laboratoire de physique statistique
laboratoire de physique statistique




Population aging through survival of the fit and stable - Brotto, Tommaso and Bunin, Guy and Kurchan, Jorge

Abstract : Motivated by the wide range of known self-replicating systems, some far from genetics, we study a system composed by individuals having an internal dynamics with many possible states that are partially stable, with varying mutation rates. Individuals reproduce and die with a rate that is a property of each state, not necessarily related to its stability, and the offspring is born on the parent's state. The total population is limited by resources or space, as for example in a chemostat or a Petri dish. Our aim is to show that mutation rate and fitness become more correlated, even if they are completely uncorrelated for an isolated individual, underlining the fact that the interaction induced by limitation of resources is by itself effcient for generating collective effects.
Approximate message passing with restricted Boltzmann machine priors - Tramel, Eric W. and Dremeau, Angelique and Krzakala, Florent

Abstract : Approximate message passing (AMP) has been shown to be an excellent statistical approach to signal inference and compressed sensing problems. The AMP framework provides modularity in the choice of signal prior; here we propose a hierarchical form of the Gauss-Bernoulli prior which utilizes a restricted Boltzmann machine (RBM) trained on the signal support to push reconstruction performance beyond that of simple i.i.d. priors for signals whose support can be well represented by a trained binary RBM. We present and analyze two methods of RBM factorization and demonstrate how these affect signal reconstruction performance within our proposed algorithm. Finally, using the MNIST handwritten digit dataset, we show experimentally that using an RBM allows AMP to approach oracle-support performance.
Finite size corrections in the random energy model and the replica approach - Derrida, Bernard and Mottishaw, Peter

Abstract : We present a systematic and exact way of computing finite size corrections for the random energy model, in its low temperature phase. We obtain explicit (though complicated) expressions for the finite size corrections of the overlap functions. In its low temperature phase, the random energy model is known to exhibit Parisi's broken symmetry of replicas. The finite size corrections given by our exact calculation can be reproduced using replicas if we make specific assumptions about the fluctuations (with negative variances!) of the number and sizes of the blocks when replica symmetry is broken. As an alternative we show that the exact expression for the non-integer moments of the partition function can be written in terms of coupled contour integrals over what can be thought of as `complex replica numbers'. Parisi's one step replica symmetry breaking arises naturally from the saddle point of these integrals without making any ansatz or using the replica method. The fluctuations of the `complex replica numbers' near the saddle point in the imaginary direction correspond to the negative variances we observed in the replica calculation. Finally our approach allows one to see why some apparently diverging series or integrals are harmless.
Approximate message-passing with spatially coupled structured operators, with applications to compressed sensing and sparse superposition codes - Barbier, Jean and Schuelke, Christophe and Krzakala, Florent

Abstract : We study the behavior of approximate message-passing (AMP), a solver for linear sparse estimation problems such as compressed sensing, when the i.i.d matrices-for which it has been specifically designed-are replaced by structured operators, such as Fourier and Hadamard ones. We show empirically that after proper randomization, the structure of the operators does not significantly affect the performances of the solver. Furthermore, for some specially designed spatially coupled operators, this allows a computationally fast and memory efficient reconstruction in compressed sensing up to the information-theoretical limit. We also show how this approach can be applied to sparse superposition codes, allowing the AMP decoder to perform at large rates for moderate block length.
Large deviation function of a tracer position in single file diffusion - Sadhu, Tridib and Derrida, Bernard

Abstract : Diffusion of impenetrable particles in a crowded one-dimensional channel is referred as the single file diffusion. The particles do not pass each other and the displacement of each individual particle is sub-diffusive. We analyse a simple realization of this single file diffusion problem where one dimensional Brownian point particles interact only by hard-core repulsion. We show that the large deviation function which characterizes the displacement of a tracer at large time can be computed via a mapping to a problem of non-interacting Brownian particles. We confirm recently obtained results of the one time distribution of the displacement and show how to extend them to the multi-time correlations. The probability distribution of the tracer position depends on whether we take annealed or quenched averages. In the quenched case we notice an exact relation between the distribution of the tracer and the distribution of the current. This relation is in fact much more general and would be valid for arbitrary single file diffusion. It allows in particular to get the full statistics of the tracer position for the symmetric simple exclusion process (SSEP) at density 1/2 in the quenched case.
Blind sensor calibration using approximate message passing - Schuelke, Christophe and Caltagirone, Francesco and Zdeborova, Lenka

