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>> Publications at DIEP

DIEP aims at performing groundbreaking interdisciplinary research that connects different fields and shines light on aspects of emergence. Below you find a list of publications of some of the researchers associated with DIEP starting in late 2020.

>> A Novel Method for Holographic Transport

by Tuna Demircik, Domingo Gallegos, Umut Gürsoy, Matti Järvinen, Ruben Lier | October 2023

We introduce a novel and effective method to compute transport coefficients in strongly interacting plasma states in holographic QFTs. Our method is based on relating the IR limit of fluctuations on a gravitational background to its variations providing a previously overlooked connection between boundary and near horizon data. We use this method to derive analytic formulas for the viscosities of an ansiotropic plasma state in the presence of an external magnetic field or another isotropy breaking external source. We then apply our findings to holographic QCD.

>> Strategy Revision Phase with Payoff Threshold in the Public Goods Game

by Marco Alberto Javarone, Shaurya Pratap Singh | October 2023

Commonly, the strategy revision phase in evolutionary games relies on payoff comparison. Namely, agents compare their payoff with the opponent, assessing whether changing strategy can be potentially convenient. Even tiny payoff differences can be crucial in this decision process. In this work, we study the dynamics of cooperation in the Public Goods Game, introducing a threshold ϵ in the strategy revision phase. In doing so, payoff differences narrower than ϵ entail the decision process reduces to a coin flip. 
Interestingly, with ordinary agents, results show that payoff thresholds curb the emergence of cooperation. Yet, the latter can be sustained by these thresholds if the population is composed of conformist agents, which replace the random-based revision with selecting the strategy of the majority. To conclude, agents sensible only to consistent payoff differences may represent 'real-world' individuals unable to properly appreciate advantages or disadvantages when facing a dilemma. These agents may be detrimental to the emergence of cooperation or, on the contrary, supportive when endowed with a conformist attitude.

>> Molecular modelling of odd viscoelastic fluids

by Paweł Matus, Ruben Lier, Piotr Surówska | October 2023

We consider an active, stochastic microscopic model of particles suspended in a fluid and show that the coarse-grained description of this model renders odd viscoelasticity. The model is made up of odd dumbbells, each featuring a robotic device as the bead, which exhibits a particular torque response. We analytically compute the stress-stress correlator and corroborate the results using molecular dynamics simulations. We also provide a unified analytical framework for several experimental and numerical setups designed to elucidate odd effects in fluids.

>> Probe particles in odd active viscoelastic fluids: how activity and dissipation determine linear stability

by Charlie Duclut, Stefano Bo, Ruben Lier, Jay Armas, Piotr Surówska, Frank Jülicher | October 2023

Odd viscoelastic materials are constrained by fewer symmetries than their even counterparts. The breaking of these symmetries allow these materials to exhibit different features, which have attracted considerable attention in recent years. Immersing a bead in such complex fluids allows for probing their physical properties, highlighting signatures of their oddity and exploring consequences of these broken symmetries. We present the conditions under which the activity of an odd viscoelastic fluid can give rise to linear instabilities in the motion of the probe particle and unveil how the features of the probe particle dynamics depend on the oddity and activity of the viscoelastic medium in which it is immersed.

>> Quantifying High-Order Interdependencies in Entangled Quantum States

Here, we leverage recent advances in information theory to develop a novel method to characterise the dominant character of the high-order dependencies of quantum systems. To this end, we introduce the Q-information: an information-theoretic measure capable of distinguishing quantum states dominated by synergy or redundancy. We illustrate the measure by investigating the properties of paradigmatic entangled Qubit states and find that -- in contrast to classical systems -- quantum systems need at least four variables to exhibit high-order properties. Furthermore, our results reveal that unitary evolution can radically affect the internal information organisation in a way that strongly depends on the corresponding Hamiltonian. Overall, the Q-information sheds light on novel aspects of the internal organisation of quantum systems and their time evolution, opening new avenues for studying several quantum phenomena and related technologies.

>> Schwinger-Keldysh effective field theory for stable and causal relativistic hydrodynamics

by Akash Jain, Pavel Kovtun | September 2023

We construct stable and causal effective field theories (EFTs) for describing statistical fluctuations in relativistic diffusion and relativistic hydrodynamics. These EFTs are fully non-linear, including couplings to background sources, and enable us to compute n-point time-ordered correlation functions including the effects of statistical fluctuations. The EFTs we construct are inspired by the Maxwell-Cattaneo model of relativistic diffusion and Müller-Israel-Stewart model of relativistic hydrodynamics respectively, and have been derived using both the Martin-Siggia-Rose and Schwinger-Keldysh formalisms. The EFTs non-linearly realise the dynamical Kubo-Martin-Schwinger (KMS) symmetry, which ensures that n-point correlation functions and interactions in the theory satisfy the appropriate fluctuation-dissipation theorems. Since these EFTs typically admit ultraviolet sectors that are not fixed by the low-energy infrared symmetries, we find that they simultaneously admit multiple realisations of the dynamical KMS symmetry. We also comment on certain obstructions to including statistical fluctuations in the recently-proposed stable and causal Bemfica-Disconzi-Noronha-Kovtun model of relativistic hydrodynamics.

>> Carrollian fluids and spontaneous breaking of boost symmetry

by Jay Armas, Emil Have | August 2023

In the hydrodynamic regime, field theories typically have their boost symmetry spontaneously broken due to the presence of a thermal rest frame although the associated Goldstone field does not acquire independent dynamics. We show that this is not the case for Carrollian field theories where the boost Goldstone field plays a central role. This allows us to give a first-principles derivation of the equilibrium currents and dissipative effects of Carrollian fluids. We also demonstrate that the limit of vanishing speed of light of relativistic fluids is a special case of this class of Carrollian fluids. Our results shine light on the thermodynamic properties and thermal partition functions of Carrollian field theories.

>> Data-driven dynamical coarse-graining for condensed matter systems

by Mauricio J. del Razo, Daan Crommelin, Peter G. Bolhuis | June 2023

Simulations of condensed matter systems often focus on the dynamics of a few distinguished components but require integrating the dynamics of the full system. A prime example is a molecular dynamics simulation of a (macro)molecule in solution, where both the molecules(s) and the solvent dynamics needs to be integrated. This renders the simulations computationally costly and often unfeasible for physically or biologically relevant time scales. Standard coarse graining approaches are capable of reproducing equilibrium distributions and structural features but do not properly include the dynamics. In this work, we develop a stochastic data-driven coarse-graining method inspired by the Mori-Zwanzig formalism. This formalism shows that macroscopic systems with a large number of degrees of freedom can in principle be well described by a small number of relevant variables plus additional noise and memory terms. Our coarse-graining method consists of numerical integrators for the distinguished components of the system, where the noise and interaction terms with other system components are substituted by a random variable sampled from a data-driven model. Applying our methodology on three different systems -- a distinguished particle under a harmonic potential and under a bistable potential; and a dimer with two metastable configurations -- we show that the resulting coarse-grained models are not only capable of reproducing the correct equilibrium distributions but also the dynamic behavior due to temporal correlations and memory effects. Our coarse-graining method requires data from full-scale simulations to be parametrized, and can in principle be extended to different types of models beyond Langevin dynamics.

