Biomedical subjects

Pierre Le Doussal

Publications and source records attributed to Pierre Le Doussal.

13 recordsLinked to original sources

PubMed2006

Can nonlinear elasticity explain contact-line roughness at depinning?

We examine whether cubic nonlinearities, allowed by symmetry in the elastic energy of a contact line, may result in a different universality class at depinning. Standard linear elasticity predicts a roughness exponent zeta = 1/3 (one loop), zeta = 0.388 +/- 0.002 (numerics) while experiments give zeta approximately = 0.5. Within functional renormalization group methods we find that a nonlocal Kardar-Parisi-Zhang-type term is generated at depinning and grows under coarse graining. A fixed point with zeta approximately = 0.45 (one loop) is identified, showing that large enough cubic terms increase the roughness. This fixed point is unstable, revealing a rough strong-coupling phase. Experimental study of contact angles theta near pi/2, where cubic terms in the energy vanish, is suggested.

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PubMed2005

Two-loop functional renormalization for elastic manifolds pinned by disorder in N dimensions.

We study elastic manifolds in an N -dimensional random potential using a functional renormalization group. We extend to N>1 our previous construction of a field theory renormalizable to two loops. For isotropic disorder with O (N) symmetry we obtain the fixed point and roughness exponent to next order in epsilon=4-d , where d is the internal dimension of the manifold. Extrapolation to the directed polymer limit d=1 allows some handle on the strong coupling phase of the equivalent N -dimensional Kardar-Parisi-Zhang growth equation, and eventually suggests an upper critical dimension d(u) approximately 2.5.

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PubMed2004

Aging in the glass phase of a two-dimensional random periodic elastic system.

Using the renormalization group method we investigate the nonequilibrium relaxation of the (Cardy-Ostlund) 2D random sine-Gordon model, which describes pinned arrays of lines. Its statics exhibit a marginal (theta = 0) glass phase for T < Tg described by a line of fixed points. We obtain the universal scaling functions for two-time dynamical response and correlations near Tg for various initial conditions, as well as the autocorrelation exponent. The fluctuation dissipation ratio is found to be nontrivial and continuously dependent on T.

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PubMed2004

Broad relaxation spectrum and the field theory of glassy dynamics for pinned elastic systems.

We study thermally activated, low-temperature equilibrium dynamics of elastic systems pinned by disorder using one loop functional renormalization group (FRG). Through a series of increasingly complete approximations, we investigate how the field theory reveals the glassy nature of the dynamics, in particular divergent barriers and barrier distributions controling the spectrum of relaxation times. First, we naively assume a single relaxation time tau(k) for each wave vector k, leading to analytical expressions for equilibrium dynamical response and correlations. These exhibit two distinct scaling regimes (scaling variables T k(theta) ln t and t/ tau(k), respectively, with T the temperature, theta the energy fluctuation exponent, and tau(k) approximately e(c k(-theta) /T) ) and are easily extended to quasiequilibrium and aging regimes. A careful study of the dynamical operators encoding for fluctuations of the relaxation times shows that this first approach is unsatisfactory. A second stage of approximation including these fluctuations, based on a truncation of the dynamical effective action to a random friction model, yields a size (L) dependent log-normal distribution of relaxation times (effective barriers centered around Ltheta and of fluctuations approximately L(theta/2) ) and some procedure to estimate dynamical scaling functions. Finally, we study the full structure of the running dynamical effective action within the field theory. We find that relaxation time distributions are nontrivial (broad but not log normal) and encoded in a closed hierarchy of FRG equations divided into levels p=0,1, em leader, corresponding to vertices proportional to the pth power of frequency omega(p). We show how each level p can be solved independently of higher ones, the lowest one (p=0) comprising the statics. A thermal boundary layer ansatz (TBLA) appears as a consistent solution. It extends the one discovered in the statics which was shown to embody droplet thermal fluctuations. Although perturbative control remains a challenge, the structure of the dynamical TBLA which encodes barrier distributions opens the way for deeper understanding of the field theory approach to glasses.

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PubMed2004

Functional renormalization group and the field theory of disordered elastic systems.

We study elastic systems, such as interfaces or lattices, pinned by quenched disorder. To escape triviality as a result of "dimensional reduction," we use the functional renormalization group. Difficulties arise in the calculation of the renormalization group functions beyond one-loop order. Even worse, observables such as the two-point correlation function exhibit the same problem already at one-loop order. These difficulties are due to the nonanalyticity of the renormalized disorder correlator at zero temperature, which is inherent to the physics beyond the Larkin length, characterized by many metastable states. As a result, two-loop diagrams, which involve derivatives of the disorder correlator at the nonanalytic point, are naively "ambiguous." We examine several routes out of this dilemma, which lead to a unique renormalizable field theory at two-loop order. It is also the only theory consistent with the potentiality of the problem. The beta function differs from previous work and the one at depinning by novel "anomalous terms." For interfaces and random-bond disorder we find a roughness exponent zeta=0.208 298 04epsilon+0.006 858epsilon(2), epsilon=4-d. For random-field disorder we find zeta=epsilon/3 and compute universal amplitudes to order O(epsilon(2)). For periodic systems we evaluate the universal amplitude of the two-point function. We also clarify the dependence of universal amplitudes on the boundary conditions at large scale. All predictions are in good agreement with numerical and exact results and are an improvement over one loop. Finally we calculate higher correlation functions, which turn out to be equivalent to those at depinning to leading order in epsilon.

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PubMed2003

Higher correlations, universal distributions, and finite size scaling in the field theory of depinning.

