Fluctuation-dissipation relations in critical coarsening: crossover from unmagnetized to magnetized initial states

0.44 MB·English

by  Alessia Annibale

Fluctuation-dissipation relations in critical
8 coarsening: crossover from unmagnetized to
0
0 magnetized initial states
2

n
a Alessia Annibale , Peter Sollich
J † ‡
9 King’s College London, Department of Mathematics, London WC2R 2LS, UK

]
n Abstract. We study the non-equilibrium dynamics of the spherical ferromagnet
n quenched to its critical temperature, as a function of the magnetization of the initial
-
s state. The two limits of unmagnetized and fully magnetized initial conditions can be
i
d understood as corresponding to times that are respectively much shorter and much
. longerthanamagnetizationtimescale,asinarecentfieldtheoreticalanalysisofthen-
t
a
vectormodel. Wecalculateexactlythecrossoverfunctionsinterpolatingbetweenthese
m
twolimits,forthemagnetizationcorrelatorandresponseandtheresultingfluctuation-
-
d dissipation ratio (FDR). For d > 4 our results match those obtained recently from
n a Gaussian field theory. For d < 4, non-Gaussian fuctuations arising from the
o
sphericalconstraintneedtobeaccountedfor. Weextendourframeworkfromthefully
c
[ magnetized case to achieve this, providing an exact solution for the relevant integral
kernel. The resulting crossover behaviour is very rich, with the asymptotic FDR X∞
1
v depending non-monotonically on the scaled age of the system. This is traced back
1 to non-monotonicities of the two-time correlator, themselves the consequence of large
8
magnetizationfluctuationsonthecrossovertimescale. Wecorrectatrivialerrorinour
3
1 earlier calculation for fully magnetized initial states; the corrected FDR is consistent
. with renormalization group expansions to first order in 4 d for the longitudinal
1 −
0 fluctuations of the O(n) model in the limit n .
→∞
8
0
:
v
Xi 1. Introduction
r
a
The use of fluctuation-dissipation ratios (FDR) has proved very fruitful in the last
decade or so for quantifying the non-equilibrium dynamics of glasses and other systems
exhibiting aging. In the context of mean-field spin glass models with infinite-range
interactions, the FDR, commonly denoted X, has been used to formulate a generalized
fluctuation-dissipation theorem (FDT) where X is interpreted in terms of an effective
temperature, T = T/X for the slow, non-equilibrated modes of the system [1]. The
eff
properties of X and T have attracted much attention, based on the hope they might
eff
allowageneralizedstatisticalmechanicaldescription forabroadclassofnon-equilibrium
phenomena [2, 3, 4].
Email [email protected]
†
Email [email protected]
‡
Fluctuation-dissipation relations in critical coarsening 2
However, the generalized FDT can be shown to hold exactly only for infinite-
range models. A matter of recent intense interest has been whether the appealing
features of this mean-field scenario survive in more realistic systems with finite-range
interactions [2]. A class of systems that has proved useful in this context is represented
byferromagnetsquenched fromhightemperature tothecriticaltemperatureT orbelow
c
(see e.g. Refs. [5, 6, 7, 8, 9] and the recent review [10]). The non-equilibrium dynamics
in these systems is due to coarsening, i.e. the growth of domains with the equilibrium
magnetization (for T < T ) or equilibrium correlation structure (for T = T ), and slows
c c
down as domain sizes increase. In an infinite system, equilibrium is never reached,
leading to aging; the age-dependence of two-time quantities has a simple physical
interpretation interms ofthegrowthof thedomainlengthscale [11]. Coarsening systems
therefore provide a physically intuitive setting for the study of aging phenomena as
observed e.g. in glasses, polymers and colloids. They are, of course, not completely
generic; compared to e.g. glasses they lack features such as thermal activation over
energetic or entropic barriers.