Abstract : The ubiquity of approximately sparse data has led a variety of communities to take great interest in compressed sensing algorithms. Although these are very successful and well understood for linear measurements with additive noise, applying them to real data can be problematic if imperfect sensing devices introduce deviations from this ideal signal acquisition process, caused by sensor decalibration or failure. We propose a message passing algorithm called calibration approximate message passing (Cal-AMP) that can treat a variety of such sensor-induced imperfections. In addition to deriving the general form of the algorithm, we numerically investigate two particular settings. In the first, a fraction of the sensors is faulty, giving readings unrelated to the signal. In the second, sensors are decalibrated and each one introduces a different multiplicative gain to the measurements. Cal-AMP shares the scalability of approximate message passing, allowing us to treat large sized instances of these problems, and experimentally exhibits a phase transition between domains of success and failure.
Exact theory of dense amorphous hard spheres in high dimension. III. The full replica symmetry breaking solution - Charbonneau, Patrick and Kurchan, Jorge and Parisi, Giorgio and Urbani, Pierfrancesco and Zamponi, Francesco

Abstract : In the first part of this paper, we derive the general replica equations that describe infinite-dimensional hard spheres at any level of replica symmetry breaking (RSB) and in particular in the fullRSB scheme. We show that these equations are formally very similar to the ones that have been derived for spin glass models, thus showing that the analogy between spin glasses and structural glasses conjectured by Kirkpatrick, Thirumalai and Wolynes is realized in a strong sense in the mean-field limit. We also suggest how the computation could be generalized in an approximate way to finite-dimensional hard spheres. In the second part of the paper, we discuss the solution of these equations and we derive from it a number of physical predictions. We show that, below the Gardner transition where the 1RSB solution becomes unstable, a fullRSB phase exists and we locate the boundary of the fullRSB phase. Most importantly, we show that the fullRSB solution predicts correctly that jammed packings are iso-static, and allows one to compute analytically the critical exponents associated with the jamming transition, which are missed by the 1RSB solution. We show that these predictions compare very well with numerical results.
Genealogies in simple models of evolution - Brunet, Eric and Derrida, Bernard

Abstract : We review the statistical properties of the genealogies of a few models of evolution. In the asexual case, selection leads to coalescence times which grow logarithmically with the size of the population, in contrast with the linear growth of the neutral case. Moreover for a whole class of models, the statistics of the genealogies are those of the Bolthausen-Sznitman coalescent rather than the Kingman coalescent in the neutral case. For sexual reproduction in the neutral case, the time to reach the first common ancestors for the whole population and the time for all individuals to have all their ancestors in common are also logarithmic in the population size, as predicted by Chang in 1999. We discuss how these times are modified by introducing selection in a simple way.
The simplest maximum entropy model for collective behavior in a neural network - Tkacik, Gasper and Marre, Olivier and Mora, Thierry and Amodei, Dario and Berry, II, Michael J. and Bialek, William

Abstract : Recent work emphasizes that the maximum entropy principle provides a bridge between statistical mechanics models for collective behavior in neural networks and experiments on networks of real neurons. Most of this work has focused on capturing the measured correlations among pairs of neurons. Here we suggest an alternative, constructing models that are consistent with the distribution of global network activity, i.e. the probability that K out of N cells in the network generate action potentials in the same small time bin. The inverse problem that we need to solve in constructing the model is analytically tractable, and provides a natural `thermodynamics' for the network in the limit of large N. We analyze the responses of neurons in a small patch of the retina to naturalistic stimuli, and find that the implied thermodynamics is very close to an unusual critical point, in which the entropy (in proper units) is exactly equal to the energy.
A simple method for estimating the entropy of neural activity - Berry, II, Michael J. and Tkacik, Gasper and Dubuis, Julien and Marre, Olivier and da Silveira, Rava Azeredo

