Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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and frequency scales: k  2=, !  2=)soasto

take advantage of the physical cancellation of

mathematical divergences. In any case, it turns the

bare parameters of the original singular expansion

into renormalized parameters and yields a renorma-

lized regular expansion. Writing that the resulting

large-scale behavior does not depend on the chosen

cutoff (, ) yields renormalization equations,

expressing quantitatively the very consistency of

the procedure (‘‘renormalizability’’ of the expan-

sion). Renormalization provides alternative technical

tools in instances treated above with the multiple-

scale method. Its main advantage is its recursive

structure: introducing a sequence (

, 

)

of cutoffs

(what is called momentum-shell RG), the whole

procedure can be iterated to integrate recursively the

influence of small-scale features on the asymptotic

behavior, allowing as to handle situations exhibiting

a hierarchy or even a continuum of scales.

Renormalization also refers to an asymptotic

analysis allowing as to classify critical behaviors, to

determine quantitatively the critical exponents and to

handle the associated divergences. Indeed, the above-

mentioned multiscale approaches fail near bifurcation

points or critical points. In this case, scale separation is

replaced by scale invariance. The key idea, underlying

RG techniques is to shift the focus on the scaling

procedure itself. The basic point is to construct a

renormalization transformation, consisting in joint

coarse-grainings and rescalings, thus relating the two

models describing the same phenomenon at different

scales (Lesne 1998); it puts forward their self-similar

properties and associated scaling laws, while eliminat-

ing specific small-scale details having no consequences

on the asymptotic, large-scale behavior. The set of

renormalization transformations has a semigroup

structure with respect to the rescaling factor (or plainly

with respect to iteration) justifying to speak of RG. It

generates a flow in the space of models, whose fixed

points correspond either to trivial or to critical

situations according to their stability. It can be shown

that the linear analysis of the renormalization trans-

formation around a critical fixed point gives access to

the critical exponents. Moreover, this analysis allows

us to split the space of models into universality classes,

each associated to the basin of attraction of a critical

fixed point. Let us emphasize that scale invariance

leads to a deep change in the modeling and investiga-

tions, shifting from a ‘‘physics focusing on the

prediction of amplitudes’’ to a ‘‘physics of the

exponents,’’ focusing on less specific, but more

universal and above all, more intrinsic features.

Far more generally, RG is associated with a

qualitative change in the questioning, since the

study takes place in a space of models. Generalized

renormalization transformation can be designed to

extract not only self-similarity properties but any

large-scale feature from a more microscopic model.

In particular, RG can be specially designed to

discriminate between essential and inessential terms

in a model: the latter do not modify the asymptotics

of the RG flow, meaning that they are of no

consequence at large scales. In other words, generic

properties of the renormalization flow in this space of

models yield universal large-scale scaling properties.

RG is thus essentially a multiscale approach, insofar

as it only retains the relations between the different

levels of descriptions, somehow ignoring the details at

each given scale. It is actually designed to capture

universal features of the multiscale organization.

Summary: The Exemplary Case

of Diffusion

Bridging the Scales

Our aim in this section is to present the whole range

of multiscale approaches in use, allowing both to

bridge models devised at different scales and to

predict the large-scale features of the phenomenon

they account for. We choose the context of diffu-

sion, Brownian motion, and transport phenomena,

where such a bridge is essential and has been much

investigated. Indeed, transport coefficients are

defined through phenomenological equations; it is

thus necessary to relate such macroscopic equations

with smaller-scale theories, so as to get an expres-

sion of the coefficients in terms of the microscopic

ingredients and to justify the validity of the

phenomenological description.

The exposition in the various subsections below,

following increasing scales, will mark out the path-

way from reversible molecular dynamics to macro-

scopic diffusion equations. We shall thus come

across the multiple-scale analysis of the Liouville

equation describing at microscopic scales a Brown-

ian grain suspended in a thermal bath of water

molecules (see the next subsection) leading to the

mesoscopic Kramers equation for the grain distribu-

tion function P(r, v, t). Next, involving higher but

still mesoscopic scales, we see that another multiple-

scale analysis leads to the reduced Smoluchowski

equation for its spatial distribution P(r, t). Random

walks offer alternative mesoscopic models, involving

effective diffusion coefficients in order to take into

account underlying features like persistence length

or other short-range correlations. Scaling limits or

more systematic renormalization methods in real

space allow to bridge discrete random-walk models

with continuous descriptions. Another RG, based on

Multiscale Approaches 477

a path-integral formulation in the framework of

field theory, allows to handle the case of self-

avoiding walks with infinite memory. Homogeni-

zation is illustrated on the case of diffusion in a

regular porous medium, whereas diffusion pro-

cesses in fractal substrates provide a counterexam-

ple, singular enough to exhibit anomalous scaling

behavior. The issue of reducing the dynamics of the

diffusion process to a simpler effective one is

encountered in many other macroscopic instances,

among which we shall mention diffusion in a

periodic medium, lending to space averaging, and

advection of a passive scalar field in a two-scale

velocity field, where a multiple-scale analysis yields

the effective diffusivity at large scale. We shall give

further technical guidelines for constructing these

steps climbing from molecular up to large macro-

scopic scales, thus providing additional illustrations

of the multiscale approaches introduced in the

previous sections on more general and abstract

grounds.