We focusinthis paper oncriticalcoarsening, i.e.coarsening atT , where interesting
c
connections to dynamical universality exist. The FDRX is determined from correlation
and response functions which, in aging systems, depend on two times: the age t of the
w
system and a later measurement time t. In contrast to mean-field spin glasses, where X
is constant within each “time sector” (e.g. t t = (1) vs t t growing with t ), in
w w w
− O −
critical coarsening the FDR is a smooth function of t/t . This makes the interpretation
w
of T/X as an effective temperature less obvious. To eliminate the time-dependence one
can consider the limit of times that are both large and well-separated. This defines an
asymptotic FDR
∞
X = lim lim X(t,t ) (1)
w
tw→∞t→∞
An important property of this quantity is that it should be universal [5, 10] in the sense
that its value is the same for different systems falling into the same universality class
of critical non-equilibrium dynamics. This makes a study of X∞ interesting in its own
right, even without an interpretation in terms of effective temperatures.
An intriguing theoretical question which has been addressed recently is whether
different initial conditions can lead to different universality classes of critical coarsening.
Due to the universality of X∞, these can be uncovered by studying the effect that
different initial conditions have on the FDR. Of particular interest has been the effect
of an initial magnetization on the ensuing coarsening. For the Ising model in high
dimension or with long-range interactions [12], one finds that magnetized initial states
doproduceadifferentvalueofX∞. Thissuggestsadifferentdynamicaluniversalityclass
from conventional coarsening from unmagnetized states, even though the magnetization
decays to zero at long times. Further steps in this direction were taken in our recent
calculation of exact FDRs for magnetized coarsening below the upper critical dimension
in the spherical model [13]. The propagation of a trivial error meant that the results
were at variance with the renormalization group (RG) result of Ref. [14] derived for the
Fluctuation-dissipation relations in critical coarsening 3
longitudinal fluctuations in the n limit of the O(n) model within an expansion
→ ∞
around d = 4. We give the corrected results in this paper, and these are consistent with
the RG calculations (see Appendix A). This suggests that the equivalence between the
dynamics of the spherical model and the large-n limit of the O(n) model extends beyond
the regime of Gaussian fluctuations, where it is trivial to establish.
Recently it was emphasized in the context of a field-theoretic analysis [15] that one
should think of the nonzero initial magnetization as introducing a new timescale in the
system. The two limits of unmagnetized and magnetized initial conditions can then
be understood as corresponding to times that are respectively much shorter and much
longer than this magnetization timescale, and one can in fact interpolate between these
two limits using a crossover function that depends on times scaled by the magnetization
timescale. This crossover function was calculated in [15] in the classical (Gaussian)
regime, i.e. above the upper critical dimension, but so far there are no predictions
for this function for lower dimensions where the critical behaviour is governed by non-
mean-fieldexponents. Weprovidethefirst resultsofthiskindinthisworkbycalculating
the relevant crossover functions exactly for the spherical model in 2 < d < 4, for the
correlator, response and FDR of the magnetization.
In Sec. 2 we recall the known crossover behaviour of the magnetization (which is
directlyrelatedtoafunctiong(t))andthegeneralrelationsencodingtheconsequences of
this for the magnetization correlation and response functions. As in [13] non-Gaussian
spin fluctuations are important and will be accounted for via the kernel L. Key to
our analysis for the more complicated functions g(t) in our current scenario is an
exact solution of the integral equation defining L that applies independently of the
time regime. In Sec. 3 we then evaluate the magnetization correlator and response
for d > 4. As expected, we find here full agreement with the Gaussian field-theoretic
calculations [15]. Sec. 4 deals with the more interesting case d < 4. Here the analysis
is more complicated but we can still derive exact results for the asymptotic FDR X∞.
The relevant crossover functions display unexpected non-monotonicities that, close to
the lower critical dimension d = 2, turn into singularities at intermediate values of the
scaled system age. We study carefully the relevant scaling regimes for d 2, and
→
investigate how they arise from the behaviour of the two-time magnetization correlator.
Our results are summarized in Sec. 5.
