Relevant bibliographies by topics / Interacting particles systems

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Interacting particles systems'

Author: Grafiati
Published: 2 November 2024
Last updated: 19 July 2025

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Journal articles on the topic "Interacting particles systems"

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Karmanov, Vladimir A. "Abnormal Bound Systems." Universe 8, no. 2 (2022): 95. http://dx.doi.org/10.3390/universe8020095.

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It is taken for granted that bound systems are made of massive constituents that interact through particle exchanges (charged particles interacting via photon exchanges, quarks in elementary particles interacting via gluon exchanges, and nucleons in nuclei interacting via meson exchanges). However, as was recently theoretically found, there exist systems dominated by exchange particles (at least for the zero exchange masses). In these systems, the contribution of massive constituents is negligible. These systems have a relativistic nature (since they are mainly made of massless particles movin
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Abadi, Noam, and Franco Ruzzenenti. "Complex Networks and Interacting Particle Systems." Entropy 25, no. 11 (2023): 1490. http://dx.doi.org/10.3390/e25111490.

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Complex networks is a growing discipline aimed at understanding large interacting systems. One of its goals is to establish a relation between the interactions of a system and the networks structure that emerges. Taking a Lennard-Jones particle system as an example, we show that when interactions are governed by a potential, the notion of structure given by the physical arrangement of the interacting particles can be interpreted as a binary approximation to the interaction potential. This approximation simplifies the calculation of the partition function of the system and allows to study the s
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Sudbury, Aidan. "The survival of various interacting particle systems." Advances in Applied Probability 25, no. 4 (1993): 1010–12. http://dx.doi.org/10.2307/1427804.

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Particles may be removed from a lattice by murder, coalescence, mutual annihilation and simple death. If the particle system is not to die out, the removed particles must be replaced by births. This letter shows that coalescence can be counteracted by arbitrarily small birth-rates and contrasts this with the situations for annihilation and pure death where there are critical phenomena. The problem is unresolved for murder.
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Sudbury, Aidan. "The survival of various interacting particle systems." Advances in Applied Probability 25, no. 04 (1993): 1010–12. http://dx.doi.org/10.1017/s0001867800025878.

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Particles may be removed from a lattice by murder, coalescence, mutual annihilation and simple death. If the particle system is not to die out, the removed particles must be replaced by births. This letter shows that coalescence can be counteracted by arbitrarily small birth-rates and contrasts this with the situations for annihilation and pure death where there are critical phenomena. The problem is unresolved for murder.
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Dolgopyat, Dmitry, Bassam Fayad, Leonid Koralov, and Shuo Yan. "Energy growth for systems of coupled oscillators with partial damping." Nonlinearity 38, no. 5 (2025): 055001. https://doi.org/10.1088/1361-6544/adc8ee.

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Abstract We consider two interacting particles on the circle. The particles are subject to stochastic forcing, which is modeled by white noise. In addition, one of the particles is subject to friction, which models energy dissipation due to the interaction with the environment. We show that, in the diffusive limit, the absolute value of the velocity of the other particle converges to the reflected Brownian motion. In other words, the interaction between the particles is asymptotically negligible in the scaling limit. The proof combines averaging for large energies with large deviation estimate
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Itoh, Yoshiaki, Colin Mallows, and Larry Shepp. "Explicit sufficient invariants for an interacting particle system." Journal of Applied Probability 35, no. 3 (1998): 633–41. http://dx.doi.org/10.1239/jap/1032265211.

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We introduce a new class of interacting particle systems on a graph G. Suppose initially there are Ni(0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution
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Itoh, Yoshiaki, Colin Mallows, and Larry Shepp. "Explicit sufficient invariants for an interacting particle system." Journal of Applied Probability 35, no. 03 (1998): 633–41. http://dx.doi.org/10.1017/s0021900200016284.

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We introduce a new class of interacting particle systems on a graph G. Suppose initially there are N i (0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distributi
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METZNER, WALTER, and CLAUDIO CASTELLANI. "TWO PARTICLE CORRELATIONS AND ORTHOGONALITY CATASTROPHE IN INTERACTING FERMI SYSTEMS." International Journal of Modern Physics B 09, no. 16 (1995): 1959–83. http://dx.doi.org/10.1142/s021797929500080x.