Microscopic Theory of Brownian Motion

The first theoretical account of Brownian motion,

namely the erratic movement of a micron-sized

pollen grain suspended in a thermal bath, for

example, water, dates back to 1905 and the famous

paper by Einstein. It took almost 60 years before a

microscopic theory was achieved; this theory has

been further worked out using multiple-scale

techniques (Cukier and Deutsch 1969). The chal-

lenge is to start from the complete deterministic

reversible dynamics of the system, described within

a probabilistic framework by the Liouville equation

@p=@t = Lp for the distribution of probability p in

the whole phase space (position and velocities of

the grain, of mass M, and all water molecules, of

mass m  M). The small parameter is the mass

ratio  =

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

m=M

measuring the efficiency of the

energy transfer upon collisions between the grain

and the bath particles, assuming a binary interac-

tion potential U =

u(jr

 rj). The Liouville

operator is decomposed into L = L

þ  L

,and

one introduces rescaled time variables 

= 

where 

= t is the timescale of the fluid particle

dynamics. Multiple-scale method is carried out

according to the general scheme, leading to the

so-called Kramers equation,

þ v:



Pðr; v; tÞ

¼ 

v þ



Pðr; v; tÞ½25

where the friction coefficient is explicitly given as

 ¼

3MkT

idt

where F

¼ e

and F

¼r

U ½26

We refer to the original, although very pedagogical,

paper by Cukier and Deutsch (1969) for a thorough

exposition and discussion of this derivation.

Mesoscopic Theory of Brownian Motion

Multiple-scale method is also of relevance to

determine the high-friction limit of the above

Kramers equation. Standard perturbation technique

with respect to the inverse of friction, 1=, fails to

describe the asymptotic regime: there is not enough

freedom to fulfill all the solubility conditions

required to avoid the appearance of secular diver-

gences (Bocquet 1997). By contrast, multiple-scale

technique yields a uniform expansion of the evolu-

tion equation still valid at long times, thus allowing

to bridge two mesoscopic levels of description,

namely the Kramers equation and the Smoluchowski

equation for the spatial density (r, t) of the

Brownian particle:

ðr; tÞ¼

M



ðr; tÞ½27

Introducing dimensionless variables  = tv

=l, R =

r=l, V = v=v

,wherel is the size and v

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

kT=M

the thermal velocity of the grain, the relevant small

parameter appears to be the dimensionless inverse of

the friction coefficient,  = v

=l; hence,



@

þ V:



PðR; V;Þ

V þ



PðR; V;Þ½28

If the friction is high (i.e.,   1), the velocity

relaxes very rapidly towards the equilibrium Max-

well distribution, and it is then enough to describe

the (slow) evolution of the spatial distribution

(r, t). Nevertheless, the relaxation stage is essential

and accordingly the -dependence is singular, as a

rule when the small perturbation parameter multi-

plies the time derivative.

According to the general procedure exposed in the

section ‘‘Multiple-scale method: principles,’’ we intro-

duce rescaled variables 

= , 

= , 

= 

, ...

considered as independent variables and look for a

solution of the Kramers equation of the form

P = P

(0)

þ P

(1)

þ 

(2)

þ, where the arguments

of all the components P

(i)

are (R, V, 

, 

, ...).

Identifying term-wise the successive powers of  yields

478 Multiscale Approaches

a hierarchy of equations. At order 0, we obtain

(0)

=(R, 

, 

, ...)e

V

. The following equa-

tions, for the [P

(i)

]

i1

, involve the linearized operator

L= @

(V þ @

). For each of them, there appears a

solubility condition, requiring that none of the additive

contributions in the equation is an eigenvector of L;

involving the components P

(j)

with j < i,itpreventsthe

appearance of a secular divergence in P

(i)

.Atorder

the solubility condition is @=@

= 0, thus determin-

ing the (trivial) 

-dependence of P

(0)

. In a similar way,

the solubility condition at order 2 allows to determine

the 

-dependence of P

(0)

. This bridges the Kramers

and Smoluchowski equations in the high-friction limit,

when retaining only the first-order term in .Werefer

to Bocquet (1997) for a pedagogical account of the

derivation and discussion of its relation with the time-

derivative expansion involved in the so-called Chap-

man–Enskog solution of the Boltzmann equation.

Random-Walk Model and Weakly

Correlated Diffusion

Random walks are discrete-time mesoscopic models,

accounting for the diffusing motion of a particle

through the statistical properties of its successive

steps, when observed at a given timescale .The

basic model (ideal random walk) assumes isotropic,

independent and identically distributed steps of var-

iance a

. Central-limit theorem straightforwardly

gives the time dependence of the mean-square dis-

placement R

(t) hjr(t)  r(0)j

i= a

t =,showing

that the motion is a normal diffusion, with diffusion

coefficient D = a

=2d in dimension d.Itistonote(see

also the next subsection) that D depends  and a, but in

a joint manner. Actually, the diffusion coefficient

associated with a diffusive motion observed at scale a

andmodeledbyarandomwalkonalatticeof

parameter a canbewrittenasD = a

,wheretherate

 depends on a (effective rate at spatial resolution a):

this is a sort of renormalization that accounts for the

rate (a) of all microsteps backward and forward of

length far smaller than a.

