Distribution of the area enclosed by a 2D random walk in a disordered medium

4 Pages·0.12 MB·English

by  K. V. Samokhin

Distribution of the area enclosed by a 2D random walk in a disordered medium
K.V. Samokhin∗
Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK
(February 1, 2008)
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The asymptotic probability distribution for a Brownian particle wandering in a 2D plane with
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random trapsto enclose thealgebraic area Aby timet is calculated usingtheinstanton technique.
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1
PACS numbers: 02.50.-r, 05.40.Fb, 05.40.Jc

n
a
J

8 It is well known that the properties of random walks B,andusingtheWienerpath-integralformulafortheso-
1
are dramatically changed by the presence of quenched lution of Eq. (1), we arrive at the following expression:

disorder. In particular, if a walker can be irreversibly
]
h trapped at some randomly distributed sites, then the ∞ r(t)=r
(A,tU)= dpeiBA r(τ)e−S[r(τ)], (2)
c asymptoticprobabilitytoreturntothestartingpoint(or
e P | Z−∞ Zr(0)=r′ D
the totalprobabilityto surviveuntil time t) is givenind
m
dimensionsbyp(t) exp( ctd/(d+2))[1],whilethemean
where
- square displacemen∼t decre−ases compared to a pure diffu-
t
a sion: r2 t2/(d+2) [2]. On another hand, if a walker t 1 i
t h i ∼ S = dτ r˙2(τ)+U(r(τ))+ Br2(τ)θ˙(τ) .
s isadvectedbyarandomforce,thenthe totalprobability −Z (cid:18)2D 2 (cid:19)
. is conserved, but the diffusion coefficient and the mean 0
t
a square displacement acquire logarithmic corrections [3]. It is easy to see that the path integralon the right-hand
m
Further examples can be found, for instance, in Ref. [4]. side of Eq. (2) represents the Euclidean Green func-
- Much less, however, is known about the influence of a tion (r,t;r′,0)ofafictitiousquantumparticleofmass
nd quenched disorder on the topological properties of ran- m =G(B2D)−1 moving in the random potential U(r) and
o dom walks, which became a subject of theoretical inves- in the uniform magnetic field B (we choose the units in
c tigation very recently [5]. By “topological” properties which h¯ = e = c = 1). Let us now assume that the
[ we mean such characteristics as the winding number of trajectory is closed (r = r′) and average the enclosed

