laboratoire de physique statistique
laboratoire de physique statistique




On the Entropy of Protein Families - Barton, John P. and Chakraborty, Arup K. and Cocco, Simona and Jacquin, Hugo and Monasson, Remi

Abstract : Proteins are essential components of living systems, capable of performing a huge variety of tasks at the molecular level, such as recognition, signalling, copy, transport, ... The protein sequences realizing a given function may largely vary across organisms, giving rise to a protein family. Here, we estimate the entropy of those families based on different approaches, including Hidden Markov Models used for protein databases and inferred statistical models reproducing the low-order (1- and 2-point) statistics of multi-sequence alignments. We also compute the entropic cost, that is, the loss in entropy resulting from a constraint acting on the protein, such as the mutation of one particular amino-acid on a specific site, and relate this notion to the escape probability of the HIV virus. The case of lattice proteins, for which the entropy can be computed exactly, allows us to provide another illustration of the concept of cost, due to the competition of different folds. The relevance of the entropy in relation to directed evolution experiments is stressed.
JSP Special Issue on Information Processing in Living Systems - Mora, Thierry and Peliti, Luca and Rivoire, Olivier
Flocking and Turning: a New Model for Self-organized Collective Motion - Cavagna, Andrea and Del Castello, Lorenzo and Giardina, Irene and Grigera, Tomas and Jelic, Asja and Melillo, Stefania and Mora, Thierry and Parisi, Leonardo and Silvestri, Edmondo and Viale, Massimiliano and Walczak, Aleksandra M.

Abstract : Birds in a flock move in a correlated way, resulting in large polarization of velocities. A good understanding of this collective behavior exists for linear motion of the flock. Yet observing actual birds, the center of mass of the group often turns giving rise to more complicated dynamics, still keeping strong polarization of the flock. Here we propose novel dynamical equations for the collective motion of polarized animal groups that account for correlated turning including solely social forces. We exploit rotational symmetries and conservation laws of the problem to formulate a theory in terms of generalized coordinates of motion for the velocity directions akin to a Hamiltonian formulation for rotations. We explicitly derive the correspondence between this formulation and the dynamics of the individual velocities, thus obtaining a new model of collective motion. In the appropriate overdamped limit we recover the well-known Vicsek model, which dissipates rotational information and does not allow for polarized turns. Although the new model has its most vivid success in describing turning groups, its dynamics is intrinsically different from previous ones in a wide dynamical regime, while reducing to the hydrodynamic description of Toner and Tu at very large length-scales. The derived framework is therefore general and it may describe the collective motion of any strongly polarized active matter system.
Learning Probabilities From Random Observables in High Dimensions: The Maximum Entropy Distribution and Others - Obuchi, Tomoyuki and Cocco, Simona and Monasson, Remi

Abstract : We consider the problem of learning a target probability distribution over a set of N binary variables from the knowledge of the expectation values (with this target distribution) of M observables, drawn uniformly at random. The space of all probability distributions compatible with these M expectation values within some fixed accuracy, called version space, is studied. We introduce a biased measure over the version space, which gives a boost increasing exponentially with the entropy of the distributions and with an arbitrary inverse `temperature' . The choice of allows us to interpolate smoothly between the unbiased measure over all distributions in the version space () and the pointwise measure concentrated at the maximum entropy distribution (). Using the replica method we compute the volume of the version space and other quantities of interest, such as the distance R between the target distribution and the center-of-mass distribution over the version space, as functions of and for large N. Phase transitions at critical values of are found, corresponding to qualitative improvements in the learning of the target distribution and to the decrease of the distance R. However, for fixed , the distance R does not vary with , which means that the maximum entropy distribution is not closer to the target distribution than any other distribution compatible with the observable values. Our results are confirmed by Monte Carlo sampling of the version space for small system sizes ().
1/f(alpha) Low Frequency Fluctuations in Turbulent Flows Transitions with Heavy-Tailed Distributed Interevent Durations - Herault, J. and Petrelis, F. and Fauve, S.

