Showing papers by "Jean-Pierre Eckmann published in 2022"
TL;DR: In this paper , the authors studied the detailed nature of this divergence as a function of the parameters α > 0 and β ≥ 0 and showed that the divergence does not disappear even when β is very large contrary to what one might believe.
Abstract: . 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 ∂ 2 t u ( x, t ) = α ( ∂ x u ( x, t )) 2 + β∂ 2 x u ( x, t ) in 1 + 1 dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when α > 0 . We study the detailed nature of this divergence as a function of the parameters α > 0 and β ≥ 0 . The divergence does not disappear even when β is very large contrary to what one might believe. But it will take longer to appear as β increases when α is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to 3 + 1 dimensions.
2 citations
23 Dec 2022
TL;DR: In this paper , the authors studied the Abelian sandpile model on a special playground, a cylinder of width w and of circumference c, and they described a phenomenon which has not been observed in other geometries: the probability distribution of avalanche sizes has a ladder structure.
Abstract: We study the Abelian sandpile model on a special playground, a cylinder of width w and of circumference c. When c≪w , we describe a phenomenon which has not been observed in other geometries: the probability distribution of avalanche sizes has a ladder structure, with the first step consisting of avalanches of size up to w⋅c/2 that are essentially equiprobable, except for a small exponential tail of order about 10 c. We explain this phenomenon and describe subsequent steps.
1 citations
TL;DR: In this paper , the formation of images in a reflective sphere in three configurations using caustics on the field of light rays is studied, i.e., imaging of a parallel beam of light, imaging of the infinite viewed from a location outside the sphere, and imaging of an object viewed through the point of its intersection with the radial line normal to the plane.
Abstract: 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; i.e., there are two images of the source point, each with partial focus. As the source point moves, the images move on two surfaces, referred to as viewable surfaces. In our systems, one viewable surface consists of points with radial focus and the other consists of points with azimuthal focus. The problems we study are (1) imaging of a parallel beam of light, (2) imaging of the infinite viewed from a location outside the sphere, and (3) imaging of a planar object viewed through the point of its intersection with the radial line normal to the plane. We verify the existence of two images experimentally and show that the distance between them agrees with the computations.
1 citations
TL;DR: In this article , the complementary link l is defined as the link connecting two triangles sharing the link l. The authors assume that for any T ∈ T , a probabilityPT is given onL(T ), i.e., ∑ √ √ n of vertices goes to ∞.
Abstract: as the number n of vertices goes to∞. Of course, Euler’s theorem holds for such triangulations, and this means that when there are n nodes, there are also 3n− 6 links and2n− 4 triangles. For an element T ∈ T , we denote byN (T ) the set of nodes and by L(T ) the set of links. For any linkl (connecting the nodes A andB), we consider the “complementary” link l, which is defined as follows: if ( A,B,C) and (A,B,D) are the two triangles sharing the linkl, thenl is the link connectingC andD. We assume that for any T ∈ T , a probabilityPT is given onL(T ), i.e., ∑
02 May 2022
TL;DR: In this article , the authors studied the divergence of a nonlinear partial differential equation in cosmological studies and showed that the blowup is unavoidable unless the corresponding equations are modified.
Abstract: 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. It has been known for quite some time that solutions to this equation diverge in finite time, when $\alpha>0$. We study the nature of this divergence as a function of the parameters $\alpha>0 $ and $\beta\ge0$. The divergence does not disappear even when $\beta $ is very large contrary to what one might believe (note that since we consider fixed initial data, $\alpha$ and $\beta$ cannot be scaled away). But it will take longer to appear as $\beta$ increases when $\alpha$ is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to $3+1$ dimensions.
TL;DR: The proposed theory of specific binding addresses the natural question of “why are proteins so big?”, and shows that molecular discrimination is often a hard task best performed by adding more layers to the protein.
Abstract: 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 interplay governs the evolution of specific protein-ligand interactions. The model shows that there are many routes to achieving molecular discrimination – by varying degrees of flexibility and shape/chemistry complementarity – but the key ingredient is precision. Harder discrimination tasks require more collective and precise coaction of structure, forces and movements. Proteins can achieve this through correlated mutations extending far from a binding site, which fine-tune the localized interaction with the ligand. Thus, the solution of more complicated tasks is enabled by increasing the protein size, and proteins become more evolvable and robust when they are larger than the bare minimum required for discrimination. The model makes testable, specific predictions about the role of flexibility and shape mismatch in discrimination, and how evolution can independently tune affinity and specificity. Thus, the proposed theory of specific binding addresses the natural question of “why are proteins so big?”. A possible answer is that molecular discrimination is often a hard task best performed by adding more layers to the protein.
TL;DR: In this article , the authors consider simulations of a 2-dimensional gas of hard disks in a rectangular container and study the Lyapunov spectrum near the vanishing LyAPunov exponents.
Abstract: We consider simulations of a 2-dimensional gas of hard disks in a rectangular container and study the Lyapunov spectrum near the vanishing Lyapunov exp onents. To this spectrum are associated “eigen-directions”, called Lyapunov modes. We car fully analyze these modes and show how they are naturally associated with vector fields over the container. We also show that the Lyapunov exponents, and the coupled dynamics o f the modes (where it exists) follow linear laws, whose coefficients only depend on the den sity of the gas, but not on aspect ratio and very little on the boundary conditions.
TL;DR: In this paper , the authors 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.
Abstract: Abstract 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 well-known methods from differential equations. We add a brief discussion about the relevance this has on optimising the processing in modern computers.