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Mastromatteo I, Tóth B, Bouchaud JP. Anomalous impact in reaction-diffusion financial models. PHYSICAL REVIEW LETTERS 2014; 113:268701. [PMID: 25615397 DOI: 10.1103/physrevlett.113.268701] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2014] [Indexed: 06/04/2023]
Abstract
We generalize the reaction-diffusion model A+B→0 in order to study the impact of an excess of A (or B) at the reaction front. We provide an exact solution of the model, which shows that the linear response breaks down: the average displacement of the reaction front grows as the square root of the imbalance. We argue that this model provides a highly simplified but generic framework to understand the square-root impact of large orders in financial markets.
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Affiliation(s)
- I Mastromatteo
- Centre de Mathématiques Appliquées, CNRS, UMR7641, Ecole Polytechnique, 91128 Palaiseau, France
| | - B Tóth
- Capital Fund Management, 23-25 Rue de l'Université, 75007 Paris, France
| | - J-P Bouchaud
- Capital Fund Management, 23-25 Rue de l'Université, 75007 Paris, France
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Mastromatteo I, Tóth B, Bouchaud JP. Agent-based models for latent liquidity and concave price impact. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:042805. [PMID: 24827291 DOI: 10.1103/physreve.89.042805] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/26/2013] [Indexed: 06/03/2023]
Abstract
We revisit the "ɛ-intelligence" model of Tóth et al. [Phys. Rev. X 1, 021006 (2011)], which was proposed as a minimal framework to understand the square-root dependence of the impact of meta-orders on volume in financial markets. The basic idea is that most of the daily liquidity is "latent" and furthermore vanishes linearly around the current price, as a consequence of the diffusion of the price itself. However, the numerical implementation of Tóth et al. (2011) was criticized as being unrealistic, in particular because all the "intelligence" was conferred to market orders, while limit orders were passive and random. In this work, we study various alternative specifications of the model, for example, allowing limit orders to react to the order flow or changing the execution protocols. By and large, our study lends strong support to the idea that the square-root impact law is a very generic and robust property that requires very few ingredients to be valid. We also show that the transition from superdiffusion to subdiffusion reported in Tóth et al. (2011) is in fact a crossover but that the original model can be slightly altered in order to give rise to a genuine phase transition, which is of interest on its own. We finally propose a general theoretical framework to understand how a nonlinear impact may appear even in the limit where the bias in the order flow is vanishingly small.
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Affiliation(s)
| | - Bence Tóth
- Capital Fund Management, 23-25, Rue de l'Université, 75007 Paris, France
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Rongy L, Trevelyan P, De Wit A. Influence of buoyancy-driven convection on the dynamics of A+B→C reaction fronts in horizontal solution layers. Chem Eng Sci 2010. [DOI: 10.1016/j.ces.2009.09.022] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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4
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Affiliation(s)
- Haim Taitelbaum
- a Department of Physics , Bar-Ilan University , Ramat-Gan , 52900 , Israel
| | - Zbigniew Koza
- a Department of Physics , Bar-Ilan University , Ramat-Gan , 52900 , Israel
- b Institute of Theoretical Physics, University of Wroclaw , 50204 , Wroclaw , Poland
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Shipilevsky BM. Death of an A -particle island in the B -particle sea: propagation and evolution of the reaction front A+B<-->C. Phys Rev E 2009; 79:021117. [PMID: 19391716 DOI: 10.1103/physreve.79.021117] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2008] [Indexed: 11/07/2022]
Abstract
We present a systematic theory of propagation and evolution of the reaction front A+B<-->C in the reaction-diffusion system where an island of particles A is surrounded by the uniform sea of particles B . In the first part of the work we give a systematic analysis of the crossover from the irreversible to reversible regime of front propagation in terms of the quasistatic approximation (QSA) and derive the key condition for the island death in the quasiequilibrium front regime. We show that the same as in the case of pure annihilation A+B-->0 the QSA enables the description of the quasiequilibrium front propagation only to a critical point t{c} on approaching to which the QSA is violated. In the second part of the work under the assumption of a sufficiently large forward reaction constant k we derive the perturbative expansion in powers of 1k which gives the asymptotically exact description of the quasiequilibrium front evolution up to t-->infinity. We demonstrate that below some critical value of the reduced backward reaction constant g<g{c} there appear two turning points on the front trajectory, the first of which arises at the sharp localized front stage and is due to the finite number of island particles whereas the second is a consequence of radical transformation of the front structure at the passage through the critical point (delocalization of the front). We find a remarkable property of self-similarity of the passage through the critical point, we derive scaling laws for such passage and show that in the limit g-->0 these laws lead to a striking phenomenon of an abrupt delocalization of the front.
