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netic ¬eld, i.e. B’ . According to Eq. (11.287) and Eq. (11.288), this ¬eld is deter-
mined either by the sample size, through the gradient term, or the phase coherence
length. When the phase-coherence length is longer than the sample size, the cor-
relation ¬eld is therefore of order of the ¬‚ux quantum divided by the sample area,
Bc ∼ φ0 /L2 , where φ0 is the normal ¬‚ux quantum φ0 = 2π /|e|, since the typical
di¬usion loops, like those depicted in Figures 11.9 and 11.10, enclose an area of the
order of the sample, L2 . We note that in magnetic ¬elds exceeding max{φ0 /L2 ,
φ0 /L2 }, the Cooperon no longer contributes to the ¬eld dependence of the conduc-

tance ¬‚uctuations, because its dependence on magnetic ¬eld is suppressed according
to the weak-localization analysis.82
We note that the weak-localization and mesoscopic ¬‚uctuation phenomena are a
general feature of wave propagation in a random media, be the wave nature classical,
such as sound and light,83 or of quantum origin such as for the motion of electrons.
The weak-localization e¬ect was in fact originally envisaged for the multiple scat-
tering of electromagnetic waves [81].84 The coherent backscattering e¬ect has been
studied experimentally for light waves (for a review on classical wave propagation in
random media, see reference [116]). For the wealth of interesting weak-localization
and mesoscopic ¬‚uctuation e¬ects, we refer the reader to reference [1], and to the
references to review articles cited therein.
81 Another way of revealing the mesoscopic ¬‚uctuations is to change the Fermi energy (i.e. the
density of conduction electron as is feasible in an inversion layer). The typical energy scale Ec for
these ¬‚uctuations is analogously determined by the typical time „trav it takes an electron to traverse
the sample according to Ec ∼ /„trav . In the di¬usive regime we have „trav ∼ L2 /D0 .
82 For an account of the experimental discovery of conductance ¬‚uctuations, see reference [115].
83 Here we refer to conditions described by Maxwell™s equations.
84 It is telling that it took the application of Feynman diagrams in the context of electronic motion

in disordered conductors to understand the properties of classical waves in random media.
448 11. Disordered conductors


11.7 Summary
Quantum e¬ects on transport coe¬cients have been studied in this chapter, espe-
cially the weak localization e¬ect, which is the most important for practical diag-
nostics in material science as it is revealed at such small magnetic ¬elds where the
di¬usion enhancement of the electron“electron interaction is una¬ected and classical
magneto-resistance e¬ects absent. Though the weak localization e¬ect is a quantum
interference e¬ect, the kinetics of the involved trajectories were the classical ones,
be they Boltzmannian or Brownian, and we could therefore make ample use of the
quasi-classical Green™s function technique developed in Chapters 7 and 8. We calcu-
lated the phase breaking rates due to interactions, the phase relaxation of the wave
function measured in magneto-resistance measurements, thereby opening the oppor-
tunity to probe the inelastic interactions experienced by electrons. We studied how
the interactions are changed as a result of disorder. In the case of Coulomb interac-
tion a universal di¬usion enhancement or weakening of screening resulted, whereas
for the case of electron“phonon interaction, the longitudinal interaction was weak-
ened owing to the compensation mechanism of the vibrating impurities, whereas the
interaction with transverse phonons could be enhanced or weakened depending on
the temperature regime. Finally, we discussed the phenomena that sets in when the
electronic motion is coherent in the sample and the signature of mesoscopic ¬‚uctu-
ations are present in transport coe¬cients, such as the quantum ¬‚uctuations in the
conductance, the universal conductance ¬‚uctuations.
12