In case of short-range correlations between the

successive steps (namely if

1

jC(t)j < 1, where

C(t) is the statistical correlation function between

elementary steps separated by a time length t), direct

computations support a time-average-like result:

the asymptotic behavior is still described by a

normal diffusion law R

(t)  2dD

eff

t, with D

eff

1

C(t). When C(t) = e

t=

eff

Dð1 þ e

1=

1  e

1=

hence D

eff

 2D if   1.

Renormalization Analysis in Case

of Markovian Diffusion

Trying to bridge lattice random walks with a

continuous description brings out the following

difficulty: as the step size a goes to 0, one has to

obviously decrease the duration  accordingly, but

by what amount is not so obvious, since the walker

velocity is ill-defined (it depends on the observation

scale). Determination of the proper joint rescaling

can be guessed from the knowledge obtained by

another mean about the system; rather, it can also

be obtained in a systematic way, thanks to RG

methods. Let us explain the basic principle.

Let us denote by P

a, 

(x, y, t) the transition prob-

ability governing the random walk, namely the

density of probability to jump from x to y in time

t, where x, y are restricted to the lattice (aZ)

and

time to N. The renormalization transformation



k, 

should express the consequence for P

a, 

of a

joint rescaling of space (by a factor of k) and time

(by a factor of k



). Taking into account the Markov

character of the walks, we are thus led to define

½

k;

a;

ðx; y; tÞk

a;

ðkx; ky; k



tÞ

in dimension d ½29

The proper value of  is to be determined self-

consistently in order that the limit lim

k!1



k, 

a, 

exists (it is then a continuous transition probability





(x, y, t) defined on R

 R

 R). The root-

mean-square displacement

RðP; tÞ

x; y

x  y

Pðx; y; tÞ

1=2

is transformed according to

Rð

k;

a;

; tÞ¼k

1

RðP

a;

; k



tÞ½30

Accordingly, it yields the diffusion law associated

with the fixed point P





for any k; RðP





; tÞ¼k

1

RðP





; k



tÞ;

hence RðP





; tÞt

1=

½31

It is anomalous except if  = 2. In the case of ideal

random walks, the proper exponent leading to a

nontrivial limit is  = 2; this limit P



is the transition

probability of a Wiener process:

ðx; y; tÞ¼½4dDt

d=2

ðxyÞ

=4dDt

with D ¼ a

=2d ½ 32

This shows that all ideal lattice random walks

belong to the same universality class, that of the

Wiener process. This approach has been fruitfully

Multiscale Approaches 479

applied to diffusion in disordered systems, the issue

being to determine whether or not the disorder,

accounted for as a noise term in the transition

probabilities, modifies the normal diffusion law

obtained in the unperturbed situation. Similar

reasoning can also be implemented for self-similar

anomalous diffusion processes, like fractional Brow-

nian motions and Levy flights (Lesne 1998).

Renormalization Analysis for Self-Avoiding Walks

Let us only mention, for the sake of completeness, the

renormalization techniques developed for determining

the conformational statistics of linear polymer chains,

whose three-dimensional shape can be represented as

the trajectory of a self-avoiding random walk. These

techniques belong to the RG corpus developed in

statistical mechanics for critical phase transitions,

within a field-theoretic framework. A formal but

exact analogy can actually be worked out between

self-avoiding walks and a spin lattice system with

n ! 0, where n is the number of spin components.

The multiscale nature of the system is so marked

here that it should rather be qualified as an absence of

characteristic scale. In this respect, standard RG

methods developed for critical phenomena lie at the

very boundary of multiscale approaches. Scale decoup-

ling is replaced by scale invariance, which is somehow

the conjugate situation: homogeneity in real space is

replaced by homogeneity in the conjugate space (space

of characteristic scales). Scale invariance here reflects

in the self-similar property, R(N)  N



, relating the

end-to-end distance R of the chain to the number N of

elementary steps (the monomers), with an anomalous

exponent  (the Flory exponent   3=5indimension

d = 3) originating from the infinite memory of the

nonoverlapping chain. We refer to Lesne (1998) and

references therein for a more detailed exposition of the

concepts and techniques only alluded here.