1 a Brownian particle, or the linking number of a closed area distribution over all positions of the starting point:
v polymer,orthe algebraicareaenclosedbythe trajectory (...) r = (1/Ω) d2r(...) (Ω is the system volume). We
h i
5 of a random walker. In this Rapid Communication, we have: R
6 concentrate on the latter case and calculate the asymp-
1 totic distribution of the area swept by a planar random (A,tU)= ∞ dB eiBAZ(t,B U),
1
walk wandering in the presence of traps. P | Z |
0 −∞
The probability distribution for a random walk start-
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ing at some point r′ at t = 0 to end at a point r after where Z(t,B) is the partition function at inverse “tem-
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/ time t obeys the diffusion equation perature” 1/T = t. Averaging now over the positions of
at traps, we finally obtain:
∂P
m =D 2P U(r)P. (1)
∂t ∇ − ∞ ∞
- (A,t)= dB dE eiBAe−EtN(E,B), (3)
ond HReir)eisDthisetrhaenddoiffmus“iponotceonetffiiacl”ie,nwthainchdrUe(prr)es=enUts0tPheitδr(arp−- P Z−∞ Z0
ping probability per unit time (U > 0). The positions whereN(E,B)istheaveragedensityofstatesofaquan-
c 0
: Rioftrapsaredistributeduniformlyinaplaneaccording tum particle described by the Hamiltonian
v
to the Poissonlaw with meandensity ρ. The probability
i H =D( i A(r))2+U(r). (4)
X for a random walk of “length” t to enclose the algebraic
− ∇−
r area A can be obtained by averaging the corresponding
a δ-function constraint over the solutions of Eq. (1) in a We choose the cylindricalgauge for the vector potential:
A =Br/2.
given distribution of traps: θ
In an ideal case (i.e. in the absence of traps), the
1 t eigenvalues of the Hamiltonian (4) are the Landau lev-
(A,tU) δ A dτ r2(τ)θ˙(τ) ,
P | ≡(cid:28) (cid:18) − 2Z0 (cid:19)(cid:29)P(r,t;r′,0|U) els En = ωc(n+1/2), where ωc = 2DB is the cyclotron
frequency. The density of states is then given by the set
where θ(t) is the angle between the radius-vector r(t) of of equidistant δ-function peaks, and the partition func-
the particle and some fixed direction in the plane. Writ- tion can be easily evaluated, resulting in the following
ingtheδ-functionasanintegraloveranauxiliaryvariable expression for the enclosed area distribution:
1
1 A
(A,t) , x . (5) GR(r,r′;E,B)= i lim 2Φ(r)ϕ(r)ϕ¯(r′)
P ∼ cosh2x ∼ Dt − η→+0Z D
exp i d2r Φ¯(E H +iη)Φ , (6)
This result was first obtained by Levy quite a while ago × (cid:18) Z − (cid:19)
[6]. The form of the dimensionless scaling variable x is
quite natural, since the only characteristic scale with di- where the two-component superfields
mensionalityofareainacleansystemisthemeansquare
ϕ(r)
displacement r2 Dt. Φ(r)= , Φ¯(r)=(ϕ∗(r),ψ¯(r))
h i∼ (cid:18)ψ(r)(cid:19)
In order to calculate the density of states in the pres-
ence of disorder, let us first estimate what characteristic
are composed of a Bose field ϕ and a Grassmanian field
scalesofenergyandmagneticfielddeterminetheasymp- ψ, and 2Φ = (1/π) (Reϕ) (Imϕ) ψ¯ ψ. The prob-
totic behavior of (A,t). We are interested in calcula- D D D D D
P ability of having N impurities located at the points
tion of the probability distribution at large but fixed t R ,...,R in the area Ω in the plane is given by the
1 N
and A . As seen from (3), this limit corresponds to
→∞ Poissonlaw:
E 0, B 0, and ω E. Therefore, it looks natural
c
→ → ≪
tostartwiththecaseofB =0andmakesurethatweare e−ρΩ
P (R ,...,R )= (ρΩ)N.
able to treat the magnetic field as a small perturbation N 1 N N!
in the relevant range of parameters.
Using this expressionto average(6)overthe positionsof
At B = 0, the low-energy behavior of the density
traps, we end up with the following effective action:
of states is determined by the rare fluctuations of the
concentration of impurities creating the large areas free
of traps which are able to sustain the eigenstates with iS[Φ]= dDr iΦ¯ E D( i A)2+iη Φ
E 0. TheasymptoticexpressionisgivenbyN(E,B = Z n (cid:0) − − ∇− (cid:1)
0)→exp( constρD/E) at E E , where E = ρU is ρ 1 e−iU0Φ¯Φ . (7)
0 0 0
the∼mean−value of the random≪potential [7]. Quantita- − (cid:16) − (cid:17)o
tively, such exponentially small “tails” are determined Animportantpropertyoftheaction(7)isthatitisin-
by the contributions of instantons, i.e. spatially local- variantundertheglobalsupersymmetrytransformations,
ized solutions of the saddle-point equations of the effec- mixing the boson and fermion sectors:
tive field theory. The typical size of a clean area, or the
instanton diameter, grows in the time representation as Φ Φ˜ =TΦ, (8)
l (Dt/ρ)1/4 [2]. If an external magnetic field is im- →