Classical statistical dynamics

The methods of quantum ¬eld theory, originally designed to study quantum ¬‚uctua-
tions, are also the tool for studying the thermal ¬‚uctuations of statistical physics, for
example in connection with understanding critical phenomena. In fact, the methods
and formalism of quantum ¬elds are the universal language of ¬‚uctuations. In this
chapter we shall capitalize on the universality of the methods of ¬eld theory as intro-
duced in Chapters 9 and 10, and use them to study non-equilibrium phenomena in
classical statistical physics where the ¬‚uctuations are those of a classical stochastic
variable. We shall show that the developed non-equilibrium real-time formalism in
the classical limit provides the theory of classical stochastic dynamics.
Newton™s law, which governs the motion of the heavenly bodies, is not the law
that seems to govern earthly ones. They sadly seem to lack inertia, get stuck and
feebly ramble around according to Brownian dynamics as described by the Langevin
equation. Their dynamics show transient e¬ects, but if they are on short time scale
too fast to observe, dissipative dynamics is typically speci¬ed by the equation v ∝ F
where the proportionality constant could be called the friction coe¬cient. This is
Aristotelian dynamics, average velocity proportional to force, believed to be correct
before Galileo came along and did thorough experimentation. If a sponge is dropped
from the tower of Pisa, it will almost instantly reach its saturation ¬nal velocity. If
a heavier sponge is dropped simultaneously, it will fall faster reaching the ground
¬rst. If on the other hand an apple is dropped and when reaching the ground is
given its opposite velocity it will according to Newton™s equation spike back up to
the position it was dropped from, before repeating its trip to the ground. If a sponge
has its impact velocity at the ground reversed, it will ¬zzle immediately back to the
ground. Unlike Newtonian mechanics, which is time reversal symmetric, sponge or
dissipative dynamics chooses a direction of time.
We now turn to consider dissipative dynamics, in particular Langevin dynamics.
In this chapter we will study systems with the additional feature of quenched disorder,
in particular vortex dynamics in disordered superconductors. The ¬eld-theoretic
formulation of the problem will allow the disorder average to be performed exactly.
The functional methods will allow construction of a self-consistent theory for the
e¬ective action describing the in¬‚uence of thermal ¬‚uctuations and quenched disorder


449
450 12. Classical statistical dynamics


on vortex motion. This will allow the determination of the vortex response to external
forces, the vortex ¬‚uctuations, and the pinning of vortices due to quenched disorder,
and allow to consider the dynamic melting of vortex lattices.


12.1 Field theory of stochastic dynamics
In this section we shall map the stochastic problem, formulated ¬rst in terms of a
stochastic di¬erential equation, onto a path integral formulation, and obtain the ¬eld
theoretic formulation of classical statistical dynamics. We show that the resulting
formalism is equivalent to that of a quantum ¬eld theory. In particular we shall
consider quenched disorder and the resulting diagrammatics. The ¬eld theoretic
formulation will allow us to perform the average over the quenched disorder exactly.

12.1.1 Langevin dynamics
A heavy particle interacting with a gas of light particles, say a pollen dust particle
submerged in water, will viewed under a microscope execute erratic or Brownian
motion. Or in general, when a particle interacting with a heat bath, i.e. weakly with
a multitude of degrees of freedom in its environment (of high enough temperature
so that quantum e¬ects are absent), will exhibit dynamics governed by the Langevin
equation
m¨ t = F(xt , t) ’ · xt + ξt ,
™ (12.1)
x
where m is the mass of the particle, F is a possible external force, · is the viscosity
or the friction coe¬cient, and ξ t is the ¬‚uctuating force describing the thermal agi-
tation of the particle due to the interaction with the environment, the thermal noise,
or some other relevant source of noise. For a system interacting with a classical envi-
ronment assumed in thermal equilibrium at a temperature T , the ¬‚uctuating force is
a Gaussian stochastic process described by the correlation function for its Cartesian
components1
(±) (β)
= 2·kB T δ(t ’ t ) δ±β
ξt ξt (12.2)
relating friction and ¬‚uctuations according to the ¬‚uctuation“dissipation theorem,
as proper for linear response.
Being the dissipative dynamics for a system coupled to a heat bath, Langevin
dynamics is relevant for describing a vast range of phenomena, and of course not just
that of a particle as considered above. For example, randomly stirred ¬‚uids in which
case the relevant equation would be the Navier“Stokes equation with proper noise
term [117]. The ¬eld theoretic formulation of the following section runs identical for
all such cases. Also, the coordinate above need not literally be that of a particle,
but could for example describe the position of a vortex in a type-II superconductor,
as discussed in Section 12.2. However, we shall in the following keep referring to the
degree of freedom as that of a particle.
1 The quantum case and the classical limit are discussed in Appendix A.
12.1. Field theory of stochastic dynamics 451


12.1.2 Fluctuating linear oscillator
For a given realization of the ¬‚uctuating force, ξ t , there is a solution to the Langevin
equation, Eq. (12.1), specifying the realization of the corresponding motion of the
particle xt . In other words, xt is a functional of ξ t , xt = xt [ξt ], and vice versa
¯
ξt = ξt [xt ]. The properties of the ¬‚uctuating force is described by its probability
¯
distribution, Pξ [ξ t ], which is assumed to be Gaussian
’1
Pξ [ξ t ] = Dξ t e’ 2 dt1 dt2 ξt 1 Kt 1 , t 2 ξt 2
1
, (12.3)