>> Risk aversion promotes cooperation

by Jay Armas, Wout Merbis, Janusz Meylahn, Soroush Rafiee Rad, Mauricio J. del Razo | June 2023

Cooperative dynamics are central to our understanding of many phenomena in living and complex systems, including the transition to multicellularity, the emergence of eusociality in insect colonies, and the development of full-fledged human societies. However, we lack a universal mechanism to explain the emergence of cooperation across length scales, across species, and scalable to large populations of individuals. We present a novel framework for modelling cooperation games with an arbitrary number of players by combining reaction networks, methods from quantum mechanics applied to stochastic complex systems, game theory and stochastic simulations of molecular reactions. Using this framework, we propose a novel and robust mechanism based on risk aversion that leads to cooperative behaviour in population games. Rather than individuals seeking to maximise payouts in the long run, individuals seek to obtain a minimum set of resources with a given level of confidence and in a limited time span. We explicitly show that this mechanism leads to the emergence of new Nash equilibria in a wide range of cooperation games. Our results suggest that risk aversion is a viable mechanism to explain the emergence of cooperation in a variety of contexts and with an arbitrary number of individuals greater than three.

>> Low-temperature Holographic Screens Correspond to Einstein-Rosen Bridges

by Marco Alberto Javarone | June 2023

Recent conjectures on the complexity of black holes suggest that their evolution manifests in the structural properties of Einstein-Rosen bridges, like the length and volume. The complexity of black holes relates to the computational complexity of their dual, namely holographic, quantum systems identified via the Gauge/Gravity duality framework. Interestingly, the latter allows us to study the evolution of a black hole as the transformation of a qubit collection performed through a quantum circuit. In this work, we focus on the complexity of Einstein-Rosen bridges. More in detail, we start with a preliminary discussion about their computational properties, and then we aim to assess whether an Ising-like model could represent their holographic dual. In this regard, we recall that the Ising model captures essential aspects of complex phenomena such as phase transitions and, in general, is deeply related to information processing systems. To perform this assessment, which relies on a heuristic model, we attempt to describe the dynamics of information relating to an Einstein-Rosen bridge encoded in a holographic screen in terms of dynamics occurring in a spin lattice at low temperatures. We conclude by discussing our observations and related implications.

>> Efficient simulations of epidemic models with tensor networks: application to the one-dimensional SIS model

by Wout Merbis, Clélia de Mulatier, Philippe Corboz | May 2023

The contact process is an emblematic model of a non-equilibrium system, containing a phase transition between inactive and active dynamical regimes. In the epidemiological context, the model is known as the susceptible-infected-susceptible (SIS) model, and widely used to describe contagious spreading. In this work, we demonstrate how accurate and efficient representations of the full probability distribution over all configurations of the contact process on a one-dimensional chain can be obtained by means of Matrix Product States (MPS). We modify and adapt MPS methods from many-body quantum systems to study the classical distributions of the driven contact process at late times. We give accurate and efficient results for the distribution of large gaps, and illustrate the advantage of our methods over Monte Carlo simulations. Furthermore, we study the large deviation statistics of the dynamical activity, defined as the total number of configuration changes along a trajectory, and investigate quantum-inspired entropic measures, based on the second Rényi entropy.

>> Complex information dynamics of epidemic spreading in low-dimensional networks

by Wout Merbis, Manlio de Domenico | May 2023

The statistical field theory of information dynamics on complex networks concerns the dynamical evolution of large classes of models of complex systems. Previous work has focused on networks where nodes carry an information field, which describes the internal state of the node, and its dynamical evolution. In this work, we propose a more general mathematical framework to model information dynamics on complex networks, where the internal node states are vector valued, thus allowing each node to carry multiple types of information. This setup is relevant for many biophysical and socio-technological models of complex systems, ranging from viral dynamics on networks to models of opinion dynamics and social contagion. The full dynamics of these systems is described in the space of all possible network configurations, as opposed to a node-based perspective. Here, we illustrate the mathematical framework presented in an accompanying letter, while focusing on an exemplary application of epidemic spreading on a low-dimensional network.

>> Emergent information dynamics in many-body interconnected systems

by Wout Merbis, Manlio de Domenico | May 2023

The information implicitly represented in the state of physical systems allows one to analyze them with analytical techniques from statistical mechanics and information theory. In the case of complex networks such techniques are inspired by quantum statistical physics and have been used to analyze biophysical systems, from virus-host protein-protein interactions to whole-brain models of humans in health and disease. Here, instead of node-node interactions, we focus on the flow of information between network configurations. Our numerical results unravel fundamental differences between widely used spin models on networks, such as voter and kinetic dynamics, which cannot be found from classical node-based analysis. Our model opens the door to adapting powerful analytical methods from quantum many-body systems to study the interplay between structure and dynamics in interconnected systems.

>> The relationship between pathological brain activity and functional network connectivity in glioma patients

by Fernando N Santos et al | April 2023

Our results indicate that pathological activity and connectivity co-localize in a complex manner in glioma. This insight is relevant to our understanding of disease progression and cognitive symptomatology.

>> The longitudinal relation between executive functioning and multilayer network topology in glioma patients

by Fernando N Santos et al | April 2023

Many patients with glioma, primary brain tumors, suffer from poorly understood executive functioning deficits before and/or after tumor resection. We aimed to test whether frontoparietal network centrality of multilayer networks, allowing for integration across multiple frequencies, relates to and predicts executive functioning in glioma. Patients with glioma (n = 37) underwent resting-state magnetoencephalography and neuropsychological tests assessing word fluency, inhibition, and set shifting before (T1) and one year after tumor resection (T2). We constructed binary multilayer networks comprising six layers, with each layer representing frequency-specific functional connectivity between source-localized time series of 78 cortical regions. Average frontoparietal network multilayer eigenvector centrality, a measure for network integration, was calculated at both time points. Regression analyses were used to investigate associations with executive functioning. At T1, lower multilayer integration (p = 0.017) and epilepsy (p = 0.006) associated with poorer set shifting (adj. R2 = 0.269). Decreasing multilayer integration (p = 0.022) and not undergoing chemotherapy at T2 (p = 0.004) related to deteriorating set shifting over time (adj. R2 = 0.283). No significant associations were found for word fluency or inhibition, nor did T1 multilayer integration predict changes in executive functioning. As expected, our results establish multilayer integration of the frontoparietal network as a cross-sectional and longitudinal correlate of executive functioning in glioma patients. However, multilayer integration did not predict postoperative changes in executive functioning, which together with the fact that this correlate is also found in health and other diseases, limits its specific clinical relevance in glioma.