Abstract : We report the experimental observation of low frequency fluctuations with a spectrum varying as in three different turbulent flow configurations: the large scale velocity driven by a two-dimensional turbulent flow, the magnetic field generated by a turbulent swirling flow of liquid sodium and the pressure fluctuations due to vorticity filaments in a swirling flow. For these three systems, noise is shown to result from the dynamics of coherent structures that display transitions between a small number of states. The interevent duration is distributed as a power law. The exponent of this power law and the nature of the dynamics (transition between symmetric states or asymmetric ones) select the exponent of the fluctuations.
Log-periodic Critical Amplitudes: A Perturbative Approach - Derrida, Bernard and Giacomin, Giambattista

Abstract : Log-periodic amplitudes appear in the critical behavior of a large class of systems, in particular when a discrete scale invariance is present. Here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial. In this case, the log-periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated.
The Depinning Transition in Presence of Disorder: A Toy Model - Derrida, Bernard and Retaux, Martin

Abstract : We introduce a toy model, which represents a simplified version of the problem of the depinning transition in the limit of strong disorder. This toy model can be formulated as a simple renormalization transformation for the probability distribution of a single real variable. For this toy model, the critical line is known exactly in one particular case and it can be calculated perturbatively in the general case. One can also show that, at the transition, there is no fixed distribution accessible by renormalization which corresponds to a disordered fixed point. Instead, both our numerical and analytic approaches indicate a transition of infinite order (of the Berezinskii-Kosterlitz-Thouless type). We give numerical evidence that this infinite order transition persists for the problem of the depinning transition with disorder on the hierarchical lattice.
Stochastic Perturbation of Integrable Systems: A Window to Weakly Chaotic Systems - Khanh-Dang Nguyen Thu Lam and Kurchan, Jorge

Abstract : Integrable non-linear Hamiltonian systems perturbed by additive noise develop a Lyapunov instability, and are hence chaotic, for any amplitude of the perturbation. This phenomenon is related, but distinct, from Taylor's diffusion in hydrodynamics. We develop expressions for the Lyapunov exponents for the cases of white and colored noise. The situation described here being `multi-resonance'aEuro''by nature well beyond the Kolmogorov-Arnold-Moser regime, it offers an analytic glimpse on the regime in which many near-integrable systems, such as some planetary systems, find themselves in practice. We show with the aid of a simple example, how one may model in some cases weakly chaotic deterministic systems by a stochastically perturbed one, with good qualitative results.
Entanglement Between Demand and Supply in Markets with Bandwagon Goods - Gordon, Mirta B. and Nadal, Jean-Pierre and Denis Phan and Semeshenko, Viktoriya

Abstract : Whenever customers' choices (e.g. to buy or not a given good) depend on others choices (cases coined `positive externalities' or `bandwagon effect' in the economic literature), the demand may be multiply valued: for a same posted price, there is either a small number of buyers, or a large one-in which case one says that the customers coordinate. This leads to a dilemma for the seller: should he sell at a high price, targeting a small number of buyers, or at low price targeting a large number of buyers? In this paper we show that the interaction between demand and supply is even more complex than expected, leading to what we call the curse of coordination: the pricing strategy for the seller which aimed at maximizing his profit corresponds to posting a price which, not only assumes that the customers will coordinate, but also lies very near the critical price value at which such high demand no more exists. This is obtained by the detailed mathematical analysis of a particular model formally related to the Random Field Ising Model and to a model introduced in social sciences by T.C. Schelling in the 70's.
Sustainable Development and Spatial Inhomogeneities - Weisbuch, Gerard