’1
where Kt1 ,t2 is the inverse of the correlator of the stochastic force
Kt,t = ξ t ξ t (12.4)
and we have used dyadic notation to express the matrix structure of the force corre-
lations in Cartesian space. This structure is, however, irrelevant as the quantity is
diagonal.
Using the one-to-one map between the ¬‚uctuating force and the particle path,
xt ←’ ξ t , the probability of given paths, xt s, equals that of the corresponding
forces, ξ t s,
Px [xt ] Dxt = Pξ [ξt ] Dξ t . (12.5)
In general, this does not allow us to state proportionality between the two probability
distributions, since the volume change in the transformation from Dξ t to Dxt must
be taken into account, the change in measure described by the Jacobian. Only if the
Langevin equation, Eq. (12.1), is linear is this a trivial matter, restricting the force
in Eq. (12.1) to that of a harmonic oscillator, F(xt , t) = ’mω0 xt (and a possible
2

external space-independent force, F(t), which we suppress in the following), i.e. the
equation of motion is that of a harmonic oscillator in the presence of a ¬‚uctuating
force
’1
’DR x = ξ (12.6)
where we have introduced the retarded Green™s function for the damped harmonic
oscillator
’1
DRd. h. o. (t, t ) = ’(m‚t + ·‚t + mω0 ) δ(t ’ t )
2 2
(12.7)
and suppressed the time variable and used matrix multiplication notation in the time
variable.2 In the considered linear case the Jacobian is a constant and
Px [xt ] ∝ Pξ [ξ t ] = Pξ [m¨ t + · xt + mω0 xt ] ,
2
™ (12.8)
x
where the last equality is obtained by using the equation of motion, Eq. (12.6). Using
the fact that the ¬‚uctuations are Gaussian, gives for the probability distribution of
paths,
’1
e’ 2 dt1 dt2 ξt 1 [xt 1 ] Kt 1 , t 2 ξt 2 [xt 2 ]
1
Px [xt ] ∝ ¯ ¯



’1
e’ 2 dt1 dt2 (m¨ t 1 +· xt 1 +mω0 xt 1 )Kt 1 , t 2 (m¨ t 2 +· xt 2 +mω0 xt 2 )
x x
2 2
1
™ ™
= (12.9)
2 It could of course be considered a matrix in Cartesian coordinates, but since it would be diagonal
it is super¬‚uous.
452 12. Classical statistical dynamics


for the case of a harmonic oscillator coupled to a heat bath.
Completing the square in the following Gaussian path integral
∞˜∞ ˜ ∞˜
D˜ t e’ 2 xt Kt , t xt ’i dt xt ·(m¨ t +· xt +mω0 xt )
x˜ 2
P [xt ] = N ’1
1
˜˜ ˜ ˜ ˜˜ ˜˜ ™˜
dt ’∞dt ˜
(12.10)
x ’∞ ’∞




gives the previous expression, and the path integral representation for the probabil-
ity distribution of paths has been obtained. The proportionality factor is ¬xed by
normalization of the probability distribution.
We can also arrive at the expression for the probability distribution of paths in
the following way. For a given realization of the ¬‚uctuating force, ξ t , the probability
distribution for the particle path corresponds with certainty to the one ful¬lling the
equation of motion as expressed by the delta functional

P [xt ] = N ’1 δ[m¨ t + · xt + mω0 xt ’ ξ t ] ,
2
™ (12.11)
x

where N ’1 is the constant resulting from the Jacobian. Introducing the functional
integral representation of the delta functional we get

D˜ e’i dt x·(m¨ +· x+mω0 x’ξ)
x 2
P [x] = N ’1 ˜ ™
. (12.12)
x ’∞




The average over the thermal noise, being Gaussian, can now be performed and we
obtain
∞ ˜∞
D˜ e’i dt ’∞dt x·(m¨ +· x+mω0 x’˜ i
x x ξξ x)
2
= N ’1 ˜ ™ ˜
Px [xt ] = P [x] . (12.13)
x ’∞
ξ


Realizing that the correlation function, Eq. (12.2), is the high-temperature classi-
cal limit of the inverse of the correlation function for a harmonic quantum oscillator
coupled to a heat bath (see Appendix A), i.e. the kinetic component of the real-time
matrix Green™s function, we introduce the notation
’1
’iDK (t, t ) = K ’1 (t, t ) = ξt ξt = 2·kB T δ(t ’ t ) (12.14)

where reference to the irrelevant Cartesian coordinates is left out, i.e. K ’1 now
denotes the scalar part in Eq. (12.2).
In addition the advanced inverse Green™s function is introduced
’1 ’1
DA (t, t ) = DR (t , t) (12.15)

and both functions are diagonal matrices in Cartesian space and will therefore be
treated as scalars. We can then rewrite for the path probability distribution (rein-
troducing an external force F(x, t))

