Bhattacharjee J.K., Bhattacharyya S. Non-Linear Dynamics Near and Far from Equilibrium
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5.8 Generalized n-Vector Model 141
where the exponents δ
1
, δ
2
are positive, which makes these terms vanish as k →0.
Consequently, terms of the kind k
δ
1
and k
δ
2
are called correction-to-scaling terms.
We now demonstrate how the correction-to-scaling terms can be calculated. To do
so, one introduces the above forms for
ψ
and λ
S
in Eqns.(5.8.15) and (5.8.16)
with κ =0 and we have
a
ψ
k
−/2
=
(n −1)
(1 +b
ψ
k
δ
+..)
d
D
p
C
D
1
(
k −p)
2
×
1
a
s
p
D/2
[1 +b
s
p
δ
+...]+a
ψ
|
k −p|
D/2
[1 +b
ψ
|
k −p|
δ
+...]
=
(n −1)
(1 +b
ψ
k
δ
+..)
d
D
p
C
D
1
(
k −p)
2
1
a
s
p
D/2
+a
ψ
|
k −p|
D/2
−
a
s
b
s
p
δ+D/2
+a
ψ
b
ψ
|
k −p|
δ+D/2
[a
s
p
D/2
+a
ψ
|
k −p|
D/2
]
2
+...
(5.8.30)
and
a
s
k
−/2
(1 +b
s
k
δ
+..) =
d
D
p
C
D
[p
2
−|
k −p|
2
]
2
p
2
|
k −p|
2
1
a
ψ
p
D/2
+a
ψ
|
k −p|
D/2
−
a
ψ
b
ψ
(p
δ+D/2
+|
k −p|
δ+D/2
)
a
2
ψ
[(p
D/2
+|
k −p|
D/2
)]
2
+...
(5.8.31)
leading to
a
s
a
ψ
=(n −1)
d
D
p
C
D
1
|
1 −p|
2
1
p
D/2
+w|
1 −p|
D/2
=
d
D
p
C
D
[p
2
−|
1 −p|
2
]
2
p
2
|
1 −p|
2
[p
D/2
+|
1 −p|
D/2
]
2
(5.8.32)
a
s
a
ψ
b
ψ
=−(n −1)b
s
d
D
p
C
D
1
|
1 −p|
2
p
δ+D/2
[p
D/2
+w|
1 −p|
D/2
]
2
−(n −1)b
ψ
w
d
D
p
C
D
1
|
1 −p|
2
|
1 −p|
δ+D/2
[p
D/2
+w|
1 −p|
D/2
]
2
+.....
(5.8.33)
and
a
s
a
ψ
b
s
=−b
ψ
d
D
p
C
D
[p
2
−|
1 −p|
2
]
2
p
2
|
1 −p|
2
p
δ+D/2
+|
1 −p|
δ+D/2
[p
D/2
+|
1 −p|
D/2
]
2
(5.8.34)
Making the reasonable assumption that δ vanishes at D =4, all our integrals have
a logarithmic divergence at D =4 (i.e. =0) and we evaluate them in the pole
approximation i.e. pick up the
−1
part. From Eq.(5.8.32), this yields
142 5 Critical Dynamics in Fluids
a
s
a
ψ
=
1
=
2(n −1)
(1 +w)
(5.8.35)
while
d
D
p
C
D
1
|
1 −p|
2
p
δ+D/2
[p
D/2
+w|
1 −p|
D/2
]
2
=
d
D
p
C
D
1
|
1 −p|
2
|
1 −p|
δ+D/2
[p
D/2
+w|
1 −p|
D/2
]
2
=
1
(1 +w)
2
2
−2δ
+O(1)
(5.8.36)
and
d
D
p
C
D
1
|
1 −p|
2
[p
2
−|
1 −p|
2
]
2
p
D/2
+|
1 −p|
D/2
]
2
×[p
δ+D/2
+|
1 −p|
δ+D/2
]
=
4
D
1
−2δ
+O(1) =
1
−2δ
+O(1) (5.8.37)
Using the above integrals, Eqns. (5.8.33) and (5.8.34) can be rewritten as
b
ψ
=−
−2δ
1
(1 +w)
(b
s
+wb
ψ
)
b
s
=−
−2δ
b
ψ
(5.8.38)
leading to
1 +w =−
−2δ
w +
−2δ
2
(5.8.39)
yielding
δ =
w/2
1 +w
, (5.8.40)
The vital point of the above calculation is that one of the correction-to-scaling
exponents is proportional to w.Ifw is small, then this exponent is small and the
correction-to-scaling term is going to persist even for very small values of k. Near
D =4, w is small for n =3/2, but detailed two loop calculations show that D =3,
w is small for n =2, i.e. it is small for the lambda transition. We also note that
for small w the amplitudes of b
ψ
and b
s
differ in sign. For negative values of the
amplitude b
s
, we have for the k-dependent thermal diffusion (we are working with
w 1, so that 1 +w 1)
λ
s
a
s
k
−/2
[1 −|b
s
|k
w/2
+....]
