long-time limit $s\to \infty$, with y and $\xi$ being kept fixed. In Appendix B, we show that (20) is solved by

$$\begin{array}{cc}\hfill C(\tau ,s;x)& =1-{\displaystyle \frac{1}{\pi }}{e}^{-x^2/\left(2\tau \right)}{\left({\displaystyle \frac{2\tau }{s}}\right)}^{1/2}{\Psi }_1\left(1,{\displaystyle \frac{1}{2}};{\displaystyle \frac{3}{2}},{\displaystyle \frac{1}{2}};-{\displaystyle \frac{\tau }{2s}},{\displaystyle \frac{x^2}{2\tau }}\right)\hfill \end{array}$$

(22a)

$$\begin{array}{cc}\hfill & =1-{\displaystyle \frac{2}{\pi }}{e}^{-{\xi }^2/(y-1)}{\left({\displaystyle \frac{y-1}{2}}\right)}^{1/2}{\Psi }_1\left(1,{\displaystyle \frac{1}{2}};{\displaystyle \frac{3}{2}},{\displaystyle \frac{1}{2}};-{\displaystyle \frac{y-1}{2}},{\displaystyle \frac{{\xi }^2}{y-1}}\right)\hfill \end{array}$$

(22b)

where ${\Psi }_1$ is a Humbert confluent hypergeometric function of two variables [58,59]. In the scaled expression (22b), we observe the independence from the waiting time s. In what follows, with a slight change of notation with respect to (2), we shall write more shortly $F_C(y,\xi ):=C(\tau ,s;x)$ in terms of the scaling variables y and $\xi$, defined above in (21). To the best of our knowledge, previous efforts have concentrated on the two-time auto-correlator $C(\tau ,s;0)$ (When considering the scaling influence of the temperature through $\gamma$, the two-time auto-correlator can be expressed via another Humbert function ${\Phi }_1$ [42]. Taking into account all three scaling variables will likely involve a three-argument Lauricella/Horn hypergeometric series).

To recover the known two-time auto-correlator $C(ys,s)=F_C(y,0)$, we now recall that ${\Psi }_1(a,b;c,c^{\prime };x,0)={}_{2}F_1(a,b;c;x)$ reduces to Gauß’ hypergeometric function, which, in turn, is related to the elementary functions in our example [58] (7.3.1.123), [58] (7.3.2.83). This gives

$$F_C(y,0)=1-{\displaystyle \frac{\sqrt{2}}{\pi }}{\left({\displaystyle \frac{\tau }{s}}\right)}^{{\displaystyle \frac{1}{2}}}{\left(-{\displaystyle \frac{\tau }{2s}}\right)}^{-{\displaystyle \frac{1}{2}}}\mathrm{artanh}\sqrt{-{\displaystyle \frac{\tau }{2s}}}={\displaystyle \frac{2}{\pi }}arctan\sqrt{{\displaystyle \frac{2s}{\tau }}}={\displaystyle \frac{2}{\pi }}arctan\sqrt{{\displaystyle \frac{2}{y-1}}}$$

(23)

where [57] (4.4.42) was also used. The last two expressions reproduce the well-known [42] (Equation (4.24)), [43] (Equation (17)), as expected. Specifically, for $y\gg 1$, one has $F_C(y,0)\simeq {\displaystyle \frac{2\sqrt{2}}{\pi }}{y}^{-1/2}$, and by comparison with the standard expected asymptotics (3a), one reads off ${\displaystyle \frac{\lambda }{z}}={\displaystyle \frac{1}{2}}$, or, since $z=2$ [5], the auto-correlation exponent $\lambda =1$ [40,42,43], as it should be. In the left panel of Figure 2, the auto-correlator is the full black line labelled $\xi =0.0$.