D˜ eiS0 [˜ ,x]+i˜ ·F+ix·j ,
x x
Px [xt ] = (12.16)
x

where
1
x ’1 ’1
˜ ˜ ’1 ˜
S0 [˜ , x] = (˜ DR x + xDA x + xDK x) (12.17)
x
2
12.1. Field theory of stochastic dynamics 453


and we have absorbed the normalization factor in the path integral notation, it is
¬xed by the normalization of the probability distribution

Dxt Px [xt ] = 1 . (12.18)

We have in addition to the physical external force F(t) introduced a source, J(t),
and have the generating functional

D˜ eiS0 [˜ ,x]+i˜ ·F+ix·J
x x
Dx
Z[F, J] = (12.19)
x

with the normalization
Z[F, J = 0] = 1 . (12.20)
The source is introduced in order to generate the correlation functions of interest,
for example

δZ[J]
Dx xt Px [xt ] = ’i =’ ¯ ¯ ¯
dt DR (t, t) F(t) ,
= (12.21)
xt ξ
δJ(t) ’∞
J=0

where the last equality follows from the equation of motion, and the retarded prop-
agator DR is thus the linear response function (for the considered linear oscillator,
the linear response is the exact response).
Because of the normalization condition, Eq. (12.20), all correlation functions of
˜
the auxiliary ¬eld x (generated by di¬erentiation with respect to the physical external
force F), vanish when the source J vanishes.
We now realize that the above theory is equivalent to the celebrated Martin“
Siggia“Rose formulation of classical statistical dynamics [118], here in its path inte-
gral formulation [119, 120, 121], albeit for the moment only for the case of a damped
harmonic oscillator.3 This restriction was of course self-in¬‚icted and the formalism
has numerous general applications, such as to critical dynamics for example studying
critical relaxation [122].
We note that whereas equilibrium quantum statistical physics is described by Eu-
clidean ¬eld theory (recall Section 1.1 and see Exercise A.1 on page 506 in Appendix
A), non-equilibrium classical stochastic phenomena are described by a ¬eld theory
formally equivalent to real-time quantum ¬eld theory.
We hasten to consider a nontrivial situation, viz. that of the presence of quenched
disorder. The corresponding ¬eld theory will be of the most complicated form, in
diagrammatic terms it will have vertices of arbitrarily high connectivity.
3 Inother words, the Martin“Siggia“Rose formalism is simply the classical limit of the real-time
technique for non-equilibrium states, where the doubling of the degrees of freedom necessary to
describe non-equilibrium situations is provided by the dynamics of the system. We note that the
presented ¬eld theoretic formulation of the Langevin dynamics is the classical limit of Schwinger™s
closed time path formulation of quantum statistical mechanics of a particle coupled linearly to, for
the considered type of damping, an Ohmic environment. Equivalently, it is the classical limit of
the Feynman“Vernon path integral formulation of a particle coupled linearly to a heat bath, as
discussed in Appendix A.
454 12. Classical statistical dynamics


12.1.3 Quenched disorder
We now return to the in general nonlinear classical stochastic problem speci¬ed by
the Langevin equation

m¨ t + · xt = ’∇V (xt ) + Ft + ξ t ,
™ (12.22)
x

where V eventually will be taken to describe quenched disorder. Owing to the pres-
ence of the nonlinear term V (xt ), the argument for the Jacobian being a constant is
less trivial. However, by using forward discretization,4 one obtains the result that,
owing to the presence of a ¬nite mass term, the Jacobian can be chosen as a constant
and the analysis of the previous section can be taken over giving5

D˜ e’i dt x·(m¨ +· x+∇V
x ’F’ξ)
P [x] = N ’1 ˜ ™
. (12.23)
x ’∞




The averages over the thermal noise and the disorder can now be performed and
we obtain the following expression for the path probability density

= N ’1 D˜ eiS[˜ ,x]+i˜ ·F ,
x x
Px [xt ] = P [x] (12.24)
x

where the action, S = S0 + SV , is a sum of a part owing to the quenched disorder
and the quadratic part
1
x ’1 ’1
(˜ DR x + xDA x + xi ξξ x) ,
˜˜ ˜
S0 [˜ , x] = (12.25)
x
2
where we have introduced the inverse propagator for the problem in the absence of
the disorder
’1
DR (t, t ) = ’(m‚t + ·‚t )δ(t ’ t ) ,
2
(12.26)
i.e. the retarded free propagator satis¬es

’(m‚t + ·‚t )Dtt = δ(t ’ t )
2 R
(12.27)

with the boundary condition
R
Dtt = 0, t < t . (12.28)
The corresponding inverse advanced Green™s function
’1 ’1

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