Effective Diffusion in a Porous Medium

(Homogenization)

Describing the diffusion in a porous medium appears

as a formidable task at the pore level: it would

require us to account for all the boundary conditions

at the border of the hollow domain V2V

actually

accessible to diffusion. When the pores have a finite

characteristic size a, a homogenization approach can

be developed at scales far larger than a.Itallowsto

account for the slowing down of the motion due to

obstacles in an effective diffusion coefficient (in plain

words, the black and white medium made of matter

and holes of size a appears as a grey homogeneous

medium at larger scales). More specifically, a diffus-

ing tracer of random trajectory r(t) experiences a

varying coefficient D[r(t)] (it equals D inside the

pores, whereas it vanishes in the nonaccessible region

V). The idea is to replace this fluctuating

realization of the transport coefficient by its spatial

average (independent of the trajectory), in what

concerns macroscopic properties:

eff

ðrÞd

r ¼

D½rd

ðwhere n

ðrÞ¼1 iff r 2VÞ ½33

Rigorous mathematical theorems ensure that the

large-scale motion can actually be described by a

Fick law and associated plain diffusion equation

(Bensoussan et al. 1978).

Anomalous Diffusion in a Fractal Medium

The above homogenization for diffusion in a porous

medium works well only if the pores have a finite

characteristic size; by contrast, diffusion in a fractal

substrate (e.g., a porous medium with pores of all

sizes) generically leads to anomalous diffusion, asso-

ciated with a time dependence of the mean-square

displacement R

(t)  t



with <1. In a fractal

substrate, the existence of obstacles and pores of all

sizes introduces spatial fluctuations at all scales and

long-range correlations in the spatial dependence of D.

This case corresponds to a critical situation and

homogenization fails to give a relevant description of

the macroscopic behavior, in the same way as mean-

field methods fail to account for critical phase transi-

tions. It reflects in the anomalous exponent <1ofthe

diffusion law, that can be related to the fractal

characteristics of the substrate ( = d

,whered

the spectral dimension and d

the fractal dimension).

Effective Diffusion in a Periodic Potential

(Averaging Method)

In case of a periodic medium, where D[r(t)] oscillates

with a small spatial period, an averaging procedure

can be developed as in the subsection ‘‘Effective

diffusion in a porous medium (homogenization),’’ to

determine an effective diffusion equation accounting

for the large-scale motion. Explicit computations

within a multiple-scale approach yield

eff

hDi

½34

where hDi denotes a space average over the

elementary cell (Givon et al. 2004).

Let us rather detail the case of diffusion of a

Brownian particle in a periodic potential U, with

U(x þ L) = U(x) for any x (restricting to dimension

1 for simplicity), at equilibrium at temperature T.

Let D be the coefficient of this particle in the

480 Multiscale Approaches

absence of the potential. At large scales dx  L,

the substrate appears to be spatially uniform. The

influence of the periodic bias exerted by the

potential on the diffusive motion (superimposition

of a modulated deterministic drift) can be described

in an average way. The result is a normal diffusion

with a reduced effective diffusion coefficient

eff

ðUÞ¼D inf

f 2C

ðLS

j1  f

ðxÞj

ðxÞ

with dm

ðxÞ¼

UðxÞ=kT

Uðx

Þ=kT

½35

where the infimum is taken over the set of smooth

periodic functions of period L and the average involves

the equilibrium distribution m

of the particle in the

potential landscape U( . ). So doing, one sees in

particular that no oriented motion can arise at

equilibrium, even if U is asymmetric. The procedure

extends to dimension d with only technical differences.

Effective Diffusivity for a Passively

Advected Scalar

Still another fruitful implementation of multiple-

scale method is encountered in the context of

diffusion and transport phenomena, in the study of

the advection by a given incompressible velocity

field v (r, t) of a passive scalar field (r, t ), for

example, the density of small inert ‘‘tracer’’ particles

advected by the fluid flow without modifying it

back. We consider the case when the fluid motion

can be decomposed into a large-scale, slowly varying

component and a small-scale, rapidly varying fluc-

tuation: v(r, t) = U(r, t) þ u(r, t). The parameter 

controls the relative strength of these components.

Another small parameter  is involved in this

problem: the ratio  = l=L  1 of the typical length

scales L and l of U and u, respectively. Here the

issue is to bridge two macroscopic descriptions: the

full hydrodynamic equation describing the evolution

of the scalar field (r, t)

ðr; tÞþvðr; tÞ:rðr; tÞ¼Dðr; tÞ½36

and a large-scale effective transport equation for an

average scalar field 

(r, t),



ðr; tÞþUðr; tÞ:r

ðr; tÞ

eff

ðr; tÞ

ðr; tÞ



½37

This procedure, amounting to account in an average

way for the small-scale contributions to the

complete hydrodynamic description, relies on a

spatio-temporal generalization of the multiple-scale

method: it involves rescaled space and time vari-

ables, X =  x,  =  t, T = 

t The different charac-

teristic scales of the velocity components are directly

reflected in their arguments: u(x, t ) and U(X, T). The

passive scalar field now expresses (x, t, X, , T) and

it is expanded as  = 

þ 

þ 



. The standard

multiple-scale procedure leads to introduce an

auxiliary field :



þ½ðu þ UÞ:@

 D @



¼u

½38

yielding the effective diffusivity tensor (where hiis a

space average)



eff

 D

eff

¼ D



i½39

Advection enhances transport, and eddy diffusivity

is larger than molecular diffusivity. In realistic cases,

there is a continuum of scales u =

n = 0

, where

has a characteristic scale l

 2

n

. Multiple-

scale method is to be iterated into an RG analysis,

achieving a recursive integration of the small and

fast scales into D

starting from the smallest and

fastest ones.

