<?xml version="1.0" encoding="utf-8"?>
<export-example>
  <doc>
    <id>9903</id>
    <completedYear>2024</completedYear>
    <publishedYear>2025</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume>18, 001</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2025-01-06</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Dynamics of systems with varying number of particles: from Liouville equations to general master equations for open systems</title>
    <abstract language="eng">A varying number of particles is one of the most relevant characteristics of systems of interest in nature and technology, ranging from the exchange of energy and matter with the surrounding environment to the change of particle number through internal dynamics such as reactions. The physico-mathematical modeling of these systems is extremely challenging, with the major difficulty being the time dependence of the number of degrees of freedom and the additional constraint that the increment or reduction of the number and species of particles must not violate basic physical laws. Theoretical models, in such a case, represent the key tool for the design of computational strategies for numerical studies that deliver trustful results. In this manuscript, we review complementary physico-mathematical approaches of varying number of particles inspired by rather different specific numerical goals. As a result of the analysis on the underlying common structure of these models, we propose a unifying master equation for general dynamical systems with varying number of particles. This equation embeds all the previous models and can potentially model a much larger range of complex systems, ranging from molecular to social agent-based dynamics.</abstract>
    <parentTitle language="eng">SciPost Physics</parentTitle>
    <identifier type="doi">10.21468/SciPostPhys.18.1.001</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="SourceTitle">SciPost Physics</enrichment>
    <enrichment key="AcceptedDate">2024-12-04</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <author>Mauricio del Razo</author>
    <submitter>Mauricio del Razo</submitter>
    <author>Luigi Delle Site</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="institutes" number="MSoCP">Modeling and Simulation of Complex Processes</collection>
    <collection role="persons" number="delrazo">del Razo Sarmina, Mauricio</collection>
    <collection role="projects" number="DFG-OpenMultiscaleBiochem">DFG-OpenMultiscaleBiochem</collection>
  </doc>
  <doc>
    <id>9931</id>
    <completedYear>2025</completedYear>
    <publishedYear>2025</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue>14</issue>
    <volume>58</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Open reaction-diffusion systems: bridging probabilistic theory and simulations across scales</title>
    <abstract language="eng">Reaction-diffusion processes are the foundational model for a diverse range of complex systems, ranging from biochemical reactions to social agent-based phenomena. The underlying dynamics of these systems occur at the individual particle/agent level, and in realistic applications, they often display interaction with their environment through energy or material exchange with a reservoir. This requires intricate mathematical considerations, especially in the case of material exchange since the varying number of particles/agents results in ``on-the-fly'' modification of the system dimension. In this work, we first overview the probabilistic description of reaction-diffusion processes at the particle level, which readily handles varying number of particles. We then extend this model to consistently incorporate interactions with macroscopic material reservoirs. Based on the resulting expressions, we bridge the probabilistic description with macroscopic concentration-based descriptions for linear and nonlinear reaction-diffusion systems, as well as for an archetypal open reaction-diffusion system. Using these mathematical bridges across scales, we finally develop numerical schemes for open reaction-diffusion systems, which we implement in two illustrative examples. This work establishes a methodological workflow to bridge particle-based probabilistic descriptions with macroscopic concentration-based descriptions of reaction-diffusion in open settings, laying the foundations for a multiscale theoretical framework upon which to construct theory and simulation schemes that are consistent across scales.</abstract>
    <parentTitle language="eng">Journal of Physics A: Mathematical and Theoretical</parentTitle>
    <identifier type="doi">10.1088/1751-8121/adc520</identifier>
    <enrichment key="opus.source">publish</enrichment>
    <enrichment key="PeerReviewed">yes</enrichment>
    <author>Mauricio del Razo</author>
    <submitter>Mauricio del Razo</submitter>
    <author>Margarita Kostré</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="persons" number="kostre">Kostre, Margarita</collection>
    <collection role="institutes" number="MSoCP">Modeling and Simulation of Complex Processes</collection>
    <collection role="persons" number="delrazo">del Razo Sarmina, Mauricio</collection>
    <collection role="projects" number="DFG-OpenMultiscaleBiochem">DFG-OpenMultiscaleBiochem</collection>
  </doc>
  <doc>
    <id>8405</id>
    <completedYear/>
    <publishedYear>2022</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue>49</issue>
    <volume>112</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A probabilistic framework for particle-based reaction–diffusion dynamics using classical Fock space representations</title>
    <parentTitle language="eng">Letters in Mathematical Physics</parentTitle>
    <identifier type="arxiv">arXiv:2109.13616</identifier>
    <identifier type="doi">10.1007/s11005-022-01539-w</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <enrichment key="AcceptedDate">2022-04-21</enrichment>
    <author>Mauricio del Razo</author>
    <submitter>Stefanie Winkelmann</submitter>
    <author>Daniela Frömberg</author>
    <author>Arthur Straube</author>