2. Setup of calculation and exact solution for L(2)
Westartbyrecapitulatingbrieflytherelevantelementsofourpreviousanalysisofcritical
coarsening in the spherical ferromagnet [13]. The model consists of N spins S on a d-
i
dimensional cubic lattice, with sites r and Hamiltonian H = 1 (S S )2 [16]. The
i 2 (ij) i− j
spins are real-valued but subject to the spherical constraint S2 = N. Langevin
Pi i
dynamics leads to a simple equation of motion for the Fourier components S =
P q
S exp( iq r ) of the spins, ∂ S = (ω +z(t))S +ξ where ω = 2 d (1 cosq )
i i − · i t q − q q q q a=1 − a
is abbreviated to ω below and ξ is independent Gaussian noise on each wavevector
P q P
Fluctuation-dissipation relations in critical coarsening 4
q = (q ,...,q ), with ξ (t)ξ∗(t′) = 2NTδ(t t′). The Lagrange multiplier z(t)
1 d h q q i −
enforces the spherical constraint; as explained in [13], it is in reality not just a simple
function of time but a dynamical variable with fluctuations of (N−1/2) that cause all
O
the non-trivial effects in the behaviour of global observables. In terms of the function
g(t) = exp 2 tdt′z(t′) the Fourier mode response is
0
(cid:16) (cid:17)
R
g(t ) m(t)
R (t,t ) = w e−ω(t−tw) = e−ω(t−tw) (2)
q w v g(t) m(t )
u w
u
t
In the second equality we have used that the time-dependent magnetization can be
written as m(t) = (1/N) S (t) = R (t,0)(1/N) S (0) = m / g(t) with m =
0 0 0 0 0
h i h i
(1/N) S (0) the initial magnetization. The full, unsubtracted tqwo-time correlator
0
h i
C (t,t ) = (1/N) S (t)S∗(t ) can be related to its equal-time value by the response
q w q q w
h i
function,
C (t,t ) = R (t,t )C (t ,t ) (3)
q w q w q w w
The relevant equal-time value is given by
C (0,0) t g(t′)
C (t,t) = q e−2ωt +2T dt′ e−2ω(t−t′) (4)
q
g(t) g(t)
Z0
The function g(t) is determined from the spherical constraint, which imposes
(dq)C (t,t) = 1. Here and below we abbreviate (dq) dq/(2π)d, where the integral
q
≡
runs over the first Brillouin zone of the hypercubic lattice, i.e. q [ π,π]d. The
R
∈ −
resulting integral equation for g(t) is
t
g(t) = (dq)C (0,0)e−2ωt +2T dt′g(t′)f(t t′) (5)
q
−
Z Z0
with f(t) = (dq) e−2ωt. Our first task will be to understand how the solution of this
crosses over between the magnetized and unmagnetized cases. In terms of the Laplace
R
transform gˆ(s) = ∞dtg(t)e−st, equation (5) reads
0
R 1 C (0,0)
q
gˆ(s) = (dq) (6)
1 2Tfˆ(s) s+2ω
Z
−
We take as the initial condition the standard choice [17, 15] of a small magnetization
m but otherwise uncorrelated spin fluctuations. The initial equal-time Fourier mode
0
correlator can then be written as
C (0,0) = δ Nm2 +(1 m2) (7)
q q,0 0 − 0
Thisunsubtractedcorrelatoris (1)forq = 0but (N)forq = 0. (Forthefluctuation-
O 6 O
dissipation behaviour we will need to look at the connected correlator C˜ , which is
q
discussed below.) Equation (7) yields, bearing in mind that the integral (dq) is really a
sum over the N discrete wavevectors with weight 1/N each,
C (0,0) m2
(dq) q = 0 +(1 m2)fˆ(s) (8)
s+2ω s − 0
Z
Fluctuation-dissipation relations in critical coarsening 5
Using this in (6) one has at criticality, where T = T = [ (dq)1/ω]−1 = [2fˆ(0)]−1,
c
1 m2 R
gˆ(s) = Kˆ−1(s) 0 +(1 m2)fˆ(s) (9)
eq s " s − 0 #
with
1
ˆ
K (s) = T (dq) (10)
eq c
ω(2ω+s)
Z
the Laplace transform of the equilibrium form (38) of the kernel K defined below. As
before [13] we want to look at the long-time limit of g(t), corresponding to small s in (9).