>> Asymptotic dynamics of three dimensional supergravity and higher spin gravity revisited

by Wout Merbis, Turmoli Neogi, Arash Ranjbar | April 2023

We reconsider the Hamiltonian reduction of the action for three dimensional AdS supergravity and W3 higher spin AdS gravity in the Chern-Simons formulation under asymptotically anti-de Sitter boundary conditions. We show that the reduction gives two copies of chiral bosons on the boundary. In particular, we take into account the holonomy of the Chern-Simons connection which manifests itself as zero mode of the momentum of the boundary chiral boson. We provide an equivalent formulation of the boundary action which we claim to be the geometric action on symplectic leaves of a (super-)Virasoro or a higher spin WN Poisson manifold in the case of supergravity or higher spin gravity respectively, where the intersection of leaves (given in terms of leaves representatives) can be identified as the bulk holonomy. This concludes the extension to non-linear algebras where the notion of coadjoint representation is not well-defined. The boundary Hamiltonian depends on a choice of boundary conditions and is equivalent to the Schwarzian action for corresponding Brown-Henneaux boundary conditions. We make this connection explicit in the extended supersymmetric case. Moreover, we discuss the geometric action in the case of W3 AdS3 gravity in both 𝔰𝔩(3) highest weight representations based on principal and diagonal 𝔰𝔩(2) embeddings.

>> Dipole superfluid hydrodynamics

by Akash Jain, Kristan Jensen, Ruochuan Liu, Eric Mefford | April 2023

We construct a theory of hydrodynamic transport for systems with conserved dipole moment, U(1) charge, energy, and momentum. These models have been considered in the context of fractons, since their elementary and isolated charges are immobile by symmetry, and have two known translation-invariant gapless phases: a ``p-wave dipole superfluid'' phase where the dipole symmetry is spontaneously broken and a ``s-wave dipole superfluid'' phase where both the U(1) and dipole symmetries are spontaneously broken. We argue on grounds of symmetry and thermodynamics that there is no transitionally-invariant gapless fluid with unbroken dipole symmetry. In this work, we primarily focus on the hydrodynamic description of p-wave dipole superfluids, including leading dissipative corrections. That theory has, in a sense, a dynamical scaling exponent z=2, and its spectrum of fluctuations includes novel subdiffusive modes ω∼−ik4 in the shear sector and magnon-like sound mode ω∼±k2−ik2. By coupling the fluid to background fields, we find response functions of the various symmetry currents. We also present a preliminary generalization of our work to s-wave dipole superfluids, which resemble z=1 fluids and feature sound waves and diffusive shear modes, as in an ordinary fluid. However, the spectrum also contains a magnon-like second-sound mode ω∼±k2±k4−ik4 with subdiffusive attenuation.

>> Ideal fracton superfluids

by Jay Armas and Emil Have | April 2023

We investigate the thermodynamics of equilibrium thermal states and their near-equilibrium dynamics in systems with fractonic symmetries in arbitrary curved space. By explicitly gauging the fracton algebra we obtain the geometry and gauge fields that field theories with conserved dipole moment couple to. We use the resultant fracton geometry to show that it is not possible to construct an equilibrium partition function for global thermal states unless part of the fractonic symmetries is spontaneously broken. This leads us to introduce two classes of fracton superfluids with conserved energy and momentum, namely p-wave and s-wave fracton superfluids. The latter phase is an Aristotelian superfluid at ideal order but with a velocity constraint and can be split into two separate regimes: the U(1) fracton superfluid and the pinned s-wave superfluid regimes. For each of these classes and regimes we formulate a hydrodynamic expansion and study the resultant modes. We find distinctive features of each of these phases and regimes at ideal order in gradients, without introducing dissipative effects. In particular we note the appearance of a sound mode for s-wave fracton superfluids. We show that previous work on fracton hydrodynamics falls into these classes. Finally, we study ultra-dense p-wave fracton superfluids with a large kinetic mass in addition to studying the thermodynamics of ideal Aristotelian superfluids.

>> An Abelian Higgs model for disclinations in nematics

by A. Santos, F. Moraes, Fernando N. Santos, S. Fumeron | March 2023

Topological defects in elastic media may be described by a geometric field akin to three-dimensional gravity. From this point of view, disclinations are line defects of zero width corresponding to a singularity of the curvature in an otherwise flat background. On the other hand, in two dimensions, the Frank free energy of a nematic liquid crystal may be interpreted as an Abelian Higgs Lagrangian. In this work, we construct an Abelian Higgs model coupled to "gravity" for the nematic phase, with the perspective of finding more realistic disclinations. That is, a cylindrically symmetric line defect of finite radius, invariant under translations along its axis. Numerical analysis of the equations of motion indeed yield a +1 winding number "thick" disclination. The defect is described jointly by the gauge and the Higgs fields, that compose the director field, and the background geometry. Away from the defect, the geometry is conical, associated to a dihedral deficit angle. The gauge field, confined to the defect, gives a structure to the disclination while the Higgs field, outside, represents the nematic order.

>> Solution formula for the general birth-death chemical diffusion master equation

by Marucio J. del Razo, Berk Tan Perçin, Alberto Lanconelli | February 2023

We propose a solution formula for chemical diffusion master equations of birth and death type. These equations, proposed and formalized in the recent paper [5], aim at incorporating the spatial diffusion of molecules into the description provided by the classical chemical master equation. We start from the general approach developed in [20] and perform a more detailed analysis of the representation found there. This leads to a solution formula for birth-death chemical diffusion master equations which is expressed in terms of the solution to the reaction-diffusion partial differential equation associated with the system under investigation. Such representation also reveals a striking analogy with the solution to the classical birth-death chemical master equations. The solutions of our findings are also illustrated for several examples.