    <author>Christof Schütte</author>
    <author>Felix Höfling</author>
    <author>Stefanie Winkelmann</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="projects" number="SFB1114-C3">SFB1114-C3</collection>
    <collection role="persons" number="winkelmann">Winkelmann, Stefanie</collection>
    <collection role="persons" number="hoefling">Höfling, Felix</collection>
    <collection role="persons" number="straube">Straube, Arthur</collection>
    <collection role="institutes" number="MSoCP">Modeling and Simulation of Complex Processes</collection>
    <collection role="projects" number="MathPlusAA1-5">MathPlusAA1-5</collection>
    <collection role="persons" number="delrazo">del Razo Sarmina, Mauricio</collection>
    <collection role="projects" number="DFG-OpenMultiscaleBiochem">DFG-OpenMultiscaleBiochem</collection>
  </doc>
  <doc>
    <id>8788</id>
    <completedYear/>
    <publishedYear>2023</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue>1</issue>
    <volume>64</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2023-01-20</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Chemical diffusion master equation: formulations of reaction-diffusion processes on the molecular level</title>
    <abstract language="eng">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.</abstract>
    <parentTitle language="eng">Journal of Mathematical Physics</parentTitle>
    <identifier type="doi">10.1063/5.0129620</identifier>
    <identifier type="arxiv">2210.02268</identifier>
    <enrichment key="AcceptedDate">2023-01-02</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <enrichment key="PeerReviewed">yes</enrichment>
    <author>Mauricio del Razo</author>
    <submitter>Stefanie Winkelmann</submitter>
    <author>Stefanie Winkelmann</author>
    <author>Rupert Klein</author>
    <author>Felix Höfling</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="projects" number="SFB1114-C3">SFB1114-C3</collection>
    <collection role="persons" number="winkelmann">Winkelmann, Stefanie</collection>
    <collection role="persons" number="hoefling">Höfling, Felix</collection>
    <collection role="institutes" number="MSoCP">Modeling and Simulation of Complex Processes</collection>
    <collection role="projects" number="MathPlusAA1-5">MathPlusAA1-5</collection>
    <collection role="persons" number="delrazo">del Razo Sarmina, Mauricio</collection>
    <collection role="projects" number="DFG-OpenMultiscaleBiochem">DFG-OpenMultiscaleBiochem</collection>
  </doc>
  <doc>
    <id>9835</id>
    <completedYear>2025</completedYear>
    <publishedYear>2026</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume>98</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2026-01-22</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Field theories and quantum methods for stochastic reaction-diffusion systems</title>
    <abstract language="eng">Complex systems are composed of many particles or agents that move and interact with one another. In most real-world applications, these systems involve a varying number of particles/agents that change due to interactions with the environment or their internal dynamics. The underlying mathematical framework to model these systems must incorporate the spatial transport of particles/agents and their interactions, as well as changes to their copy numbers, all of which can be formulated in terms of stochastic reaction-diffusion processes. However, the standard probabilistic representation of these processes can be overly complex because of the combinatorial aspects arising due to the non-linear interactions and varying particle numbers. In this manuscript, we review the main field theory representations of stochastic reaction-diffusion systems, which handle these issues "under–the–hood’’. First, we focus on bringing techniques familiar to theoretical physicists —such as second quantization, Fock space, path integrals and quantum field theory— back into the classical domain of reaction-diffusion systems. We demonstrate how various field theory representations, which have evolved historically, can all be unified under a single basis-independent representation. We then extend existing quantum-based methods and notation to work directly on the level of the unifying representation, and we illustrate how they can be used to consistently obtain previous known results in a more straightforward manner, such as numerical discretizations and relations between model parameters at multiple scales. Throughout the work, we contextualize how these representations mirror well-known models of chemical physics depending on their spatial resolution, as well as the corresponding macroscopic (large copy number) limits. The framework presented here may find applications in a diverse set of scientific fields, including physical chemistry, theoretical ecology, epidemiology, game theory and socio-economical models of complex systems, specifically in the modeling and multi-scale simulation of complex systems with varying numbers of particles/agents. The presentation is done in a self-contained educational and unifying manner such that it can be followed by researchers across several fields.</abstract>
    <parentTitle language="eng">Rev. Mod. Phys.</parentTitle>
    <identifier type="arxiv">2409.13377</identifier>
    <identifier type="doi">10.1103/9qlw-gyd7</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <enrichment key="AcceptedDate">09.10.2025</enrichment>
    <author>Mauricio del Razo</author>
    <submitter>Mauricio del Razo</submitter>
    <author>Tommaso Lamma</author>
    <author>Wout Merbis</author>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="institutes" number="compsys">Computational Systems Biology</collection>
    <collection role="institutes" number="MSoCP">Modeling and Simulation of Complex Processes</collection>
    <collection role="persons" number="delrazo">del Razo Sarmina, Mauricio</collection>
    <collection role="projects" number="DFG-OpenMultiscaleBiochem">DFG-OpenMultiscaleBiochem</collection>
  </doc>
</export-example>
