# Jean-Pierre EckmannUniversity of Geneva | UNIGE · Department of Theoretical Physics

Jean-Pierre Eckmann

PhD

## About

283

Publications

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19,814

Citations

Citations since 2016

## Publications

Publications (283)

We discuss the Monge problem of mass transportation in the framework of stochastic thermodynamics and revisit the problem of the Landauer limit for finite-time thermodynamics, a problem that got the interest of Krzysztof Gawedzki in the last years. We show that restricted to one dimension, optimal transportation is efficiently solved numerically by...

Proteins need to selectively interact with specific targets among a multitude of similar molecules in the cell. But despite a firm physical understanding of binding interactions, we lack a general theory of how proteins evolve high specificity. Here, we present such a model that combines chemistry, mechanics and genetics, and explains how their int...

We discuss the Monge problem of mass transportation in the framework of stochastic thermodynamics and revisit the problem of the Landauer limit for finite-time thermodynamics, a problem that got the interest of Krzysztof Gawedzki in the last years. We show that restricted to one dimension, optimal transportation is efficiently solved numerically by...

We study the formation of images in a reflective sphere in three configurations using caustics on the field of light rays. The optical wavefront emerging from a source point reaching a subject following passage through the optical system is, in general, a Gaussian surface with partial focus along the two principal directions of the Gaussian surface...

We study the formation of images in a reflective sphere in three configurations using caustics of the field of light rays. The optical wavefront emerging from a source point reaching a subject following passage through the optical system is, in general, a Gaussian surface with partial focus along the two principal directions of the Gaussian surface...

Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories (EFT) originated from cosmological studies. In particular, we are interested in the equation $\partial_t^2 u(x,t) = \alpha (\partial_x u(x,t))^2 +\beta \partial_x^2 u(x,t)$ in $1+1$ dimensions....

Proteins need to selectively interact with specific targets among a multitude of similar molecules in the cell. But despite a firm physical understanding of binding interactions, we lack a general theory of how proteins evolve high specificity. Here, we present such a model that combines chemistry, mechanics and genetics, and explains how their int...

Mitchell Feigenbaum discovered an intriguing property of viewing images through cylindrical mirrors or looking into water. Because the eye is a lens with an opening of about 5 mm, many different rays of reflected images reach the eye and need to be interpreted by the visual system. This has the surprising effect that what one perceives depends on t...

The unprecedented prowess of measurement techniques provides a detailed, multi‐scale look into the depths of living systems. Understanding these avalanches of high‐dimensional data—by distilling underlying principles and mechanisms—necessitates dimensional reduction. We propose that living systems achieve exquisite dimensional reduction, originatin...

Mitchell Feigenbaum discovered an intriguing property of viewing images through cylindrical mirrors or looking into water. Because the eye is a lens with an opening of about 5mm, many different rays of reflected images reach the eye, and need to be interpreted by the visual system. This has the surprising effect that what one perceives depends on t...

The unprecedented prowess of measurement techniques provides a detailed, multi-scale look into the depths of living systems. But understanding these avalanches of high-dimensional data -- by distilling underlying principles of more general nature -- necessitates dimensional reduction. We will explain how some geometric insights by mathematicians al...

We study metastable behavior in a discrete nonlinear Schrödinger equation from the viewpoint of Hamiltonian systems theory. When there are n<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69...

Cosmological N-body simulations are done on massively parallel computers. This necessitates the use of simple time integrators and, additionally, of mesh-grid approximations of the potentials. Recently, Adamek et al. (Nat Phys 12:346, 2016), Barrera-Hinojosa and Li (GRAMSES: a new route to general relativistic N-body simulations in cosmology—I. Met...

We study the sensitivity of the computed orbits for the Kepler problem, both for continuous space, and discretizations of space. While it is known that energy can be very well preserved with symplectic methods, the semi-major-axis is in general not preserved. We study this spurious shift, as a function of the integration method used, and also as a...

Allocation of goods is a key feature in defining the connection between the individual and the collective scale in any society. Both the process by which goods are to be distributed, and the resulting allocation to the members of the society may affect the success of the population as a whole. One of the most striking natural examples of a highly s...

Proteins are a matter of dual nature. As a physical object, a protein molecule is a folded chain of amino acids with multifarious biochemistry. But it is also an instantiation along an evolutionary trajectory determined by the function performed by the protein within a hierarchy of interwoven interaction networks of the cell, the organism and the p...

Protein is matter of dual nature. As a physical object, a protein molecule is a folded chain of amino acids with diverse biochemistry. But it is also a point along an evolutionary trajectory determined by the function performed by the protein within a hierarchy of interwoven interaction networks of the cell, the organism, and the population. A phys...

