Exact Solution of the Glauber–Ising Model on the Finite-Length Semi-Open Chain

Abstract

The exact time–space correlation function of the $1D$ Glauber–Ising model, quenched to temperature $T=0$ and on a semi-open lattice of finite size N, is obtained. This also enables deducing the exact empty-interval probability of the dual $1D$ coagulation–diffusion process on a periodic finite ring and reproducing the long-time decay of the particle concentration. These results are consistent with the generic expectations of dynamical finite-size scaling theory.

1. Introduction

Physical ageing phenomena [1,2] may arise in a many-body system after a quench, typically from a disordered initial state, either onto a critical point where at least two physical phases become indistinguishable or else into a phase coexistence region where two macroscopic physical phases coexist. In both cases, the after-quench dynamics is a slow one, which may come from the effects of critical-point fluctuations or else 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 $\ell \left(t\right)$ is growing with time. The phenomenology of physical ageing is contained in its three defining properties, namely [3]

1.

Slow dynamics (relaxations are slower than might be described by simple exponentials);

2.

Absence of time-translation invariance;

3.

Dynamical scaling.

• These are reflected in the properties of the (coarse-grained) order parameter $\varphi =\varphi (t,\mathit{r})$), which depends on the time t and the space coordinates $\mathit{r}$. For a disordered initial state, the average order parameter vanishes, viz. $\langle \varphi (t,\mathit{r})\rangle =0$. One of the most frequently studied observables is the correlation function Its Fourier transform, the structure function, can be measured in scattering experiments.)

  $$C(t;\mathit{r})=〈\varphi (t,\mathit{r})\varphi (t,\mathbf{0})〉=t^{-b}{F}_C\left({\displaystyle \frac{\left|\mathit{r}\right|}{t^{1/z}}}\right)$$

  (1)

  whose scaling is specified here for an algebraically growing domain size $\ell \left(t\right)\sim t^{1/z}$ and then defines the dynamical exponent $z$. The exponent b is an ageing exponent, but in what follows we shall restrict ourselves to cases where $b=0$. For a non-conserved order parameter and short-ranged interactions, one has $z=2$ [4,5,6]. (For a conserved order parameter, one speaks of phase separation and z takes different values [5,6]. Long-range interactions lead to further modifications [5,7,8,9,10,11].) Up to metric scale factors, the form of the scaling function $F_C$ is generically expected to be universal, hence independent of most of the ‘details’ of the underlying microscopic physics; see [4,12,13,14,15,16] for reviews. The theoretical task of finding the form of $F_C$ is also of practical importance since a priori knowledge of $F_C$ permits long-time predictions on the basis of short-time data.

The universality of functions such as $F_C$ permits their study via well-chosen and mathematically extremely simplified models. In this context, the celebrated $1D$ Ising model continues to play an important role. It is defined on a chain $\mathrm{\Lambda }\subset \mathbb{Z}$ with spin variables ${\sigma }_n=\pm 1$ attached to each of its sites. At equilibrium, it is specified through the Hamiltonian [17,18]

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

(2)

where the exchange coupling was normalized to unity. Because of its short-ranged interactions, in one dimension, there is no equilibrium phase transition at any non-vanishing temperature $T>0$ [18]. We shall be interested here in some of its non-equilibrium dynamical properties (a slow Glauber–Ising dynamics in a ferrimagnetic chain made from Co⁺ ions and organic radical spins that are strongly antiferromagnetically coupled was studied experimentally, e.g., [19,20,21,22]). Such a dynamics, at temperature T, may be created in a heat-bath formulation by selecting randomly, at each time step $\mathrm{\Delta }t$, a site $n\in \mathrm{\Lambda }$ whose spin ${\sigma }_n$ is updated according to the Glauber rule [23], with the rate [24]

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

(3)

Clearly, the temperature T is a property of the heat bath. In one spatial dimension, Glauber’s rule has the remarkable and attractive feature that local spin-observables, such as local magnetization, local correlators and so on, satisfy closed equations of motion, which can be studied analytically, rather than infinite uncoupled hierarchies of equations of motion [25], which arise generically. Specifically, 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 over the thermal histories defined by Equation (3). The correlator obeys the (rescaled) equation of motion [23,24,26,27,28], with the short-hand $0\le \gamma =tanh(2/T)\le 1$

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

(4)

which only contains correlators $C_n\left(t\right)$ at different sites but does not contain any reference to higher multi-point correlators. For an initially disordered system, one has $C_n\left(0\right)={\delta }_{n,0}$ [23,24].

