Biomedical subjects

Cécile Monthus

Publications and source records attributed to Cécile Monthus.

7 recordsLinked to original sources

PubMed2006

Freezing transition of the directed polymer in a 1 + d random medium: location of the critical temperature and unusual critical properties.

In dimension d > or =3, the directed polymer in a random medium undergoes a phase transition between a free phase at high temperature and a low-temperature disorder-dominated phase. For the latter phase, Fisher and Huse have proposed a droplet theory based on the scaling of the free-energy fluctuations Delta F(l) approximately l theta at scale l. On the other hand, in related growth models belonging to the Kardar-Parisi-Zhang universality class, Forrest and Tang have found that the height-height correlation function is logarithmic at the transition. For the directed polymer model at criticality, this translates into logarithmic free-energy fluctuations Delta FTc(l) approximately (lnl)sigma with sigma = 1/2. In this paper, we propose a droplet scaling analysis exactly at criticality based on this logarithmic scaling. Our main conclusion is that the typical correlation length xi(T) of the low-temperature phase diverges as ln xi(t) approximately [-ln(Tc-T)]1/sigma approximately [-ln(Tc-T)]2, instead of the usual power law xi(T) approximately (Tc-T)-v. Furthermore, the logarithmic dependence of Delta FTc(l) leads to the conclusion that the critical temperature Tc actually coincides with the explicit upper bound T2 derived by Derrida and co-workers, where T2 corresponds to the temperature below which the ratio Z 2 L/(ZL)2 diverges exponentially in L. Finally, since the Fisher-Huse droplet theory was initially introduced for the spin-glass phase, we briefly mention the similarities with and differences from the directed polymer model. If one speculates that the free energy of droplet excitations for spin glasses is also logarithmic at Tc, one obtains a logarithmic decay for the mean square correlation function at criticality, C2(r) approximately 1/(ln r)sigma, instead of the usual power law l/rd-2+eta.

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PubMed2006

Statistics of low energy excitations for the directed polymer in a random medium.

We consider a directed polymer of length L in a random medium of space dimension d = 1,2,3. The statistics of low energy excitations as a function of their size l is numerically evaluated. These excitations can be divided into bulk and boundary excitations, with respective densities rho(bulk)(L) (E = 0,l) and rho(boundary)(L)(E=0,l). We find that both densities follow the scaling behavior rho(bulk, boundary)(L)(E = 0,l)=L(-1-theta)(d)R(bulk,boundary)(x = l/L), where theta(d) is the exponent governing the energy fluctuations at zero temperature (with the well-known exact value theta(1)= 1/3 in one dimension). In the limit x = l/L --> 0, both scaling functions R(bulk)(x) and R(boundary)(x) behave as R(bulk,boundary)(x) approximately x(-1-theta)(d), leading to the droplet power law rho(bulk, boundary)(L) (E = 0,l) approximately l(-1-theta)(d) in the regime 1 << l << L. Beyond their common singularity near x --> 0, the two scaling functions R(bulk,boundary)(x) are very different: whereas R(bulk)(x) decays monotonically for 0 < x < 1, the function R(boundary)(x) first decays for 0 < x < x(min), then grows for x(min) < x < 1, and finally presents a power law singularity R(boundary)(x) approximately (1-x)(-sigma)(d) near x -->1. The density of excitations of length l = L accordingly decays as rho(boundary)(L)(E = 0,l = L) approximately L(-lambda)(d) where gamma(d) = 1+ theta(d) - lambda(d). We obtain lambda(1) approximately 10.67, lambda(2) = 0.53, and lambda(3) approximately 0.39, suggesting the possible relation lambda(d) = 2theta(d).

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PubMed2004

Directed polymers and interfaces in random media: free-energy optimization via confinement in a wandering tube.

We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that the optimal strategy for the polymer in dimension 1+d with 0 1/2 for the wandering of the best favorable tube available. The corresponding free energy then scales as F approximately Lomega with omega=2nu-1 and the left tail of the probability distribution involves a stretched exponential of exponent eta=(4-d)/2. These results generalize the well known exact exponents nu=2/3, omega=1/3, and eta=3/2 in d=1, where the subleading transverse length R(S) approximately L(1/3) is known as the typical distance between two replicas in the Bethe ansatz wave function. We then extend our approach to correlated disorder in transverse directions with exponent alpha and/or to manifolds in dimension D+d= d(t) with 0 0 ). In particular, for an interface of dimension ( d(t) -1) in a space of total dimension 5/3< d(t) <3 with random-bond disorder, our approach yields the confinement exponent nu(S) =( d(t) -1)(3- d(t) )/(5 d(t) -7). Finally, we study the exponents in the presence of an algebraic tail 1/ V1+micro in the disorder distribution, and obtain various regimes in the (micro,d) plane.

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PubMed2004

Nonlinear response of the trap model in the aging regime: exact results in the strong-disorder limit.