We study metastable behavior in a discrete nonlinear Schr\"odinger equation from the viewpoint of Hamiltonian systems theory. When there are $n < \infty$ sites in this equation, we consider initial conditions in which almost all the energy is concentrated in one end of the system. We are interested in understanding how energy flows through the syst...

Allocation of goods is a
key feature in defining the connection between the individual and the collective scale in any society. Both the process by which goods are to be distributed, and the resulting allocation to the members of the society may affect the success of the population as a whole. One of the most striking natural examples of a highly s...

Collective behaviours in societies such as those formed by ants are thought to be the result of distributed mechanisms of information processing and direct decision-making by well-informed individuals, but their relative importance remains unclear. Here we tracked all ants and brood movements to investigate the decision strategy underlying brood tr...

In an earlier paper, we proved the validity of large deviations theory for the particle approximation of quite general chemical reaction networks (CRNs). In this paper, we present a more geometric insight into the mechanism of that proof, exploiting the notion of spherical image of the reaction polytope. This allows to view the asymptotic behavior...

Significance
Many protein functions involve large-scale motion of their amino acids, while alignment of their sequences shows long-range correlations. This has motivated search for physical links between genetic and phenotypic collective behaviors. The major challenge is the complex nature of protein: nonrandom heteropolymers made of 20 species of...

There has been growing evidence that cooperative interactions and configurational rearrangements underpin protein functions. But in spite of vast genetic and structural data, the information-dense, heterogeneous nature of protein has held back the progress in understanding the underlying principles. Here we outline a general theory of protein that...

There has been growing evidence that cooperative interactions and configurational rearrangements underpin protein functions. But in spite of vast genetic and structural data, the information-dense, heterogeneous nature of protein has held back the progress in understanding the underlying principles. Here we outline a general theory of protein that...

Non-equilibrium steady states for chains of oscillators (masses) connected by harmonic and anharmonic springs and interacting with heat baths at different temperatures have been the subject of several studies. In this paper, we show how some of the results extend to more complicated networks. We establish the existence and uniqueness of the non-equ...

We study metastable motions in weakly damped Hamiltonian systems. These are believed to inhibit the transport of energy through Hamiltonian, or nearly Hamiltonian, systems with many degrees of freedom. We investigate this question in a very simple model in which the breather solutions that are thought to be responsible for the metastable states can...

How DNA is mapped to functional proteins is a basic question of living matter. We introduce and study a physical model of protein evolution which suggests a mechanical basis for this map. Many proteins rely on large-scale motion to function. We therefore treat protein as learning amorphous matter that evolves towards such a mechanical function: Gen...

How DNA is mapped to functional proteins is a basic question of living matter. We introduce and study a physical model of protein evolution which suggests a mechanical basis for this map. Many proteins rely on large-scale motion to function. We therefore treat protein as learning amorphous matter that evolves towards such a mechanical function: Gen...

We discuss, in the context of energy flow in high-dimensional systems and Kolmogorov-Arnol'd-Moser (KAM) theory, the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numeric...

We prove a sample path Large Deviation Principle (LDP) for a class of jump processes whose rates are not uniformly Lipschitz continuous in phase space. Building on it we further establish the corresponding Wentzell-Freidlin (W-F) (infinite time horizon) asymptotic theory. These results apply to jump Markov processes that model the dynamics of chemi...

We treat proteins as amorphous learning matter: A `gene' encodes bonds in an `amino acid' network making a `protein'. The gene is evolved until the network forms a shear band across the protein, which allows for long-range soft modes required for protein function. The evolution projects the high-dimensional sequence space onto a low-dimensional spa...

We study a chain of four interacting rotors (rotators) connected at both ends
to stochastic heat baths at different temperatures. We show that for
non-degenerate interaction potentials the system relaxes, at a stretched
exponential rate, to a non-equilibrium steady state (NESS). Rotors with high
energy tend to decouple from their neighbors due to f...

We consider a chain of three rotors (rotators) whose ends are coupled to
stochastic heat baths. The temperatures of the two baths can be different, and
we allow some constant torque to be applied at each end of the chain. Under
some non-degeneracy condition on the interaction potentials, we show that the
process admits a unique invariant probabilit...

We introduce, test and discuss a method for classifying and clustering data
modeled as directed graphs. The idea is to start diffusion processes from any
subset of a data collection, generating corresponding distributions for
reaching points in the network. These distributions take the form of
high-dimensional numerical vectors and capture essentia...

We rely on a recent method for determining edge spectra and we use it to
compute the Chern numbers for Hofstadter models on the honeycomb lattice having
rational magnetic flux per unit cell. Based on the bulk-edge correspondence,
the Chern number $\sigma_H$ is given as the winding number of an eigenvector of
a $2 \times 2$ transfer matrix, as a fun...