Conclusions

Multiscale approaches allow to predict large-scale

behavior generated by a given model; even more,

they offer constructive tools to bridge models at

different scales for the same phenomenon. They

provide systematic and mathematically well-

controlled tools to turn faithful but intractable

models into effective reduced ones, thus lying at

the core of statistical mechanics, many-body dyna-

mical systems, and, more generally, at all issues of

the still-in-progress complex systems science. Indeed,

in a complex system (that might be their very

definition), levels are so interrelated that it is

essential to investigate jointly all the scales, from

elementary units up to the whole system, and its

emergent properties; neither theoretical nor numer-

ical approaches can alone consider all the levels

together, showing the relevance, if not the necessity,

of multiscale approaches.

Basic preliminary issues are to determine the

proper elementary level, the proper collective vari-

ables, and the relevant small parameters. Let us

remark that the implementation of a multiscale

technique rapidly faces the fundamental issue of

defining a macroscopic variable; it offers some clues,

indicating that a macroscopic variable might be a

Multiscale Approaches 481

phenomenological quantity observable at our scale,

a slow mode, or collective variable.

Multiscale approaches take benefit of the separa-

tion of scales involved in the different mechanisms

at work in the phenomenon under consideration.

The basic idea, seen above at work in various

instances and different ways, is to somehow decou-

ple the different scales and to solve several simpler

single-scale problems. Any multiscale implementa-

tion actually involves, at some stage and more or

less explicitly, a limiting process in which the scale

separation ratio 1= tends to 1: this limiting process

has to be carefully controlled in order that the

method can be applied to real situation. Finally, to

be successful, multiscale approaches should achieve

a trade-off between:



accuracy (minimizing the loss of information

involved in the reduction or projection technique),



efficiency and tractability (this is, e.g., one of the

major successes of hydrodynamics)



robustness of the resulting reduced model (to be

checked a posteriori),



flexibility (extending to heterogeneous systems

involving different components), and



scope (bridging many different levels in order to

capture the whole hierarchical structure).

Let us conclude by emphasizing a much fruitful

benefit of multiscale approaches: they allow to

investigate structural stability of a model, in parti-

cular to evidence relevant parameters and essential

mechanisms controlling large-scale features. In this

respect, they lead beyond the (necessarily restricted)

scope of a specific model and give an explicit account

of the observer biased view, related to its scale of

observation. They hence contribute to capture a more

complete and controlled understanding of the real

physical systems.

Finally, a note on bibliographic guide to multi-

scale approaches may be useful. Technical details

and several applications of multiscale perturba-

tive expansions, in particular multiple-timescale

method, with references to the original papers,

can be found in Nayfeh (1973). Applications of

multiple-scale method, fully worked out in a very

pedagogical way, can be found in the work of

Cukier and Deutsch (1969), Piasecki (1993),

Bocquet (1997),andMazzino et al. (2004).An

acknowledged reference on homogenization tech-

niques and multiscale analysis in periodic media

is Bensoussan et al. (1978); see also the mono-

graphs by Lochak and Meunier (1988) and

Berdichersky et al. (1999). Two recent review

papers on multiscale approaches and reduction

techniques are Givon et al. (2004) and Gorban et al.

(2004). Basic principles and technical aspects of

scaling theories and RG approaches from a multiscale

viewpoint can be found in Lesne (1998).

See also: Adiabatic Piston; Averaging Methods;

Bifurcations in Fluid Dynamics; Boltzmann Equation

(Classical and Quantum); Central Manifolds, Normal

Forms; Interacting Particle Systems and Hydrodynamic

Equations; Korteweg–de Vries Equation and Other

Modulation Equations; Localization for Quasiperiodic

Potentials; Singularity and Bifurcation Theory; Stability

Problems in Celestial Mechanics; Stationary Phase

Approximation; Universality and Renormalization.

Further Reading

Auger P and Bravo de la Parra R (2000) Methods of

aggregation of variables in population dynamics. Comptes

Rendus de l’Acade´mie des Sciences, Pari s, Lif e Sc iences

323: 665–674.

Bensoussan A, Lions JL, and Papanicolaou G (1978) Asymptotic

Analysis for Periodic Structures. Amsterdam: North-Holland.

Berdichersky V, Jikov V, and Paparieolasu G (1999) Homo-

genization. Singapore: World Scientific.

Bocquet L (1997) High-friction limit of the Kramers equation:

the multiple time-scale approach. American Journal of Physics

65: 140–144.

Chen LY, Goldenfeld N, and Oono Y (1996) Renormalization

group and singular perturbations: multiple scales, boundary

layers, and reductive perturbation theory. Physical Review E

54: 376–394.

Cukier RI and Deutch JM (1969) Microscopic theory of

Brownian motion: the multiple-time-scale point of view.

Physical Review 177: 240–244.

Gaveau B, Lesne A, and Schulman LS (1999) Spectral signatures

of hierarchical relaxation. Physics Letters A 258: 222–228.