>> Multicenter Evaluation of AI-generated DIR and PSIR for Cortical and Juxtacortical Multiple Sclerosis Lesion Detection

by Fernando N. Santos et al | February 2023

Artificial intelligence–generated double inversion-recovery and phase-sensitive inversion-recovery images performed well compared with their MRI-acquired counterparts and can be considered reliable in a multicenter setting, with good between-reader and between-center interpretative agreement.

>> Emergence of High-Order Functional Hubs in the Human Brain

by Fernando N. Santos et al | February 2023

Network theory is often based on pairwise relationships between nodes, which is not necessarily realistic for modeling complex systems. Importantly, it does not accurately capture non-pairwise interactions in the human brain, often considered one of the most complex systems. In this work, we develop a multivariate signal processing pipeline to build high-order networks from time series and apply it to resting-state functional magnetic resonance imaging (fMRI) signals to characterize high-order communication between brain regions. We also propose connectivity and signal processing rules for building uniform hypergraphs and argue that each multivariate interdependence metric could define weights in a hypergraph. As a proof of concept, we investigate the most relevant three-point interactions in the human brain by searching for high-order “hubs” in a cohort of 100 individuals from the Human Connectome Project. We find that, for each choice of multivariate interdependence, the high-order hubs are compatible with distinct systems in the brain. Additionally, the high-order functional brain networks exhibit simultaneous integration and segregation patterns qualitatively observable from their high-order hubs. Our work hereby introduces a promising heuristic route for hypergraph representation of brain activity and opens up exciting avenues for further research in high-order network neuroscience and complex systems.

>> Multimodal multilayer network centrality relates to executive functioning

by Fernando N. Santos et al | January 2023

Higher-form symmetries are a valuable tool for classifying topological phases of matter. However, emergent higher-form symmetries in interacting many-body quantum systems are not typically exact due to the presence of topological defects. In this paper, we develop a systematic framework for building effective theories with approximate higher-form symmetries, i.e. higher-form symmetries that are weakly explicitly broken. We focus on a continuous U(1) q-form symmetry and study various patterns of symmetry breaking. This includes spontaneous or explicit breaking of higher-form symmetries, as well as pseudo-spontaneous symmetry breaking patterns where the higher-form symmetry is both spontaneously and explicitly broken. We uncover a web of dualities between such phases and highlight their role in describing the presence of dynamical higher-form vortices. In order to study the out-of-equilibrium dynamics of these phases of matter, we formulate respective hydrodynamic theories and study the spectra of excitations exhibiting higher-form charge relaxation and Goldstone relaxation effects. We show that our framework is able to describe various phase transitions due to proliferation of vortices or defects. This includes the melting transition in smectic crystals, the plasma phase transition from polarised gases to magnetohydrodynamics, the spin-ice transition, the superfluid to neutral fluid transition and the Meissner effect in superconductors, among many others.

>> Approximate higher-form symmetries, topological defects, and dynamical phase transitions

by Jan Armas and Akash Jain | January 2023

Higher-form symmetries are a valuable tool for classifying topological phases of matter. However, emergent higher-form symmetries in interacting many-body quantum systems are not typically exact due to the presence of topological defects. In this paper, we develop a systematic framework for building effective theories with approximate higher-form symmetries, i.e. higher-form symmetries that are weakly explicitly broken. We focus on a continuous U(1) q-form symmetry and study various patterns of symmetry breaking. This includes spontaneous or explicit breaking of higher-form symmetries, as well as pseudo-spontaneous symmetry breaking patterns where the higher-form symmetry is both spontaneously and explicitly broken. We uncover a web of dualities between such phases and highlight their role in describing the presence of dynamical higher-form vortices. In order to study the out-of-equilibrium dynamics of these phases of matter, we formulate respective hydrodynamic theories and study the spectra of excitations exhibiting higher-form charge relaxation and Goldstone relaxation effects. We show that our framework is able to describe various phase transitions due to proliferation of vortices or defects. This includes the melting transition in smectic crystals, the plasma phase transition from polarised gases to magnetohydrodynamics, the spin-ice transition, the superfluid to neutral fluid transition and the Meissner effect in superconductors, among many others.

>> The Euler characteristic as a topological marker for outbreaks in vector-borne disease

by D. Souza E.  Santos and Fernando N. Santos | December 2022

Epidemic outbreaks represent a significant concern for the current state of global health, particularly in Brazil, the epicentre of several vector-borne disease outbreaks and where epidemic control is still a challenge for the scientific community. Data science techniques applied to epidemics are usually made via standard statistical and modelling approaches, which do not always lead to reliable predictions, especially when the data lacks a piece of reliable surveillance information needed for precise parameter estimation. In particular, dengue outbreaks reported over the past years raise concerns for global health care, and thus novel data-driven methods are necessary to predict the emergence of outbreaks. In this work, we propose a parameter-free approach based on geometric and topological techniques, which extracts geometrical and topological invariants as opposed to statistical summaries used in established methods. Specifically, our procedure generates a time-varying network from a time-series of new epidemic cases based on synthetic time-series and real dengue data across several districts of Recife, the fourth-largest urban area in Brazil. Subsequently, we use the Euler characteristic (EC) to extract key topological invariant of the epidemic time-varying network and we finally compared the results with the effective reproduction number (Rt) for each data set. Our results unveil a strong correlation between epidemic outbreaks and the EC. In fact, sudden changes in the EC curve preceding and/or during an epidemic period emerge as a warning sign for an outbreak in the synthetic data, the EC transitions occur close to the periods of epidemic transitions, which is also corroborated. In the real dengue data, where data is intrinsically noise, the EC seems to show a better sign-to-noise ratio once compared to Rt. In analogy with later studies on noisy data by using EC in positron emission tomography scans, the EC estimates the number of regions with high connectivity in the epidemic network and thus has potential to be a signature of the emergence of an epidemic state. Our results open the door to the development of alternative/complementary topological and geometrical data-driven methods to characterise vector-borne disease outbreaks, specially when the conventional epidemic surveillance methods are not effective in a scenario of extreme noise and lack of robustness in the data.

>> On black hole interior reconstruction, singularities and the emergence of time

by Jan de Boer, Daniel Jafferis and Lampros Lamprou| December 2022

We propose a CFT definition of local observables in both the exterior and interior of bulk black holes, whenever such an interior exists. We achieve this by introducing a small microcanonical black hole as a "probe" and using its modular flow to propagate operators from the asymptotic boundary to the interior of other black holes along its worldline, elaborating on the ideas of [2009.04476]. The key conceptual advance is a CFT criterion for selecting states whose modular flow acts as geometric proper time translation in the bulk, which we dub "local equilibrium" states. Our interior reconstruction depends on the choice of code subspace but not on the specific black hole microstate and does not suffer from the "frozen vacuum" problem of other approaches. By virtue of our construction, the question of firewall typicality reduces to a technical problem we articulate and we identify a CFT correlator that is expected to signal the approach to the black hole singularity via a universal divergence. We end with comments on the utility of our framework to the quest for a quantum description of de Sitter cosmologies.