Several mathematical identities, listed in Appendix E, can be used to check the expressions (22) and to extract a few consequences. First, the limit relation given in Lemma A1 allows us to confirm that in the limit $\tau \to 0$ or $y\to 1$, the expressions (22) indeed reduce to the single-time correlator (17), without, however, the irrelevant and sub-dominant terms coming from the initial conditions. This is illustrated in the left panel of Figure 2, where the curves become more close to the auto-correlator when $\xi \to 0$. Second, the asymptotic relation given in Lemma A2, especially the independence of the two leading terms on the second argument, describes what happens when $y=t/s$ becomes large. In Figure 2, the left panel shows that the asymptotic behaviour is

$$F_C(y,\xi )\stackrel{y\gg 1}{\simeq }{\displaystyle \frac{\sqrt{8}}{\pi }}{\displaystyle \frac{1}{y^{1/2}}}\left(1+\mathrm{O}\left(y^{-1/2}\right)\right)$$

(24)

and that this leading behaviour is universal in the sense that it is independent of the scaled distance ${\xi }^2={\displaystyle \frac{x^2}{2s}}$. This is reproduced from (A35), since the leading term cancels, but the next order does give (24). Third, one may also investigate the leading behaviour of the correlator when $\xi$ becomes large but y is kept fixed. This is shown in the right panel of Figure 2. While the main figure merely displays a rather rapid decay with increasing $\xi$, the inset shows that the plot of the square root of the logarithm of $F_C$, namely $-\sqrt{-ln{F}_C(y,\xi )}$ against $\xi$, produces an almost straight line as a function of $\xi$, but with a y-dependent slope. Lemma A3 gives the mathematical justification of this observation and, again, the second term in (A36) establishes that for $\xi \gg 1$

$$F_C(y,\xi )\stackrel{\xi \gg 1}{\simeq }{\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle \frac{y+1}{\xi }}exp\left(-{\displaystyle \frac{{\xi }^2}{y+1}}\right)$$

(25)

The dashed lines in the inset represent these asymptotic predictions for several values of $y\ge 1$. Again, if one takes the limit $y\to 1$ in (25), one recovers the leading large-distance asymptotics of the single-time correlator $\mathrm{erfc}(\xi /\sqrt{2})$, as it should be.

3. Critical Relaxations in Finite-Size Systems

Having seen for the infinite system that our technique of analytical continuation, via (9) or (16), reproduces the known results, we shall now apply it to extract the finite-size scaling properties of the ageing dynamics.

1. Consider the Glauber–Ising model on a periodic ring with N sites (see Figure 3). In the continuum limit, the equation of motion for the single-time correlator is now

$${\partial }_{t}C(t;x)={\partial }_x^{2}C(t;x),C(t;0)=C(t;N)=1$$

(26)

The first of these boundary conditions will be treated as before by analytic continuation. For the second one, in the case of periodic boundary conditions, we expect to have

$$C(t;x)=C(t;N-x,C(t;-x)=2-C(t;x)$$

(27)

The first of these is an inversion relation, as suggested by Figure 3. Both together analytically continue $C(t;x)$ from the physical region $x\in [0,N]$ towards $x\in \mathbb{R}$, via (27). Together, they imply the periodicity relation

$$\begin{array}{ccc}\hfill C\left(t;2N+x\right)& =& C\left(t;N-(-N-x)\right)=C\left(t;-N-x\right)\hfill \\ & =& 2-C\left(t;N+x\right)=2-C\left(t;N-(-x)\right)=2-C\left(t;-x\right)\hfill \\ & =& C\left(t;x\right)\hfill \end{array}$$

(28)

where, in the first line, we used the inversion condition (27). In the second line, we applied the second continuation condition (27) and then the inversion once more. Finally, a last application of the second condition (27) brought us to the end result in the third line.

Figure 3. Periodic ring with N sites. Starting from an arbitrary site labelled 0, the property $C(t;x)=C(t;N-x)$ becomes intuitive for $x\ge 1$.

Consequently, while the physical correlator is obtained for values $0\le x\le N$, the analytically continued function $C(t;x)$ is a function of period $2N$ in x, but only half of it has a physical meaning. We therefore have the Fourier representation

$$C(t;x)=\sum _{k=-\infty }^{\infty }\tilde{C}(t;k)e^{\mathrm{i}\pi k{\displaystyle \frac{x}{N}}},\tilde{C}(t;k)={\displaystyle \frac{1}{2N}}{\int }_{-N}^N\mathrm{d}xC(t;x)e^{-\mathrm{i}\pi k{\displaystyle \frac{x}{N}}}$$

(29)

Because of the periodicity condition (28), we specifically have

$$C(t;N)=C(t;-N),{\partial }_{x}C(t;N)={\partial }_{x}C(t;-N)$$

(30)

which are, of course, compatible with the required boundary conditions in (26).