We study the dynamics in the one-dimensional disordered trap model with a broad distribution of trapping times p(tau) approximately 1/tau(1+mu), when an external force is applied from the very beginning at t=0, or only after a waiting time t(w), in the linear as well as in the nonlinear response regime. Using a real-space renormalization procedure that becomes exact in the limit of strong disorder mu-->0, we obtain explicit results for many observables, such as the diffusion front, the mean position, the thermal width, the localization parameters and the two-particle correlation function. In particular, the scaling functions for these observables give access to the complete interpolation between the unbiased case and the directed case. Finally, we discuss in detail the various regimes that exist for the average position in terms of the two times and the external field.

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PubMed2003

Anomalous diffusion, localization, aging, and subaging effects in trap models at very low temperature.

We study in detail the dynamics of the one-dimensional symmetric trap model via a real-space renormalization procedure which becomes exact in the limit of zero temperature. In this limit, the diffusion front in each sample consists of two delta peaks, which are completely out of equilibrium with each other. The statistics of the positions and weights of these delta peaks over the samples allows to obtain explicit results for all observables in the limit T-->0. We first compute disorder averages of one-time observables, such as the diffusion front, the thermal width, the localization parameters, the two-particle correlation function, and the generating function of thermal cumulants of the position. We then study aging and subaging effects: our approach reproduces very simply the two different aging exponents and yields explicit forms for scaling functions of the various two-time correlations. We also extend the real-space renormalization group method to include systematic corrections to the previous zero temperature procedure via a series expansion in T. We then consider the generalized trap model with parameter alpha in [0,1] and obtain that the large scale effective model at low temperature does not depend on alpha in any dimension, so that the only observables sensitive to alpha are those that measure the "local persistence," such as the probability to remain exactly in the same trap during a time interval. Finally, we extend our approach at a scaling level for the trap model in d=2 and obtain the two relevant time scales for aging properties.

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PubMed2003

Localization properties of the anomalous diffusion phase in the directed trap model and in the Sinai diffusion with a bias.

We study the localization properties of the anomalous diffusion phase x approximately t(micro) with 0 0 and corresponds to the Golosov localization. For finite micro we thus generalize the usual real space renormalization method to allow for the spreading of the thermal packet over many renormalized valleys. Our construction allows one to compute exact series expansions in micro for all observables: to compute observables at order micro (n), it is sufficient to consider in each sample a spreading of the thermal packet onto at most (1+n) traps. So our approach provides a description of the structure of the thermal packet sample by sample, and a full statistical characterization of the important traps at a given order in micro. For the directed trap model, we show explicitly up to order micro(2) how to recover the exact expressions for the diffusion front, the thermal width, and the localization parameter Y2. We then use our method to derive exact results for the localization parameters Y(k) for arbitrary k, the correlation function of two particles, and the generating function of thermal cumulants. We then explain how these results apply to the Sinai diffusion with bias by deriving the quantitative mapping between the large-scale renormalized descriptions of the two models. Finally we study the internal structure of the effective "traps" for the Sinai model via path-integral methods.

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PubMed2002

Localization of thermal packets and metastable states in the Sinai model.

We consider the Sinai model describing a particle diffusing in a one-dimensional random force field. As shown by Golosov, this model exhibits a strong localization phenomenon for the thermal packet: all thermal trajectories starting from the same initial condition in the same sample remain within a finite distance of each other even in the limit of infinite time. More precisely, he has proved that the disorder average P(t)(y) of the distribution of the relative distance y=x(t)-m(t) with respect to the (disorder-dependent) most probable position m(t), converges in the limit t--> infinity, towards a distribution P(G)(y) defined as a functional of two independent Bessel processes. In this paper, we revisit this question of the localization of the thermal packet. We first generalize the result of Golosov by computing explicitly the joint distribution P( infinity )(y,u) of relative position y=x(t)-m(t) and relative energy u=U(x(t))-U(m(t)) for the thermal packet. Next, we compute the localization parameters Y(k), representing the disorder-averaged probabilities that k particles of the thermal packet are at the same place in the infinite-time limit, and the correlation function C(l) representing the disorder-averaged probability density that two particles of the thermal packet are at a distance l from each other. We, moreover, prove that our results for Y(k) and C(l) exactly coincide with the thermodynamic limit L--> infinity of the analog quantities computed for independent particles at equilibrium in a finite sample of length L. So even if the Sinai dynamics on the infinite line is always out-of-equilibrium since it consists in jumps in deeper and deeper wells, the particles of the same thermal packet can nevertheless be considered asymptotically as if they were at thermal equilibrium in a Brownian potential. Finally, we discuss the properties of the finite-time metastable states that are responsible for the localization phenomenon and compare with the general theory of metastable states in glassy systems, in particular as a test of the Edwards conjecture.

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