In this regime Kˆ (s) is given for d > 4 by Kˆ (0) Kˆ (s) = as(d−4)/2 and for d < 4 by
eq eq eq
−
Kˆ (s) = bs(d−4)/2, with a and b some d-dependent constants. In the remaining square
eq
bracket of (9) only the first term is present for a fully magnetized initial state (m = 1);
0
conversely, only the second survives for the unmagnetized case (m = 0). To see the
0
crossover between these limits the two terms need to be of the same order. Because we
are interested in small s and fˆ(0) is nonzero, this implies that m2 and s must be of the
0
same order. We then find to leading order in these small quantities
m2 fˆ(0)
Kˆ−1(0) 0 + (d > 4)
 eq " s2 s #
gˆ(s) = (11)
 s(4−d)/2 m20 + fˆ(0) (d < 4)
b " s2 s #
or in the time domain
Kˆ−1(0) m2t+fˆ(0) (d > 4)
eq 0
g(t) =  1 m2h i fˆ(0) (12)
 b "Γ(d/02)t(d−2)/2 + Γ((d 2)/2)t(d−4)/2# (d < 4)
−
One can combine these two expressions as
1 c t
g(t) = t−κ(m2t+c) = t−κ +1 (13)
µ 0 µ τ
d d (cid:18) m (cid:19)
where we have defined
4−d (d < 4) bΓ(d/2) (d < 4)
κ = 2 µ = (14)
( 0 (d > 4) d ( Kˆeq(0) (d > 4)
ˆ
and c = (1 κ)f(0). In the second equality of (13) we have taken out the factor of c to
−
identify the crossover timescale
c
τ = (15)
m m2
0
which as anticipated in the introduction depends on the initial magnetization of the
system. In the time domain, our statement of the relevant long-time scaling m2 s
0 ∼
can now be phrased as follows: we will be considering the limit of large t, t and τ
w m
(corresponding to small m ) at fixed time ratios u = t /τ and u = t/τ . For ease of
0 w w m t m
comparison with the work of [15] we will write simply u u and mostly work with u
w
≡
Fluctuation-dissipation relations in critical coarsening 6
and the time ratio x = t/t = u /u instead of u and u . In terms of these variables one
w t t
can write the function g(t) as
c
g(t) = t−κ(ux+1) (16)
µ
d
For the magnetization one then finds
m c µ tκ/2 µ1/2 ux
m(t) = 0 = d = d (17)
g(t) sτm c √ux+1 tα/2sux+1
q
with the exponent α defined as α = 1 κ as in [13]. The last square root equals unity
−
for long times if the initial magnetization is kept finite and nonzero (so that u 1).
≫
Otherwise it gives the well-known correction to the fully magnetized result when the
initial magnetization is small, i.e. when t τ [17]. In particular, for u = ux 1,
m t
∼ ≪
the magnetization displays critical initial slip, increasing as m(t) tκ/2, before crossing
∼
over to the t−α/2 decay around u = 1. Our analysis for the fully magnetized case in [13]
t
is now recognized as relating to the limit t,t τ , and accordingly all results in this
w m
≫
paper should reduce to the ones in [13] in the limit u . (Loosely speaking, one can
→ ∞
think of this limit as corresponding to τ 0, i.e. “m = ” [15].) In the opposite
m 0
→ ∞
limit t,t τ we should get back the results for the unmagnetized case m = 0. In
w m 0
≪
termsofour scaling variables, this limitcorresponds tou 0atfixedx. Notethatthere
→
is in principle a third, “mixed” regime where the earlier time t τ but the later time
w m
≪
t τ , i.e. u 1 and ux 1. We will see, however, that essentially no new behaviour
m
≫ ≪ ≫
arises here and the crossover between the magnetized and unmagnetized cases, which
the analysis below will allow us to elucidate explicitly, is governed principally by u.
We next explore how the crossover effects in g(t) modify the expressions for the
long-time behaviour of the connected Fourier mode correlator C˜ (t,t ) = C (t,t )
q w q w
−
(1/N) S (t) S∗(t ) = C (t,t ) Nδ m(t)m(t ) and the response function R (t,t ).
