arXiv subjects

J. Unterberger

Publications and source records attributed to J. Unterberger.

9 recordsLinked to original sources

arXiv2026

Hierarchical models for large chemical reaction networks

The quest for the origin of life, especially in the metabolism-first scenario inspired by the celebrated Miller-Urey experiment, has triggered a research program dedicated to studying the emergence of complex dynamical behaviors in large chemical mixtures. Though autocatalysis, understood as the capacity of a reaction network to grow exponentially, has been recognized as a potential driver of instability and multistability, no quantitative theory has yet emerged, partly because of the lack of available kinetic data. We introduce a computational tool for large chemical reaction networks based on a scale-splitting algorithm inspired by Wilson's renormalization group. We focus on dilute regimes, where species of interest have low concentration, non-unimolecular reactions may be neglected, and the dynamics is close to linear. Depending on parameter thresholds, such networks can exhibit autocatalytic behavior. Our algorithm takes as input a network structure and outputs (1) a simplified effective graph containing the dominant reaction pathways, obtained through recursive coarse-graining; and (2) analytical formulas for the dynamics in terms of kinetic rates, called hierarchical formulas. These formulas are approximate but interpretable, accurate when scale separation is effective, and provide a reliable multiscale description of the dynamics. Their domains of validity define kinetic phases, each typically associated with a distinct pattern of chemical composition. We show on a simple example that this approach enables fast and reliable inference of kinetic rates from concentration time series. Hierarchical formulas have been implemented as a Python package and are illustrated on a simplified model of the formose reaction.

q-bio.MN↗

arXiv2022

Exact computation of growth-rate fluctuations in random environment

We consider a general class of Markovian models describing the growth in a randomly fluctuating environment of a clonal biological population having several phenotypes related by stochastic switching. Phenotypes differ e.g. by the level of gene expression for a population of bacteria. The time-averaged growth rate of the population, $Λ$, is self-averaging in the limit of infinite times; it may be understood as the fitness of the population in a context of Darwinian evolution. The observation time $T$ being however typically finite, the growth rate fluctuates. For $T$ finite but large, we obtain the variance of the time-averaged growth rate as the maximum of a functional based on the stationary probability distribution for the phenotypes. This formula is general. In the case of two states, the stationary probability was computed by Hufton, Lin and Galla \cite{HufLin2}, allowing for an explicit expression which can be checked numerically.

q-bio.PE↗

arXiv2020

Phase transitions in optimal strategies for betting

Kelly's criterion is a betting strategy that maximizes the long term growth rate, but which is known to be risky. Here, we find optimal betting strategies that gives the highest capital growth rate while keeping a certain low value of risky fluctuations. We then analyze the trade-off between the average and the fluctuations of the growth rate, in models of horse races, first for two horses then for an arbitrary number of horses, and for uncorrelated or correlated races. We find an analog of a phase transition with a coexistence between two optimal strategies, where one has risk and the other one does not. The above trade-off is also embodied in a general bound on the average growth rate, similar to thermodynamic uncertainty relations. We also prove mathematically the absence of other phase transitions between Kelly's point and the risk free strategy.

cond-mat.stat-mech↗

arXiv2019

A mathematical derivation of zero-temperature 2D superconductivity from microscopic Bardeen-Cooper-Schrieffer model

Starting from H. Fröhlich's second-quantized Hamiltonian for a $d$-dimensional electron gas in interaction with lattice phonons describing the quantum vibrations of a metal, we present a rigorous mathematical derivation of the superconducting state, following the principles laid out originally in 1957 by J. Bardeen, L. Cooper and J. Schrieffer. As in the series of papers written on the subject in the 90es, of which the present paper is a continuation, the representation of ions as a uniform charge background allows for a $(1+d)$-dimensional fermionic quantum-field theoretic reformulation of the model at equilibrium. For simplicity, we restrict in this article to $d=2$ dimensions and zero temperature, and disregard effects due to electromagnetic interactions. Under these assumptions, we prove transition from a Fermi liquid state to a superconducting state made up of Cooper pairs of electrons at an energy level $Γ_ϕ\sim \hbarω_D e^{-π/mλ}$ equal to the mass gap, expressed in terms of the Debye frequency $ω_D$, electron mass $m$ and coupling constant $λ$. The dynamical $U(1)$-symmetry breaking produces at energies lower than the energy gap $Γ_ϕ$ a Goldstone boson, a non-massive particle described by an effective $(2+1)$-dimensional non-linear sigma-model, whose parameters and correlations are computed. The proof relies on a mixture of general concepts and tools (multi-scale cluster expansions, Ward identities), adapted to this quantum many-body problem with its extended infra-red singularity located on the Fermi circle, and a specific $1/N$-expansion giving the leading diagrams at intermediate energies. Ladder diagrams are proved to provide the leading behavior in the infra-red limit, in agreement with mean-field theory predictions.