Various aspects of this model have been thoroughly analyzed many times [23,24,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43], notably for $T=0$, where updates that would lead to an increase in $\mathcal{H}$ are forbidden. Because of the competition between the two stationary and absorbing states where all spins are either $+1$ or $-1$, the $T=0$-dynamics becomes slow and obeys dynamical scaling with the exponent $z=2$ (a rigorous new bound on the spectral gap implies for all $T\le T_c$ the improved bound $z\ge 2$ [44]). In doing so, a main issue is the appropriate treatment of the constraint $C_0\left(t\right)=1$, which precludes the immediate application of Fourier series. A recently introduced possibility to treat this uses spatial symmetry properties [45,46], which provides a convenient way to treat the model in a finite and periodic lattice. Here we shall consider how to extend this idea to a chain that is open on one end (following an idea from [23]), find the exact correlation function in this case, and analyze how the finite-size effects will modify the scaling (1) of the infinite-size system, especially for a case when spatial translation invariance does not hold. This will also enable studying the influence of the boundary conditions on the form of the scaling function. One of the aims of this study is to provide an explicitly worked out case to serve as a background for more generic studies. One issue will be how to insert the results to be obtained in the context of finite-size scaling [47,48], especially for dynamics [49].

Another aspect of this problem arises from the link of the Glauber–Ising model with stochastic reaction–diffusion processes [50]. Here we shall focus on the $1D$ coagulation–diffusion process, mainly studied via the so-called empty-interval method, e.g., [29,35,42,45,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68]. Each site of the lattice is either empty or occupied by a single particle A. Particles can randomly hop to a nearest-neighbor site, and, if that site already happens to be occupied, the two particles undergo, with probability one, a coagulation reaction $A+A\to A$. In one dimension, the particle concentration decays (For brief overviews on experimental results in $1D$ we refer to [62,64] and Refs. therein.) for long times as $c\left(t\right)\sim t^{-1/2}$ [66], which makes it (i) a slow process (of which in principle the ageing can be analysed) with anomalous transport and (ii) is distinct from mean-field theories that hold for dimensions $d>2$ and would give $c_{\mathrm{MF}}\left(t\right)\sim t^{-1}$ [13]. It is long established that this process is dual to the Glauber–Ising model at $T=0$ [50,69]. We are interested in extending this to finite lattices with N sites. For technical simplicity, we shall admit a continuum limit throughout, which in principle holds for sufficiently large distances $\left|x\right|=n\mathfrak{a}$ with respect to the lattice constant $\mathfrak{a}$. On finite systems, this continuum assumption will work for sufficiently large lattice sizes N that the large-distance limit mentioned above can sensibly be taken. As we shall show, the correlation function $C(t;x)$ on a semi-open lattice in the Glauber–Ising model corresponds to the empty-interval probability $E(t,x)$ on a periodic ring in the diffusion–coagulation process, from which observables such as the concentration $c\left(t\right)$ can be found.

This paper is organized as follows. Section 2 first recalls the treatment of a finite periodic chain in the Glauber–Ising model before the semi-open geometry is defined and then solved through an extension of the spatial symmetry method. We also discuss if/when a natural-looking short-cut towards the scaling function is applicable. In Section 3 we shall show that the correlation function $C(t;x)$ of the semi-open Glauber–Ising model can be reinterpreted as the empty-interval probability $E(t,x)$ of diffusion–coagulation on a periodic ring. The conclusions are given in Section 4. Three appendices contain the technical details of the calculations.

2. Correlation Functions on a Finite Chain

We now describe the calculation of the spin correlation function, which will be conducted throughout in the continuum limit such that $C_n\left(t\right)\mapsto C(t;x)$. We shall also restrict to $T=0$ since this is the only situation where dynamical scaling holds and slow dynamics occurs.