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The wave function of two fermions, repulsively interacting in the presence of a Fermi sea, is evaluated in detail. We consider large but finite systems in order to obtain an unambiguous picture of the two-particle correlations. As recently pointed out by Anderson, in d≤2 dimensions the particles may be correlated even when situated on the Fermi surface. The “partial exclusion principle” for two particles with opposite spin on the same Fermi point is discussed, and related to results from the T-matrix approximation. Particles on different Fermi points are shown to be uncorrelated in d>1. Usi
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Morvan, A., T. I. Andersen, X. Mi, et al. "Formation of robust bound states of interacting microwave photons." Nature 612, no. 7939 (2022): 240–45. http://dx.doi.org/10.1038/s41586-022-05348-y.

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AbstractSystems of correlated particles appear in many fields of modern science and represent some of the most intractable computational problems in nature. The computational challenge in these systems arises when interactions become comparable to other energy scales, which makes the state of each particle depend on all other particles1. The lack of general solutions for the three-body problem and acceptable theory for strongly correlated electrons shows that our understanding of correlated systems fades when the particle number or the interaction strength increases. One of the hallmarks of in
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SKOROHOD, A. V. "Infinite systems of randomly interacting particles." Random Operators and Stochastic Equations 1, no. 1 (1993): 1–14. http://dx.doi.org/10.1515/rose.1993.1.1.1.

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Dissertations / Theses on the topic "Interacting particles systems"

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Glass, K. "Dynamics of systems of interacting particles." Thesis, University of Cambridge, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.599435.

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In this thesis, we bring together three different problems in studying the equations where u<I><SUB>i</SUB></I> is a vector of length <I>m</I>, and <I>β</I> is a real parameter restricted to <I>β </I>≥ -1. The <I>N</I>-body problem concerns <I>N</I> masses attracting one another according to a (1)/(<I>r<SUP>2</SUP></I>) gravitational force. Much work has been done in finding <I>central configurations </I>of the <I>N</I> masses. If a system of masses released from a central configuration, it will remain similar to itself for all time, and can exhibit periodic behaviour.
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Franz, Benjamin. "Recent modelling frameworks for systems of interacting particles." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:ac76d159-4cdd-40c9-b378-6ea1faf48aed.

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In this thesis we study three different modelling frameworks for biological systems of dispersal and combinations thereof. The three frameworks involved are individual-based models, group-level models in the form of partial differential equations (PDEs) and robot swarms. In the first two chapters of the thesis, we present ways of coupling individual based models with PDEs in so-called hybrid models, with the aim of achieving improved performance of simulations. Two classes of such hybrid models are discussed that allow an efficient simulation of multi-species systems of dispersal with reaction
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Romanovsky, Igor Alexandrovich. "Novel properties of interacting particles in small low-dimensional systems." Diss., Available online, Georgia Institute of Technology, 2006, 2006. http://etd.gatech.edu/theses/available/etd-07102006-041659/.

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Thesis (Ph. D.)--Physics, Georgia Institute of Technology, 2007.<br>Landman, Uzi, Committee Member ; Yannouleas, Constantine, Committee Member ; Bunimovich, Leonid, Committee Member ; Chou, Mei-Yin, Committee Member ; Pustilnik, Michael, Committee Member.
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Jacquot, Stéphanie Mireille. "Large systems of interacting particles : the Marcus-Lushnikov process and the β-Laguerre ensemble". Thesis, University of Cambridge, 2012. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.610327.

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Geiger, Benjamin [Verfasser], and Klaus [Akademischer Betreuer] Richter. "From few to many particles: Semiclassical approaches to interacting quantum systems / Benjamin Geiger ; Betreuer: Klaus Richter." Regensburg : Universitätsbibliothek Regensburg, 2020. http://d-nb.info/1215906064/34.