Abstract : Historical data, theory and computer simulations support a connection between growth and economic inequality. Our present world with large regional differences in economic activity is a result of fast economic growth during the last two centuries. Because of limits to growth we might expect a future world to develop differently with far less growth. The question that we here address is: ``Would a world with a sustainable economy be less unequal?'' We then develop integrated spatial economic models based on limited resources consumption and technical knowledge accumulation and study them by the way of computer simulations. When the only coupling between world regions is diffusion we do not observe any spatial unequality. By contrast, highly localized economic activities are maintained by global market mechanisms. Structures sizes are determined by transportation costs. Wide distributions of capital and production are also predicted in this regime.
Finite Size Corrections to the Large Deviation Function of the Density in the One Dimensional Symmetric Simple Exclusion Process - Derrida, Bernard and Retaux, Martin

Abstract : The symmetric simple exclusion process is one of the simplest out-of-equilibrium systems for which the steady state is known. Its large deviation functional of the density has been computed in the past both by microscopic and macroscopic approaches. Here we obtain the leading finite size correction to this large deviation functional. The result is compared to the similar corrections for equilibrium systems.
Adaptive Cluster Expansion for the Inverse Ising Problem: Convergence, Algorithm and Tests - Cocco, S. and Monasson, R.

Abstract : We present a procedure to solve the inverse Ising problem, that is, to find the interactions between a set of binary variables from the measure of their equilibrium correlations. The method consists in constructing and selecting specific clusters of spins, based on their contributions to the cross-entropy of the Ising model. Small contributions are discarded to avoid overfitting and to make the computation tractable. The properties of the cluster expansion and its performances on synthetic data are studied. To make the implementation easier we give the pseudo-code of the algorithm.
Critical Exponents in Zero Dimensions - Alexakis, A. and Petrelis, F.

Abstract : In the vicinity of the onset of an instability, we investigate the effect of colored multiplicative noise on the scaling of the moments of the unstable mode amplitude. We introduce a family of zero dimensional models for which we can calculate the exact value of the critical exponents beta (m) for all the moments. The results are obtained through asymptotic expansions that use the distance to onset as a small parameter. The examined family displays a variety of behaviors of the critical exponents that includes anomalous exponents: exponents that differ from the deterministic (mean-field) prediction, and multiscaling: non-linear dependence of the exponents on the order of the moment.
A Branching Random Walk Seen from the Tip - Brunet, Eric and Derrida, Bernard

Abstract : We show that all the time-dependent statistical properties of the rightmost points of a branching Brownian motion can be extracted from the traveling wave solutions of the Fisher-KPP equation. The distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the Fisher-KPP traveling waves when the initial condition is modified. The limiting measure exhibits the surprising property of superposability: the statistical properties of the distances between the rightmost points of the union of two realizations of the branching Brownian motion shifted by arbitrary amounts are the same as those of a single realization. We discuss the extension of our results to more general branching random walks.
Phase Fluctuations in the ABC Model - Bodineau, T. and Derrida, B.

Abstract : We analyze the fluctuations of the steady state profiles in the modulated phase of the ABC model. For a system of L sites, the steady state profiles move on a microscopic time scale of order L (3). The variance of their displacement is computed in terms of the macroscopic steady state profiles by using fluctuating hydrodynamics and large deviations. Our analytical prediction for this variance is confirmed by the results of numerical simulations.
Optimal Stability of Advection-Diffusion Lattice Boltzmann Models with Two Relaxation Times for Positive/Negative Equilibrium - Ginzburg, Irina and d'Humieres, Dominique and Kuzmin, Alexander