2.1. Periodic Chain

We begin by recalling the result in the $1D$ Glauber–Ising model on a periodic ring with N sites [46]. One purpose of this subsection is to briefly recall the technique we use here, and we shall emphasize later the differences with respect to the semi-open chain. In the continuum limit, one has from (4) for the correlation function $C(t;x)$ the following equation of motion, together with the boundary conditions

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

(5)

and where $0\le x\le N$. The first of these constraints comes from the Ising constraint in (4) and the second one comes from the periodicity. Rather than dealing with these directly, we recognize first that the physical correlation function $C(t;x)$ is even in x. This means that we can restrict to positive values of x only such that $C(t;-x)$ with negative spatial arguments becomes available for purely mathematical purposes. This permits treating the two constraints in (5) by using an analytic continuation to negative values of x. From now on, $C(t;x)$ will denote that analytically continued function. Only at the very end do we revert to the physical correlation function by making the substitution $x\mapsto \left|x\right|$. The analytic continuation is expressed explicitly as follows [46]:

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

(6)

Clearly, the spatial symmetries (6) reproduce the constraints in (5). Together, these can be shown to imply that the analytically continued function $C(t;x)$ is periodic in the spatial coordinate x, with period $2N$; see (6). In this way, the physically motivated constraints are embedded into spatial symmetry properties of the analytically continued function $C(t;x)$. Hence, for the analytically continued function, one has the Fourier series representation [70]

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

(7)

and now the equation of motion (5) can indeed be solved in Fourier space. A straightforward calculation leads, for the physical correlation function, to [46]

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

(8)

where ${\vartheta }_3$ is a Jacobi theta function [71]. It follows that both the physical constraints as well as the required periodicity properties (6) are indeed satisfied if they only hold for the initial correlator $C(0;x)$. If the initial correlation function decays with x, for example, $C(0;x)\sim {\left|x\right|}^{-\aleph }$ for large $\left|x\right|$ and with $\aleph >0$, the term is irrelevant in the sense that it merely gives rise to corrections to the leading finite-size scaling limit behavior [26,46]. For example, for a fully disordered initial state with $C(0;x)\sim \delta \left(x\right)$, this correction term in the second line of (8) vanishes. The finite-size scaling behavior is obtained by simultaneously taking the limits $t\to \infty$, $\left|x\right|\to \infty$ and $N\to \infty$ but such that the (finite-size) scaling variables

$$\mathfrak{u}={\displaystyle \frac{\left|x\right|}{t^{1/2}}},\mathfrak{v}={\displaystyle \frac{N}{t^{1/2}}}$$

(9)

are kept finite. This scaling limit also arises naturally for a fully disordered initial state (In general, an initial correlator $C(0;x)$ in (8) breaks dynamical scaling.) where $C(0;x)\sim \delta \left(x\right)$. Then one may re-cast the physical single-time correlator (1) as

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

(10)

where ${\vartheta }_2$ is another Jacobi theta function [71] that is distinct from ${\vartheta }_3$. This form is in generic agreement with the expectation of dynamical finite-size scaling [49]. The expression (10) gives the explicit finite-size scaling functions for the single-time correlator in terms of the finite-size scaling variables $\left|x\right|/N$ and $t/N^2$ for a periodic chain of length N (The generic form is quite analogous to existing analytical results in the spherical model in $2<d<4$ dimensions, with periodic boundary conditions and quenched to $T<T_c$ [72]; see the discussion in Section 2.2 below).

2.2. Semi-Open Chain

Our focus shall be on the semi-open chain; see Figure 1. In contrast to the periodic ring considered in Section 2.1, spatial translation invariance no longer holds true. In one spatial dimension, previous results with open boundary conditions include finite-size scaling studies on the critical relaxation times in the Glauber–Ising model [36] or on the particle density in asymmetric exclusion models via algebraic techniques, e.g., [42,73,74,75]. To our knowledge, this is the first study of finite-size scaling for free boundary conditions on time–space correlation functions in the Glauber–Ising model. In Figure 1, we indicate that the correlation function $C_n\left(t\right)=C_{-n}\left(t\right)$ should be considered as being of a central spin at fixed position $n=0$ and another one n sites away. This is symmetric in n, and we can therefore consider $n\ge 0$ without restriction on the generality (the only restriction we admit is that the site $n=0$ is at the center of the interval $[-N,N]$). Then we can also speak of the correlation function $C_n\left(t\right)=\langle {\sigma }_0\left(t\right){\sigma }_n\left(t\right)\rangle$ between an Ising spin fixed at the leftmost edge of the interval $[0,N]$ and another spin at site n to the right. We shall use this picture from now on. Specifically, consider that, at the leftmost edge, an Ising spin is fixed, and we look for the correlator $C_n\left(t\right)=\langle {\sigma }_0\left(t\right){\sigma }_n\left(t\right)\rangle$ with another spin at site n to the right. The chain is open at the right end because of the requirement $C_N\left(t\right)=0$. In the continuum limit, we have (with $0\le x<N$)