In this way, the boundary conditions are taken into account, such that the analytically continued function $C(t;x)$ is a periodic solution of period $2N$, of the simple diffusion equation ${\partial }_{t}C(t;x)={\partial }_x^{2}C(t;x)$. In Appendix C, we show that for $\left|x\right|\le N$, the physical correlator can be expressed in terms of a Jacobi theta function ${\vartheta }_3(z,q)$

$$\begin{array}{ccc}\hfill C(t;x)& =& {\displaystyle \frac{1}{2N}}{\int }_0^N\mathrm{d}{x}^{\prime }\left\{2{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{\left|x\right|+x^{\prime }}{N}},e^{-{\pi }^{2}t/N^2}\right)\right.\hfill \\ & & \left.+C(0;x^{\prime })\left[{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{\left|x\right|-x^{\prime }}{N}},e^{-{\pi }^{2}t/N^2}\right)-{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{\left|x\right|+x^{\prime }}{N}},e^{-{\pi }^{2}t/N^2}\right)\right]\right\}\hfill \end{array}$$

(31)

and that the required periodicity properties (27) are indeed satisfied, if they only hold for the initial correlator $C(0;x)$. The term depending on the initial conditions $C(0;x)$ should be irrelevant. Then, or for a fully disordered initial state, one may alternatively re-write the physical single-time correlator (2) as

$$C(t;x)=F_C\left(1,{\displaystyle \frac{\left|x\right|}{2\sqrt{t}}}\right)={\int }_0^1\mathrm{d}u{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}u+{\displaystyle \frac{\pi }{2}}{\displaystyle \frac{\left|x\right|}{N}},e^{-{\pi }^{2}t/N^2}\right)=1-{\displaystyle \frac{2}{\pi }}{\int }_0^{\left|x\right|/N}\mathrm{d}v{\vartheta }_2\left(\pi v,e^{-4{\pi }^{2}t/N^2}\right)$$

(32)

where ${\vartheta }_{2,3}$ are distinct Jacobi theta functions [57]. The expressions (32) give finite-size scaling expressions for the single-time correlator in terms of the finite-size scaling variables $x/N$ and $t/N^2$.

In Figure 4a, we show the analytically continued function $C(t;x)$ as computed in Appendix C. By construction, for $t>0$, it is an analytic function of x and periodic with period $2N$, but it only has a physical meaning in the interval $0\le x\le N$. That function is quite distinct from the physical scaling function $F_C(1,\xi )$ of the single-time correlator (2), with the scaling variables (21), which is shown in Figure 4b on the same scale. It becomes periodic in x with period N, as expected intuitively for a periodic lattice of N sites, but its derivative has jumps at $x=nN$ with $n\in \mathbb{Z}$. For $N\to \infty$, it converges towards the infinite-size correlator; however, at $\left|x\right|={\displaystyle \frac{N}{2}}$, it has a minimum value which converges exponentially to zero as $N\to \infty$. For a fully disordered initial state, the correlator is non-vanishing only at $x=nN$.

Figure 4. (a) Analytically continued function $C(t;x)$, as computed in Appendix C, for $t=5$ and $N=[5,10,20,30]$ from top to bottom, in the interval $0\le x\le N$. It satisfies the periodicity conditions (27). The full black line gives the initial function $C(0;x)$ for a completely disordered initial lattice. The thin horizontal lines indicate the values $C(t;x)=0$ and $C(t;x)=1$, respectively. (b) Physical scaling function $F_C(1,\xi )$ of Equation (2), for the same values of t and N. The full black line corresponds to a completely disordered initial state.

2. A finite-size generalisation of the second moment can be given as follows

$$\begin{array}{ccc}\hfill {\mathsf{\ell }}^2\left(t\right)& =& {\displaystyle \frac{{\int }_0^N\mathrm{d}x8{\left(\frac{N}{\pi }sin\frac{\pi x}{N}\right)}^{2}C(t;x)}{{\int }_0^N\mathrm{d}xC(t;x)}}={\displaystyle \frac{2N^2}{{\pi }^2}}\left(1-{\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{{\int }_0^1\mathrm{d}wsin\left(2\pi w\right){\vartheta }_3\left(\frac{\pi }{2}w,q\right)}{{\int }_0^1\mathrm{d}ww{\vartheta }_3\left(\frac{\pi }{2}w,q\right)}}\right)\hfill \\ & =& \left\{\begin{array}{cc}{\displaystyle \frac{2}{{\pi }^2}}{N}^2\hfill & \mathrm{if}t\gg N^2\hfill \\ {\displaystyle \frac{4}{3}}t\hfill & \mathrm{if}t\ll N^2\hfill \end{array}\right.\hfill \end{array}$$

(33)

with the only variable $q=e^{-\pi \left(\pi t/N^2\right)}$ (see Appendix C for details). This reproduces the infinite lattice result (18) and has the expected finite-size scaling form ${\mathsf{\ell }}^2\left(t\right)=N^2{F}_{\mathsf{\ell }}\left(t{N}^{-2}\right)$. Deep into the finite-size saturation regime, one has $\mathsf{\ell }\left(t\right)\approx 0.45N$.