Abstract : Despite the growing popularity of Lattice Boltzmann schemes for describing multi-dimensional flow and transport governed by non-linear (anisotropic) advection-diffusion equations, there are very few analytical results on their stability, even for the isotropic linear equation. In this paper, the optimal two-relaxation-time (OTRT) model is defined, along with necessary and sufficient (easy to use) von Neumann stability conditions for a very general anisotropic advection-diffusion equilibrium, in one to three dimensions, with or without numerical diffusion. Quite remarkably, the OTRT stability bounds are the same for any Peclet number and they are defined by the adjustable equilibrium parameters. Such optimal stability is reached owing to the free (''kinetic'') relaxation parameter. Furthermore, the sufficient stability bounds tolerate negative equilibrium functions (the distribution divided by the local mass), often labeled as ``unphysical''. We prove that the non-negativity condition is (i) a sufficient stability condition of the TRT model with any eigenvalues for the pure diffusion equation, (ii) a sufficient stability condition of its OTRT and BGK/SRT sub-classes, for any linear anisotropic advection-diffusion equation, and (iii) unnecessarily more restrictive for any Peclet number than the optimal sufficient conditions. Adequate choices of the two relaxation rates and the free-tunable equilibrium parameters make the OTRT sub-class more efficient than the BGK one, at least in the advection-dominant regime, and allow larger time steps than known criteria of the forward time central finite-difference schemes (FTCS/MFTCS) for both, advection and diffusion dominant regimes.
A Diffusive System Driven by a Battery or by a Smoothly Varying Field - Bodineau, T. and Derrida, B. and Lebowitz, J. L.

Abstract : We consider the steady state of a one dimensional diffusive system, such as the symmetric simple exclusion process (SSEP) on a ring, driven by a battery at the origin or by a smoothly varying field along the ring. The battery appears as the limiting case of a smoothly varying field, when the field becomes a delta function at the origin. We find that in the scaling limit the long range pair correlation functions of the system driven by a battery are very different from the ones known in the steady state of the SSEP maintained out of equilibrium by contact with two reservoirs, even when the steady state density profiles are identical in both models.
Anomalous Fourier's Law and Long Range Correlations in a 1D Non-momentum Conserving Mechanical Model - Gerschenfeld, A. and Derrida, B. and Lebowitz, J. L.

Abstract : We study by means of numerical simulations the velocity reversal model, a one-dimensional mechanical model of heat transport introduced in 1985 by Ianiro and Lebowitz. Our numerical results indicate that this model, which does not conserve momentum, exhibits nevertheless an anomalous Fourier's law similar to the ones previously observed in momentum-conserving models. This disagrees with what can be expected by solving the Boltzmann equation (BE) for this system. The pair correlation velocity field also looks very different from the correlations usually seen in diffusive systems, and shares some similarity with those of momentum-conserving heat transport models.
Current Fluctuations of the One Dimensional Symmetric Simple Exclusion Process with Step Initial Condition - Derrida, Bernard and Gerschenfeld, Antoine

Abstract : For the symmetric simple exclusion process on an infinite line, we calculate exactly the fluctuations of the integrated current Q(t) during time t through the origin when, in the initial condition, the sites are occupied with density rho(a) on the negative axis and with density rho(b) on the positive axis. All the cumulants of Q(t) grow like root t. In the range where Q(t) similar to root t, the decay exp[-Q(t)(3)/t] of the distribution of Q(t) is non-Gaussian. Our results are obtained using the Bethe ansatz and several identities derived recently by Tracy and Widom for exclusion processes on the infinite line.
Current Fluctuations in One Dimensional Diffusive Systems with a Step Initial Density Profile - Derrida, Bernard and Gerschenfeld, Antoine

Abstract : We show how to apply the macroscopic fluctuation theory (MFT) of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim to study the current fluctuations of diffusive systems with a step initial condition. We argue that one has to distinguish between two ways of averaging (the annealed and the quenched cases) depending on whether we let the initial condition fluctuate or not. Although the initial condition is not a steady state, the distribution of the current satisfies a symmetry very reminiscent of the fluctuation theorem. We show how the equations of the MFT can be solved in the case of non-interacting particles. The symmetry of these equations can be used to deduce the distribution of the current for several other models, from its knowledge (Derrida and Gerschenfeld in J. Stat. Phys. 136, 1-15, 2009) for the symmetric simple exclusion process. In the range where the integrated current Q(t) similar to root t, we show that the non-Gaussian decay exp [-Q (t) (3) /t] of the distribution of Q(t) is generic.