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

(11)

Since the left spin is considered fixed, changes in $C(t;x)$ only arise from the motion of the right spin. This leads to a reduction in the diffusion constant by a factor 2 in comparison to the periodic case (5) where both spins are mobile (consideration of two mobile spins would require using correlators with two spatial variables, in the spirit of [57], but is beyond the scope of the present work). Rather than dealing with these two constraints directly, we shall implement them via spatial symmetries in an analytically continued function $C(t;x)$ in the same spirit as for the periodic case above. But, in contrast to Section 2.1, these conditions are chosen to be

$$C(t;-x)=2-C(t;x),C(t;x)=1-{\displaystyle \frac{x}{N}}+B(t;x)$$

(12)

The first of these solves the first constraint. From the definition (12) of the function $B(t;x)$, the second constraint implies

$$B(t;\pm N)=B(t;0)=0$$

(13)

In addition, combination with the first property (12) proves that, on the interval $[-N,N]$, the function B is anti-symmetric (as shown in Appendix A)

$$B(t;-x)=-B(t;x)$$

(14)

Since $B(t;\pm N)=0$ vanishes at the extremities of the interval $[-N,N]$, it can be considered to be of spatial period $2N$ (in spite of the absence of spatial translation invariance, we have satisfied once more the conditions for a Fourier analysis to be applicable [70] (Kap. 4)). Furthermore, it admits a Fourier representation

$$B(t;x)={\displaystyle \sum _{k=1}^{\infty }}{b}_k\left(t\right)sin\left({\displaystyle \frac{\pi }{N}}kx\right),b_k\left(t\right)={\displaystyle \frac{1}{N}}{\int }_{-N}^N\mathrm{d}xB(t;x)sin\left({\displaystyle \frac{\pi }{N}}kx\right)$$

(15)

which is the analogue of (7) above. Then the equation of motion (11) can be solved in Fourier space, and we find (see Appendix A for the details)

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

(16)

where finally the analytically continued function is reduced to the physical correlation function $C(t;x)$ by making at the very end the substitution $x\mapsto \left|x\right|$. This gives the exact expression for the correlation function $C(t;x)$ between an Ising spin at the center of the open segment $[-N,N]$ and another spin at the position $-N\le x\le N$ subject to Glauber dynamics and such that the correlation function is forced to vanish $C(t;\pm N)=0$ at the end of the segment; see Figure 1. Up to a trivial re-scaling in time (which comes from the rescaled equations of motion) the corrections to the leading scaling forms for both the periodic and semi-open lattices are the same. It follows that the discussion of the irrelevance of spatially decaying initial correlators can be taken over from the periodic case (8) in Section 2.1 and hence also holds for the semi-open chain. The leading scaling contributions can be cast into their final forms

$$\begin{array}{cc}\hfill C^{\mathrm{semi}}(t;x)& =1-{\int }_0^{\left|x\right|/N}\mathrm{d}v{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}v,e^{-{\displaystyle \frac{{\pi }^2}{2}}{\displaystyle \frac{t}{N^2}}}\right)\hfill \end{array}$$

(17a)

$$\begin{array}{cc}\hfill C^{\mathrm{per}}(t;x)& =1-{\displaystyle \frac{2}{\pi }}{\int }_0^{\left|x\right|/N}\mathrm{d}v{\vartheta }_2\left(\pi v,e^{-4{\pi }^2{\displaystyle \frac{t}{N^2}}}\right)\hfill \end{array}$$

(17b)

which depend on the finite-size scaling variables $\left|x\right|/N$ and $t/N^2$. They are both consistent with the generic expectations of dynamical finite-size scaling [49] in the sense that we can write the correlation functions (17) as $C(t;x;N)=F_C\left({\displaystyle \frac{\left|x\right|}{\sqrt{t}}},{\displaystyle \frac{N}{\sqrt{t}}}\right)$, which generalizes (1), and where the universal scaling functions $F_C$ are boundary-condition-dependent. The existence of a second argument is the new feature of finite geometries.