Abstract : The number of possible activity patterns in a population of neurons grows exponentially with the size of the population. Typical experiments explore only a tiny fraction of the large space of possible activity patterns in the case of populations with more than 10 or 20 neurons. It is thus impossible, in this undersampled regime, to estimate the probabilities with which most of the activity patterns occur. As a result, the corresponding entropy which is a measure of the computational power of the neural population cannot be estimated directly. We propose a simple scheme for estimating the entropy in the undersampled regime, which bounds its value from both below and above. The lower bound is the usual `naive' entropy of the experimental frequencies. The upper bound results from a hybrid approximation of the entropy which makes use of the naive estimate, a maximum entropy fit, and a coverage adjustment. We apply our simple scheme to artificial data, in order to check their accuracy; we also compare its performance to those of several previously defined entropy estimators. We then apply it to actual measurements of neural activity in populations with up to 100 cells. Finally, we discuss the similarities and differences between the proposed simple estimation scheme and various earlier methods.
Ising models for neural activity inferred via selective cluster expansion: structural and coding properties - Barton, John and Cocco, Simona

Abstract : We describe the selective cluster expansion (SCE) of the entropy, a method for inferring an Ising model which describes the correlated activity of populations of neurons. We re-analyze data obtained from multielectrode recordings performed in vitro on the retina and in vivo on the prefrontal cortex. Recorded population sizes N range from N = 37 to 117 neurons. We compare the SCE method with the simplest mean field methods (corresponding to a Gaussian model) and with regularizations which favor sparse networks (L-1 norm) or penalize large couplings (L-2 norm). The network of the strongest interactions inferred via mean field methods generally agree with those obtained from SCE. Reconstruction of the sampled moments of the distributions, corresponding to neuron spiking frequencies and pairwise correlations, and the prediction of higher moments including three-cell correlations and multi-neuron firing frequencies, is more difficult than determining the large-scale structure of the interaction network, and, apart from a cortical recording in which the measured correlation indices are small, these goals are achieved with the SCE but not with mean field approaches. We also find differences in the inferred structure of retinal and cortical networks: inferred interactions tend to be more irregular and sparse for cortical data than for retinal data. This result may reflect the structure of the recording. As a consequence, the SCE is more effective for retinal data when expanding the entropy with respect to a mean field reference S - S-MF, while expansions of the entropy S alone perform better for cortical data.
Trajectory entanglement in dense granular materials - Puckett, James G. and Lechenault, Frederic and Daniels, Karen E. and Thiffeault, Jean-Luc

Abstract : The particle-scale dynamics of granular materials have commonly been characterized by the self-diffusion coefficient D. However, this measure discards the collective and topological information known to be an important characteristic of particle trajectories in dense systems. Direct measurement of the entanglement of particle space-time trajectories can be obtained via the topological braid entropy S-braid, which has previously been used to quantify mixing efficiency in fluid systems. Here, we investigate the utility of S-braid in characterizing the dynamics of a dense, driven granular material at packing densities near the static jamming point phi(J). From particle trajectories measured within a two-dimensional granular material, we typically observe that S-braid is well defined and extensive. However, for systems where phi greater than or similar to 0.79, we find that S-braid (like D) is not well defined, signifying that these systems are not ergodic on the experimental timescale. Both S-braid and D decrease with either increasing packing density or confining pressure, independent of the applied boundary condition. The related braiding factor provides a means to identify multi-particle phenomena such as collective rearrangements. We discuss possible uses for this measure in characterizing granular systems.
Microscopic versus macroscopic approaches to non-equilibrium systems - Derrida, Bernard

Abstract : The one-dimensional symmetric simple exclusion process (SSEP) is one of the very few exactly soluble models of non-equilibrium statistical physics. It describes a system of particles which diffuse with hard core repulsion on a one-dimensional lattice in contact with two reservoirs of particles at unequal densities. The goal of this paper is to review the two main approaches which lead to the exact expression of the large deviation functional of the density of the SSEP in its steady state: a microscopic approach (based on the matrix product ansatz and an additivity property) and a macroscopic approach (based on the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim).
Statistical distributions in the folding of elastic structures - Adda-Bedia, Mokhtar and Boudaoud, Arezki and Boue, Laurent and Deboeuf, Stephanie

Abstract : The behaviour of elastic structures undergoing large deformations is the result of the competition between confining conditions, self-avoidance and elasticity. This combination of multiple phenomena creates a geometrical frustration that leads to complex fold patterns. By studying, both experimentally and numerically, the case of a rod confined isotropically into a disc, we show that the emergence of the complexity is associated with a well-defined underlying statistical measure that determines the energy distribution of sub-elements, `branches', of the rod. This result suggests that branches act as the `microscopic' degrees of freedom laying the foundations for a statistical mechanical theory of this athermal and amorphous system.
Dynamic force spectroscopy of DNA hairpins: II. Irreversibility and dissipation - Manosas, M. and Mossa, A. and Forns, N. and Huguet, J. M. and Ritort, F.

