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