Figure 2 illustrates these scaling functions: the semi-open correlator (17a) in Figure 2a and the periodic correlator (17b) in Figure 2b over against the finite-size scaling variable $\left|x\right|/N$. A common aspect is that the functional forms of these correlators not only depend on the second finite-size scaling variable $N/\sqrt{t}$ but on the boundary conditions as well. At first sight, their behavior is clearly quite distinct (notice that the qualitative shape of the curves in Figure 2b is similar to that found for the spherical model in $2<d<4$ dimensions, quenched to temperature $T<T_c$ [72]). Upon closer inspection, it appears that the behavior of the semi-open correlator (17a) in the interval $0\le {\displaystyle \frac{\left|x\right|}{N}}\le 1$ is quite analogous, although not identical, to the behavior of the periodic correlator (17b) in the interval $0\le {\displaystyle \frac{\left|x\right|}{N}}\le {\displaystyle \frac{1}{2}}$ (this also illustrates that the condition $C(t;N)=0$ does not eliminate the possibilities of short-range order, analogously to the periodic case). Here the distinct boundary conditions $C^{\mathrm{semi}}(t;\pm N)=0$ and $C^{\mathrm{per}}(t;\pm N)=1$ are of essential influence. For example, at $x=N$, the semi-open correlator vanishes exactly, whereas, in the periodic case, at $x={\displaystyle \frac{1}{2}}N$, the correlator tends to zero exponentially fast with increasing N. On the other hand, if one considers, as in the inset of Figure 2a, the dependence of $C(t;x)$ on the bulk scaling variable $x/\sqrt{t}$, the scaling functions are close to those of the spatially infinite system, viz. $C_{\infty }^{\mathrm{semi}}(t;x)=\mathrm{erfc}\left({\displaystyle \frac{\left|x\right|}{\sqrt{2t}}}\right)$, if $N/\sqrt{t}$ is large enough (analogously, for periodic boundary conditions in the limit $N/\sqrt{t}\to \infty$, the curves converge towards the correlator $C_{\infty }^{\mathrm{per}}(t;x)=\mathrm{erfc}\left({\displaystyle \frac{\left|x\right|}{2\sqrt{t}}}\right)$ [31]. Conversely, if ${\displaystyle \frac{\left|x\right|}{N}}\ll 1$, one finds a linear decay whose slope approaches that of the infinite-size system for sufficiently large $N/\sqrt{t}$. Deviations from the infinite-size curve first become visible for large values of ${\displaystyle \frac{\left|x\right|}{\sqrt{t}}}$. Since the model is made from ‘hard’ Ising spins, one finds a cusp at $x=0$ (see Figure 2), usually referred to as Porod’s law [4]. In contrast, such a cusp does not exist for ‘soft’ spins, as in, for example, the spherical model, where the correlator is rounded off at $x=0$ ([72], Figure 1a).

2.3. A Short-Cut Towards Dynamical Scaling?

Given that scaling approaches often allow for a rapid and simple derivation of (universal) scaling functions, it is tempting to try such a scaling approach for the solution of the equation of motion (11) and with its associated boundary conditions. One might try one’s hand at a simple scaling ansatz of the form

$$C^{\mathrm{scal}}(t;x)=F\left({\displaystyle \frac{\left|x\right|}{\sqrt{t}}}\right);\mathfrak{u}={\displaystyle \frac{\left|x\right|}{\sqrt{t}}}$$

(18)

which should hold in the scaling limit where simultaneously $t\to \infty$, $x\to \infty$ but $\mathfrak{u}$ is kept finite [76]. Inserting the ansatz into (11) readily gives the differential equation

$$F^{″}\left(\mathfrak{u}\right)+\mathfrak{u}{F}^{\prime }\left(\mathfrak{u}\right)=0⟹F\left(\mathfrak{u}\right)=F_0+F_1{\int }_0^{\mathfrak{u}}\mathrm{d}{u}^{\prime }{e}^{-{u^{\prime }}^2/2}$$

(19)

and where the two constants $F_{0,1}$ are to be found from the boundary conditions $F\left(0\right)=1$ and $F(N/\sqrt{t})=0$, implied by (11). This yields

$$C^{\mathrm{scal}}(t;x)=1-{\displaystyle \frac{{\int }_0^{x/\sqrt{t}}\mathrm{d}{u}^{\prime }{e}^{-{u^{\prime }}^2/2}}{{\int }_0^{N/\sqrt{t}}\mathrm{d}{u}^{\prime }{e}^{-{u^{\prime }}^2/2}}}=1-{\displaystyle \frac{\mathrm{erf}\frac{x}{\sqrt{2t}}}{\mathrm{erf}\frac{N}{\sqrt{2t}}}}={\displaystyle \frac{\mathrm{erfc}\frac{x}{\sqrt{2t}}-\mathrm{erfc}\frac{N}{\sqrt{2t}}}{1-\mathrm{erfc}\frac{N}{\sqrt{2t}}}}$$

