## >> 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.

## >> 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

## by Mauricio J. del Razo, Daniela Frömberg, Arthur V. Straube, Christof Schütte, Felix Höfling and Stefanie Winkelmann | September 2021

###### 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

## by Ruben Lier, Jay Armas, Stefano Bo, Charlie Duclut, Frank Jülicher and Piotr Surówka | September 2021

###### 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

## by M. J. del Razo, M. Dibak, C. Schütte, F. Noé | March 2021

###### 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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