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\strut\hfill September 1994
\vfil
{\bf
\centerline {Non-equilibrium dynamics of the}
\centerline {one-dimensional Glauber model}
}

\bl\bl\bl
\centerline{Stefan Van Gulck$^\dagger$ and Jan Naudts}

\bl\bl
\parindent 80pt
\item {} Department of Physics, Antwerpen University,
\item {} Universiteitsplein 1, B-2610 Antwerpen, Belgium.
\item {} Internet: uia.ac.be
\parindent 0pt

\bl\bl\vfil
Abstract:
We study the relaxation of the one-dimensional Glauber model
towards equilibrium at finite temperature, taking averages over
all possible initial configurations. Let $C(q,t)$ denote the
spatial Fourier transform of the equal-time two-point
correlation function $\langle\sigma_0\sigma_n\rangle_t$. It can
be written as the Laplace transform of a relaxation spectrum
$\rho_q(\omega)$. The latter turns out to contain a
non-integrable singularity at a $q$-dependent frequency
$\omega=\theta(q)$. The singularity originates from the
self-term in the equations of motion and has important
consequences. E.g., it produces the dominating finite-size
correction to $C(q,t)$. As a consequence, we are able to
formulate a finite-size scaling theory in the limit of zero
temperature, long times and small wave vectors.

\bl\bl\vfil
Keywords: Ising model, Glauber dynamics, non-equilibrium dynamics,
relaxation spectrum, finite-size scaling.

PACS: 64.60.Ht, 05.50.+q

\bl
$^\dagger$bursaal IWONL

\eject
\centerline{\bf 1. Introduction}

\bl
Most problems in the field of non-equilibrium statistical
mechanics appear to be rather unaccessible for theoretical
analysis. The present paper deals with the simple looking
question how a system in thermodynamical equilibrium at an
initial temperature $T_0$ is relaxing in time when it is
suddenly cooled towards a second order phase transition
temperature $T_c$. Exact calculations on this problem are
possible for the one-dimensional Ising model with Glauber
dynamics, altough in this case the transition occurs at $T_c=0$.
In particular, scaling of the non-equilibrium equal-time
correlation function $C(q,t)$ has been described recently by
Bray$^{[BA]}$. These results are reproduced here starting from
an exact expression for $C(q,t)$ instead of an approximate one.

\bl
As a technical tool we introduce the relaxation spectrum
$\rho_q(\omega)$ which is in fact the non-equilibrium analogue
of the frequency-dependent structure function (which itself is
the Fourier transform with respect to space and time of the
two-point correlation function). In systems in equilibrium the
frequency-dependent structure function cannot be negative. Here,
the relaxation spectrum will be shown to be positive for small
$\omega$ and negative for large $\omega$, both regions being
separated by a non-integrable singularity. In order to clarify
the important r\^ole of this singularity we show that it
determines the leading order finite size corrections to
scaling.

\bl
According to Pandit et al.$^{[PF]}$, dynamical finite-size
scaling theory cannot be applied in the one-dimensional Ising
model because the gap in the hamiltonian vanishes when the
temperature tends to zero, even if the system is still finite.
The corresponding slowly relaxing mode can be identified to be
the uniform magnetisation. If however averages are taken over
all possible initial configurations then by symmetry the average
magnetisation will remain zero at all times and the slowly
relaxing mode is absent. It is then feasible and meaningful to
calculate the finite-size corrections to the equal-time pair
correlation function. The interest of doing so is that the
one-dimensional Ising model offers the unique opportunity of
exact calculations.

\bl\bl
\centerline{\bf 2. Model}

\bl
A {\sl configuration} $\eta$ of the system with size $N$
is a map from $\{0,1,\cdots,N-1\}$ into $\{-1,1\}$.
The {\sl spin variables} $\sigma_0,\sigma_1,\cdots\sigma_{N-1}$
are defined by $\sigma_n(\eta)=\eta(n)$.