(20)

where $\mathrm{erf}\left(x\right)$ is the error function [71] (7.1.1) and $\mathrm{erfc}\left(x\right)=1-\mathrm{erf}\left(x\right)$ is the complementary error function. Of course, such a scaling solution will be independent of any initial correlator $C(0;x)$. Since, for large arguments $\mathrm{erfc}\left(x\right)\stackrel{x\to \infty }{\to }0$ that are exponentially fast [71], the solution (20) certainly has the attractive feature that, for $N/\sqrt{t}\gg 1$, one recovers the exactly known scaling function $C_{\infty }^{\mathrm{scal}}(t;x)\to \mathrm{erfc}\left({\displaystyle \frac{x}{\sqrt{2t}}}\right)$ of the spatially infinite system [31]. We also observe that the final result (20) agrees once more with the generic expectations of dynamical finite-size scaling [49] since it depends on both variables ${\displaystyle \frac{x}{\sqrt{t}}}$ and ${\displaystyle \frac{N}{\sqrt{t}}}$, although from the original ansatz (18) one might have expected a different result.

Does the approach leading to (20) represent a useful short-cut in order to obtain the exact physical correlation function? If that were so, comparison with the exact solution (17a) would imply the mathematical identity

$$f(x,y):={\displaystyle \frac{\mathrm{erf}x}{\mathrm{erf}y}}\stackrel{?}{=}{\int }_0^{x/y}\mathrm{d}u{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}u,e^{-{\pi }^2/4y^2}\right)={\displaystyle \frac{2y}{\sqrt{\pi }}}{\int }_0^{x/y}\mathrm{d}u{e}^{-y{u}^2/2}{\vartheta }_3\left(\mathrm{i}uy,e^{-4y^2}\right)$$

(21)

where the last relation is a consequence of the modular transformation

$${\vartheta }_3\left(\pi u,e^{-\pi t}\right)=t^{-1/2}exp\left(-\pi {\displaystyle \frac{u^2}{t}}\right){\vartheta }_3\left(\mathrm{i}\pi {\displaystyle \frac{u}{t}},e^{-\pi /t}\right)$$

(22)

which follows from Poisson’s resummation formula [77,78] of the Jacobi theta function ${\vartheta }_3\left({\displaystyle \frac{\pi }{2}}u,q\right)={\vartheta }_3\left({\displaystyle \frac{\pi }{2}}(2-u),q\right)={\sum }_{k\in \mathbb{Z}}{q}^{k^2}cos\left(\pi uk\right)$ [71] (16.27.3).

In Figure 3 numerical tests of the conjectured relation (21) are shown. In the left panel, essentially the dependence on x of the function $f(x,y)$ is displayed for several values of y. The full curves show the left-hand side of (21) as it follows from $C^{\mathrm{scal}}$, whereas the points show the right-hand side of (21) as it follows from $C^{\mathrm{semi}}$. Clearly, for sufficiently large y, excellent agreement is found and the points fall very nicely onto the full lines of the same color. However, when $y\lesssim 2$, notable deviations appear, which are particularly notable around $y\approx 1$. The right panel illustrates this further by showing $f(x,y)$ as a function of y for several values of $x/y$. Again, for sufficiently large y, one observes very good agreement (the points fall clearly onto the full lines of the same color and deviations should be exponentially small), but $C^{\mathrm{scal}}$ and $C^{\mathrm{semi}}$ lead to different results for small values of y, which appears to be most strong around $y\approx 1$. The same effect can also be seen in Figure 2a, where the dashed gray lines give $C^{\mathrm{scal}}(t;x)$ for two values of $N/\sqrt{t}$. For the larger one (blue curve), there is very good agreement with the exact result (17a). However, for the smaller one (green curve), deviations are notable, although the curves are qualitatively similar.

Hence the proposed identity (21) only holds approximately in the region $y\gtrsim 2$. The intriguing and simple short-cut towards a finite-size scaling function only produces an approximate result, probably since the ansatz (18) merely depends on the single scaling variable $\mathfrak{u}$. Comparison with the exact result (17a) shows this to be an over-simplification [49]. Still, the quantitative agreement of the simple form (20) with the exact result (17a) is not so bad (see Figure 2a), and the simplicity of its derivation might become useful for a quick orientation in more complicated models. Implicitly, in the scaling approach described here, one has admitted that $N/\sqrt{t}\gg 1$, but Figure 3 shows that the cross-over, when the length scale $\ell \left(t\right)\sim \sqrt{t}$ becomes comparable to N, is not completely captured.