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Lafleche, Laurent. "Dynamique de systèmes à grand nombre de particules et systèmes dynamiques." Thesis, Paris Sciences et Lettres (ComUE), 2019. http://www.theses.fr/2019PSLED010.

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On étudie dans cette thèse le comportement en temps long de solutions d’équations aux dérivées partielles. Celles-ci modélisent des systèmes à grand nombre de particules dont la dynamique est due à des forces externes, internes et à l’interaction entre ces particules. Cependant, on considère différentes échelles. On voyage ainsi du niveau quantique des atomes au niveau macroscopique des étoiles, et l’on voit que des différences apparaissent bien que certaines propriétés soient conservées. Dans ce voyage, on croise le chemin de diverses applications telles que l’astrophysique, les plasmas,les s
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Gracar, Peter. "Random interacting particle systems." Thesis, University of Bath, 2018. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.761028.

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Consider the graph induced by Z^d, equipped with uniformly elliptic random conductances on the edges. At time 0, place a Poisson point process of particles on Z^d and let them perform independent simple random walks with jump probabilities proportional to the conductances. It is well known that without conductances (i.e., all conductances equal to 1), an infection started from the origin and transmitted between particles that share a site spreads in all directions with positive speed. We show that a local mixing result holds for random conductance graphs and prove the existence of a special pe
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Deshayes, Aurélia. "Modèles de croissance aléatoire et théorèmes de forme asymptotique : les processus de contact." Thesis, Université de Lorraine, 2014. http://www.theses.fr/2014LORR0168/document.

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Cette thèse s'inscrit dans l'étude des systèmes de particules en interaction et plus précisément dans celle des modèles de croissance aléatoire qui représentent un quantité qui grandit au cours du temps et s'étend sur un réseau. Ce type de processus apparaît naturellement quand on regarde la croissance d'un cristal ou bien la propagation d'une épidémie. Cette dernière est bien modélisée par le processus de contact introduit en 1974 par Harris. Le processus de contact est un des plus simples systèmes de particules en interaction présentant une transition de phase et l'on connaît maintenant bien
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Wang, Hao Carleton University Dissertation Mathematics and Statistics. "Interacting branching particle systems and superprocesses." Ottawa, 1995.

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Deshayes, Aurélia. "Modèles de croissance aléatoire et théorèmes de forme asymptotique : les processus de contact." Electronic Thesis or Diss., Université de Lorraine, 2014. http://www.theses.fr/2014LORR0168.

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Cette thèse s'inscrit dans l'étude des systèmes de particules en interaction et plus précisément dans celle des modèles de croissance aléatoire qui représentent un quantité qui grandit au cours du temps et s'étend sur un réseau. Ce type de processus apparaît naturellement quand on regarde la croissance d'un cristal ou bien la propagation d'une épidémie. Cette dernière est bien modélisée par le processus de contact introduit en 1974 par Harris. Le processus de contact est un des plus simples systèmes de particules en interaction présentant une transition de phase et l'on connaît maintenant bien
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Books on the topic "Interacting particles systems"

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Kipnis, Claude. Scaling limits of interacting particle systems. Springer, 1999.

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Salabura, Piotr. Vector mesons in strongly interacting systems. Wydawn. Uniwersytetu Jagiellońskiego, 2003.

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Liggett, Thomas M. Interacting particle systems. Springer, 2005.

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Liggett, Thomas M. Interacting Particle Systems. Springer New York, 1985. http://dx.doi.org/10.1007/978-1-4613-8542-4.

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Liggett, Thomas M. Interacting Particle Systems. Springer Berlin Heidelberg, 2005. http://dx.doi.org/10.1007/b138374.

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Liggett, Thomas M. Interacting Particle Systems. Springer New York, 1985.

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1938-, Arenhövel H., ed. Many body structure of strongly interacting systems: Refereed and selected contributions of the symposium "20 years of physics at the Mainz Microtron MAMI," Mainz, Germany, October 19-22, 2005. Societá italiana di fisica, 2006.