q q w q w q,0 w q w
h ih i −
(Here, as previously, we will not write explicitly the dependence on τ .) From [13]
m
we know that the equal-time connected correlator has the same expression as the
unsubtracted correlator
C˜ (t ,t ) = 1 [C˜ (0,0)e−2ωtw +2T tw dt′e−2ω(tw−t′)g(t′)] (18)
q w w q c
g(t )
w Z0
except for the appropriately modified initial condition C˜ (0,0) = 1 m2 which – in
q − 0
contrast to the unsubtracted C – is (1) for all q. For the zero Fourier mode one sees
q
O
that in the long-time limit the first term is subleading and the integral diverges at the
upper end so that one can use the asymptotics of g(t′), giving
1 tw u/(2 κ)+1/(1 κ)
C˜ (t ,t ) = [C˜ (0,0)+2T dt′g(t′)] = 2T t − − (19)
0 w w 0 c c w
g(t ) u+1
w Z0
Similarly in the ratio of nonzero and zero mode correlators, expressed in terms of the
scaling variable w = ωt ,
w
C˜ (t ,t ) (1 m2)e−2w +2T t 1dze−2w(1−y)g(zt )
q w w = − 0 c w 0 w (20)
C˜0(tw,tw) 1+2Tctw 01Rdzg(ztw)
R
Fluctuation-dissipation relations in critical coarsening 7
one can neglect the non-integral terms for long times and gets
C˜ (t ,t ) 1dze−2w(1−y)z−κ(zu+1) 1dze−2w(1−z)z−κ(uz +1)
q w w = 0 = 0 (21)
C˜0(tw,tw) R 01dzz−κ(uz +1) R u/(2−κ)+1/(1−κ)
Putting the last twRo results together yields the general scaling
T 2w 1
C˜ (t ,t ) = c (w,u), (w,u) = dze−2w(1−z)z−κ(uz +1) (22)
q w w C C
ω F F u+1
Z0
One checks easily that (w,u) reduces to the analogous scaling functions for the
C
F
unmagnetized and fully magnetized cases [13] in the appropriate limits u 0 and
→
u . The magnetization response function is the zero mode response R . From (2),
0
→ ∞
using the scaling of the magnetization found in (17), it is given by
m(t) u+1
R (t,t ) = = xκ/2 (23)
0 w
m(t ) sux+1
w
The results above are valid within the Gaussian approximation for the spin
dynamics inthe spherical model, where thesmall fluctuations inthe Lagrangemultiplier
z(t) are neglected. As we saw in [13], in order to study the FD behaviour of the
magnetization (i.e. of the zero Fourier mode, which is a global observable) when
an initial nonzero magnetization is present, we need to account for non-Gaussian
corrections arising from these Lagrange multiplier fluctuations. Fortunately our earlier
expressions [13] for the resulting magnetization correlator and response are valid for
arbitrary initial conditions and can be used directly. The magnetization correlator
including non-Gaussian effects is [13]
C(t,t ) = C(1)(t,t )+C(2)(t,t ) (24)
w w w
with
C(1)(t,t ) = C˜ (t,t ) dt′[M(t,t′)C˜ (t ,t′)+M(t ,t′)C˜ (t,t′)]m(t′)
w 0 w 0 w w 0
−
Z
+ dt′dt′ M(t,t′)M(t ,t′ )m(t′)m(t′ )C˜ (t′,t′ ) (25)
w w w w 0 w
Z
′ ′ ′ ′ ′
= dt dt [δ(t t) M(t,t)m(t)]
w − −
Z
[δ(t t′ ) M(t ,t′ )m(t′ )]C˜ (t′,t′ ) (26)
× w − w − w w w 0 w
and
1
C(2)(t,t ) = dt′dt′ M(t,t′)M(t ,t′ )C˜C˜(t′,t′ ) (27)
w 2 w w w w
Z
where C˜C˜(t′,t′ ) = (dq)C˜2(t′,t′ ). The corresponding expression for the global
w q w
magnetization response including non-Gaussian effects is [13]
R
′ ′ ′ ′ ′
R(t,t ) = dt [δ(t t) M(t,t)m(t)]R (t,t ) (28)
w 0 w
− −
Z
The key function M appearing here is defined as follows. One starts from the kernel
K(t,t ) = (dq)R (t,t )C (t,t ) = (dq)R2(t,t )C (t ,t ) (29)
w q w q w q w q w w
Z Z
Fluctuation-dissipation relations in critical coarsening 8
and its inverse L defined by
′ ′ ′
dt K(t,t)L(t,t ) = δ(t t ) (30)
w w
−
Z
The behaviour of K(t,t ) near t = t can be shown to imply the following structure for
w w
L
L(t,t ) = δ′(t t )+2T δ(t t ) L(2)(t,t ) (31)
w w c w w
− − −
where the first term arises from the fact that K(t,t ) is causal (i.e. it vanishes for
w
t > t) and has a unit jump at t = t. Finally, M is defined to be proportional to the
w w
integral of L:
t
′ ′
M(t,t ) = m(t) dt L(t,t ) (32)
w w
Z
In our previous analysis [13] we had found long-time scaling forms of L(2) separately
for the unmagnetized and magnetized cases, with different methods needed for d > 4
and d < 4. With the function g(t) no longer being a simple power law, it seems
difficult if not impossible to adapt these methods to our current crossover calculation.