We study how desert ants, Cataglyphis niger, a species that lacks pheromone-based recruitment mechanisms, inform each other about the presence of food. Our results are based on automated tracking that allows us to collect a large database of ant trajectories and interactions. We find that interactions affect an ant's speed within the nest. Fast ant...

We consider networks of massive particles connected by non-linear springs.
Some particles interact with heat baths at different temperatures, which are
modeled as stochastic driving forces. The structure of the network is
arbitrary, but the motion of each particle is 1D. For polynomial interactions,
we give sufficient conditions for H\"ormander's "...

Studies of the graph of dictionary definitions (DD) (Picard et al., 2009; Levary et al., 2012) have revealed strong semantic coherence of local topological structures. The techniques used in these papers are simple and the main results are found by understanding the structure of cycles in the directed graph (where words point to definitions). Based...

Dictionaries link a given word to a set of alternative words (the definition) which in turn point to further descendants. Iterating through definitions in this way, one typically finds that definitions loop back upon themselves. We demonstrate that such definitional loops are created in order to introduce new concepts into a language. In contrast t...

Based on the work of Durhuus-J{\'o}nsson and Benedetti-Ziegler, we revisit
the question of the number of triangulations of the 3-ball. We introduce a
notion of nucleus (a triangulation of the 3-ball without internal nodes, and
with each internal face having at most 1 external edge). We show that every
triangulation can be built from trees of nuclei...

Motivated by a stochastic differential equation describing the dynamics of interfaces, we study the bifurcation behaviour of a more general class of such equations. These equations are characterized by a two-dimensional phase space (describing the position of the interface and an internal degree of freedom). The noise accounts for thermal fluctuati...

In this paper we continue the study of a topological glassy system. The state
space of the model is given by all triangulations of a sphere with $N$ nodes,
half of which are red and half are blue. Red nodes want to have 5 neighbors
while blue ones want 7. Energies of nodes with other numbers of neighbors are
supposed to be positive. The dynamics is...

Dictionaries are inherently circular in nature. A given word is linked to a
set of alternative words (the definition) which in turn point to further
descendants. Iterating through definitions in this way, one typically finds
that definitions loop back upon themselves. The graph formed by such
definitional relations is our object of study. By elimin...

We consider a heat conduction model introduced by Collet and Eckmann (2009 Commun. Math. Phys. 287 1015-38). This is an open system in which particles exchange momentum with a row of (fixed) scatterers. We assume simplified bath conditions throughout, and give a qualitative description of the dynamics extrapolating from the case of a single particl...

We present a theoretical framework using quorum percolation for describing the initiation of activity in a neural culture. The cultures are modeled as random graphs, whose nodes are excitatory neurons with k(in) inputs and k(out) outputs, and whose input degrees k(in) = k obey given distribution functions p(k). We examine the firing activity of the...

Most of our considerations will be done for
Ã\wp
1 unimodal functions, but in fact some results hold for unimodal maps, and some hold for any continuous map of an interval
to itself. But the ideas of the proofs seem more transparent in the
Ã\wp
1 case. It will also be easy to see that the particular choice of the interval [-1,1] and fixing the...

The iterations of maps of an interval into itself certainly present one of the easiest models or examples of nonlinear (dissipative) dynamical systems. In fact, iterations of the form \({\rm x}_{{\rm n}} \rightarrow {\rm x}_{{\rm n + 1}} = {\rm f}({\rm x}_{{\rm n}})\), where f maps [-1,1] into itself, can be viewed as a discrete time version of a c...

In this section, we present a part of the beautiful theory of Misiurewicz [1980] on the existence of an invariant measure
which is absolutely continuous with respect to Lebesgue measure and which is ergodic, for certain s–unimodal maps. We assume
some familiarity with general ergodic theory.

We add now an important element to our discussion, namely a parameter. The parameter will always be called μ, and we want
to study maps f, depending on μ. Before doing this, we illustrate in a physical example how one should think of μ and of the
map. The example how one should think of μ and of the map. The example we choose is the Reyleigh-Bénard...

We shall now leave maps for a moment, and draw some combinatorial conclusions from the definition of the ordering of itineraries.
The definitions given below are of course motivated through the connections between itineraries and maps. We shall indicate
these connections for illustration, but they do not enter our arguments.

Before we study parametrized families of maps, we want to analyze individual maps. We are interested in the possible behavior
of the successive images of an initial point x0 on the interval [-1,1] for a fixed map f. For this we first outline a graphical method for determining the iterates
xn = fn(x0){\rm x}_{{\rm n}} = {\rm f}^{{\rm n}}({\rm x}_0)...

Infinite sequences of period doubling bifurcations have been observed in higher dimensional systems. One of them is the Henon
map in
\mathbbR2\mathbb{R}^2
:
${rcl}\left({rl}{{\rm x}}\\ {{\rm y}}\right) \mapsto \left({rl}1 - &\mu{\rm x}^2 + y\\ &{{\rm bx}}\right).$\begin{array}{rcl}\left(\begin{array}{rl}{{\rm x}}\\ {{\rm y}}\end{array}\right) \m...