>> Holographic duals of the N=1* gauge theory

by Jay Armas, Giorgos Batzios, Jan Pieter van der Schaar| December 2022

We use the long-wavelength effective theory of black branes (blackfold approach) to perturbatively construct holographic duals of the vacua of the N=1* supersymmetric gauge theory. Employing the mechanism of Polchinski and Strassler, we consider wrapped black five-brane probes with D3-brane charge moving in the perturbative supergravity backgrounds corresponding to the high and low temperature phases of the gauge theory. Our approach recovers the results for the brane potentials and equilibrium configurations known in the literature in the extremal limit, while away from extremality we find metastable black D3-NS5 configurations with horizon topology ℝ3×𝕊2×𝕊3 in certain regimes of parameter space, which cloak potential brane singularities. We uncover novel features of the phase diagram of the N=1* gauge theory in different ensembles and provide further evidence for the appearance of metastable states in holographic backgrounds dual to confining gauge theories.

>> Hydrodynamics of plastic deformations in electronic crystals

by Jay Armas, Erik van Heumen, Akash Jain and Ruben Lier | November 2022

We construct a new hydrodynamic framework describing plastic deformations in electronic crystals. The framework accounts for pinning, phase, and momentum relaxation effects due to translational disorder, diffusion due to the presence of interstitials and vacancies, and strain relaxation due to plasticity and dislocations. We obtain the hydrodynamic mode spectrum and correlation functions in various regimes in order to identify the signatures of plasticity in electronic crystal phases. In particular, we show that proliferation of dislocations de-pins the spatially resolved conductivity until the crystal melts, after which point a new phase of a pinned electronic liquid emerges. In addition, the mode spectrum exhibits a competition between pinning and plasticity effects, with the damping rate of some modes being controlled by pinning-induced phase relaxation and some by plasticity-induced strain relaxation. We find that the recently discovered damping-attenuation relation continues to hold for pinned-induced phase relaxation even in the presence of plasticity and dislocations. We also comment on various experimental setups that could probe the effects of plasticity. The framework developed here is applicable to a large class of physical systems including electronic Wigner crystals, multicomponent charge density waves, and ordinary crystals.

>> Determining maximal entropy functions for objective Bayesian inductive logic

by Soroush Rafiee Rad, J. Landes, Jon Williamson | October 2022

According to the objective Bayesian approach to inductive logic, premisses inductively entail a conclusion just when every probability function with maximal entropy, from all those that satisfy the premisses, satisfies the conclusion. However, when premisses and conclusion are constraints on probabilities of sentences of a first-order predicate language, it is by no means obvious how to determine these maximal entropy functions. This paper makes progress on the problem in the following ways. Firstly, we introduce the concept of an entropy limit point and show that, if the set of probability functions satisfying the premisses contains an entropy limit point, then this limit point is unique and is the maximal entropy probability function. Next, we turn to the special case in which the premisses are simply sentences of the logical language. We show that if the uniform probability function gives the premisses positive probability, then the maximal entropy function can be found by simply conditionalising this uniform prior on the premisses. We generalise our results to demonstrate agreement between the maximal entropy approach and Jeffrey conditionalisation in the case in which there is a single premiss that specifies the probability of a sentence of the language. We show that, after learning such a premiss, certain inferences are preserved, namely inferences to inductive tautologies. Finally, we consider potential pathologies of the approach: we explore the extent to which the maximal entropy approach is invariant under permutations of the constants of the language, and we discuss some cases in which there is no maximal entropy probability function.

>> Probabilistic Entailment on First Order Languages and Reasoning with Inconsistencies

by Soroush Rafiee Rad | October 2022

We investigate an approach for drawing logical inference from inconsistent premisses. The main idea in this approach is that the inconsistencies in the premisses should be interpreted as uncertainty of the information. We propose a mechanism, based on Kinght’s study of inconsistency, for revising an inconsistent set of premisses to a minimally uncertain, probabilistically consistent one. We will then generalise the probabilistic entailment relation introduced in for propositional languages to first order case to draw logical inference from a probabilistic set of premisses. We will then argue how this combination can allow us to limit the effect of uncertainty introduced by inconsistent premisses to only the reasoning on the part of the premise set that is relevant to the inconsistency

>> Chemical diffusion master equation: formulations of reaction--diffusion processes on the molecular level

by Mauricio del Razo, S. Winkelmann, R. Klein, F. Hofling | October 2022

The chemical diffusion master equation (CDME) describes the probabilistic dynamics of reaction--diffusion systems at the molecular level [del Razo et al., Lett. Math. Phys. 112:49, 2022]; it can be considered the master equation for reaction--diffusion processes. The CDME consists of an infinite ordered family of Fokker--Planck equations, where each level of the ordered family corresponds to a certain number of particles and each particle represents a molecule. The equations at each level describe the spatial diffusion of the corresponding set of particles, and they are coupled to each other via reaction operators --linear operators representing chemical reactions. These operators change the number of particles in the system, and thus transport probability between different levels in the family. In this work, we present three approaches to formulate the CDME and show the relations between them. We further deduce the non-trivial combinatorial factors contained in the reaction operators, and we elucidate the relation to the original formulation of the CDME, which is based on creation and annihilation operators acting on many-particle probability density functions. Finally we discuss applications to multiscale simulations of biochemical systems among other future prospects.

>> Artificial Collusion: Examining Supracompetitive Pricing by Q-Learning Algorithms

by A. den Boer, Janusz Meylahn, M. Schinkel | September 2022

We examine recent claims that a particular Q-learning algorithm used by competitors ‘autonomously’ and systematically learns to collude, resulting in supracompetitive prices and extra profits for the firms sustained by collusive equilibria. A detailed analysis of the inner workings of this algorithm reveals that there is no immediate reason for alarm. We set out what is needed to demonstrate the existence of a colluding price algorithm that does form a threat to competition.