\bl
The time evolution of the kinetic Ising model is described
by the master equation
$${\hbox{d}\ \over\hbox{d}t}P(\eta;t)=\sum_{n=0}^{N-1}
\left[c_n(\eta^n)P(\eta^n;t)-c_n(\eta)P(\eta;t)\right]
\eqno (2.1)$$
where $\eta^n$ is defined by $\eta^n(n)=-\eta(n)$,
and $\eta^n(m)=\eta(m)$ otherwise,
and where $c_n(\eta)$ is the {\sl rate}
with which the sign of $\eta(n)$ changes.
In the Glauber model these transition rates are given by
$$c_n={1\over 2}
\left[1-{\gamma\over 2}\sigma_n(\sigma_{n-1}+\sigma_{n+1})\right]
\eqno(2.2)$$
(periodic boundary conditions are used in case of finite $N$).
The parameter $\gamma$ is related to the inverse of the temperature
$T$ in such a way that $\gamma\rightarrow 1$ when $T\rightarrow 0$
($\gamma=\tanh(2J/k_BT)$ where $J>0$ is the interaction strength of
the Ising model and $k_B$ is Boltzmann's constant).

\bl
The non-equilibrium equal-time two-point correlation function is
the expectation $\langle\sigma_0\sigma_n\rangle_t$ of the
product of the two spin variables $\sigma_0$ and $\sigma_n$ with
respect to the distribution of configurations at time $t$.
It satisfies the equations of motion:
$${\partial\ \over\partial t} \langle\sigma_0\sigma_n\rangle_t
=-2\langle\sigma_0\sigma_n\rangle_t
+\gamma\left(\langle\sigma_0\sigma_{n-1}\rangle_t
+\langle\sigma_0\sigma_{n+1}\rangle_t\right),
\quad\quad (n\not=0)
\eqno(2.3)$$
Introduce the Fourier transformed correlation function
$$C(q,t)=\sum_{n=0}^{N-1}e^{iqn}\langle\sigma_0\sigma_n\rangle_t
\eqno(2.4)$$
Then (2.3) becomes (using periodic boundary conditions)
$${\partial\ \over\partial t} C(q,t)=
-\theta(q)C(q,t)+{1\over N}\sum_{q^*}\theta(q^*)C(q^*,t)
\eqno(2.5)$$
with $\theta(q)=2(1-\gamma\cos(q))$.

\bl\bl
\centerline{\bf 3. Infinite system}

\bl
The equations of motion (2.5) can be solved by taking the
Laplace transform ${\tilde C}(q,z)$ of $C(q,t)$. One obtains
in the limit $N\rightarrow\infty$
$${\tilde C}(q,z)
={\sqrt {z^2+4z+4(1-\gamma^2)}\over z(z+\theta(q))}
\eqno(3.1)$$
((16) of ref.\ [BA] coincides with (3.1) for $z\rightarrow 0$,
$q\rightarrow 0$).