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Affiliation(s)
- Boris M Shipilevsky
- Institute of Solid State Physics, Chernogolovka, Moscow district 142432, Russia
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Polanowski P, Koza Z. Reaction-diffusion fronts in systems with concentration-dependent diffusivities. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:036103. [PMID: 17025704 DOI: 10.1103/physreve.74.036103] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/23/2006] [Indexed: 05/12/2023]
Abstract
We examine properties of a reaction front that forms in irreversible reaction-diffusion systems with concentration-dependent diffusivities. We study two different models of such systems and find that in the limit of a vanishingly small diffusivity of the reaction product, the reaction front dynamics enters a separate universality class, with the front width asymptotically tending to a constant value, and the reaction rate at the reaction front center diminishing with time t as t(-1/2). This behavior can be also observed in systems with nonvanishing (but small) diffusivity of the reaction product at intermediate times.
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Affiliation(s)
- Piotr Polanowski
- Department of Molecular Physics, Technical University of Łódź, Zeromskiego 116, 90-924 Łódź, Poland
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Hecht I, Moran Y, Taitelbaum H. Reaction-diffusion front width anomalies in disordered media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:051109. [PMID: 16802920 DOI: 10.1103/physreve.73.051109] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2005] [Revised: 02/03/2006] [Indexed: 05/10/2023]
Abstract
We study the front characteristics of the A + B --> C reaction-diffusion system with initially separated reactants in disordered media, exemplified by two-dimensional (2D) percolation. We investigate the front characteristics as a function of the disorder degree in this system, in particular close to criticality. We show that the front width exponent is larger than the mean-field (MF) exponent of 1/6, and at criticality it approaches 1/4, which is the one-dimensional (1D) exponent. We show that previous predictions in the literature for the 2D percolation cluster at criticality are wrong. The results are discussed in the context of other systems with attenuated transport where the front width exponent is smaller than the MF exponent. We also study the short-time behavior of the front width exponent, and discuss the validity of the scaling relations between the relevant exponents.
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Affiliation(s)
- Inbal Hecht
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
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Täuber UC, Howard M, Vollmayr-Lee BP. Applications of field-theoretic renormalization group methods to reaction–diffusion problems. ACTA ACUST UNITED AC 2005. [DOI: 10.1088/0305-4470/38/17/r01] [Citation(s) in RCA: 217] [Impact Index Per Article: 11.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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9
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Yuste SB, Acedo L, Lindenberg K. Reaction front in an A+B-->C reaction-subdiffusion process. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:036126. [PMID: 15089380 DOI: 10.1103/physreve.69.036126] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2003] [Indexed: 05/24/2023]
Abstract
We study the reaction front for the process A+B-->C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular, its unusual behavior at the center of the reaction zone.
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Affiliation(s)
- S B Yuste
- Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
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Sinder M, Pelleg J, Sokolovsky V, Meerovich V. Competing reactions with initially separated components in the asymptotic time region. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:022101. [PMID: 14525025 DOI: 10.1103/physreve.68.022101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2003] [Indexed: 05/24/2023]
Abstract
Two competing irreversible reactions with initially separated components and with essentially different reaction constants are theoretically studied in the asymptotic time region. The description of the two simultaneous reactions is reduced to the consideration of two reactions separated in space. It is shown that the reaction rate profile can have two maxima and their ratio is independent of time. The location and relative value of the maxima are functions of the reaction constants and initial concentrations.
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Affiliation(s)
- Misha Sinder
- Department of Materials Engineering, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel
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11
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Shipilevsky BM. Diffusion-controlled annihilation A+B-->0 with initially separated reactants: the death of an A particle island in the B particle sea. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:060101. [PMID: 16241185 DOI: 10.1103/physreve.67.060101] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/21/2003] [Indexed: 05/04/2023]
Abstract
We consider the diffusion-controlled annihilation dynamics A+B-->0 with equal species diffusivities in the system where an island of particles A is surrounded by the uniform sea of particles B. We show that once the initial number of particles in the island is large enough, then at any system's dimensionality d the death of the majority of particles occurs in the universal scaling regime within which approximately 4/5 of the particles die at the island expansion stage and the remaining approximately 1/5 at the stage of its subsequent contraction. In the quasistatic approximation, the scaling of the reaction zone has been obtained for the cases of mean-field (d>or=d(c)) and fluctuation (d<d(c)) dynamics of the front.
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12
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Koza Z. Asymptotic expansion for reversible A+B<-->C reaction-diffusion process. ACTA ACUST UNITED AC 2002; 66:011103. [PMID: 12241337 DOI: 10.1103/physreve.66.011103] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2002] [Indexed: 11/07/2022]
Abstract
We study long-time properties of reversible reaction-diffusion systems of type A+B<-->C by means of the perturbation expansion in powers of 1/t (inverse of time). For the case of equal diffusion coefficients we present exact formulas for the asymptotic forms of reactant concentrations and a complete, recursive expression for an arbitrary term of the expansions. Taking an appropriate limit we show that by studying reversible reactions one can obtain "singular" solutions typical of irreversible reactions.
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Affiliation(s)
- Zbigniew Koza
- Institute of Theoretical Physics, University of Wrocław, plac Maxa Borna 9, PL-50204 Wrocław, Poland.