>> Intrinsic fluctuations of reinforcement learning promote cooperation

by Wolfram Barfuss, Janusz Meylahn | September 2022

In this work, we ask for and answer what makes classical reinforcement learning cooperative. Cooperating in social dilemma situations is vital for animals, humans, and machines. While evolutionary theory revealed a range of mechanisms promoting cooperation, the conditions under which agents learn to cooperate are contested. Here, we demonstrate which and how individual elements of the multi-agent learning setting lead to cooperation. Specifically, we consider the widely used temporal-difference reinforcement learning algorithm with epsilon-greedy exploration in the classic environment of an iterated Prisoner's dilemma with one-period memory. Each of the two learning agents learns a strategy that conditions the following action choices on both agents' action choices of the last round. We find that next to a high caring for future rewards, a low exploration rate, and a small learning rate, it is primarily intrinsic stochastic fluctuations of the reinforcement learning process which double the final rate of cooperation to up to 80\%. Thus, inherent noise is not a necessary evil of the iterative learning process. It is a critical asset for the learning of cooperation. However, we also point out the trade-off between a high likelihood of cooperative behavior and achieving this in a reasonable amount of time. Our findings are relevant for purposefully designing cooperative algorithms and regulating undesired collusive effects.

>> Ellipticity control of terahertz high-harmonic generation in a Dirac semimetal

by Piotr Surówka et al | August 2022

We report on terahertz high-harmonic generation in a Dirac semimetal as a function of the driving-pulse ellipticity and on a theoretical study of the field-driven intraband kinetics of massless Dirac fermions.Very efficient control of third-harmonic yield and polarization state is achieved in electron-doped Cd3As2 thin films at room temperature. The observed tunability is understood as resulting from terahertz-field driven intraband kinetics of the Dirac fermions. Our study paves the way for exploiting nonlinear optical properties of Dirac matter for applications in signal processing and optical communications.

>> Fractonic Berezinskii-Kosterlitz-Thouless transition from a renormalization group perspective

by K. Grosvenor, R. Lier, Piotr Surówka | July 2022

Proliferation of defects is a mechanism that allows for topological phase transitions. Such a phase transition is found in two dimensions for the XY-model, which lies in the Berezinskii-Kosterlitz-Thouless (BKT) universality class. The transition point can be found using renormalization group analysis. We apply renormalization group arguments to determine the nature of BKT transitions for the three-dimensional plaquette-dimer model, which is a model that exhibits fractonic mobility constraints. We show that an important part of this analysis demands a modified dimensional analysis that changes the interpretation of scaling dimensions upon coarse-graining. Using this modified dimensional analysis we compute the beta functions of the model and predict a finite critical value above which the fractonic phase melts, proliferating dipoles. Importantly, the transition point and its value are found unequivocally within the formalism of renormalization group.

>> Lift force in odd compressible fluids

by R. Lier, C. Duclut, S. Bo, Jay Armas, F. Julicher, Piotr Surówka | May 2022

We compute the response matrix for a tracer particle in a compressible fluid with odd viscosity living on a two-dimensional surface. Unlike the incompressible case, we find that an odd compressible fluid can produce an odd lift force on a tracer particle. Using a "shell localization" formalism, we provide analytic expressions for the drag and odd lift forces acting on the tracer particle in a steady state and also at finite frequency. Importantly, we find that the existence of an odd lift force in a steady state requires taking into account the non-conservation of the fluid mass density due to the coupling between the two-dimensional surface and the three-dimensional bulk fluid.

>> Non-local microwave electrodynamics in ultra-pure PdCoO2

by Piotr Surówka et al | April 2022

The motion of electrons in the vast majority of conductors is diffusive, obeying Ohm's law. However, the recent discovery and growth of high-purity materials with extremely long electronic mean free paths has sparked interest in non-ohmic alternatives, including viscous and ballistic flow. Although non-ohmic transport regimes have been discovered across a range of materials, including two-dimensional electron gases, graphene, topological semimetals, and the delafossite metals, determining their nature has proved to be challenging. Here, we report on a new approach to the problem, employing broadband microwave spectroscopy of the delafossite metal PdCoO2 in three distinct sample geometries that would be identical for diffusive transport. The observed differences, which go as far as differing power laws, take advantage of the hexagonal symmetry of PdCoO2. This permits a particularly elegant symmetry-based diagnostic for non-local electrodynamics, with the result favouring ballistic over strictly hydrodynamic flow. Furthermore, it uncovers a new effect for ballistic electron flow, owing to the highly facetted shape of the hexagonal Fermi surface. We combine our extensive dataset with an analysis of the Boltzmann equation to characterize the non-local regime in PdCoO2. More broadly, our results highlight the potential of broadband microwave spectroscopy to play a central role in investigating exotic transport regimes in the new generation of ultra-high conductivity materials.

>> Bose-Hubbard realisation of fracton defects

by K. Giergel, R. Lier, Piotr Surówka, A. Kosior | May 2022

Bose-Hubbard models are simple paradigmatic lattice models used to study dynamics and phases of quantum bosonic matter. We combine the extended Bose-Hubbard model in the hard-core regime with ring-exchange hoppings. By investigating the symmetries and low-energy properties of the Hamiltonian we argue that the model hosts fractonic defect excitations. We back up our claims with exact numerical simulations of defect dynamics exhibiting mobility constraints. Moreover, we confirm the robustness of our results against fracton symmetry breaking perturbations. Finally we argue that this model can be experimentally realized in recently proposed quantum simulator platforms with big time crystals, thus paving a way for the controlled study of many-body dynamics with mobility constraints.

>> Approximate symmetries, pseudo-Goldstones, and the second law of thermodynamics

by Jay Armas, Akash Jain and Ruben Lier | December 2021

We propose a general hydrodynamic framework for systems with spontaneously broken approximate symmetries. The second law of thermodynamics naturally results in relaxation in the hydrodynamic equations, and enables us to derive a universal relation between damping and diffusion of pseudo- Goldstones. We discover entirely new physical effects sensitive to explicitly broken symmetries. We focus on systems with approximate U(1) and translation symmetries, with direct applications to pinned superfluids and charge density waves. We also comment on the implications for chiral perturbation theory.

>> A stable and causal model of magnetohydrodynamics

by Jay Armas and Filippo Camilloni | January 2022

We formulate the theory of first-order dissipative magnetohydrodynamics in an arbitrary hydrodynamic frame under the assumption of parity-invariance and discrete charge symmetry. We study the mode spectrum of Alfvén and magnetosonic waves as well as the spectrum of gapped excitations and derive constraints on the transport coefficients such that generic equilibrium states with constant magnetic fields are stable and causal under linearised perturbations. We solve these constraints for a specific equation of state and show that there exists a large family of hydrodynamic frames that renders the linear fluctuations stable and causal. This theory does not require introducing new dynamical degrees of freedom and therefore is a promising and simpler alternative to Müller-Israel-Stewart-type theories. Together with a detailed analysis of transport, entropy production and Kubo formulae, the theory presented here is well suited for studying dissipative effects in various contexts ranging from heavy-ion collisions to astrophysics.