\bl
By inverse Laplace transform of ${\tilde C}(q,z)$ one obtains
$C(q,t)$. It is however easier to make the {\sl ansatz} that
${\tilde C}(q,z)$ is the Stieltjes transform of a relaxation
spectrum $\rho_q(\omega)$
$${\tilde C}(q,z)
={1\over z}C(q,\infty)
+\int_0^\infty\hbox{ d}\omega\ \rho_q(\omega){1\over z+\omega}
\eqno(3.2)$$
with
$$C(q,\infty)={2\sqrt {1-\gamma^2}\over \theta(q)}\eqno(3.3)$$
and then to calculate the inverse Stieltjes transform
$$\rho_q(\omega)=-{1\over \pi}\hbox{Im}
\left[{\tilde C}(q,-\omega+i0)-{C(q,\infty)\over-\omega+i0}\right],
\quad\quad\omega>0\eqno(3.4)$$
One obtains
$$\eqalign{
\rho_q(\omega)
&={1\over\pi}{\sqrt {4\gamma^2-(\omega-2)^2}
\over \omega\left[\theta(q)-\omega\right]}
\quad\hbox{ if }\quad 2(1-\gamma)\le\omega\le 2(1+\gamma)\cr
&=0\quad\hbox{ otherwise }\cr
}\eqno(3.5)$$
See fig.\ 1.
Note that $\rho_q(\omega)$ contains a non-integrable singularity
at $\omega=\theta(q)$. Now $C(q,t)$ is obtained as the Laplace
transform of $\rho_q(\omega)$:
$$C(q,t)=C(q,\infty)+{\bf P}\int_0^\infty
\rho_q(\omega)e^{-\omega t}\hbox{ d}\omega\eqno(3.6)$$
The symbol ${\bf P}$ indicates that the principal value has to
be taken when integrating through the singularity. By a change
of integration variable one obtains
$$C(q,t)=C(q,\infty)+{1\over\pi}
{\bf P}\int_0^\pi\ {X_t(q^*,q)\over q-q^*}
\hbox{ d}q^*\eqno(3.7)$$
with
$$X_t(q^*,q)={4\gamma^2-(\theta(q^*)-2)^2\over \theta(q^*)}
\ e^{-\theta(q^*)t}
\ {q-q^*\over \theta(q)-\theta(q^*)}\eqno(3.8)$$
Because $\rho_q(\omega)\ge 0$ for small $\omega$ one has always
for long times that $C(q,t)>C(q,\infty)$. In fact, there exists a
time-dependent wave vector $q_0(t)$ which tends to zero as
$t\rightarrow\infty$ in such a way that $C(q,t)>C(q,\infty)$ if
and only if $q>q_0(t)$. For small $q\not=0$ this implies that
$C(q,t)$ will first rise and cross the equilibrium value
$C(q,\infty)$ and then, for long times, will converge by
decreasing. See fig.\ 2.

\bl\bl
\centerline{\bf 4. Finite-size corrections}

\bl
Now we calculate $C(q,t)$ for a finite ring with $N$
spins. Then, by comparison with the result for the
infinite system, we can understand the origin of the singularity
in $\rho_q(\omega)$. In addition, finite-size corrections can be
calculated.

\bl
We use the method of Reiss$^{[RH]}$. It was introduced to
generalise Glauber's method$^{[GR]}$ to the case of temperature
varying with time. Here we assume an initial configuration
corresponding to infinite temperature, and evolution of the
system with Glauber dynamics at fixed temperature. The result of
the calculations is
$$\langle\sigma_0\sigma_n\rangle_t
={2\gamma\over N}\sum_{m\hbox{\small \ odd},=1}^{N-1}
\sin(\pi {mn\over N})\sin(\pi{m\over N}){1-e^{-\theta(q_m)t}\over\theta(q_m)}
\eqno(4.1)$$
with $q_m\equiv \pi m/N$.
After Fourier-transformation one obtains
$$C_N(q_{2n},t)=C_N(q_{2n},\infty)+
{2\over N}\sum_{m\hbox{\small \ odd},=1}^{N-1}
{X_t(q_m,q_{2n})\over q_{2n}-q_{m}}
\eqno(4.2)$$
where $X_t$ has been defined in (3.8).
Hence a near-singularity is already present in the finite system
and obviously, (4.2) converges to (3.7) as $n\rightarrow\infty$,
$N\rightarrow\infty$ in such a way that $q=q_{2n}\equiv 2\pi
n/N$ remains constant.

\bl
Note that the longest relaxation time is $1/\theta(\pi/N)$ which
becomes $1/4\sin^2(\pi/2N)$ at zero temperature ($\gamma=1$).
Hence, in contrast to the situation of ref.\ [PF], in the finite
system the gap in the spectrum of the generator of the time
evolution does not vanish in the limit of zero temperature
(see the introduction).

