9. TRATAMIENTO DE LOS PACIENTES CON CÁLCULOS URETERALES
9.8 Comentario
A rather natural approach to broaden the analytical insight beyond scaling arguments is to apply the already mentioned replica technique[59]. This means creating a theory of multiple interacting particles, solving this for arbitrary particle number n and finally continuating this result to the rather peculiar case of vanishing particle number, n→0. As is often the case with interacting many particle systems a closed solution is usually not feasible for elastic manifolds in disordered potentials, with the one notable and very prominent exemption of then exact solution[60] of the directed polymer in 1+1dimensions.
In this section we will follow the conceptually identical treatment of di- rected manifolds by Mezard and Parisi, see Ref. 65. As the bulk of this work has been presented before[4] the treatment here will remain limited and we refer to that prior work for any omitted details. In an attempt to not obfuscate things unnecessarily we restrict ourselves to one transversal dimension, d=1, for the time being and comment later on how to adapt
this to higher dimensions.
The very basis of replica approaches to disordered systems is the following
representation35of the logarithm of a number Z 35The proof of this is very straightfor-
ward when done in the right direction. Application of L’Hospital’s rule to the right hand side yields limn→0 Z
n−1 n = limn→0Z nln Z 1 =ln Z. ln Z= lim n→0 Zn−1 n .
This identity is often referred to as thereplica trick. Its importance for the
statistical physics of disordered systems is that it offers a means to compute the quenched average of the free energy
F= −β−1ln Z
in a manner that performs the actual averaging procedure before the evalua- tion of the partition function. We will briefly dwell on this point to elucidate the appeal of the replica method. The partition function of an elastic line in a particular realisationVof the disorder is given as the integral over phase
space36of the Boltzmann weights 36The phase space consists here of all pos-
sible conformations z(x)of the line and thus a functional integration.
ZV = Z
Dz(x) e−βHV[z(x)]
where we introduced subscripts to emphasise the explicit dependence on the disorder realisation. Then the quenched average of the free energy is given
by integration over disorder realisations37 37Each realisation contributes with some
statistical weight P(V ).
F= −β−1
Z
DVP(V )ln ZV.
In the replica approach the relevant computation38is thereplicated partition 38This is also the step where the effective attractive replica interaction is generated. Before averaging over the disorder we have an n-particle system with a factorising par- tition function Znindicating statistical in-
dependence and therefore no interaction. function Zrep=Zn =
∏
α Z Dzα e−β∑αH[zα] =∏
α Z Dzα e−βHrepwhere the replica index39αgoes over the n system replicas and the last term
introduces theeffective replica Hamiltonian. This is generated by interpreting
the disorder averaged Boltzmann weight as the Boltzmann weight of a new effective energy functional.
The elastic part of the initial energy functional is independent of the
39Every occurrence of small Greek letters
as summation index is a replica index go- ing over 1, . . . ,n. These limits are therefore omissible.
disorder which leaves us at evaluating40 the disorder contribution to the
40Suppose we have a random quantity X
which we write as X =ikYwith known cumulants Xn c, then we find[61] ln e−ikY=ln eX=
∑
∞ n=1 (−ik)n n! Y n c =∑
∞ n=1 (−ik)n n! i k n Xnc=∑
∞ n=1 1 n!Xnc and therefore eX=e∑∞n=1 (−1)n+1 n! Xnc. replicated Hamiltonian e−β∑α RL 0dx V(x,zα(x))=eβ2/2∑α,β RL 0 dx RL 0 dy V(x,zα(x))V(y,zβ(y)) =eβ2 g2/2∑α,βR0Ldx δ(zα(x)−zβ(x)).In its entirety the replica Hamiltonian41thus reads
41For the directed line we would have a
first derivative in the first term. Then, by variation (for a more detailed presentation see for example Ref. 4) we see that the re- stricted partition function Z(x,z)obeys a differential equation that essentially is the Schrödinger equation for the Lieb-Liniger (LL) model[62] HLL= − N
∑
j=1 2xj+2c∑
1≤i<j≤N δ(xi−xj)of interacting bosons. Thus the replicated partition function is given by the ground state energy[60, 62]
E(n) =E1n+E3n3
of this quantum model, which implies ω=
1/3. Hrep=
∑
α L Z 0 dx(2xzα)2− βg2 2∑
α,β L Z 0 dx δ(zα(x) −zβ(x)).And there we have it, the second term is nothing but an attractive short-range pair interaction of two replicated lines. However, we have to keep in mind that there are n interacting lines and n ultimately is going to zero. Thus, it is unclear a priori how much knowledge of binding problems of two lines can be transferred. Nevertheless, the replica method enabled us to study the average behaviour of a disordered system using one effective, deterministic system.