>> Momentum-dependent scaling exponents of nodal self-energies measured in strange metal cuprates and          modelled using semi-holography

by Mark S. Golden et al | November 2021

The anomalous strange metal phase found in high-Tc cuprates does not follow the conventional condensed-matter principles enshrined in the Fermi liquid and presents a great challenge for theory. Highly precise experimental determination of the electronic self-energy can provide a test bed for theoretical models of strange metals, and angle-resolved photoemission can provide this as a function of frequency, momentum, temperature and doping. Here we show that constant energy cuts through the nodal spectral function in (Pb,Bi)2Sr2−xLaxCuO6+δ have a non-Lorentzian lineshape, meaning the nodal self-energy is k dependent. We show that the experimental data are captured remarkably well by a power law with a k-dependent scaling exponent smoothly evolving with doping, a description that emerges naturally from AdS/CFT-based semi-holography. This puts a spotlight on holographic methods for the quantitative modelling of strongly interacting quantum materials like the cuprate strange metals.

>> Fractons in curved space

by Akash Jain and Kristan Jensen | November 2021

We consistently couple simple continuum field theories with fracton excitations to curved spacetime backgrounds. We consider homogeneous and isotropic fracton field theories, with a conserved U(1) charge and dipole moment. Coupling to background fields allows us to consistently define a stress-energy tensor for these theories and obtain the respective Ward identities. Along the way, we find evidence for a mixed gauge-gravitational anomaly in the symmetric tensor gauge theory which naturally couples to conserved dipoles. Our results generalise to systems with arbitrarily higher conserved moments, in particular, a conserved quadrupole moment.

>> Reduced plasticity in coupling strength in the SCN clock in aging as revealed by Kuramoto modelling

by Anouk W. van Beurden, Janusz M. Meylahn, Stefan Achterhof, Johanna H. Meijer, Jos H. T. Rohling | September 2021

Circadian clocks drive daily rhythms in physiology and behavior. In mammals the clock resides in the suprachiasmatic nucleus (SCN) of the hypothalamus. The SCN consist of a network of coupled neurons which are synchronized to produce a coherent rhythm. Due to plasticity of the network, seasonal adaptation to short winter days and long summer days occurs. Disturbances in circadian rhythmicity of the elderly have negative health effects, such as neurodegenerative diseases. With the rise in life expectancy this is becoming a major issue. In our paper, we used a model to compare the neuronal coupling in the SCN between young and old animals. We investigated whether exposure to short photoperiod can strengthen coupling among clock cells, and thereby clock function, in old animals. We observed that this is not possible, indicating that simple environmental manipulations are not an option. We suggest that receptor targeted interventions are required, setting the path for further investigation.

>> Notes on symmetries in particle physics

by Akash Jain | September 2021

These are introductory notes on symmetries in quantum field theory and how they apply to particle physics. The notes cover the fundamentals of group theory, their representations, Lie groups, and Lie algebras, along with an elaborate discussion of the representations of SU(N), Lorentz, and Poincare groups and their respective algebras. We spend a lot of time on the realisation of these symmetry groups in quantum field theory, as both global and gauge symmetries, as well as their spontaneous breaking and the Higgs mechanism. In the end, we culminate all the lessons from the course to enumerate the symmetries and field content of the Standard Model of particle physics and write down the Standard Model Lagrangian. Special consideration is given to how the weak-force gauge bosons and the matter fields obtain their mass via the Higgs mechanism.

>> Tracking probabilistic truths: a logic for statistical learning

by Soroush Rafiee Rad, Alexandru Baltag and Sonja Smeets | September 2021

We propose a new model for forming and revising beliefs about unknown probabilities. To go beyond what is known with certainty and represent the agent’s beliefs about probability, we consider a plausibility map, associating to each possible distribution a plausibility ranking. Beliefs are defined as in Belief Revision Theory, in terms of truth in the most plausible worlds (or more generally, truth in all the worlds that are plausible enough). We consider two forms of conditioning or belief update, corresponding to the acquisition of two types of information: (1) learning observable evidence obtained by repeated sampling from the unknown distribution; and (2) learning higher-order information about the distribution. The first changes only the plausibility map (via a ‘plausibilistic’ version of Bayes’ Rule), but leaves the given set of possible distributions essentially unchanged; the second rules out some distributions, thus shrinking the set of possibilities, without changing their plausibility ordering.. We look at stability of beliefs under either of these types of learning, defining two related notions (safe belief and statistical knowledge), as well as a measure of the verisimilitude of a given plausibility model. We prove a number of convergence results, showing how our agent’s beliefs track the true probability after repeated sampling, and how she eventually gains in a sense (statistical) knowledge of that true probability. Finally, we sketch the contours of a dynamic doxastic logic for statistical learning.

>> A probabilistic framework for particle-based reaction-diffusion dynamics using classical Fock space representations

The modeling and simulation of stochastic reaction-diffusion processes is a topic of steady interest that is approached with a wide range of methods. For the highly resolved level of particle-based dynamics there exist comprehensive numerical simulation schemes, while the corresponding mathematical formalization is not yet fully developed. The aim of this paper is to derive the probabilistic evolution equation for chemical reaction kinetics that is coupled to the spatial diffusion of individual particles, as well as to develop a framework for systematically formulating, analyzing, and coarse-graining their stochastic dynamics. To account for the non-conserved and unbounded particle number of this type of open systems, we employ a classical analogue of the quantum mechanical Fock space that contains the symmetrized probability densities of the many-particle configurations in space. Following field-theoretical ideas of second quantization, we introduce creation and annihilation operators that act on single-particle states and that provide natural representations of symmetrized probability densities as well as of reaction and diffusion operators. The resulting evolution equation, termed chemical diffusion master equation (CDME), serves as the foundation to derive more coarse-grained descriptions of reaction-diffusion dynamics. In this regard, we show that a discretization of the evolution equation by projecting onto a Fock subspace generated by a finite set of single-particle densities leads to a generalized form of the well-known reaction-diffusion master equation, which supports non-local reactions between grid cells and which converges properly in the continuum limit.

>> Passive odd viscoelasticity

Active chiral viscoelastic materials exhibit elastic responses perpendicular to the applied stresses, referred to as odd elasticity. We use a covariant formulation of viscoelasticity combined with an entropy production analysis to show that odd elasticity is not only present in active systems but also in broad classes of passive chiral viscoelastic fluids. In addition, we demonstrate that linear viscoelastic chiral solids do require activity in order to manifest odd elastic responses. In order to model the phenomenon of passive odd viscoelasticity we propose a chiral extension of Jeffreys model. We apply our covariant formalism in order to derive the dispersion relations of hydrodynamic modes and obtain clear imprints of odd viscoelastic behavior.