= a
s
k
−/2
1 −
k
k
c
w/2
(5.8.41)
5.8 Generalized n-Vector Model 143
Since the correction term does not disappear until k k
c
, we have for not so small
values of k, an effective exponent
X
λ
=−
d ln λ
s
d ln k
=
2
+
(
k
k
c
)
w/2
1 −(
k
k
c
)
w/2
w
2
=
2
1 +w
(
k
k
c
)
w/2
1 −(
k
k
c
)
w/2
(5.8.42)
For
k
k
c
10
−3
and w 0.1, the effective exponent is about 24% greater than the
dynamic scaling exponent of /2. Note that for
k
k
c
→0, we would see the exponent
/2, but real experiments can never get down to this asymptotic range, and hence in
the k-range accessible to the experiments,the asymptotic dynamic scaling exponent
is still not accessible due to the existence of a small correction-to-scaling expo-
nent. Note that our discussion in k-space simply carries over to κ-space when we
discuss the thermal conductivity measurements as a function of temperature near
the lambda point. While the above gives a correct qualitative feel for the way the
things work, to get quantitative results one needs to investigate the renormalization
group flow equations.
References
Critical Dynamics in Ordinary Fluids
1. K. Kawasaki, Ann. Phys. (N.Y.) 61 1 (1970)
2. J. D. Gunton, in Dynamical Critical Phenomena and Related Topics, Ed. C. P. Enz
(Springer, New York), 1, (1979)
3. R. A. Ferrell, Phys. Rev. Lett. 24 1169 (1970)
4. R. F. Chang, H. C. Burstyn and J. V. Sengers, Phys. Rev. A19 866 (1979)
5. H. L. Swinney and D. L. Henry, Phys. Rev. A8 2486 (1973)
6. J. K. Bhattacharjee and R. A. Ferrell, Phys. Rev. A23 1511 (1981)
For the Superfluid Transition in
4
He
1. L. Sasvari, F. Schwabl and P. Sz
´
epfalusy Physica 81A 108 (1975)
2. C. DeDominicis and L. Peliti, Phys. Rev. Lett. 38 505 (1977); Phys. Rev. B18 353 (1978)
3. R. A. Ferrell et.al., Ann. Phys. 47 565 (1968)
4. B. I. Halperin and P. C. Hohenberg, Phys. Rev. 177 952 (1977)
5. R. A. Ferrell and J. K. Bhattacharjee, Phys. Rev. Lett. 42 1638 (1979)
6. R. A. Ferrell and J. K. Bhattacharjee, JLTP 36 184 (1979)
7. V. Dohm and R. Folk, Z. Phys. B40 79 (1980)
8. V. Dohm and R. Folk, Phys. Rev. Lett. 46 349 (1981); Z. Phys. B41 251 (1981); 45, 129
(1981);
9. G. Ahlers, P. C. Hohenberg and A. Kornblit, Phys. Rev. Lett. 46 493 (1981); Phys. Rev.
B25 3136 (1982)
10. R. A. Ferrell and J. K. Bhattacharjee, Phys. Rev. Lett. 44 403 (1980)
6
Systems Far from Equilibrium
6.1 Introduction
In this chapter our focus will be on situations which are not close to thermal equilib-
rium. To understand the problems associated with it, we first discuss the situation
close to equilibrium in a somewhat generalized setting. What we have repeatedly
seen in the previous chapters is that small excursions from the equilibrium distribu-
tion relax to the equilibrium with time scales set by relaxation rates which we are
familiar with in hydrodynamics - the mass or concentration diffusivity, the thermal
diffusivity, momentum diffusivity. The balance between the fluctuating forces (ob-
tained by averaging over small scale processes in the critical phenomena) and the
dissipative processes ensured that the equilibrium distribution would be attained at
t →∞. Describing the system in terms of a fluctuating field ψ(x), we have repeat-
edly used the free energy functional F(ψ) to describe the probability distribution
e
−F(ψ)
for the weighting of the fluctuations and to describe the dynamics we have
used a Langevin equation of the form
˙
ψ =−
δF
δψ
+N (6.1.1)
Where is the relaxation rate and N is the noise with the correlation
N(t
1
)N(t
2
)=2δ(t
1
−t
2
) (6.1.2)
This structure ensures that the equilibrium distribution is always attained as t →∞.
In adding the reversible terms to the above dissipative equation, we have always
taken care to ensure that the terms are such that they preserve the equilibrium
distribution. Thus we always ensured that the system will attain equilibrium. If we
probe the equilibrium fluctuations by applying a very weak external force, then it
146 6 Systems Far from Equilibrium
should be possible to get at the transport coefficients by studying the response of
the system. This is what we do now.