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Kipnis, Claude, and Claudio Landim. Scaling Limits of Interacting Particle Systems. Springer Berlin Heidelberg, 1999. http://dx.doi.org/10.1007/978-3-662-03752-2.

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Papanicolaou, George, ed. Hydrodynamic Behavior and Interacting Particle Systems. Springer US, 1987. http://dx.doi.org/10.1007/978-1-4684-6347-7.

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George, Papanicolaou, and University of Minnesota. Institute for Mathematics and its Applications., eds. Hydrodynamic behavior and interacting particle systems. Springer-Verlag, 1987.

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Book chapters on the topic "Interacting particles systems"

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Liverani, C. "Interacting Particles." In Hard Ball Systems and the Lorentz Gas. Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-662-04062-1_8.

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Nolting, Wolfgang, and William D. Brewer. "Systems of Interacting Particles." In Fundamentals of Many-body Physics. Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-540-71931-1_4.

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Nolting, Wolfgang. "Systems of Interacting Particles." In Theoretical Physics 9. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-98326-4_4.

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Cichocki, B. "Interacting Brownian Particles." In Dynamics: Models and Kinetic Methods for Non-equilibrium Many Body Systems. Springer Netherlands, 2000. http://dx.doi.org/10.1007/978-94-011-4365-3_5.

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Skorohod, A. V. "Randomly Interacting Systems Of Particles." In Stochastic Equations for Complex Systems. Springer Netherlands, 1988. http://dx.doi.org/10.1007/978-94-009-3767-3_2.

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Guo, M. Z., and G. Papanicolaou. "Bulk Diffusion for Interacting Brownian Particles." In Statistical Physics and Dynamical Systems. Birkhäuser Boston, 1985. http://dx.doi.org/10.1007/978-1-4899-6653-7_3.

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Mikhailov, Alexander S., and Gerhard Ertl. "Systems with Interacting Particles and Soft Matter." In Chemical Complexity. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-57377-9_11.

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Spohn, Herbert. "Interacting Brownian Particles: A Study of Dyson’s Model." In Hydrodynamic Behavior and Interacting Particle Systems. Springer US, 1987. http://dx.doi.org/10.1007/978-1-4684-6347-7_13.

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Chaikin, P. M., W. D. Dozier, and H. M. Lindsay. "Experiments on Suspensions of Interacting Particles in Fluids." In Hydrodynamic Behavior and Interacting Particle Systems. Springer US, 1987. http://dx.doi.org/10.1007/978-1-4684-6347-7_2.

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Sergeev, Y. A. "Nonlinear Concentration Waves in Fluidized Beds of Interacting Particles." In Mobile Particulate Systems. Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-015-8518-7_15.

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Conference papers on the topic "Interacting particles systems"

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Rokidi, Stamatia G., Petros G. Koutsoukos, Libardo A. Perez, Zahid Amjad, and Robert W. Zuhl. "Iron Oxide Colloidal Suspension Stabilization by Polymeric Dispersants." In CORROSION 2016. NACE International, 2016. https://doi.org/10.5006/c2016-07530.

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Abstract Stabilizing colloidal suspensions helps prevent scale deposition and inorganic particulate matter fouling in aqueous systems. Colloidal suspension stabilization was modeled using iron oxide (Fe2O3) suspensions in synthetic tap water containing a mixed electrolyte system that included a variety of ions interacting with suspended particles. Zeta potential measurements of suspended particles were conducted over a wide pH range (3.5 to 10) both in the absence and in the presence of deposit control polymers (DCPs). Results suggest that both polymer functional group type(s) and molecular we
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Algis, David, Berenger Bramas, Emmanuelle Darles, and Lilian Aveneau. "Efficient GPU Implementation of Particle Interactions with Cutoff Radius and Few Particles per Cell." In 2024 International Symposium on Parallel Computing and Distributed Systems (PCDS). IEEE, 2024. http://dx.doi.org/10.1109/pcds61776.2024.10743892.

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Zeisel, H., and F. Durst. "Computations of Erosion-Corrosion Processes in Separated Two-Phase Flows." In CORROSION 1990. NACE International, 1990. https://doi.org/10.5006/c1990-90029.