\bl
At this point we need a trick to calculate the difference
between $C(q,t)$ and $C_N(q,t)$ to first order in $1/N$.
The essence is that the function $q^*\rightarrow X_t(q^*,q)$
is analytic also in the point $q^*=q$. In fact,
from the definition (3.8) follows
$$X_t(q,q)={\sqrt{4\gamma^2-(\theta(q)-2)^2}\over\theta(q)}
\ e^{-\theta(q)t}\eqno(4.3)$$
For convenience we take $N$ even. The integral in (3.7) can be split into
two parts, one containing the singularity, the other
containing the complicated but regular contribution.
The latter integral can be approximated by a sum.
$$\eqalign{
&C(q,t)-C(q,\infty)\cr
&=X_t(q,q){1\over\pi}{\bf P}\int_0^\pi\ {1\over q-q^*}
\hbox{ d}q^*
+{1\over\pi}\int_0^\pi\ {X_t(q^*,q)-X_t(q,q)\over q-q^*}
\hbox{ d}q^*\cr
&=-X_t(q,q){1\over\pi}\ln{\pi-q\over q}
+{2\over N}\sum_{m\hbox{\small \ odd},=1}^{N-1}
{X_t(q_m,q)-X_t(q,q)
\over q-q_m}+\hbox{O}\left(N^{-2}\right)\cr
&=C_N(q,t)-C_N(q,\infty)
+{1\over N}X_t(q,q)\left[{1\over q}-{1\over \pi-q}\right]
+\hbox{O}\left(N^{-2}\right)\cr
}\eqno(4.4)$$
In the last line we used that
$${2\over N}\sum_{m\hbox{\small \ odd},=1}^{N-1}
{1\over q_{2n}-q_{m}}=-{1\over\pi}\ln{\pi-q\over q}
+{1\over N}\left[{1\over \pi-q}-{1\over q}\right]
+\hbox{O}\left(N^{-2}\right)
\eqno(4.5)$$

\bl
The second term in the r.h.s.\ of
(2.5) is sometimes called the {\sl self-}contribution,
because it arises from the fact that (2.3) is not valid for $n=0$
(note that $N^{-1}\sum_q\theta(q)C(q,t)=2-\gamma\langle\sigma_0\sigma_1\rangle_t
-\gamma\langle\sigma_0\sigma_{N-1}\rangle_t$).
If one omits the self-term in (2.5) then $C(q,t)$
relaxes in a pure exponential way with relaxation time
$1/\theta(q)$. Hence the structure of the relaxation spectrum $\rho_q(\omega)$
originates fully from this self-contribution. In this
light it is not unexpected that the leading-order correction term
to $C(q,t)$ relaxes as $\exp(-\theta(q)t)$, as follows from (4.3)
and (4.4).

\bl
Finally, note that the finite-size analogue of (3.1) is
$$\tilde C_N(q,z)=\tilde C(q,z){1-K(z)^N\over 1+K(z)^N}
\eqno(4.6)$$
with
$$K(z)={1\over 2\gamma}\left(2+z-\sqrt {z^2+4z+4(1-\gamma)^2}\right)
\eqno(4.7)$$

\bl\bl
\centerline{\bf 5. Finite-size scaling}

\bl
Introduce the scaling variables
$$a=2(1-\gamma)t
\quad\quad\hbox{ and }\quad\quad
b=2\gamma(1-\cos(q))t
\eqno(5.1)$$
Note that the thermal correlation length $\xi$ is approximately
equal to $1/\sqrt{2(1-\gamma)}$. Hence the choice of the scaling
variable $a$ implies a dynamic exponent $z=2$.
By a change of integration variable, (3.6) becomes, in the
limit of $t\rightarrow\infty$, $q\rightarrow 0$, and
$\gamma\rightarrow 1$,
$$C(q,t)=C(q,\infty)+\sqrt {t}\ g_0(a,b)+\cdots
\eqno(5.2)$$
with the scaling function $g_0$ given by
$$g_0(a,b)
=2e^{-a}\pi^{-1}{\bf P}\int_0^\infty\hbox{ d}y
\ e^{-y}{\sqrt{y} \over (y+a)(b-y)}
\eqno(5.3)$$
Note that $g_0$ can be rewritten as
$$g_0(a,b)=-{2\sqrt a\over a+b}
+{2\over a+b}{1\over\sqrt\pi}\int_0^1\hbox{ d}y{1\over\sqrt y}
\ \left\{a\ e^{-ay}
+b\ e^{-a-b(1-y)}\right\}
\eqno(5.4)$$
and hence coincides with the scaling function proposed by
Bray$^{[BA]}$.