Alas, we are not aware of a way computing the partition function exactly. We therefore make use of a variational approach in Fourier space, but before we can do that, we have to take of some calamities of the replica Hamiltonian. For one the bending energy introduces infrared singularities∼ k−4, k being the wave number in Fourier space, in the Green’s function, which have to be regularised. The method we choose is to introduce a parabolic potential (or “mass” term)R
dxµz2(x). This effectively confines the system to a finite size in the transversal dimension and we are then ultimately interested in taking the limit µ→0an unbounded system.
Secondly, an infinitely small disorder correlation length in the transversal dimension, just as a point interaction in a binding problem, is problematic. We therefore introduce a finite correlation length λ, that is we write
V(x,z)V(x0,z0) =g2f
λ((z−z
0)2)δ(x−x0),
where fλis a sufficiently fast decaying function42depending on the length λ 42We use the Gaussian √2π f
λ(x2) = λe−x2/(2λ2), which does in deed give a rep- resentation of the Dirac delta-distribution for λ→ 0as it converges to zero every- where but for x=0and is normalised for every λ.
that gives the original δ-correlator (see eq. (4)) in the limit λ→0. So, we
are really using the following replica Hamiltonian in Fourier space
Hrep= 1 2L
∑ ∑
k (κk 4+ µ)z2α− βg 2 2∑
α,β L Z 0 dx f((zα−zβ)2). (6)We use Feynman variation[63, 64] in replica space with the one kind of trial Hamiltonian we are confident will lead us to solvable realms: a Gaussian functional, i.e. one that is quadratic in the fields zα, HV =
(2L)−1∑
k∑α,βzαG−1αβzβ with Gαβ−1 = (κk4+µ)δαβ+σαβ and the self-
energy matrix σ providing variational parameters. Extremising43the free
43In standard statistical physics the varia-
tional estimate is an upper bound to the free energy, which means that the varia- tional free energy has to be minimised. However, we are interested in the limit of vanishing particle number, n→0, which de- mands that the variational free energy has to be maximised.
energy estimate44 nF ≡ FV+ hHrep− HViV with respect to the self-energy
44All quantities that are subscriptedVare
to be computed usingHV.
for the self-energy matrix [65] σαγ= −∑α06=ασαα0 α=β −2βg2˜f0R dk 2πβ(Gαα+Gγγ−2Gαγ α6=γ with ˜fλ(x) ≡ R dy√2π−1e−y2/2fλ(y2x) = √ 2π−1√λ2+x−1.
Following Mezard and Parisi we choose a one-step hierarchical replica
symmetry breaking Ansatz for σ. In d=1, the variation45of the free energy 45We refer to our earlier in-detail presenta-
tion in Ref. 4 for explanations, the actual calculation and the definition of the vari- ables used here.
estimate yields two self-consistent equations
uc∝ S1(Σ)(λ2+S1(Σ))−1 and Σ ∝(β20g16κ3u20c )−1
for ucand Σ, where we omitted numerical constants. We are not able to give
a closed solution, because combining these two equations, we find that the solutions would require finding the root of a polynomial of degree 15 which in general cannot be expressed in terms of radicals,
Σ1/20∼ λ
2+
Σ−3/4 Σ−3/4
∼λ2Σ15/20+const.
Nonetheless, using the condition uc<1, we can show that there is no solu-
tion unless the potential strength g and correlation length λ are above finite values.
Figure 17: Matrix with one-step breaking.
Within the equally marked areas the ma- trix elements are identical. This is the first step to the iterator hierarchical breaking of replica symmetry[66]. This figure is taken from and the reader for more explanation referred to Ref. 4.
Although variation in replica space is known[65] to fail in reproducing the exact solution[60] for the problem of the DL in disorder in finite di- mensions, we interpret this as an indication for the existence of a critical disorder strength or, correspondingly, a critical temperature. The variational replica approach leads, however, to the thermal roughness exponent also in the low-temperature phase, which should come to little surprise as there is Gaussian energy statistics involved just as in the thermal case without (relevant) disorder, cp. Fig. 33 on page 54.