inst
∼
posedontheinstanton,thenitseffectontheenergyspec- where T is a 2 2 unitary supermatrix [9]. Due to this
×
trumcanbecalculatedperturbativelyaslongasthemag- symmetry,theproblemoffindingthesaddlepointsof(7)
neticlengthl = 1/Bconsiderablyexceedstheinstan- can be considerably simplified. Indeed, since the saddle
B
ton dimension l p, which is the case if A (Dt/ρ)1/2. pointmanifoldisinvariantunderthetransformations(8),
inst
It is this condition that determines the limi≫ts of applica- weareabletoseektheinstantonsolutioninthefollowing
bility of our theory. Note also that if we were interested form:
aintcmaolcduelraattioenAo,ftthheeninwteercmouedldiauteseatshyemepxtaoctitcsexopfrPes(sAio,nt)s Φ (r)= ϕinst(r) , (9)
inst
(cid:18) 0 (cid:19)
for the density of states at E ω /2 ω [8] (since the
c c
− ≪
random potential is positive everywhere, the density of whereϕ (r)=ϕ (r)isacylindricallysymmetricspa-
inst inst
states vanishes at E <ω /2).
c tially localized function. If to write the boson fields as
Let us now calculate the effect of an external mag- ϕ=ϕ +iϕ , ϕ∗ =ϕ iϕ , where ϕ are real on the
1 2 1 2 1,2
netic field on instantons explicitly. The density of states initialfunctionalintegra−tioncontour,thentheimaginary
canbeexpressedintermsoftheretardedGreenfunction partofthe Greenfunctionis determinedbyanon-trivial
of the Schr¨odinger equation with the Hamiltonian (4): saddle point of the action in the complex plane of ϕ
1,2
N(E,B) = (πΩ)−1 dDr Im GR(r,r;E,B) U, where [10,11], and
GR(r,r′;E,B−)= r(ER H+i0h)−1 r′ . TheGireenfunc-
h | − | i
tioncan,inturn,becalculatedbystandardmeansofthe N(E,B) e−Sinst(E,B) (10)
quantumfieldtheory. Inordertocarryouttheaveraging ∼
overthepositionsoftraps,weresorttothesupersymme- withtheexponentialaccuracy. Thefermionsectorcanbe
try approach, in which the cancellation of denominators neglected as long as we are not interested in calculation
is achieved by doubling the degrees of freedom and in- of the pre-exponential factor in (10).
troducing the commuting and anticommuting fields on In order to make Eq. (7) in the boson sector real, we
equalfooting(see,forinstance,Ref.[9]). Beforedisorder rotate the integration contour: ϕ e−iπ/4ϕ . As
1,2 1,2
→
averaging,theretardedGreenfunctioncanbewrittenas resultof this, the exponentonthe right-handside of Eq.
the following functional integral: (6) changes: iS[Φ] S[Φ], where the action S is
→−
2
S = d2r ϕ E+D( i A)2 ϕ and (10), the Lifshitz result N(E) exp( constρD/E)
i i ∼ −
Z n (cid:0)− − ∇− (cid:1) is recovered.
+ρ 1 e−U0ϕ2i (11) In principle, one can find the exact solutions of the
(cid:16) − (cid:17)o Schr¨odinger equations inside and outside the potential
well at B =0 (they are expressedin terms of the conflu-
(i=1,2). Introducing the dimensionless variables: 6
ent hypergeometric functions), match them at the point
x = x and finally end up with a transcendental equa-
r =ξx, ϕ (r)= U f(x), 1
inst 0 tionforx (α,β),whichcanbesolvedatβ 0. However,
1
p →
we prefer not to follow this procedure here, because the
whereξ = D/E,weobtain,fromEq. (11),anon-linear
sameresultscanbeobtainedusingmuchmorephysically
differentialpequation for the saddle-point solution f:
apparent reasoning in the spirit of the Lifshitz’s original
derivation [7]. First, we note that, in the limit α ,
1 d x d f +β2x2f +α2e−f2f =f, (12) the instanton solution satisfies the Schr¨odinger equ→ati∞on
− xdx(cid:18) dx(cid:19)
foraparticleconfinedinthepotentialwellwithinfinitely
high walls in a magnetic field. The ground state energy
where α = E /E 1 and β = ω /4E 1. Similar
equations wpitho0ut m≫agnetic field werce obta≪ined in Refs. ǫ0 is equalto unity (inthe units ofE). Inthe absenceof
magnetic field this condition fixes the radius of the well
[12,13], using different techniques.
at x (0) = a. At β = 0, the lowest order perturbative
Due to complexity of the equation (12), we are able 1 6
correction to the ground state energy is
to find only an approximate solution. To this end, we
replace the “potential” V(f) = α2e−f2 in Eq. (12) by a a
β2 dxx3f2(x)
piecewise constant potential:
Z
δǫ = 0 , (18)
0 a
α2 , at f <1, dxxf2(x)
Veff(f)=(cid:26)0 , at f >1. (13) Z0
where f(x) J (x) is the unperturbed ground state
Then, Eq. (12) becomes effectively linear andreduces to ∼ 0
wave function (16). Calculating the integrals with the
a couple of the Schr¨odinger equations, whose solutions
Bessel functions, we obtain: δǫ =cβ2, where c 1.261.
f (x) satisfy the following conditions: 0 ≃
1,2 To keep the ground state energy fixed, this correction