Fortunately, however, there is a general and fully exact solution for L(2) which applies
in any dimension and for any g(t). To obtain this, we essentially integrate by parts
in (30). In the derivative of K with respect to the earlier time argument we separate
off the contribution from the unit step and write
′
∂ K(t,t ) = δ(t t )+K (t,t ) (33)
tw w − − w w
where K′, like K, vanishes for t > t and is finite elsewhere. Correspondingly we split
w
off the first term from (31) and write
t
′ ′
dt L(t,t ) = δ(t t )+N(t,t ) (34)
w w w
−
Z
where explicitly
t
N(t,t ) = 2T dt′L(2)(t′,t ) (35)
w c w
−
Ztw
and N(t,t ) also vanishes for t > t. Integrating by parts in (30) and substituting these
w w
definitions then yields
t
′ ′ ′ ′ ′
K (t,t )+ dt K (t,t)N(t,t ) N(t,t ) = 0 (36)
w w w
−
Ztw
The point of this transformation is that non-equilibrium effects manifest themselves
in K′ in a very simple form. To see this, note from (4) for the unsubtracted
correlator that ∂ C (t ,t ) = [g′(t )/g(t ) + 2ω]C (t ,t ) + 2T , while from (2)
tw q w w − w w q w w c
∂ R2(t,t ) = [g′(t )/g(t )+2ω]R2(t,t ). Inserting into (29) gives
tw q w w w q w
g(t ) g(t )
K′(t,t ) = 2T (dq)R2(t,t ) = w 2T (dq)e−2ω(t−tw) = w K′ (t t ) (37)
w c q w g(t) c − g(t) eq − w
Z Z
where
T
K (t t ) = (dq) ce−2ω(t−tw) (38)
eq w
− ω
Z
Fluctuation-dissipation relations in critical coarsening 9
(with Laplace transform given by (10)) is the equilibrium form of K(t,t ). With the
w
simple multiplicative structure of (37) one can now solve the integral equation (36) for
N by inspection:
g(t )
w
N(t,t ) = N (t t ) (39)
w eq w
− g(t)
where N (t t ) is the solution of the equilibrium version of (36), which is related to
eq w
−
the corresponding L(2) by
eq
t−tw 2λd(t t )(d−4)/2 (d < 4)
N (t t ) = 2T dτ L(2)(τ) 4−d − w (40)
eq − w c −Z0 eq ≈ ( µ1d + d2−λd4 (t−tw)(4−d)/2 (d > 4)
The last approximation gives the scalings for large time differences t t , derived
w
−
from the corresponding asymptotic behaviour of L(2). The latter is L(2)(t t ) =
eq eq − w
λ (t t )(d−6)/2 in d < 4 and L(2)(t t ) = λ (t t )(2−d)/2 in d > 4, with λ a d-
d − w eq − w d − w d
dependent coefficient [13]. This behaviour can be derived from the Laplace transform
of L(2), which from the equilibrium versions of (30,31) follows as
eq
Lˆ(2)(s) = s+T 1/Kˆ (s) (41)
eq c − eq
Note that N decays to zero for d < 4 because Lˆ(2)(0) = ∞dτL(2)(τ) = 2T exactly,
eq eq 0 eq c
while for d > 4 it approaches the nonzero limit 2T Lˆ(2)(0) = 1/µ [13].
c − eq R d
The kernel M is directly related to N from (32) and (35):
M(t,t ) = m(t)[δ(t t )+N(t,t )] (42)
w w w
−
and in our current context we do not then need to compute L(2) explicitly. Briefly,
though, the general solution for L(2) is
g(t ) g′(t)g(t )
L(2)(t,t ) = ∂ N(t,t ) = w L(2)(t t )+ w N (t t ) (43)
w − t w g(t) eq − w g2(t) eq − w
and we outline in Appendix B how this retrieves all of our previous results in the
appropriate limits. The key advantage of the above solution method is that it
automatically accounts for all non-equilibrium effects by reducing the problem to an
equilibrium calculation at criticality, where allfunctions depend only ontime differences
and the relevant integral equation can easily be solved by Laplace transform as shown
in (41) above.
