Statistical stability and linear response for random hyperbolic dynamics

We consider families of random products of close-by Anosov diffeomorphisms, and show that statistical stability and linear response hold for the associated families of equivariant and stationary measures. Our analysis relies on the study of the top Oseledets space of a parametrized transfer operator cocycle, as well as ad-hoc abstract perturbation statements. As an application, we show that, when the quenched central limit theorem (CLT) holds, under the conditions that ensure linear response for our cocycle, the variance in the CLT depends differentiably on the parameter.

Deep learning the slow modes for rare events sampling

The development of enhanced sampling methods has greatly extended the scope of atomistic simulations, allowing long-time phenomena to be studied with accessible computational resources. Many such methods rely on the identification of an appropriate set of collective variables. These are meant to describe the system’s modes that most slowly approach equilibrium under the action of the sampling algorithm. Once identified, the equilibration of these modes is accelerated by the enhanced sampling method of choice. An attractive way of determining the collective variables is to relate them to the eigenfunctions and eigenvalues of the transfer operator. Unfortunately, this requires knowing the long-term dynamics of the system beforehand, which is generally not available. However, we have recently shown that it is indeed possible to determine efficient collective variables starting from biased simulations. In this paper, we bring the power of machine learning and the efficiency of the recently developed on the fly probability-enhanced sampling method to bear on this approach. The result is a powerful and robust algorithm that, given an initial enhanced sampling simulation performed with trial collective variables or generalized ensembles, extracts transfer operator eigenfunctions using a neural network ansatz and then accelerates them to promote sampling of rare events. To illustrate the generality of this approach, we apply it to several systems, ranging from the conformational transition of a small molecule to the folding of a miniprotein and the study of materials crystallization.

Using the group analysis transfer operator in the method of similarity for studying the fuel combustion process in engines

In this paper, we study modelling of the process of combustion of small quantities of a pyrotechnic mixture based on potassium perchlorate and aluminium powder in impulse devices (engines). The analysis of combustion processes includes an introduced general criterion (prime criterion) of similarity composed of Wobbe numbers that are determined for the engine and its model in the course of theoretical studies or experiments. Research data show that the transfer operator in the field of group theory is mostly used for describing the process of combustion of metallized pyrotechnic mixtures.

Hecke Triangle Groups, Transfer Operators and Hausdorff Dimension

We consider the family of Hecke triangle groups $$ \Gamma _{w} = \langle S, T_w\rangle $$ Γ w = ⟨ S , T w ⟩ generated by the Möbius transformations $$ S : z\mapsto -1/z $$ S : z ↦ - 1 / z and $$ T_{w} : z \mapsto z+w $$ T w : z ↦ z + w with $$ w > 2.$$ w > 2 . In this case, the corresponding hyperbolic quotient $$ \Gamma _{w}\backslash {\mathbb {H}}^2 $$ Γ w \ H 2 is an infinite-area orbifold. Moreover, the limit set of $$ \Gamma _w $$ Γ w is a Cantor-like fractal whose Hausdorff dimension we denote by $$ \delta (w). $$ δ ( w ) . The first result of this paper asserts that the twisted Selberg zeta function $$ Z_{\Gamma _{ w}}(s, \rho ) $$ Z Γ w ( s , ρ ) , where $$ \rho : \Gamma _{w} \rightarrow \mathrm {U}(V) $$ ρ : Γ w → U ( V ) is an arbitrary finite-dimensional unitary representation, can be realized as the Fredholm determinant of a Mayer-type transfer operator. This result has a number of applications. We study the distribution of the zeros in the half-plane $$\mathrm {Re}(s) > \frac{1}{2}$$ Re ( s ) > 1 2 of the Selberg zeta function of a special family of subgroups $$( \Gamma _w^N )_{N\in {\mathbb {N}}} $$ ( Γ w N ) N ∈ N of $$\Gamma _w$$ Γ w . These zeros correspond to the eigenvalues of the Laplacian on the associated hyperbolic surfaces $$X_w^N = \Gamma _w^N \backslash {\mathbb {H}}^2$$ X w N = Γ w N \ H 2 . We show that the classical Selberg zeta function $$Z_{\Gamma _w}(s)$$ Z Γ w ( s ) can be approximated by determinants of finite matrices whose entries are explicitly given in terms of the Riemann zeta function. Moreover, we prove an asymptotic expansion for the Hausdorff dimension $$\delta (w)$$ δ ( w ) as $$w\rightarrow \infty $$ w → ∞ .

Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems

Transfer operators such as Perron–Frobenius and Koopman operator play a key role in modeling and analysis of complex dynamical systems, which allow linear representations of nonlinear dynamics by transforming the original state variables to feature spaces. However, it remains challenging to identify the optimal low-dimensional feature mappings from data. The variational approach for Markov processes (VAMP) provides a comprehensive framework for the evaluation and optimization of feature mappings based on the variational estimation of modeling errors, but it still suffers from a flawed assumption on the transfer operator and therefore sometimes fails to capture the essential structure of system dynamics. In this paper, we develop a powerful alternative to VAMP, called kernel embedding based variational approach for dynamical systems (KVAD). By using the distance measure of functions in the kernel embedding space, KVAD effectively overcomes theoretical and practical limitations of VAMP. In addition, we develop a data-driven KVAD algorithm for seeking the ideal feature mapping within a subspace spanned by given basis functions, and numerical experiments show that the proposed algorithm can significantly improve the modeling accuracy compared to VAMP.

Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

Given any smooth Anosov map, we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that, in the case of expanding maps, it reduces exactly to the usual space of functions of bounded variation which has proved to be particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations, which provides new information that could prove useful in treating hyperbolic systems with singularities.

Surface and deep ocean connectivity inferred from robust extraction of coherent sets in ocean flow using models and sparse, scattered, and incomplete float data with transfer operator and dynamic Laplacian methods.

<p>Transport and mixing properties of the ocean's circulation is crucial to dynamical analyses, and often have to be carried out with limited observed information. Finite-time coherent sets are regions of the ocean that minimally mix (in the presence of small diffusion) with the rest of the ocean domain over the finite period of time considered. In the purely advective setting (in the zero diffusion limit) this is equivalent to identifying regions whose boundary interfaces remain small throughout their finite-time evolution. Finite-time coherent sets thus provide a skeleton of distinct regions around which more turbulent flow occurs. Well known manifestations of finite-time coherent sets in geophysical systems include rotational objects like ocean eddies, ocean gyres, and atmospheric vortices. In real-world settings, often observational data is scattered and sparse, which makes the difficult problem of coherent set identification and tracking challenging. I will describe mesh-based numerical methods [3] to efficiently approximate the recently defined dynamic Laplace operator [1,2], and rapidly and reliably extract finite-time coherent sets from models or scattered, possibly sparse, and possibly incomplete observed data. From these results we can infer new chemical and physical ocean connectivities at global and intra-basin scales (at the surface and at depth), track series of eddies, and determine new oceanic barriers.</p><p>[1] G. Froyland. Dynamic isoperimetry and the geometry of Lagrangian coherent structures. <em>Nonlinearity</em>, 28:3587-3622, 2015</p><p>[2] G. Froyland and E. Kwok. A dynamic Laplacian for identifying Lagrangian coherent structures on weighted Riemannian manifolds. <em>Journal of Nonlinear Science</em>, 30:1889–1971, 2020.</p><p>[3] Gary Froyland and Oliver Junge. Robust FEM-based extraction of finite-time coherent sets using scattered, sparse, and incomplete trajectories. <em>SIAM J. Applied Dynamical Systems</em>, 17:1891–1924, 2018.</p>

Efficient Numerical Evaluation of Thermodynamic Quantities on Infinite (Semi-)classical Chains

This work presents an efficient numerical method to evaluate the free energy density and associated thermodynamic quantities of (quasi) one-dimensional classical systems, by combining the transfer operator approach with a numerical discretization of integral kernels using quadrature rules. For analytic kernels, the technique exhibits exponential convergence in the number of quadrature points. As demonstration, we apply the method to a classical particle chain, to the semiclassical nonlinear Schrödinger (NLS) equation and to a classical system on a cylindrical lattice. A comparison with molecular dynamics simulations performed for the NLS model shows very good agreement.