3. The critical two-time correlator $C(\tau ,s;x)$ satisfies the equation of motion

$${\partial }_{\tau }C(\tau ,s;x)={\displaystyle \frac{1}{2}}{\partial }_x^{2}C(\tau ,s;x),C(0,s;x)=C(s;x)$$

(34)

with the single-time correlator, restricted to the relevant term, from (32). In Appendix D, we show that

$$C(\tau ,s;x)={\displaystyle \frac{1}{2}}\underset{0}{\overset{1}{\int \int }}\mathrm{d}v\mathrm{d}u{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}(v+u),e^{-{\displaystyle \frac{{\pi }^2}{N^2}}s}\right)\left[{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{\left|x\right|}{N}}-{\displaystyle \frac{\pi }{2}}v,e^{-{\displaystyle \frac{{\pi }^2}{N^2}}{\displaystyle \frac{\tau }{2}}}\right)+{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{\left|x\right|}{N}}+{\displaystyle \frac{\pi }{2}}v,e^{-{\displaystyle \frac{{\pi }^2}{N^2}}{\displaystyle \frac{\tau }{2}}}\right)\right]$$

(35)

and hence, the auto-correlator becomes

$$C(\tau ,s;0)={\int }_0^1\mathrm{d}v{\int }_0^1\mathrm{d}u{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}(v+u),e^{-{\displaystyle \frac{{\pi }^2}{N^2}}s}\right){\vartheta }_3\left({\displaystyle \frac{\pi }{2}}v,e^{-{\displaystyle \frac{y-1}{2}}{\displaystyle \frac{{\pi }^2}{N^2}}s}\right)$$

(36)

which only depends on the finite-size scaling variables $s/N^2$ and $\tau /N^2={\displaystyle \frac{s}{N^2}}(y-1)$.

In Figure 5a, the scaled two-time auto-correlator $C(ys,s)=F_C(y,0)$ given in (36) is shown for several values of N and $s=5$ fixed. Analogous with the generic expectation formulated in Section 1, we find a cross-over to a plateau when $y\gg 1$, with its height $C_{\infty }^{\left(2\right)}$ decreasing as N increases for fixed s. However, in the $1D$ Glauber–Ising model, the cross-over towards to the plateau occurs directly, and we do not initially see the more rapid decay observed in the spherical model in $2<d<4$ dimensions and $T<T_c$ [19] or in the $2D$ Glauber–Ising model at $T<T_c$ [37] (see Figure 1). From the heuristics of Section 1, this plateau should be found when the observation time $t\gg N^2$ is deeply into the finite-size regime, but the waiting time s is not, thus $s\ll N^2$. When we apply this to the auto-correlator (36) and let $y\gg 1$, the second theta function should converge rapidly towards unity, whereas the evaluation of the first one requires us to take many terms of the defining series [57] into account. This is treated by using the modular identity (A23), such that the auto-correlator becomes, in the plateau region