Recently we constructed a renormalizable field theory up to two loops for the quasistatic depinning of elastic manifolds in a disordered environment. Here we explore further properties of the theory. We show how higher correlation functions of the displacement field can be computed. Drastic simplifications occur, unveiling much simpler diagrammatic rules than anticipated. This is applied to the universal scaled width distribution. The expansion in d=4-epsilon predicts that the scaled distribution coincides to the lowest orders with the one for a Gaussian theory with propagator G(q)=1/q(d+2 zeta), zeta being the roughness exponent. The deviations from this Gaussian result are small and involve higher correlation functions, which are computed here for different boundary conditions. Other universal quantities are defined and evaluated: We perform a general analysis of the stability of the fixed point. We find that the correction-to-scaling exponent is omega=-epsilon and not -epsilon/3 as used in the analysis of some simulations. A more detailed study of the upper critical dimension is given, where the roughness of interfaces grows as a power of a logarithm instead of a pure power.

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PubMed2003

Exact multilocal renormalization of the effective action: application to the random sine Gordon model statics and nonequilibrium dynamics.

We extend the exact multilocal renormalization group (RG) method to study the flow of the effective action functional. This important physical quantity satisfies an exact RG equation which is then expanded in multilocal components. Integrating the nonlocal parts yields a closed exact RG equation for the local part, to a given order in the local part. The method is illustrated on the O(N) model by straightforwardly recovering the eta exponent and scaling functions. Then it is applied to study the glass phase of the Cardy-Ostlund, random phase sine Gordon model near the glass transition temperature. The static correlations and equilibrium dynamical exponent z are recovered and several results are obtained, such as the equilibrium two-point scaling functions. The nonequilibrium, finite momentum, two-time t,t' response and correlations are computed. They are shown to exhibit scaling forms, characterized by exponents lambda(R) not equal lambda(C), as well as universal scaling functions that we compute. The fluctuation dissipation ratio is found to be nontrivial and of the form X[q(z)(t-t'),t/t']. Analogies and differences with pure critical models are discussed.

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PubMed2003

Universal interface width distributions at the depinning threshold.

We compute the probability distribution of the interface width at the depinning threshold, using recent powerful algorithms. It confirms the universality classes found previously. In all cases, the distribution is surprisingly well approximated by a generalized Gaussian theory of independent modes which decay with a characteristic propagator G(q)=1/q(d+2zeta); zeta, the roughness exponent, is computed independently. A functional renormalization analysis explains this result and allows one to compute the small deviations, i.e., a universal kurtosis ratio, in agreement with numerics. We stress the importance of the Gaussian theory to interpret numerical data and experiments.

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PubMed2003

Specific heat of classical disordered elastic systems.

We study the thermodynamics of disordered elastic systems, applied to vortex lattices in the Bragg glass phase. Using the replica variational method we compute the specific heat of pinned vortons in the classical limit. We find that the contribution of disorder is positive, linear at low temperature, and exhibits a maximum. It is found to be important compared to other contributions, e.g., core electrons, mean field, and nonlinear elasticity that we evaluate. The contribution of droplets is subdominant at weak disorder in d=3.

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PubMed2003

Variable-range hopping and quantum creep in one dimension.

We study the quantum nonlinear response to an applied electric field E of a one-dimensional pinned charge-density wave or Luttinger liquid in the presence of disorder. From an explicit construction of low-lying metastable states and of bounce instanton solutions between them, we demonstrate quantum creep v=e(-c/E(1/2)) as well as a sharp crossover at E=E(*) towards a linear response form consistent with variable-range hopping arguments, but dependent only on electronic degrees of freedom.

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PubMed2003

Functional renormalization group for anisotropic depinning and relation to branching processes.

Using the functional renormalization group, we study the depinning of elastic objects in presence of anisotropy. We explicitly demonstrate how the Kardar-Parisi-Zhang (KPZ) term is always generated, even in the limit of vanishing velocity, except where excluded by symmetry. This mechanism has two steps. First a nonanalytic disorder-distribution is generated under renormalization beyond the Larkin length. This nonanalyticity then generates the KPZ term. We compute the beta function to one loop taking properly into account the nonanalyticity. This gives rise to additional terms, missed in earlier studies. A crucial question is whether the nonrenormalization of the KPZ coupling found at 1-loop order extends beyond the leading one. Using a Cole-Hopf-transformed theory we argue that it is indeed uncorrected to all orders. The resulting flow equations describe a variety of physical situations: We study manifolds in periodic disorder, relevant for charge density waves, as well as in nonperiodic disorder. Further the elasticity of the manifold can either be short range (SR) or long range (LR). A careful analysis of the flow yields several nontrivial fixed points. All these fixed points are transient since they possess one unstable direction towards a runaway flow, which leaves open the question of the upper critical dimension. The runaway flow is dominated by a Landau-ghost mode. For LR elasticity, relevant for contact line depinning, we show that there are two phases depending on the strength of the KPZ coupling. For SR elasticity, using the Cole-Hopf transformed theory we identify a nontrivial 3-dimensional subspace which is invariant to all orders and contains all above fixed points as well as the Landau mode. It belongs to a class of theories which describe branching and reaction-diffusion processes, of which some have been mapped onto directed percolation.

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PubMed2002

Functional renormalization group at large N for disordered systems.

We introduce a method, based on an exact calculation of the effective action at large N, to bridge the gap between mean-field theory and renormalization in complex systems. We apply it to a d-dimensional manifold in a random potential for large embedding space dimension N. This yields a functional renormalization group equation valid for any d, which contains both the O(epsilon=4-d) results of Balents-Fisher and some of the nontrivial results of the Mezard-Parisi solution, thus shedding light on both. Corrections are computed at order O(1/N). Applications to the Kardar-Parisi-Zhang growth model, random field, and mode coupling in glasses are mentioned.

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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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