3. Coagulation–Diffusion Process

The $1D$ Glauber–Ising model at temperature $T=0$ is dual [50,69] to the coagulation–diffusion process of particles of a single species A, and provided diffusion $A+Ø\stackrel{D}{⟷}Ø+A$ and coagulation $A+A\stackrel{D}{⟶}A+Ø,Ø+A$ occur with the same rate (if the rates are different, the universal long-time exponent $c\left(t\right)\sim t^{-1/2}$ is kept but the associated amplitude will be modified, e.g., [79]. This is in agreement with experimental results, e.g., [62] and Refs. therein). In the exact solution, a central quantity is the empty-interval probability $E_n\left(t\right)$ [45,52,55,56,57,60], which is the probability to find an interval of n subsequent empty sites. Under the stated conditions, $E_n\left(t\right)$ obeys a closed set of equations of motion. In the continuum limit, one rather deals with a function $E(t,x)$, which obeys the well-known equation of motion [35,53]

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

(23)

Herein, the last condition holds if the particles are moving on a ring of N sites. If initially there is at least one particle in the system, the last particle that has survived the coagulation reactions cannot decay because of the lack of a reaction partner. On a ring of N sites, the largest empty interval can have $N-1$ sites. The other constraint follows since only nearest-neighbor particles can undergo a coagulation reaction. In what follows, we shall always scale to $D=1$.

Clearly, the equations of motion (11) and (23), along with their constraints, are identical up to a trivial re-scaling $t\mapsto 4t$ when going from the semi-open Glauber–Ising model to the coagulation–diffusion process. On a periodic ring of size N, the empty-interval probability can be read off from the previous discussion

$$\begin{array}{ccc}\hfill E(t,x)& =& 1-{\int }_0^{x/N}\mathrm{d}u{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}u,e^{-{\displaystyle \frac{{\pi }^2}{2}}{\displaystyle \frac{4t}{N^2}}}\right)\hfill \\ & & +{\displaystyle \frac{1}{2N}}{\int }_0^N\mathrm{d}{x}^{\prime }E(0,x^{\prime })\left[{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{x-x^{\prime }}{N}},e^{-{\displaystyle \frac{{\pi }^2}{2}}{\displaystyle \frac{4t}{N^2}}}\right)-{\vartheta }_3\left({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{x+x^{\prime }}{N}},e^{-{\displaystyle \frac{{\pi }^2}{2}}{\displaystyle \frac{4t}{N^2}}}\right)\right]\hfill \end{array}$$

(24)

This is the precise statement of the duality with Glauber–Ising chain mentioned in Section 1 for the case of finite chains with N sites. One of the quantities of interest is the time-dependent particle concentration, which follows directly once $E(t,x)$ is known [35,53] and reads

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

(25)

Herein, the first term does reproduce the well-known analytic result by Krebs et al. in their Equation (6.7) [60]. We point out that their result was derived on a discrete lattice (see their Equation (6.6) [60]) and a continuum limit was merely taken at the very end of their calculation (their solution also contains corrections to scaling that come from the discreteness of the lattice) [60]. This observation already serves as a useful cross-check of our calculational technique in Section 2.2, including the use of the continuum limit right from the beginning.

The empty-interval probability can be expressed as

$$E(t,x)={\int }_x^N\mathrm{d}{x}^{\prime }P(t,x^{\prime })$$

(26)

where $P(t,x)=\mathrm{Pr}\left(•\boxed{x};t\right)$ is the probability to find an empty interval of size x bounded on the left by a particle. Carrying out the partial integration in (25) leads to a more compact expression for the time-dependent density (as derived in Appendix B)

$$c\left(t\right)={\int }_0^1\mathrm{d}uP(0,Nu){\vartheta }_3\left({\displaystyle \frac{\pi }{2}}u,e^{-2{\pi }^2{\displaystyle \frac{t}{N^2}}}\right)$$

(27)

If initially the particles are uncorrelated and have the infinite-volume concentration $c_{\mathrm{eq}}$, the initial probabilities on a finite ring may be chosen as

$$P(0,x)={\displaystyle \frac{c_{\mathrm{eq}}{e}^{-c_{\mathrm{eq}}x}}{1-e^{-c_{\mathrm{eq}}N}}},E(0,x)={\displaystyle \frac{e^{-c_{\mathrm{eq}}x}-e^{-c_{\mathrm{eq}}N}}{1-e^{-c_{\mathrm{eq}}N}}}$$