cond-mat.supr-con↗

arXiv2018

Global fluctuations for 1D log-gas dynamics

We study in this article the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, \begin{equation} dλ_t^i=\frac{1}{\sqrt{N}} dW_t^i - V'(λ_t^i) dt+ \fracβ{2N} \sum_{j\not=i} \frac{dt}{λ^i_t-λ^j_t}, \qquad i=1,\ldots,N, \end{equation} with $β>1$, sometimes called generalized Dyson's Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $β$-ensemble, with sufficiently regular convex potential $V$. The limit $N\to\infty$ is known to satisfy a mean-field Mac-Kean-Vlasov equation. We prove that, for suitable initial conditions, fluctuations around the limit are Gaussian and satisfy an explicit PDE. The proof is very much indebted to the harmonic potential case treated in Israelsson \cite{Isr}. Our key argument consists in showing that the time-evolution generator may be written in the form of a transport operator on the upper half-plane, plus a bounded non-local operator interpreted in terms of a signed jump process. As an essential technical argument ensuring the convergence of the above scheme, we give an $N$-independent large-deviation type estimate for the probability that $\sup_{t\in[0,T]}\max_{i=1,\ldots,N} |λ_t^i|$ is large, based on a multi-scale argument and entropic bounds.

math.PR↗

arXiv2015

Generalized PDE estimates for KPZ equations through Hamilton-Jacobi-Bellman formalism

We study in this series of articles the Kardar-Parisi-Zhang (KPZ) equation $$ \partial_t h(t,x)=νΔh(t,x)+λV(|\nabla h(t,x)|) +\sqrt{D}\, η(t,x), \qquad x\in{\mathbb{R}}^d $$ in $d\ge 1$ dimensions. The forcing term $η$ in the right-hand side is a regularized white noise. The deposition rate $V$ is assumed to be isotropic and convex. Assuming $V(0)\ge 0$, one finds $V(|\nabla h|)\ltimes |\nabla h|^2$ for small gradients, yielding the equation which is most commonly used in the literature. The present article, a continuation of [24], is dedicated to a generalization of the PDE estimates obtained in the previous article to the case of a deposition rate $V$ with polynomial growth of arbitrary order at infinity, for which in general the Cole-Hopf transformation does not allow any more a comparison to the heat equation. The main tool here instead is the representation of $h$ as the solution of some minimization problem through the Hamilton-Jacobi-Bellman formalism. This sole representation turns out to be powerful enough to produce local or pointwise estimates in ${\cal W}$-spaces of functions with "locally bounded averages", as in [24], implying in particular global existence and uniqueness of solutions.

math.AP↗

arXiv2011

Mode d'emploi de la théorie constructive des champs bosoniques (A user's guide to bosonic constructive field theory)

We develop in this article the principal constructive arguments used in quantum field theory, limiting us to bosonic theories, for which there does not exist any recent general presentation. The article is primarily written for mathematicians or mathematical physicists knowing the basic arguments of quantum field theory, and desiring to discover a general framework in which they can be made rigorous. It also provides a glimpse of a recent series of articles \cite{MagUnt1,MagUnt2} whose aim is to give a constructive definition of rough paths and fractionary stochastic calculus.

math-ph↗

arXiv2010

From constructive field theory to fractional stochastic calculus. (I) The Lévy area of fractional Brownian motion with Hurst index $α\in (1/8,1/4)$

Let $B=(B_1(t),\ldots,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $α<1/4$. Defining properly iterated integrals of $B$ is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We show in this paper how to obtain second-order iterated integrals as the limit when the ultra-violet cut-off goes to infinity of iterated integrals of weakly interacting fields defined using the tools of constructive field theory, in particular, cluster expansion and renormalization. The construction extends to a large class of Gaussian fields with the same short-distance behaviour, called multi-scale Gaussian fields. Previous constructions \cite{Unt-Holder,Unt-fBm} were of algebraic nature and did not provide such a limiting procedure.

math.PR↗

arXiv2009

Hölder-continuous rough paths by Fourier normal ordering

We construct in this article an explicit geometric rough path over arbitrary $d$-dimensional paths with finite $1/α$-variation for any $α\in(0,1)$. The method may be coined as 'Fourier normal ordering', since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest Fourier frequencies. In doing so, there appear non-trivial tree combinatorics, which are best understood by using the structure of the Hopf algebra of decorated rooted trees (in connection with the Chen or multiplicative property) and of the Hopf shuffle algebra (in connection with the shuffle or geometric property). Hölder continuity is proved by using Besov norms. The method is well-suited in particular in view of applications to probability theory (see the companion article \cite{Unt09} for the construction of a rough path over multidimensional fractional Brownian motion with Hurst index $α<1/4$, or \cite{Unt09ter} for a short survey in that case).

math.PR↗