arXiv subjects

Barbara Drossel

Publications and source records attributed to Barbara Drossel.

At least 19 recordsLinked to original sources

arXiv2026

The different localisation properties of the eigenmodes of the Laplacian and adjacency matrix of 2D random geometric graphs

We compare the spectrum and the localisation properties of the eigenmodes of the Laplacian and the adjacency matrix of 2D random geometric graphs, using numerical diagonalization of these matrices for different system sizes and connectivities. For sufficiently large ensembles of systems, we evaluate the spectrum, the probability distribution of the participation ratio and the relation between participation ratios and eigenvalues. While all eigenmodes of the adjacency matrix are localised for sufficiently large system sizes, the Laplacian matrix always leads to a small proportion of system-spanning modes due to a conservation law, and therefore to power-law tails in the probability distribution of the participation ratio and its relation to the eigenvalues. By disentangling the effects of finite system size, of mean degree, of component size distribution, and of network motifs, we provide a thorough understanding of the data.

cond-mat.dis-nn↗

arXiv2025

The stochastic discrete nonlinear Schr\"odinger equation: microscopic derivation and finite-temperature phase transition

We study a stochastic version of the one-dimensional discrete nonlinear Schr{\"o}dinger equation (DNSE), which is derived from first principles, and thus possesses all the properties required by statistical mechanics, such as detailed balance and the H-theorem. The stochastic version shows disordered and localised dynamics, and displays a corresponding phase transition at a finite temperature value. The phase transition can be captured in a quantitative way by a mean-field type approach. The corresponding coarsening dynamics shows an unexpected dependence on the noise strength, which is reminiscent of stochastic resonance. The phase transition is linked with negative temperature phase transitions, which have been reported recently for the Hamiltonian dynamics of the DNSE. Our approach gives a clue to how these negative temperature phase transitions can be implemented in experimental setups, which are inevitably coupled to a positive temperature heat bath.

cond-mat.stat-mech↗

arXiv2025

Localized and delocalized modes on random geometric graphs in 1D

We perform an extensive investigation of the localization properties of the eigenmodes of the Laplace and adjacency matrix for one-dimensional random geometric graphs. We evaluate the density of states, the probability distribution of the participation ratio and its relation to the eigenvalue. By disentangling the influence of system size, graph component size distribution, mean degree of nodes, network motifs, and degeneracy, we provide a comprehensive understanding of this system. We compare our findings to ordered graphs with the same mean degree and to one-dimensional tight-binding models.

cond-mat.dis-nn↗

arXiv2025

Dynamics of localized states in the stochastic discrete nonlinear Schr\"odinger equation

We revisit aspects of dynamics and stability of localized states in the deterministic and stochastic discrete nonlinear Schr\"odinger equation. By a combination of analytic and numerical techniques, we show that localized initial conditions disperse if the strength of the nonlinear part drops below a threshold and that localized states are unstable in a noisy environment. As expected, the constants of motion in the nonlinear Schr\"odinger equation play a crucial role. An infinite temperature state emerges when multiplicative noise is applied, while additive noise yields unbounded dynamics since conservation of normalization is violated.

nlin.PS↗

arXiv2024

The scaling behaviour of localised and extended states in one-dimensional tight-binding models with disorder

We investigate two one-dimensional tight-binding models with disorder that have extended states at zero energy. We use exact and partial diagonalisation of the Hamiltonian to obtain the eigenmodes and the associated participation ratios, and the transfer-matrix method to determine the localisation length. The first model has no on-site disorder, but random couplings. While the participation ratio remains finite at zero energy, the localisation length diverges logarithmically as the energy goes to zero. We provide an intuitive derivation of this logarithmic divergence based on the weak coupling of the two sublattices. The second model has a conserved quantity as the row sums of the Hamiltonian are zero. This model can be represented as a harmonic chain with random couplings, or as a diffusion model on a lattice with random links. We find, in agreement with existing analytical calculations, that the number of system-spanning eigenmodes increases proportionally to the square root of the system size, and we related this power law to other power laws that characterise the scaling behaviour of the eigenmodes, the participation ratio, the localisation length, and their dependence on energy and system size. When disorder is so strong that the smallest hopping terms can be arbitrarily close to zero, all these power laws change, and we show a crossover between the two scaling regimes. All these results are explained by intuitive arguments based on scaling.

cond-mat.dis-nn↗

arXiv2023

Microbiome abundance patterns as attractors and the implications for the inference of microbial interaction networks

Inferring microbial interaction networks from abundance patterns is an important approach to advance our understanding of microbial communities in general and the human microbiome in particular. Here we suggest discriminating two levels of information contained in microbial abundance data: (1) the quantitative abundance values and (2) the pattern of presences and absences of microbial organisms. The latter allows for a binary view on microbiome data and a novel interpretation of microbial data as attractors, or more precisely as fixed points, of a Boolean network. Starting from these attractors, our aim is to infer an interaction network between the species present in the microbiome samples. To accomplish this task, we introduce a novel inference method that combines the previously published ESABO (Entropy Shifts of Abundance vectors under Boolean Operations) method with an evolutionary algorithm. The key idea of our approach is that the inferred network should reproduce the original set of (observed) binary abundance patterns as attractors. We study the accuracy and runtime properties of this evolutionary method, as well as its behavior under incomplete knowledge of the attractor sets. Based on this theoretical understanding of the method we then show an application to empirical data.