(28)

The distribution $E(0,x)$ obeys the two constraints (23), as it should. On a finite lattice of size N, $c_{\mathrm{eq}}=c_{\mathrm{eq}}(c_0,N)$ must be chosen such that the initial concentration indeed takes the desired value $c_0=c_{\mathrm{eq}}/(1-e^{-c_{\mathrm{eq}}N})$, consistent with $P(0,0)=c_0$. Physical arguments for this form for initially uncorrelated particles of concentration $c_0$ are recalled in Appendix C.

Alternatively, one may also start from the initial distribution [76]

$$P(0,x)=c_0{\left(1-{\displaystyle \frac{x}{N}}\right)}^{c_{0}N-1},E(0,x)=e^{c_{0}Nln\left(1-x/N\right)}$$

(29)

which obeys the same boundary conditions as the choice (28). Then the concentration can be found via

$$c\left(t\right)=c_0{\int }_0^1\mathrm{d}u{u}^{c_{0}N-1}{\vartheta }_4\left({\displaystyle \frac{\pi }{2}}u,e^{-2{\pi }^2{\displaystyle \frac{t}{N^2}}}\right)$$

(30)

with the theta function ${\vartheta }_4$ [71]. The scaling solution (the leading part in (25)) implicitly starts with an initial value $c_0=1$, and the modular transformation (22) produces the expected long-time decay $c\left(t\right)\simeq {\left(2\pi t\right)}^{-1/2}$. If an initial concentration $c_0<1$ is chosen, the concentration $c\left(t\right)$ will initially decay more slowly than the scaling solution and will cross over to the scaling decay once the more rapidly decaying scaling solution has become close to it.

We have restricted attention here to the mere calculation of time-dependent concentrations $c\left(t\right)$. The study of many-point particle correlation functions requires the analysis of many-hole probabilities, e.g., following the lines of [45,56,57].

4. Conclusions

Even a century after its introduction [17,18], and after a long history of having fruitfully stimulated many different insights into phase transitions at and far from equilibrium (for historical reviews, see [80,81,82,83,84,85]), the “Ising model still thrives” [86]. We have studied some aspects of the celebrated Glauber–Ising dynamics [23], which in turn has become quite time-honored itself. The Glauber–Ising dynamics in $1D$ is rightly famous since it is one of the rare cases where the usually infinite hierarchy of coupled equations of motion [25,87] naturally decouples and thus becomes available to methods of analytical study. This feature has furnished explicit examples in Ising model contexts for a long time.

In the continuum limit, time–space-dependent correlation functions $C(t;x)$ then obey simple diffusion equations but are still subject to boundary conditions, which prevent a totally straightforward solution, viz. in terms of Fourier analysis. Much of the body of work in the past decades has been on how to treat these. Our main innovation in this work has been to show how to use spatial symmetries to recast the problem into one where Fourier series methods can indeed be used and then to show how this applies in the case of non-periodic boundary conditions on finite lattices of size N. The results presented here can immediately be used as initial conditions for the calculation of two-time correlators and the exploration of novel finite-size effects therein [88,89]. We also hope that these techniques may become useful in different applications in the future, which may involve more general interactions, surface magnetic fields and/or more general boundary conditions. Similarly, in the related coagulation–diffusion process, it should be possible to consider particle currents at the boundaries [57] or to extend the techniques at hand towards the analysis of correlation functions. Again, the boundary conditions that arise in the equations of motion for the empty-interval probabilities $E(t,x)$ were long considered so difficult that attention was shifted to other observables that are easier to analyze [35,53].

Explicit results were shown in Figure 2 and satisfactorily enter into the generic and expected context of dynamical finite-size scaling [49]. This also provided the opportunity to test a proposal for a short-cut towards the dynamical finite-size scaling functions, which, although not exact, still satisfy dynamical finite-size scaling and might become of heuristic value in more complicated systems. Numerical work will now be needed to further understand which aspects are specific to the $1D$ Glauber–Ising dynamics and which ones permit further generalization. At the very least, our result should serve as a benchmark for future numerical studies. Long-standing relations with integrable quantum chains [29,42] point towards possible extensions regarding quantum dynamics [68,90,91].