With the general solution for M(t,t ), and hence for the magnetization correlator
w
and response, now in hand we analyse separately the cases d > 4 and d < 4.
3. Crossover behaviour in d > 4
We first consider the situation d > 4 above the upper critical dimension. We expect
to find here the same results for universal quantities as in the Gaussian field theory
of [15]. The zero Fourier mode Gaussian correlator and response are obtained from (19)
and (23) by setting κ = 0:
u+1 u+2
R (t,t ) = , C˜ (t ,t ) = T t (44)
0 w 0 w w c w
sux+1 u+1
Fluctuation-dissipation relations in critical coarsening 10
With these, one has from (39), (40) and (42)
m2 g(t′) 1 2λ
M(t,t′)m(t′) = 0 δ(t t′)+ + d (t t′)(4−d)/2
g(t)g(t′) ( − g(t) "µd d 4 − #)
−
µqu 1
= d t−1δ(x y)
tw (ux+1)1/2(uy +1)1/2( w −
uy +1 1 2λ
+ + d t(4−d)/2(x y)(4−d)/2 (45)
ux+1 "µd d 4 w − #)
−
where we have rescaled the times with t and introduced the scaling variable y = t′/t .
w w
Inthelong-timelimitthefirstandthethirdtermsintheaboveexpression aresubleading
for d > 4 so
u(uy+1)1/2
′ ′
M(t,t)m(t) = (46)
t (ux+1)3/2
w
By inserting this expression into (28) one finds the magnetization response
′ ′ ′ ′
R(t,t ) = R (t,t ) dt M(t,t)m(t)R (t,t )
w 0 w 0 w
−
Z
u+1 1/2 u(u+1)1/2 u+1 3/2
= (x 1) = (47)
ux+1 − (ux+1)3/2 − ux+1
(cid:18) (cid:19) (cid:18) (cid:19)
The magnetization correlator is found from (25) and reads after rescaling all times
u+2 u+1 x u(u+1)1/2 1 uy(uy+2)
C(1)(t,t ) = T t dy dy
w c w(u+1 sux+1 −Z1 (ux+1)3/2−Z0 (ux+1)3/2(u+1)1/2
1  uy(uy+2) x 1 u2y (uy +2)
w w
dy + dy dy
− (u+1)3/2(ux+1)1/2 w (ux+1)3/2(u+1)3/2
Z0 Z1 Z0
1 y u2y (uy +2)
w w
+ dy dy
w (ux+1)3/2(u+1)3/2
Z0 Z0
1 1 u2y(uy+2)
+ dy dy
w
Z0 Zy (ux+1)3/2(u+1)3/2)
2+3u+2u2 + 1u3
= T t 2 (48)
c w
(u+1)3/2(ux+1)3/2
The term C(2) scales as t(4−d)/2 for 4 < d < 6, where the integral (27) that defines
∼ w
it can be shown to be dominated by aging timescales, and as t−1 for d > 6, where
∼ w
C˜C˜(t′,t′ ) in (27) behaves as a short range kernel, so it is always subleading. Thus (48)
w
represents the full long-time magnetization correlator for d > 4.
The t-dependence in the correlator C C(1) is the same as in the response
≡
R and only occurs via the overall factor (ux + 1)−3/2 = (u + 1)−3/2. It therefore
t
cancels in the resulting FDR which follows after a few lines (using ∂ [t F(x,u)] =
tw w
(1+u∂ x∂ )F(x,u) to calculate ∂ C) as
u − x tw
T R(t,t ) 4 (u+1)4
c w ∞
X(t,t ) = = X (u) (49)
w ∂ C(t,t ) 5(u+1)4 + 3 ≡
tw w 5
Thus, for d > 4 the FDR is t-independent and hence identical to the asymptotic
FDR X∞(u) = lim X(t,t ). It interpolates between 1/2 (for u 1)
t≫tw=uτm≫1 w ≪