$$\begin{array}{ccc}\hfill C(\tau ,s;0)& \stackrel{y\gg 1}{\simeq }& {\int }_0^1\mathrm{d}v{\int }_0^1\mathrm{d}u{\displaystyle \frac{1}{\sqrt{\pi }}}{\displaystyle \frac{N}{\sqrt{s}}}{e}^{-{(u+v)}^2{N}^2/4s}{\vartheta }_3\left(\mathrm{i}{\displaystyle \frac{u+v}{2}}{\displaystyle \frac{N^2}{s}},e^{-N^2/s}\right)\left(1+\dots \right)\hfill \\ & =& {\displaystyle \frac{1}{\sqrt{\pi }}}{\displaystyle \frac{N}{\sqrt{s}}}{\left({\displaystyle \frac{N}{\sqrt{s}}}\right)}^{-2}{\int }_0^{N/\sqrt{s}}\mathrm{d}\alpha {\int }_0^{N/\sqrt{s}}\mathrm{d}\beta e^{-{(\alpha +\beta )}^2/4}{\vartheta }_3\left(\mathrm{i}{\displaystyle \frac{\alpha +\beta }{2}}{\displaystyle \frac{N}{\sqrt{s}}},e^{-N^2/s}\right)\hfill \\ & \stackrel{N\gg \sqrt{s}}{\simeq }& {\displaystyle \frac{1}{\sqrt{\pi }}}{\displaystyle \frac{s^{1/2}}{N}}{\int }_0^{\infty }\mathrm{d}\alpha {\int }_0^{\infty }\mathrm{d}\beta e^{-{(\alpha +\beta )}^2/4}\left(2+\dots \right)\hfill \\ & =& {\displaystyle \frac{4}{\sqrt{\pi }}}{\displaystyle \frac{s^{1/2}}{N}}\hfill \end{array}$$

(37)

where we changed the integration variables and then see that, in the theta function ${\vartheta }_3$, only the leading terms are not strongly suppressed for $N^2\gg s$. Because of the exponential suppression in the integrand, the integration limits can be extended to infinity without significantly changing the value of the integral, which rapidly converges to 2. Then, we have the finite-size plateau height, for $s\ll N^2$

$$C_{\infty }^{\left(2\right)}=\underset{y\to \infty }{lim}C\left(s(y-1),s;0;N\right)={\displaystyle \frac{4}{\sqrt{\pi }}}{\displaystyle \frac{s^{1/2}}{N}}$$

(38)

Given that we have previously seen that $z=2$ and $\lambda =1$, we confirm the generic expectations (4). That was the main aim of this work. Both scaling laws quoted in (4) are tested in Figure 5, respectively. Figure 5a shows that $C_{\infty }^{\left(2\right)}$ decreases with N while Figure 5b shows that $C_{\infty }^{\left(2\right)}$ increases with s, as long as $s\ll N^2$. Figure 5 also serves as an illustration that the asymptotic form (38) already holds to good approximation for relatively small values of s and N. After the kinetic spherical model [19], this is only the second exactly solvable example where this kind of test was carried out exactly.

Figure 5. Two-time scaled auto-correlator $C(ys,s)=F_C(y,0)$ in the $1D$ Glauber–Ising model quenched to $T=0$, as a function of $y=t/s$ for (a) finite systems of sizes $N=[10,20,30,40]$ from top to bottom and for a waiting time $s=5$, and (b) the waiting times $s=[5,20,80]$ from bottom to top and of finite size $N=30$. The full black line is the scaled infinite-size auto-correlator $\left(23\right)$.

4. Conclusions

We evaluated the scaled single-time and two-time correlators in the $1D$ Glauber–Ising model, quenched to $T=0$, on a finite-size and periodic chain. This allows us to check for the validity of the generically expected finite-size scaling forms [26] and, in particular, to confirm the novel expectation (4) of the plateau height $C_{\infty }^{\left(2\right)}$. Both scaling laws quoted in (4) are exemplified in Figure 5a and Figure 5b, respectively. We notice that the precise form of the finite-size auto-correlator of the $1D$ Glauber–Ising model, as shown in Figure 5, does not reproduce all of the generic expectations in coarsening systems, as schematised in Figure 1.

In view of the general derivation [38], we expect the result (4) to hold in general, but further tests, including non-exactly-solvable models, are welcome. One forthcoming case is the detailed test of (4) in the $2D$ Glauber–Ising model at $T<T_c$ [37]. Since the exact solution of the $1D$ Glauber–Ising model is restricted to nearest-neighbour interactions, any tests of (4) in the currently intensively studied long-range coarsening [7,8,9,10,11,60,61,62,63,64,65,66,67,68,69,70] or phase-separating Ising models [12] will require numerical studies. Here, it might also be of interest to study non-integrable generalisations of Glauber dynamics (6), for example, by combining non-conserved Glauber-type dynamics with conserved Kawasaki-type dynamics at a different temperature, which no longer satisfies detailed balance [47]. At the least, the use of (4) gives a different computational tool for the determination of $\lambda$ and $\lambda /z$, although it still has to be seen how precise a numerical technique it will turn out to be.

In addition, there is not yet any test of the generalisation of (4) [38] for quenches onto $T=T_c$. Furthermore, eventual extensions to quantum systems have not yet been tried.