We assume a time dependent infinitesimal force F(t)applied to the system and
the field ψ describing the fluctuations responds via the response function χ(t −t
),
so that the average value in the presence of the external force is
ψ(t)
F
=
∞
−∞
χ(t −t
)f (t
)dt
(6.1.3)
The response χ(τ) is real, and satisfies the causality condition
χ(τ)=0ifτ<0 (6.1.4)
In terms of the Fourier transforms, Eq.(6.1.4) reads
ψ(ω)=χ(ω)F (ω) (6.1.5)
The response of the system will be assumed to be finite, which can be ensured by
∞
0
χ(τ)dτ <∞ .
The frequency dependent susceptibility χ(ω) of Eq.(6.1.5) is given by (keeping in
mind Eq.(6.1.4))
χ(ω)=
∞
0
dτχ(τ)e
iωτ
(6.1.6)
In the above equation, it is possible to consider ω to be complex, although only
in the upper half plane. Denoting the complex ω by z, we can have z =ω +iη
with η>0, and Eq.(6.1.6) would be well defined yielding finite values of χ(z)
everywhere in the upper half plane. Thus χ(z) has no poles in the upper half plane.
For η →∞χ(z)→0. For real ω, χ(ω) is a complex valued function of ω, and
the above considerations can be used to relate the real and the imaginary parts of
χ(ω). This is the usual Krammers-Kronig relation and we offer a quick derivation.
For the complex function
f(z)=
χ(z)
z −ω
0
(6.1.7)
where ω
0
is a real number, there are no poles of the function χ(z) in the upper half
plane and for the contour shown in Fig. 6.1
C
f(z)dz=0 (6.1.8)
6.1 Introduction 147
Figure 6.1. The contour in the complex plane
On the semicircle of infinite radius the contribution to the integral is zero since f(z)
goes to zero at large values of the argument faster than
1
|z|
. Thus
lim
→0
ω
0
−
−∞
f(ω)dω+
small
semicircle
f(z)dz+
∞
ω
0
+
f(ω)dω
=0
ω
0
−
−∞
f(ω)dω+
∞
ω
0
+
f(ω)dω=P
∞
−∞
f(ω)dω (6.1.9)
where
P denotes the principal value. For the remaining part,
semicircle
of radius
f(ω
0
+e
iθ
)d(ω
0
+e
iθ
) =−i
f(ω
0
+e
iθ
)e
iθ
dθ
=−i
χ(ω
0
+e
iθ
)
e
iθ
e
iθ
dθ
=−iπχ(ω
0
) (6.1.10)
leading to
χ(ω
0
) =
1
iπ
P
∞
−∞
χ(ω)
ω −ω
0
dω (6.1.11)
Writing
χ(ω
0
) =χ
1
(ω
0
) +iχ
2
(ω
0
)
148 6 Systems Far from Equilibrium
we obtain
χ
1
(ω
0
) =
1
π
P
∞
−∞
χ
2
(ω)
ω −ω
0
dω (6.1.12)
and
χ
2
(ω
0
) =−
1
π
P
∞
−∞
χ
1
(ω)
ω −ω
0
dω (6.1.13)
The above connections between the real and imaginary parts of χ(ω) are called
the Krammers-Kronig relations.
The time dependent response is given, according to Eq.(6.1.5) by
ψ(t)
F
=
1
2π
∞
−∞
e
iωt
χ(ω)F(ω)dω (6.1.14)
We now specialize to a particular kind of force,
F(t) = F for t<0
= 0 for t>0
For the Fourier transform of F(t),wehave
F(ω) =
∞
−∞
dte
iωt
F(t)=F
0
−∞
dte
iωt
= F lim
→0
0
−∞
dte
iωt
e
t
= lim
→0
F
+iω
= F lim
→0
2
+ω
2
−
iω
2
+ω
2
(6.1.15)
Recalling that
P
1
ω
= lim
→0
2
+ω
2
and
δ(ω)=
1
π
lim
→0
2
+ω
2
Eq.(6.1.15) can be rewritten as
F(ω)=F
P
1
iω
+π δ(ω)
(6.1.16)
6.1 Introduction 149
The response of Eq.(6.1.14) can then be written as
ψ(t)
F
=
F
2πi
P
∞
∞
e
−iωt
ω
χ(ω)dω+
F
2
χ(0) (6.1.17)
Note that for t>0, the complex function
f
1
(z) =
e
izt
χ(z)
z −ω
0
is analytic in the upper half plane and the arguments between Eqns.(6.1.7) and
(6.1.11) can be repeated to yield
e
iω
0
t
χ(ω
0
) =
1
iπ
P
∞
−∞
e
iωt
χ(ω)
ω −ω
0
dω (6.1.18)
For t<0,
e
−iω
0
t
χ(ω
0
) =
1
iπ
P
∞
−∞
e
−iωt
χ(ω)
ω −ω
0
dω (6.1.19)
Thus for t>0
χ(0) =
1
iπ
P
∞
−∞
e
iωt
χ(ω)
ω
dω
and for t<0
χ(0) =
1
iπ
P
∞
−∞
e
−iωt
χ(ω)
ω
dω
leading to (from Eq.(6.1.17))
ψ(t)
F
=Fχ(0) for t<0 (6.1.20)
and
ψ(t)
F
=
F
iπ
P
∞
−∞
dω
χ(ω)
ω
cos ωt for t>0 (6.1.21)
So long as the force acts on the system, the response is a constant - it is only when
the force is removed that the system relaxes.