Consider a once differentiable one-parameter family of unimodal maps as in the preceding section. We assume for definiteness
that the parameter varies in [0,1] and for simplicity we also assume the maps fμ are symmetric functions. If f0 (1) = 0 and f1 = −1, then all maximal admissible itineraries A with
RC £ A £ RL¥{\rm RC} \leq \underline{{\rm A}...

In this section, we temporarily relax the principle of describing only proved theorems and not any conjectures. The question
addressed in the little is a very delicate one and it has only been answered for specific, isolated one-parameter families
of maps. Our description will thus be inspired by the experience one has collect in proving these isol...

Having discovered the universlity of experimentally, Feigenbaum went on to propose an explanation for it which was inspired
by the renormalization group approach to critical phenomena in statistical mechanics. The principal result of this section
is to show that Feigenbaum’s explanation is correct, at least in a certain limiting regime to be explai...

So far, we have analyzed in some detail the relationship between itineraries and points in [-1,1]. We now ask the question
of how many points can have the same itinerary. The answer to this question will be relevant for the sections on topological
conjugacy, on sensitive behavior and on ergodic properties. It turns out that in the case of S-unimoda...

In this section, we reexamine the bifurcation diagram Fig. I.19 in the largeand we shall analyze in great detail the different
parameter values at which bifurcations occur. The following observation has been made by Feigenbaum. Consider those points μn in the bifurcation diagram where there is a bifurcation from length 2n-1 to length 2n. The follow...

We now ask how many stable periodic orbits a unimodal map can have. This question was first asked by Julia, in 1918. How showed
that for certain unimodal maps which are restrictions to [-1,1] of analytic functions, there can be at most one stable periodic
orbit. In particular, his theory applies to f(x) = 1 − μx2, 0 < μ ≤ 2. But a real breakthrough...

As we have seen in section I.1, there are many situations in which a map
\mathbbRn ® \mathbbRn\mathbb{R}^{{\rm n}} \rightarrow \mathbb{R}^{{\rm n}}
seems more appropriate than a map
\mathbbR1 ® \mathbbR1\mathbb{R}^1 \rightarrow \mathbb{R}^1
. The purpose of this section is to illustrate how the features of one-dimensional maps described in I.6...

If we look closely at the Figure 1.19, we distinguish, from left to right, more or less clearly, a stable orbit of period
1 (i.e., a fixed point), then a stable orbit of period 2, then 4, 8, then a “mess,” then another orbit of period 6, then 5,
and 3. Other stable periods are barely visible inbetween. The astonishing fact about this arrangement of...

The purpose of this section is to harvest the fruits of our preceding work. We shall see in fact that there is an almost one-to-one
relation between itineraries and points on the line. We have already seen in Section II.1, that to every x
e\epsilon
[-1,1], there is for fixed f an (admissible) itinerary I(x). We shall see now in which sense to ever...

In this section, and the next, we analyze in more detail the S-unimodal maps without stable periodic orbit. We shall see later that such maps are quite frequent, and thus their study is not only justified from a mathematical point of view but is relevant as a somehow “physical” problem one dimensional dynamical systems.

The aim of this section is to analyze the extent to which the converse is true. Does a kneading sequence already determine a map? This question was primarily addressed by milnor-Thurston [1997], and a very satisfactory answer was given by Guckenheimer [1978]. This section is mainly a presentation of his results, combined with the analysis of Misiur...

Taking into account the coupling between the position of the wall and an
internal degree of freedom, namely its phase $\phi$, we examine, in the rigid
wall approximation, the dynamics of a magnetic domain wall subject to a weak
pinning potential. We determine the corresponding force-velocity
characteristics, which display several unusual features w...

We examine bootstrap percolation in d-dimensional, directed metric graphs in the context of recent measurements of firing dynamics in 2D neuronal cultures. There are two regimes, depending on the graph size N. Large metric graphs are ignited by the occurrence of critical nuclei, which initially occupy an infinitesimal fraction, f_* -> 0, of the gra...

In this paper, we first define a deterministic particle model for heat conduction. It consists of a chain of N identical subsystems, each of which contains a scatterer and with particles moving among these scatterers. Based on this
model, we then derive heuristically, in the limit of N → ∞ and decreasing scattering cross-section, a Boltzmann equati...

Motivation and Interpretation.- One-Parameter Families of Maps.- Typical Behavior for One Map.- Parameter Dependence.- Systematics of the Stable Periods.- On the Relative Frequency of Periodic and Aperiodic Behavior.- Scaling and Related Predictions.- Higher Dimensional Systems.- Properties of Individual Maps.- Unimodal Maps and Thier Itineraries.-...