>> Logistic growth on networks: exact solutions for the SI model

by Wout Merbis and Ivan Lodato | September 2021

The SI model is the most basic of all compartmental models used to describe the spreading of information through a population. Despite its apparent simplicity, the analytic solution of this model on networks is still lacking. We address this problem here, using a novel formulation inspired by the mathematical treatment of many-body quantum systems. This allows us to organize the time-dependent expectation values for the state of individual nodes in terms of contributions from subgraphs of the network. We compute these contributions systematically and find a set of symmetry relations among subgraphs of differing topologies. We use our novel approach to compute the spreading of information on three different sample networks. The exact solution, which matches with Monte-Carlo simulations, visibly departs from the mean-field results.

>> Conversations on Quantum Gravity

by Jay Armas | August 2021

A book consisting of 37 interviews to theoretical physicists on their views about the emergence of space and time.

>> Limiting dynamics for Q-learning with memory one in two-player, two-action games

by Janusz Meylahn | July 2021

We develop a computational method to identify all pure strategy equilibrium points in the strategy space of the two-player, two-action repeated games played by Q-learners with one period memory. In order to approximate the dynamics of these Q-learners, we construct a graph of pure strategy mutual best-responses. We apply this method to the iterated prisoner's dilemma and find that there are exactly three absorbing states. By analyzing the graph for various values of the discount factor, we find that, in addition to the absorbing states, limit cycles become possible. We confirm our results using numerical simulations.

>> Probabilities with Gaps and Gluts

by D. Klein, O. Majer, S. R. Rad | April 2021

The paper studies probabilities in non-classical contexts where one is faced with incomplete or even contradictory information. In classical logic, strong completeness is assumed for information that ensures the truth of either A or ¬A (the negation of A) for any A. Even worse when dealing with contradictory information the classical setting is trivialized and does not allow for any meaningful inference. In this paper we follow up on some recent approaches in the literature to investigate non-standard probabilities in situations of incomplete or contradictory information and study the formal machinery that allows for meaningful inference in these scenarios. We do in particular study conditionalisation and learning in this setting and sketch the contours of a theory of belief aggregation for these non-standard probabilities. To apper in Journal of Philosophical Logic.

>> Independent Markov Decomposition: Towards modeling kinetics of biomolecular complexes

by T. Hempel, M. J. del Razo, C. T. Lee, B. C. Taylor, R. E. Amaro, F. Noé | March 2021

Molecular simulations of proteins are often coarse-grained into Markov state models (MSMs), in which each protein configuration is assigned to a state and the transitions between states are inferred from the molecular simulations.  As we explore larger and more complex biological systems, the number of states will face a combinatorial explosion, rendering it impossible to gather sufficient data to parametrize the MSM. In this work, we introduce an approach to decompose a system of interest into separable subsystems. We show that MSMs built for each sub-system, that are cheaper to parametrize, can be later coupled to reproduce the behaviors of the global system. To aid in the choice of decomposition we also describe a score to quantify its goodness. This decomposition strategy has the promise to enable robust modeling of complex biomolecular systems.

>> Two-community noisy Kuramoto model with general interaction strengths. I & II (two papers)

by S. Achterhof, J. Meylahn | March 2021

We study the noisy Kuramoto model on a two-community network. Making this simple adjustment to the standard noisy Kuramoto model leads to an increase in the richness of the model by making it possible for there to be up to four possible steady-state solutions in parts of the parameter space. We introduce a geometric interpretation of the self-consistency equations which allows us to identify all bifurcation points and so leads to a full classification of the phase diagram. Making this change in the underlying network structure of the interaction of the model gives rise to new phenomena like non-symmetrically synchronized solutions.

>> Multiscale molecular kinetics by coupling Markov state models and reaction-diffusion dynamics

Computing large time- and length-scale kinetics of interacting molecules is fundamental to understand biomolecular processes, such as protein-drug binding and virus capsid formation. In general, direct simulation of such processes at scales relevant to life is computationally unfeasible. Long-time dynamics of small to intermediate molecules/complexes can be estimated with Markov models parametrized with large ensembles of short simulations. However, these are still limited to small length-scales. To model multiple molecules at large lengthscales, particle-based reaction-diffusion is more suitable but lacks molecular detail. To combine the best of both, this work develops a general framework to couple Markov state models of molecular kinetics with particle-based reaction-diffusion simulations, which is capable of efficiently simulating large time and length-scales with great accuracy.

>> Exact epidemic models from a tensor product formulation

by W. Merbis  | February 2021

A general framework for the exact description of stochastic systems on networks is presented and applied to many well-known compartmental models of epidemiology. The formulation is inspired by methods from quantum mechanics and represents the state of the population as a vector in the tensor product space of N individual probability vector spaces. The transitions between different states, as specified by the compartmental model as well as the interaction network, are obtained by taking suitable linear combinations of tensor products of smaller matrices. Several mean-field approximations known in the literature are recovered from the exact formulation. In addition, we show how the exact transition rate matrix for the susceptible-infected (SI) model can be used to find analytic solutions for SI outbreaks on finite trees and the cycle graph.

>> Learning to Collude in a Pricing Duopoly

by J. Meylahn and A. den Boer  | January 2021

We design and analyze a pricing algorithm that learns to collude (charge higher than competitive prices) when it "competes" against itself and learns to price according to a best-response when playing against opponents in a certain class. This poses a threat to consumer welfare since the implementation of the algorithm would be legal under current anti-trust legislation.

>> Topology and broken Hermiticity

by C. Coulais, R. Fleury and J. van Wezel | November 2020

A review paper on topological phases of matter published in Nature, following one of the DIEP workshops. In recent years, however, there has been a considerable push to explore the consequences of topology and symmetries in non-conservative, non-equilibrium or non-Hermitian systems. A plethora of driven artificial materials has been reported, blurring the lines between a wide variety of fields in physics and engineering, including condensed matter, photonics, phononics, optomechanics, as well as electromagnetic and mechanical metamaterials. 

>> Topological waves in passive and active fluids on curved surfaces: a unified picture

by R. Green, J. Armas, Jan de Boer, Luca Giomi | November 2020

A study of hydrodynamics on curved surfaces and the emergence of topological edge modes - hydrodynamic waves confined to the equators of surfaces, such as Kelvin and Yanai waves on the surface of the Earth. 
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