If ψ(0)
F
is the expectation value at time t =0, when the force is switched
off, then this relaxation can be expressed as,
ψ(t)
F
=
ψ(0)
F
e
−t
=Fχ(0)e
−t
(6.1.22)
150 6 Systems Far from Equilibrium
where we have used Eq.(6.1.20). In an identical fashion the correlation C(t) =
ψ(t)ψ(0)
can be written as
C(t) =
ψ(t)ψ(0)
=e
−t
C(0) (6.1.23)
where C(0) is the equal time correlation function. Comparing Eqns.(6.1.23),
(6.1.22) and (6.1.21)
C(t) =C(0)χ(0)
−1
1
iπ
P
∞
−∞
dω
χ(ω)
ω
cos ωt (6.1.24)
The zero frequency susceptibility χ(0) is simply the static susceptibility which is
related to the equal time correlation function C(0) by the standard relation (hψ is
the additional term in the free energy when an external field h is present making
the weighting function
e
β(F(h=0 )+hψ )
and the susceptibility χ is
∂ψ
∂h
)
χ(0) =
1
k
B
T
C(0) (6.1.25)
Thus,
C(t) =
k
B
T
iπ
P
∞
−∞
dω
χ(ω)
ω
cos ωt (6.1.26)
which is the well known fluctuation dissipation theorem (FDT).
6.2 Ginzburg-Landau Model
We return to one of the models we looked at briefly in Chapter 2 - the time de-
pendent Ginzburg-Landau model with a conserved current. The Ginzburg-Landau
free energy (equilibrium) is
F =
d
D
x
m
2
2
φ
2
+
1
2
(
∇φ)
2
+
λ
4
φ
4
(6.2.1)
with the mass term m
2
proportional to T −T
0
. The chemical potential
j =−
∇
δF
δφ
(6.2.2)
6.2 Ginzburg-Landau Model 151
The mesoscopic models as written down in Eq.(6.2.1) involve coarse graining over
microscopic scales and this averaging over the small scale degrees of freedom gives
a random contribution
ξ to the current
j =−
∇
δF
δφ
+
ξ (6.2.3)
with the correlation of ξ prescribed as
ξ
i
= 0 and
ξ
i
(x,t)ξ
j
(
x
,t
)
= 2k
B
Tδ
ij
δ(x −
x
)δ(t −t
) (6.2.4)
The Langevin equation is given by
∂φ
∂t
=−
∇.
j =[(m
2
−∇
2
)∇
2
φ +λ∇
2
φ
3
]+
∇.
ξ (6.2.5)
and with the correlation of ξ specified as in Eq.(6.2.4), FDT will be satisfied.
we now want to add a driving field. The current thus generated will be propor-
tional to
E.Ifφ is uniform in a region of space (the uniform value being normalized
to unity), the current will be
j
E
=(1 −φ
2
)
E (6.2.6)
In writing down the Langevinequation with j
E
included in the current the symmetry
breaking among the different spatial directions has to be taken into account. The
direction along
E will be termed the parallel direction and the other directions will
be perpendicular directions. The gradient in the parallel direction is denoted by ∂
and in the perpendicular direction by ∇
⊥
. The generalization of Eq.(6.2.5) taking
into account this breaking of symmetry is
∂φ
∂t
=
(m
2
⊥
−α
⊥
∇
2
⊥
)∇
2
⊥
φ +(m
2
−α
δ
2
)δ
2
φ
−2α∇
2
⊥
∂
2
φ +λ(∇
2
⊥
φ
3
+β∂
2
φ
3
) +E∂φ
2
−(
∇
⊥
.
ξ
⊥
+∂ξ
) (6.2.7)
with the noise correlation given by
∇
⊥
.
ξ
⊥
(x,t)
∇
⊥
.
ξ
⊥
(
x
,t
)
= N
⊥
(−∇
2
⊥
)δ(x −
x
)δ(t −t
) (6.2.8)
∂ξ
(x,t)∂
ξ
(
x
,t
)
= N
(−∂
2
)δ(x −
x
)δ(t −t
) (6.2.9)