Abstract : We investigate irreversibility and dissipation in single molecules that cooperatively fold/unfold in a two-state manner under the action of mechanical force. We apply path thermodynamics to derive analytical expressions for the average dissipated work and the average hopping number in two-state systems. It is shown how these quantities only depend on two parameters that characterize the folding/unfolding kinetics of the molecule: the fragility and the coexistence hopping rate. The latter has to be rescaled to take into account the appropriate experimental set-up. Finally we carry out pulling experiments with optical tweezers in a specifically designed DNA hairpin that shows two-state cooperative folding. We then use these experimental results to validate our theoretical predictions.
The ideas behind self-consistent expansion - Schwartz, Moshe and Katzav, Eytan

Abstract : In recent years we have witnessed a growing interest in various non-equilibrium systems described in terms of stochastic nonlinear field theories. In some of those systems, like KPZ and related models, the interesting behavior is in the strong coupling regime, which is inaccessible by traditional perturbative treatments such as dynamical renormalization group (DRG). A useful tool in the study of such systems is the self-consistent expansion (SCE), which might be said to generate its own `small parameter'. The self-consistent expansion (SCE) has the advantage that its structure is just that of a regular expansion, the only difference is that the simple system around which the expansion is performed is adjustable. The purpose of this paper is to present the method in a simple and understandable way that hopefully will make it accessible to a wider public working on non-equilibrium statistical physics.
Depinning exponents of the driven long-range elastic string - Duemmer, Olaf and Krauth, Werner

Abstract : We perform a high-precision calculation of the critical exponents for the long-range elastic string driven through quenched disorder at the depinning transition, at zero temperature. Large-scale simulations avoid finite-size effects and improve accuracy. We explicitly demonstrate the equivalence of fixed-velocity and fixed-driving-force simulations. The roughness, growth, and velocity exponents are calculated independently, and the dynamic and correlation length exponents are derived. The critical exponents satisfy known scaling relations and agree well with analytical predictions.
Non-equilibrium steady states: fluctuations and large deviations of the density and of the current - Derrida, Bernard

Abstract : These lecture notes give a short review of methods such as the matrix ansatz, the additivity principle or the macroscopic fluctuation theory, developed recently in the theory of non-equilibrium phenomena. They show how these methods allow us to calculate the. fluctuations and large deviations of the density and the current in non-equilibrium steady states of systems like exclusion processes. The properties of these fluctuations and large deviation functions in non-equilibrium steady states ( for example, non-Gaussian. fluctuations of density or non-convexity of the large deviation function which generalizes the notion of free energy) are compared with those of systems at equilibrium.
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Noise-induced bifurcations, multiscaling and on-off intermittency - Aumaitre, Sebastien and Mallick, Kirone and Petrelis, Francois

Abstract : We present recent results on noise-induced transitions in a nonlinear oscillator with randomly modulated frequency. The presence of stochastic perturbations drastically alters the dynamical behaviour of the oscillator: noise can wash out a global attractor but can also have a constructive role by stabilizing an unstable fixed point. The random oscillator displays a rich phenomenology but remains elementary enough to allow for exact calculations: this system is thus a useful paradigm for the study of noise-induced bifurcations and is an ideal testing ground for various mathematical techniques. We show that the phase is determined by the sign of the Lyapunov exponent ( which can be calculated non-perturbatively for white noise), and we derive the full phase diagram of the system. We also investigate the effect of time correlations of the noise on the phase diagram and show that a smooth random perturbation is less efficient than white noise. We study the critical behaviour near the transition and explain why noise-induced transitions often exhibit intermittency and multiscaling: these effects do not depend on the amplitude of the noise but rather on its power spectrum. By increasing or filtering out the low frequencies of the noise, intermittency and multiscaling can be enhanced or eliminated.