q-bio.PE↗

arXiv2023

The passage of time and top-down causation

It is often claimed that the fundamental laws of physics are deterministic and time-symmetric and that therefore our experience of the passage of time is an illusion. This paper will critically discuss these claims and show that they are based on the misconception that the laws of physics are an exact and complete description of nature. I will argue that all supposedly fundamental deterministic and time-symmetric laws have their limitations and are supplemented by stochastic and irreversible elements. In fact, a deterministic description of a system is valid only as long as interactions with the rest of the world can be ignored. The most famous example is the quantum measurement process that occurs when a quantum system interacts with a macroscopic environment such as a measurement apparatus. This environment determines in a top-down way the possible outcomes of the measurement and their probabilities. I will argue that more generally the possible events that can occur in a system and their probabilities are the result of top-down influences from the wider context. In this way the microscopic level of a system is causally open to influences from the macroscopic environment. In conclusion, indeterminism and irreversibility are the result of a system being embedded in a wider context.

quant-ph↗

arXiv2022

Explicit expressions for stationary states of the Lindblad equation for a finite state space

The Lindblad equation describes the time evolution of a density matrix of a quantum mechanical system. Stationary solutions are obtained by time-averaging the solution, which will in general depend on the initial state. We provide an analytical expression for the steady states of the Lindblad equation using the quantum jump unraveling, a version of an ergodic theorem, and the stationary probabilities of the corresponding discret-time Markov chains. Our result is valid when the number of states appearing the in quantum trajectory is finite. The classical case of a Markov jump-process is recovered as a special case, and differences between the two are discussed.

quant-ph↗

arXiv2022

Master Stability Functions for metacommunities with two types of habitats

Current questions in ecology revolve around instabilities in the dynamics on spatial networks and particularly the effect of node heterogeneity. We extend the Master Stability Function formalism to inhomogeneous biregular networks having two types of spatial nodes. Notably, this class of systems also allows the investigation of certain types of dynamics on higher-order networks. Combined with the Generalized Modelling approach to study the linear stability of steady states, this is a powerful tool to numerically asses the stability of large ensembles of systems. We analyze the stability of ecological metacommunities with two distinct types of habitats analytically and numerically in order to identify several sets of conditions under which the dynamics can become stabilized by dispersal. Our analytical approach allows general insights into stabilizing and destabilizing effects in metapopulations. Specifically, we show that maladaptive dispersal may be stable under certain conditions.

q-bio.PE↗

arXiv2021

Dephasing versus collapse: Lessons from the tight-binding model with noise

Condensed matter physics at room temperature usually assumes that electrons in conductors can be described as spatially narrow wave packets - in contrast to what the Schrödinger equation would predict. How a finite-temperature environment can localize wave functions is still being debated. Here, we represent the environment by a fluctuating potential and investigate different unravellings of the Lindblad equation that describes the one-dimensional tight-binding model in the presence of such a potential. While all unravellings show a fast loss of phase coherence, only part of them lead to narrow wave packets, among them the quantum-state diffusion unravelling. Surprisingly, the decrease of the wave packet width for the quantum state diffusion model with increasing noise strength is slower than that of the phase coherence length. In addition to presenting analytical and numerical results, we also provide phenomenological explanations for them. We conclude that as long as no feedback between the wave function and the environment is taken into account, there will be no unique description of an open quantum system in terms of wave functions. We consider this to be an obstacle to understanding the quantum-classical transition.

quant-ph↗

arXiv2021

Obtaining the long-term behavior of Master equations with finite state space from the structure of the associated state transition network

The Master equation describes the time evolution of the probabilities of a system with a discrete state space. This time evolution approaches for long times a stationary state that will in general depend on the initial probability distribution. Conditions under which the stationary state is unique are usually given as remarks appended to more comprehensive theories in the mathematical literature. We provide a direct and complete derivation of a necessary and sufficient criterion for when this steady state is unique. We translate this problem into the language of graph theory and show that there is a one-to-one correspondence between minimal absorbing sets within the state-transition network and linearly independent stationary states of the Master equation.

math-ph↗

arXiv2020

Bifurcation and chaos in nonlinear Lindblad equations

The Lindblad equation describes the dissipative time evolution of a density matrix that characterizes an open quantum system in contact with its environment. The widespread ensemble interpretation of a density matrix requires its time evolution to be linear. However, when the density matrix is obtained from a mean field theory of interacting quantum systems or from a top-down control by a changing classical environment, the ensemble interpretation is inappropriate and nonlinear dynamics arise naturally. We therefore study the dynamical behavior of nonlinear Lindblad equations using the example of a two-level system. By using techniques developed for classical dynamical systems we show that various types of bifurcations and even chaotic dynamics can occur. We also discuss experimental situations for which our results could be relevant.