\bl
The relaxation spectrum corresponding with (5.2), (5.3) is
$$\eqalign{
\rho_q(\omega)
&={2\over\pi}{\sqrt {\omega-\omega_0}
\over\omega(k^2+\omega_0-\omega)}\qquad\hbox{ if }\omega\ge \omega_0\cr
&= 0\qquad\hbox{ if }\omega\le \omega_0\cr
}\eqno(5.5)$$
with the gap frequency $\omega_0=2(1-\gamma)\simeq \xi^{-2}$ and the scaled
wave vector $k$ such that $k^2=2\gamma(1-\cos q)\simeq q^2$.
In fact, (5.5) can be obtained immediately from (3.5)
by assuming that $\omega$ and $\omega_0$ are small.

\bl
In finite-size scaling$^{[BM]}$ the extra scaling variable that
one expects is $t/N^2$ (note that $z=2$). Let
$$c=\sqrt{t}/ N\eqno(5.6)$$
>From (4.4) and (5.2) follows
in the limit of $t\rightarrow\infty$, $q\rightarrow 0$,
and $\gamma\rightarrow 1$
$$C_N(q,t)-C_N(q,\infty)=\sqrt{t} \left[g_0(a,b)+ c\ g_1(a,b)
+\hbox{O}(c^2)\right]+\cdots
\eqno(5.7)$$
with
$$g_1(a,b)=2{e^{-(a+b)}\over a+b}\eqno(5.8)$$
This result is compatible with the existence of a scaling function
$g$ such that
$$C_N(q,t)-C_N(q,\infty)=\sqrt{t} \ g(a,b,c)+\cdots
\eqno(5.9)$$
The leading order correction to the relaxation spectrum (5.5) is
a Dirac-delta function at the frequency $\omega_0+k^2$
$${2\over \omega N}\delta(\omega-\omega_0-k^2)
\eqno(5.10)$$
with $k$ and $\omega$ as in (5.5).

\bl\bl
\centerline{\bf 6. Epilogue}

\bl
We expect some of our formulas to be more generally valid.
However, already a modest generalisation like replacing the
Glauber dynamics by some other dynamics satisfying detailed
balance with respect to the Ising Hamiltonian makes the present
approach very hard, if not impossible. Indeed, in general the
equations of motion (2.3) are not closed but involve products
of more than 2 spin variables. Hence it is not clear how to
obtain (3.1) in closed form. Therefore, more abstract methods
will be needed.

\bl\bl
\centerline{\bf References}
{\raggedright\parskip 2pt\parindent 24pt

\item{[BM]} see e.g. M.N. Barber, {\sl Finite-size Scaling,} in:
Phase Transitions and Critical Phenomena, Vol. 8, ed. C. Domb and
J.L. Lebowitz (Academic Press, 1983), p. 145 --- 266.

\item{[BA]} A.J. Bray, J. Phys. A{\bf 23}, L67-L72 (1990).

\item{[GR]} R.J. Glauber, J. Math. Phys. {\bf 4}, 294-307 (1963).

\item{[PF]} R. Pandit, G. Forgacs, and P. Rujan, Phys. Rev.
B{\bf 24}, 1576 (1981).

\item{[RH]} H. Reiss, Chem. Phys. {\bf 47}, 15-24 (1980).
}

\bl\bl
\centerline{\bf Figure Captions}

\bl
Fig.\ 1:
Spectral density $\rho_q(\omega)$ as a function of $\omega$
for $\gamma=0.9$ and $\theta(q)=0.4$ (see 3.5).

\bl
Fig.\ 2:
$C(q,t)$ as a function of $q$ for $t=0.2,1.0,5.0$ and $\gamma=0.8$
obtained by numerical integration of (3.7); the dashed line is the
$t=\infty$ result.


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