Givon D, Kupferman R, and Stuart A (2004) Extracting

macroscopic dynamics: model problems and algorithms.

Nonlinearity 17: R55–R127.

Gorban A, Karlin I, and Zinovyev A (2004) Constructive methods

of invariant manifolds for kinetic problems. Physics Reports

396: 197–403.

Haken H (1996) Slaving principle revisited. Physica D 97:

95–103.

Lesne A (1998) Renormalization Methods. New York: Wiley.

Lochak P and Meunier C (1988) Multiphase Averaging for

Classical Systems. Berlin: Springer.

Mazzino A, Musacchio S, and Vulpiani A (2004) Multiple-scale

analysis and renormalization for preasymptotic scalar trans-

port. Physical Review E 71: 011113.

Murray JD (2002) Mathematical Biology, 3rd edn. Berlin:

Springer.

Nayfeh AH (1973) Perturbation Methods. New York: Wiley.

Piasecki J (1993) Time scales in the dynamics of the Lorentz

electron gas. American Journal of Physics 61: 718–722.

482 Multiscale Approaches

Negative Refraction and Subdiffraction Imaging

S O’Brien, Tyndall National Institute, Cork,

Republic of Ireland

S A Ramakrishna, Indian Institute of Technology,

Kanpur, India

Introduction

The concept of negative refraction has caused a

revolution in classical optics and electromagnetic

theory in the past few years (Pendry 2004,

Ramakrishna 2005). If a material has negative

dielectric permittivity (") and negati ve magnetic

permeability () simultaneously at a given frequency

!, then it can be said to have a negative refractive

index defined as

n ¼

ﬃﬃﬃﬃﬃﬃ

"

½1

Several peculiar consequences of Maxwell’s equations

for the propagation of radiation in such a material

were originally pointed out by Veselago (1968).But

the lack of such natural materials failed to create much

enthusiasm until recently when composite structured

photonic materials have been shown to have negative

refractive index (Smith et al. 2000, Shelby et al. 2001).

The question then boils down to what constitutes

materials with negative " and ? Where the structure

varies spatially on a scale much less than the

wavelength of the incident radiation, composite

electromagnetic materials can be regarded effectively

as homogene ous media. A set of effective response

functions: the effective permittivity, "

eff

, and the

effective permeability, 

eff

, can then be ascribed to

these materials. To develop a homogeneous view of

the electromagnetic properties of a mediu m com-

posed of discrete atoms and molecules was the

motivation for defining a permittivity " and permea-

bility . The simplicity provided by such a descrip-

tion cannot be understated. Provided the radiation

cannot resolve the underlying structure, replicating

the atoms of a material with structure on a larger

scale therefore represents a straightforward exten-

sion of the original concept.

If we consider arrays of structures defined by a

unit cell of dimensions, d, then our effective

description of the response of the medium to

electromagnetic radiation of angular frequency !

will be valid provided that

d   ¼ 2c=! ½2

This restriction ensures that the underlying structure

of the medium will merely refract and not scatter the

incident radiation, in which case an effective

permittivity and permeability for the medium

become valid. The above inequality defines

the long wavelength or effective medium limit

(Garland and Tanner 1978). Maxwell’s equations,

written in the absence of free charges and external

currents,

  D ¼ 0;   E ¼

½3

  B ¼ 0;   H ¼

½4

together with the constitutive relations:

Bð!Þ¼



eff

ð!ÞHð!Þ½5

Dð!Þ¼"

eff

ð!ÞEð!Þ½6

then provide us with a complete description of the

electromagnetic properties of the material over the

frequency range of interest. Note that the effective-

medium parameters are a function of the frequency

as the material polarization response depends on the

time history of the applied fields (Landau et al.

1984). These effective parameters were then general-

ized to analytic complex functions to account for

absorption, and to second-ranked tensors to describe

anisotropic responses.

The real parts of these effective material para-

meters can always be negative; there is nothing

fundamentally wrong about that. Provided that they

are dispersive, that is, they vary as a function of

frequency, and dissipative as a consequence of the

famous Kramers–Kronig relations (Landau et al.

1984), such materials are causally possible. Simulta-

neously negative values of "

eff

and 

eff

change the

nature of electromagnetic radiation in these media.

For example, the wave vector in such isotropic

media points opposite to the Poynting vector and

gives rise to many new interesting effects such as

modified refraction, negative Doppler shifts, etc.

Such materials can support a variety of surface

electromagnetic modes, which can have dramatic

effects such as the possibility of a perfect lens which

has unlimited image resolution (Pendry 2000) and is

not subject to the traditional diffrac tion limit.

New artificial electromagnetic composite struc-

tures, often referred to a s ‘‘meta-materials,’’ allow

us to access values of these material parameters

which are not found in naturally occurring materi-

als. We will show here how to obtain negative

values of "

eff

and 

eff

in meta-materials using a

variety of resonance phenomena. Then we will

look at the problem of imaging with subdiffraction

resolution using negative refractive index

materials.