quant-ph↗

arXiv2020

Predicting properties of the stationary probability currents for two-species reaction systems without solving the Fokker-Planck equation

We derive methods for estimating the topology of the stationary probability current $\vec{j}_s$ of the two-species Fokker-Planck equation (FPE) without the need to solve the FPE. These methods are chosen such that they become exact in certain limits, such as infinite system size or vanishing coupling between species in the diffusion matrix. The methods make predictions about the fixed points of $\vec{j}_s$ and their relation to extrema of the stationary probability distribution and to fixed points of the convective field, which is related to the deterministic drift of the system. Furthermore, they predict the rotation sense of $\vec{j}_s$ around extrema of the stationary probability distribution. Even though these methods cannot be proven to be valid away from extrema, the boundary lines between regions with different rotation sense are obtained with surprising accuracy. We illustrate and test these method using simple reaction systems with only one coupling term between the two species as well as a few generic reaction networks taken from literature. We use it also to investigate the shape of non-physical probability currents occurring in reaction systems with detailed balance due to the approximations involved in deriving the Fokker-Planck equation.

cond-mat.stat-mech↗

arXiv2020

Relation between the convective field and the stationary probability distribution of chemical reaction networks

We investigate the relation between the stationary probability distribution of chemical reaction systems and the convective field derived from the chemical Fokker-Planck equation (CFPE) by comparing predictions of the convective field to the results of stochastic simulations based on Gillespie's algorithm. The convective field takes into account the drift term of the CFPE and the reaction bias introduced by the diffusion term. For one-dimensional systems, fixed points and bifurcations of the convective field correspond to extrema and phenomenological bifurcations of the stationary probability distribution whenever the CFPE is a good approximation to the stochastic dynamics. This provides an efficient way to calculate the effect of system size on the number and location of probability maxima and their phenomenological bifurcations in parameter space. For two-dimensional systems, we study models that have saddle-node and Hopf bifurcations in the macroscopic limit. Here, the existence of two stable fixed points of the convective field correlates either with two peaks of the stationary probability distribution, or with a peak and a shoulder. In contrast, a Hopf bifurcation that occurs in the convective field for decreasing system size is not accompanied by the onset of a crater-shaped probability distribution; decreasing system size rather destroys craters and replaces them by local maxima.

cond-mat.stat-mech↗

arXiv2019

Emergence of time

Microphysical laws are time reversible, but macrophysics, chemistry and biology are not. This chapter explores how this asymmetry (a classic example of a broken symmetry) arises due to the cosmological context, where a non-local Direction of Time is imposed by the expansion of the universe. This situation is best represented by an Evolving Block Universe, where local arrows of time (thermodynamic, electrodynamic, gravitational, wave, quantum, biological) emerge in concordance with the Direction of Time because a global Past Condition results in the Second Law of Thermodynamics pointing to the future. At the quantum level, the indefinite future changes to the definite past due to quantum wave function collapse events.

gr-qc↗

arXiv2019

How Downwards Causation Occurs in Digital Computers

Digital computers carry out algorithms coded in high level programs. These abstract entities determine what happens at the physical level: they control whether electrons flow through specific transistors at specific times or not, entailing downward causation in both the logical and implementation hierarchies. This paper explores how this is possible in the light of the alleged causal completeness of physics at the bottom level, and highlights the mechanism that enables strong emergence (the manifest causal effectiveness of application programs) to occur. Although synchronic emergence of higher levels from lower levels is manifestly true, diachronic emergence is generically not the case; indeed we give specific examples where it cannot occur because of the causal effectiveness of higher level variables.

cs.OH↗

arXiv2019

A closed form for Jacobian reconstruction from timeseries and its application as an early warning signal in network dynamics

The Jacobian matrix of a dynamical system describes its response to perturbations. Conversely one can estimate the Jacobian matrix by carefully monitoring how the system responds to environmental noise. Here we present a closed form analytical solution for the calculation of a system's Jacobian from a timeseries. Being able to access a system's Jacobian enables us to perform a broad range of mathematical analyses by which deeper insights into the system can be gained. Here we consider in particular the computation of the leading Jacobian eigenvalue as an early warning signal for critical transition. To illustrate this approach we apply it to ecological meta-foodweb models, which are strongly nonlinear dynamical multi-layer networks. Our analysis shows that accurate results can be obtained, although the data demand of the method is still high.

nlin.CD↗

arXiv2019

The concerted emergence of well-known spatial and temporal ecological patterns in an evolutionary food web model in space

Ecological systems show a variety of characteristic patterns of biodiversity in space and time. It is a challenge for theory to find models that can reproduce and explain the observed patterns. Since the advent of island biogeography these models revolve around speciation, dispersal, and extinction, but they usually neglect trophic structure. Here, we propose and study a spatially extended evolutionary food web model that allows us to study large spatial systems with several trophic layers. Our computer simulations show that the model gives rise simultaneously to several biodiversity patterns in space and time, from species abundance distributions to the waxing and waning of geographic ranges. We find that trophic position in the network plays a crucial role when it comes to the time evolution of range sizes, because the trophic context restricts the occurrence and survival of species especially on higher trophic levels.

q-bio.PE↗