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Abstract The paper describes the outcome of development work, yielding a computer code to predict erosion-corrosion rates in CO2-steel system taking into acount the action of the flow, the latter containing solid particles. In the paper, the predicted results are compared with different experimental data and it is shown that good agreement is obtained. It is also demonstrated that the erosion-corrosion model imbetted in the numerical code reacts satisfactorly to temperature and pressure variations as well as to variations of the flow conditions, flow geometry, particle sizes and particle conce
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Chalal, Sofiane, Nina H. Amini, Gaoyue Guo, and Hamed Amini. "Observed quantum particles system with graphon interaction." In 2024 IEEE 63rd Conference on Decision and Control (CDC). IEEE, 2024. https://doi.org/10.1109/cdc56724.2024.10885856.

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Izrailev, F. M. "Regular versus chaotic dynamics in closed systems of interacting Fermi particles." In NUCLEI AND MESOSCOPIC PHYSICS: Workshop on Nuclei and Mesoscopic Physics: WNMP 2004. AIP, 2005. http://dx.doi.org/10.1063/1.1996878.

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Herrera, Dianela, and Sergio Curilef. "Numerical study of a Vlasov equation for systems with interacting particles." In PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2014 (ICNAAM-2014). AIP Publishing LLC, 2015. http://dx.doi.org/10.1063/1.4912388.

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Kim, Bongsoo, Kyozi Kawasaki, Michio Tokuyama, Irwin Oppenheim, and Hideya Nishiyama. "A FDR-Preserving Field Theory for Interacting Brownian Particles: One-Loop Theory and MCT." In COMPLEX SYSTEMS: 5th International Workshop on Complex Systems. AIP, 2008. http://dx.doi.org/10.1063/1.2897790.

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CARMONA, J. M., N. MICHEL, J. RICHERT, and P. WAGNER. "NUCLEAR FRAGMENTATION, PHASE TRANSITIONS AND THEIR CHARACTERIZATION IN FINITE SYSTEMS OF INTERACTING PARTICLES." In Proceedings of the Conference “Bologna 2000: Structure of the Nucleus at the Dawn of the Century”. WORLD SCIENTIFIC, 2001. http://dx.doi.org/10.1142/9789812810939_0023.

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Briegel, Hans. "Entanglement in quantum many-body systems far away from thermodynamic equilibrium." In Workshop on Entanglement and Quantum Decoherence. Optica Publishing Group, 2008. http://dx.doi.org/10.1364/weqd.2008.eoqs1.

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We show that quantum mechanical entanglement can prevail even in noisy open quantum many-body systems at high temperature and far away from thermodynamical equilibrium, despite the deteriorating effect of decoherence. The system consists of a number N of interacting quantum particles, and it can interact and exchange particles with some environment. The effect of decoherence is counteracted by a simple mechanism, where system particles are randomly reset to some standard initial state, e.g. by replacing them with particles from the environment.
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Ozyer, Baris, Ismet Erkmen, and Aydan M. Erkmen. "Catching Continuum Between Preshape and Grasping Based on Fluidics." In ASME 2010 10th Biennial Conference on Engineering Systems Design and Analysis. ASMEDC, 2010. http://dx.doi.org/10.1115/esda2010-24632.

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We propose a new fluidics based methodology to determine a continuum between preshaping and grasping so as to appropriately preshape a multifingered robot hand for creating an optimal initialization of grasp, with minimum energy loss towards task execution, upon landing on an object. In this paper, we investigate the effects of impact forces and momentum transfer between different hand preshapes landing on an object. Momentum transfer parameters lead to modification of object orientation and position at the very initial stage of task after that preshaped fingers land on the object. We model fi
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Reports on the topic "Interacting particles systems"

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Pullammanappallil, Pratap, Haim Kalman, and Jennifer Curtis. Investigation of particulate flow behavior in a continuous, high solids, leach-bed biogasification system. United States Department of Agriculture, 2015. http://dx.doi.org/10.32747/2015.7600038.bard.