Artificial Plasmas

From the electromagnetic viewpoint, a plasma can

be represented as a medium with dielectric permit-

tivity whose real part is negative. The Coulomb

force an d the finite mass of the electrons combine to

give an ideal plasma a dispersion in the relative

permittivity,

"(!), given by

"ð!Þ¼1 

½7

where the plasma frequency is defined by !

(e

)=("

),  is the number density of electrons,

e is the electronic charge, and m

is the electron

mass. The permittivity of the plasma is negative at

frequencies below the plasma frequency.

A plasma-like behavior characterizes the electron

gas in the noble and alkali metals, with a plasma

frequency typically at ultraviolet frequencies.

Because of the presence of dissipation, at lower

frequencies resist ive effects dominate and the plas-

mons cannot be excited. To obtain materials with

negative dielectric permittivity at low frequencies, a

lower plasma frequency is required corresponding to

more massive particles and a lower particle density

. A structure consisting of a three-dimensional

lattice of very thin wires simulates a low-density

plasma of very heavy charged particles and is shown

in Figure 1 ( Pendry et al. 1998). A simple model

allows us to describe the desired reduction in !

such a structure.

First consid er a displacement of the electrons in

the wires along one of the cubic axes. Only the wires

directed along that axis are active and thus provide a

lowered effective density of electrons, 

eff

, given by

the area occupied by the active wires. Thus,



eff

¼ 

a

½8

An even more profound effect of constraining the

electrons to run along thin wires is a result of the

induced magnetic field which wraps the wires as

the electrons are in motion. Suppose a current I

flows in the wires. The magnetic field is

HðrÞ¼

2R

a

½9

where R is the distance from the wire center, v is the

electron drift velocity, and e is the charge density in

the wire. In terms of the magnetic vector potential,

the magnetic field is

HðRÞ¼

1

  AðRÞ½10

where

AðRÞ¼



ve

lnðd=aÞ½11

and d is the lattice spacing. The importance of the

divergence of the magnetic field with the wire radius

as seen in eqn [9] is the contribution to the canonical

electronic momentum given by eA. If we neglect the

variation of the fields with distance from the wire

center, we can view this contribution as defining a

new effective mass for the electr ons given by

eff





2

lnðd=aÞ½12

Figure 1 A periodic structure composed of infinite conducting

wires arranged in a simple cubic lattice. Provided the factor a /d

is small enough, the structure responds to incident electromag-

netic waves as a plasma of very heavy charged particles.

484 Negative Refraction and Subdiffraction Imaging

Now the effective plasma frequency for the system



eff

2c

lnðd=aÞ

½13

is seen to be much reduced. As a n example, the

plasma frequency of 1 mm aluminum wires paced

by 10 mm is about 2 GHz, and the corresponding

electronic effective mass is almost 15 times that of

a proton! The factors of effective mass and charge

density cancel leaving an expression comprising

only the macroscopic system parameters. This is to

be expected as a circuit analysis in terms of a

capacitance and inductance c an also be used to

formulate the problem. However, such an

approach can obscure the true nature of the

problem which is encapsulated as a low-frequency

plasma oscillation. Inclusion of the finite resistivity

of the metal yields a finite lifetime for the plasmon

excitation. Experiments have shown that a reduc-

tion in the plasma frequency of six orders of

magnitude from the ultraviolet t o the micro wave

regioncanbeachievedinthesethin-wirecompo-

sites (Pendry et al. 1998).

Artificial Magnetism

Although the Maxwell equations [2]–[4] are sym-

metric in the electric and magnetic fields, we are yet

to discover a free magnetic pole. The magnetism we

find in natural materials is limited to spin systems

and restricts the values of 

eff

. Up to microwave

frequencies, magnetic activity is common and

certain insulating ferromagnets and antiferro-

magnetic compounds such as MgF

and FeF

can

even exhibit a negative permeability at some

frequencies. However, large losses can accompany

the magnetic activity in these materials.

Recently, it has become clear that a wide variety

of composite structures comprising resonant inclu-

sions can display magnetic activity in the effective

medium limit (Pendry et al. 1999). Efficient screen-

ing of A C magnetic fields can be achieved using a

thin cylindrical shell of metal or superconductor. In

order t o obtain a large magnetic response such that

the modulus of the magnetic susceptibility, j

j> 1,

what we require is a r esonant over-screening

material response. A collection of subwavelength-

sized structures that exhibits such an over-screening

response can constitute a negative 

eff

material.

One such resonant subwavelength structure is the

so-called split-ring resonator (SRR), which can be

scaled to form magnetic meta-materials from

microwave to optical frequencies (Pendry et al.

1999, O’Brien and Pendry 2002b). An SRR

structure which has been demonstrated experimen-

tally to have a resonant magnetic response at

microwave and THz frequencies is depicted in

Figure 2a (Smith et al .). It comprises of two planar

rings of metal on an insulating backing. The rings

couple inductively to the magnetic field normal to

the plane of the rings. Because of the large

capacitance between the rings, the structure r eso-

nates at some frequency. Driven by the back

electromotive force (emf), a large response is

expected in the vicinity of the resonance frequency

which is also antiphased in a small frequency range

above the resonant frequency. If the SRRs are much

smaller than the free-space wavelength, a collection

of such SRRs would behave as a negative 

eff

material at these frequencies.