f (x )=f (x )=1, f′(x 0)=f′(x +0), (14) should be compensated by the correspondingincrease in
1 1 2 1 1 1− 2 1 the radius of the potential well: x2(β) = x2(0)(1+δǫ ).
1 1 0
where x (α,β) is the position of the discontinuity in Substituting this in (15), we finally obtain:
1
the effective potential, which is to be determined self-
πρDa2 cD2
consistently. Going back to the dimensional variables, it S (E,B) 1+ B2 . (19)
inst ≃ E (cid:18) 4E2 (cid:19)
is easy to see from (11) and (12) that the instanton ac-
tion is proportional to the area of the effective potential
This expression is valid at E 0, B 0, ω /E 0.
well: → → c →
Theasymptoticprobabilitydistributionoftheenclosed
S =πρξ2x2. (15) areacannow be obtainedfrom(3), (10)and(19)by cal-
inst 1 culating the integrals by the steepest descent method:
Let us start with the case of B = 0, i.e. β = 0. The ∞ ∞
solution of the linearized saddle-point equations can be (A,t) dB dE eiBAe−Ete−Sinst(E,B)
P ∼Z Z
written in form −∞ 0
a√π √ρA2
f (x)=C J (x), at 0<x<x , exp( πρDa2t)exp . (20)
f(x)= 1 1 0 1 (16) ∼ − (cid:26)− c (Dt)3/2(cid:27)
(cid:26)f2(x)=C2K0(αx), at x1 <x, p
The first exponential on the right-hand side is noth-
where J0(x) and K0(x) are the Bessel functions of real ing but the asymptotic “tail” of the total probability
andimaginaryarguments,respectively. Substitutingthis p(t) = dA (A,t) for a random walker not to be
solution in the matching conditions (14), we obtain the trapped [R1]. TPhe second exponential thus represents the
equation conditionalprobability to enclose the areaA, provideda
walker has survived until time t.
J (x) K (αx)
1 =α 1 . (17) Weseethattheasymptoticbehaviorof (A,t)isdras-
J (x) K (αx) P
0 0 tically changedby the presence of disorder,comparedto
theLevy’sresult(5). ThedistributionbecomesGaussian
In the limit α , assuming that x 1 and using
→ ∞ 1 ∼ (A,t) exp( x2), with the scaling variable
the asymptotic expansions of the Bessel functions [14], P ∼ −
fiwrestobzetraoinoJf0th(xe1f)u=nct0i,oin.eJ. x(x1)=. Aa,ftwerhesurebsati≈tu2ti.o4n05inis(t1h5e) x A (ρDt)1/4, (21)
0 ∼ Dt
3
so that the standard deviation now growsslower than in ∗ Permanent address: L.D.Landau Institutefor Theoret-
the clean case: A2 1/2 t3/4 (the mean value A is, of ical Physics, Kosygina Str.2, 117940 Moscow, Russia.
h i ∼ h i
course,zero). Suchadifferentformofthescalingvariable
can be related to the presence of an extra length scale [1] B.Ya.BalagurovandV.G.Vaks,Zh.Eksp.Teor.Fiz.65,
r 1/ρ in the system, which depends on the concen- 1939 (1973) [Sov.Phys. – JETP, 38, 968 (1974)].
ρ
tra∼tiopn of traps (nothing depends on the absolute value [2] P.GrassbergerandI.Procaccia,J.Chem.Phys.77,6281
U of the random potential). Note also that a similar (1982).
G0aussian distribution was obtained in Ref. [15] for a 2D [3] D.S. Fisher, Phys.Rev.A 30, 960 (1984).
[4] J.P. Bouchaud and A. Georges, Phys. Rep. 195, 127
random walk in a box of a finite size L. In this case, the
(1990).
fictitious magnetic field was also treated perturbatively,
giving rise to A2 1/2 L(Dt)1/2. Qualitatively, such a [5] K.V.Samokhin,J.Phys.A31,8789(1998);ibid31,9455
h i ∼ (1998).
differencebetweenthetwosystemsisduetothefactthat
[6] P.Levy,Processus Stochastique et Mouvement Brownien
in our case the size of the effective potential well is not
(Gauthier-Villars, Paris, 1948).
constant, but grows with time as L(t)=l (t) t1/4.
inst ∼ [7] I.M. Lifshitz, Usp. Fiz. Nauk 83, 617 (1964) [Sov. Phys.
In conclusion, we studied the asymptotic probability – Uspekhi7, 549 (1965)].
distribution of the algebraic area enclosed by a planar [8] E. Brezin, D.Gross and C. Itzykson,Nucl.Phys. B 235
random walk in the presence of immobile random traps.
[FS11], 24 (1984).
It is shown that this probability is directly related to [9] K.B. Efetov, Supersymmetry in Disorder and Chaos
“the Lifshitz tail” in the density of states of a quantum (Cambridge University Press, 1997).
particleinaPoissondisorderanduniformmagneticfield. [10] J.S. Langer, Ann.Phys. 41, 108 (1967).
In contrast to the case of an ideal random walk, the en- [11] C.CallanandS.Coleman,Phys.Rev.D16(1977)1762.
closed area distribution turns out to be Gaussian with [12] R.FriedbergandJ.M. Luttinger,Phys.Rev.B 12, 4460
the standard deviation growing as t3/4. (1975).
[13] T.C. Lubensky,Phys.Rev.A 30, 2657 (1984).
This work was financially supported by the Engineer- [14] M.AbramowitzandI.A.Stegun,HandbookofMathemat-
ing and Physical Sciences Research Council (UK). ical Functions (Dover,New York,1965).
[15] J. Desbois and A.Comtet, J. Phys. A 25, 3097 (1992).
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