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Recent concerns regarding global warming and energy security have accelerated research and developmental efforts to produce biofuels from agricultural and forestry residues, and energy crops. Anaerobic digestion is a promising process for producing biogas-biofuel from biomass feedstocks. However, there is a need for new reactor designs and operating considerations to process fibrous biomass feedstocks. In this research project, the multiphase flow behavior of biomass particles was investigated. The objective was accomplished through both simulation and experimentation. The simulations included
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Varadhan, S. R. Interacting Particle Systems and Their Scaling Limits. Defense Technical Information Center, 1996. http://dx.doi.org/10.21236/ada308783.

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Zhang, Xingyu, Matteo Ciantia, Jonathan Knappett, and Anthony Leung. Micromechanical study of potential scale effects in small-scale modelling of sinker tree roots. University of Dundee, 2021. http://dx.doi.org/10.20933/100001235.

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When testing an 1:N geotechnical structure in the centrifuge, it is desirable to choose a large scale factor (N) that can fit the small-scale model in a model container and avoid unwanted boundary effects, however, this in turn may cause scale effects when the structure is overscaled. This is more significant when it comes to small-scale modelling of sinker root-soil interaction, where root-particle size ratio is much lower. In this study the Distinct Element Method (DEM) is used to investigate this problem. The sinker root of a model root system under axial loading was analysed, with both upw
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Anisimov, Petr Mikhaylovich. Quantum interaction of a few particle system mediated by photons. Office of Scientific and Technical Information (OSTI), 2017. http://dx.doi.org/10.2172/1356103.

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Peter J. Mucha. Final Report: Model interacting particle systems for simulation and macroscopic description of particulate suspensions. Office of Scientific and Technical Information (OSTI), 2007. http://dx.doi.org/10.2172/939459.

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Sviratcheva, K. D., and J. P. Draayer. Realistic Two-body Interactions in Many-nucleon Systems: Correlated Motion beyond Single-particle Behavior. Office of Scientific and Technical Information (OSTI), 2006. http://dx.doi.org/10.2172/885281.

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Chefetz, Benny, and Jon Chorover. Sorption and Mobility of Pharmaceutical Compounds in Soils Irrigated with Treated Wastewater. United States Department of Agriculture, 2006. http://dx.doi.org/10.32747/2006.7592117.bard.

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Research into the fate of pharmaceutical compounds (PCs) in the environment has focused on aspects of removal efficiency during sewage treatment, degradation in surface water and accumulation in soils and sediments. However, very little information is available on the binding interactions of pharmaceuticals with dissolved organic matter (DOM) originating from wastewater treatment. Such interactions can significantly affect the transport potential of PCs in soils by altering compound affinity for soil particle surfaces. Our primary hypothesis is that the transport potential of PCs in soils is s
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Chefetz, Benny, and Jon Chorover. Sorption and Mobility of Pharmaceutical Compounds in Soils Irrigated with Treated Wastewater. United States Department of Agriculture, 2006. http://dx.doi.org/10.32747/2006.7709883.bard.

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Research into the fate of pharmaceutical compounds (PCs) in the environment has focused on aspects of removal efficiency during sewage treatment, degradation in surface water and accumulation in soils and sediments. However, very little information is available on the binding interactions of pharmaceuticals with dissolved organic matter (DOM) originating from wastewater treatment. Such interactions can significantly affect the transport potential of PCs in soils by altering compound affinity for soil particle surfaces. Our primary hypothesis is that the transport potential of PCs in soils is s
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Grabowski, Wojciech. Evolution of Precipitation Particle Size Distributions within MC3E Systems and its Impact on Aerosol-Cloud-Precipitation Interactions. Office of Scientific and Technical Information (OSTI), 2016. http://dx.doi.org/10.2172/1244254.

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10

Kollias, Pavlos. Evolution of Precipitation Particle Size Distributions within MC3E Systems and its Impact on Aerosol-Cloud-Precipitation Interactions: Final Report. Office of Scientific and Technical Information (OSTI), 2017. http://dx.doi.org/10.2172/1374165.

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