Theoretical calculations (Pendry et al. 1999)

assuming a nondispersive metal show that a periodic

lattice of such structures is characterized by a

magnetic permeability given by



eff

¼ 1 

f !

 !

þ i!

½14

where f = R

is the filling factor,

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

3lc

R

ln 2w=g

½15

is the resonant frequency, an d the damping of the

resonance is determined by the factor

 ¼



R

½16

Here d is the lattice spacing, R is the inner radius of

the ring, w is the width of the rings, l is the distance

between adjacent planes of SRRs, and  is the

conductance per unit length of the rings measured

along the circumference. Orientation of planar SRRs

Frequency

Wavevector

(a)

(b)

Figure 2 (a) The split-ring resonator structure. The structure is

planar with an internal radius R. The metal rings are of width w

and are separated by a spacing g. (b) Generic dispersion

relationship, ! vs. k, for a resonant structure with an isotropic

effective permeability as in eqn [15].

Negative Refraction and Subdiffraction Imaging 485

along all three Cartesian axes allows for the creation

of an isotropic material. Figure 3 shows the generic

dispersion of the (!) given by eqn [14]. A higher

resistivity for the material of the SRR would

broaden the resonance and the freque ncy region

with Re() < 0 might vanish altogether for large

resistivity.

For isotropic homogeneous materials with a

resonant effective permeability as in eqn [14] we

can illustrate a generic di spersion relationship, ! vs.

k, shown in Figure 2b. The solid lines represent

twofold degenerate transverse modes and the

dispersionless longitudinal magnetic plasmon

mode at the magnetic plasmon frequency (!

The dashed lines are a band of propagating states

with a linear dispersion determined by the

polarizability of the SRRs and a flat band of

resonant states at the magnetic resonance fre-

quency !

. The gap in the dispersion can be

regarded as arising from the hybridization and

avoided crossing of these bands. The important

points to note are:

1. Wherever 

eff

is negative there is a gap in the

dispersion relationship. This is the case for !

!<!

, the frequency where 

eff

= 0. Only

evanescent modes with imaginary wave vector

exist in this region.

2. A longitudinal magnetic plasma mode, which

shows no dispersion, appears at ! = !

An alternative approach to obtaining a nonzero

magnetic susceptibility in composite media is pro-

vided by the zeroth-order transverse electric (TE)

Mie resonance in dielectric particles. Ferroelectric

and phonon polaritonic materials are promising

candidates for providing the necessary large dielec-

tric constants up to infrared frequencies (O’Brien

and Pendry 2002a).

The high-frequency scaling properties of the SRR

offer an interesting insight. The plasma-like dielec-

tric permittivity of noble metals

"ð!Þ¼ð"

Þ¼"



!ð! þ iÞ

½17

is essentially a large negative real number for !



!  . For a 2D array of simplified SRRs consisting

of a single conducting ring with symmetrically

placed small capacitive gaps, the quasistatic effective

magnetic permeability for a magnetic field applied

normal to the plane of the SRR is (O’Brien and

Pendry 2002b)



eff

¼ 1 

 !

þ i!

½18

where f

f (L

þL

)

1

, =L

 (L

þL

)

1

, and

=(L

þL

)

1

. In the above expressions,

=

R

is the geometrical inductance per unit

length of the structure and C="

=n

is the

capacitance per unit length of the structure for series

connection. Here it has been assumed that the

thickness of the SRR () is small compared to the

skin depth  ’c

An additional inductive impedance in the struc-

ture, the kinetic or inertial inductance, L

2R="

 = 2

R

=, determines the effective

filling fraction and damping of the resonance through

the ratio of the two contributions to the total

inductance. This contribution to the inductance arises

from the finite electron mass and implies that simply

decreasing the size of the resonators indefinitely will

not result in our being able to realize a strong

magnetic response at near-infrared or optical fre-

quencies. As the dimensions of the structure are

reduced that fraction of the energy of the displace-

ment current associated with the inertial mass of the

electrons increases. A finite  then means that

dissipative losses increase. Thus, strong damping of

the resonance will be avoided if the quantity R=2

is large. We note here that with  equal to the

London penetration depth, this ratio also determines

the screening efficiency of low-frequency magnetic

fields by a thin layer of superconductor. This result

points to a broader similarity between the low-

frequency electromagnetic properties of the super-

conducting condensate and those of a perfect plasma.

Other nanocomposites in addition to the SRR

have been proposed which may lead to a magnetic

response at optical frequencies. These include pairs

of nanometer-sized metallic sticks where simulta-

neous electric and magnetic dipole resonances lead

to a strongly dispersive effective permittivity and

permeability.

Frequency (a.u.)

–5

Re(μ)

Im(μ)

6589

Figure 3 (a) The generic magnetic response of the SRR

structure. Re() < 0 in a frequency band above the resonance

frequency.

486 Negative Refraction and Subdiffraction Imaging