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Park SH, Parus S, Kopelman R, Taitelbaum H. Gel-free experiments of reaction-diffusion front kinetics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:055102. [PMID: 11735998 DOI: 10.1103/physreve.64.055102] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/2001] [Indexed: 05/23/2023]
Abstract
We present a gel-free experimental system to study the kinetics of the reaction front in the A+B-->C reaction-diffusion system with initially-separated reactants. The experimental setup consists of a CCD camera monitoring the kinetics of the front formed in the reaction-diffusion process Cu(2+) + tetra [disodium ethyl bis(5-tetrazolylazo) acetate trihydrate] -->1:1 complex, in aqueous, gel-free solution, taking place inside a 150 microm gap between two flat microscope slides. The experimental results agree with the theoretical predictions for the time dependence of the front's width, height, and location, as well as the global reaction rate.
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Affiliation(s)
- S H Park
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, USA
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Sinder M, Pelleg J. Asymptotic properties of a reversible A+B<-->C (static) reaction-diffusion process with initially separated reactants. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:3340-8. [PMID: 11088833 DOI: 10.1103/physreve.62.3340] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/10/2000] [Indexed: 04/15/2023]
Abstract
The asymptotic properties of the reaction front formed in a reversible reaction-diffusion process A+B<-->C (static) with initially separated reactants are investigated. The case of arbitrary nonzero values of the diffusion constants D(A) and D(B) and initial concentrations a(0) and b(0) of the reactants A and B is considered. The system is studied in the limit of t-->infinity and g-->0, where t and g are the time and the backward reaction rate constant, respectively. The dynamics of the reaction front is described as a crossover between the "irreversible" regime at times t<<g(-1) and the "reversible" regime at times t>>g(-1). The general properties of the crossover are studied with the help of an extended scaling approach formulated in this work. On the basis of the mean-field equations the analytical solutions in the reversible regime t>>g(-1) inside the reaction zone are discussed. It is shown that in the immobile reaction zone the reaction rate profile has two distinct maxima. This profile differs drastically from the usual single-maximum reaction rate profile inherent in the mobile reaction zone. The two-hump reaction zone profile is the result of the influence of C on the reaction rate in the reversible regime. Numerical computation of the mean-field kinetics equations supports the results of the asymptotic consideration.
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Affiliation(s)
- M Sinder
- Department of Materials Engineering, Ben Gurion University of the Negev, Beer Sheva 84105, Israel
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15
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Sinder M, Pelleg J. Crossover from nonclassical to classical chemical kinetics in an initially separated A + B<-->C reaction-diffusion system with arbitrary diffusion constants. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:4935-4942. [PMID: 11031536 DOI: 10.1103/physreve.61.4935] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/1999] [Indexed: 05/23/2023]
Abstract
The asymptotic long-time properties of the reaction front formed in a reversible reaction-diffusion process A + B<-->C with initially separated reactants are investigated. The case of arbitrary nonzero values of the diffusion constants DA, DB, DC of the components A, B, C and the initial concentrations a0 and b0 of A and B is considered. The system is studied in the limit of g-->0, where g is the backward reaction rate constant. In accordance with previous work, the dynamics of the reaction front is described as a crossover between the "irreversible" regime at times t << g-1 and the "reversible" regime at times t >> g-1. It is shown that through this crossover the macroscopic properties of the reaction front, such as the global rate of C production, the motion of the reaction zone center, and the concentration profiles of the components outside the reaction front, are unchanged. The concentration profiles of the components inside the reaction zone are described by quasistatic equations. The results of the theoretical consideration are confirmed by computing the mean-field kinetics equations.
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Affiliation(s)
- M Sinder
- Department of Materials Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel
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Sinder M, Pelleg J. Properties of the crossover from nonclassical to classical chemical kinetics in a reversible A+B<-->C reaction diffusion process. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:R6259-62. [PMID: 11970609 DOI: 10.1103/physreve.60.r6259] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/03/1999] [Indexed: 04/18/2023]
Abstract
We study the properties of the reaction front formed in a reversible reaction diffusion process A+B<-->C, with initially separated reactants. The case of the mobile C component is considered. In accordance with Chopard et al. [Phys. Rev. E 47, R40 (1993)] the dynamics of the front is described as a crossover between the "irreversible" regime at short times and the "reversible" regime at long times. A refined definition for the rate of C production is suggested, taking into account both the forward and the backward reaction rates. By this definition within the framework of the mean-field equations it is shown that the reversible regime is characterized by scaling of the local rate of C production as R(local) approximately t(-1) and by scaling of the global rate of C production as R(global) approximately t(-1/2). It is also established that in the considered special case of equal diffusion coefficients and equal initial concentrations, the macroscopic properties of the reaction front, such as the global rate of the C production R(global) and the concentration profiles of the components outside the front reaction, are unchanged through this crossover.
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Affiliation(s)
- M Sinder
- Department of Materials Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
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Sánchez AD, Bouzat S, Wio HS. Reaction kinetics of diffusing particles injected into a reactive substrate. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:2677-83. [PMID: 11970069 DOI: 10.1103/physreve.60.2677] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/1999] [Indexed: 11/07/2022]
Abstract
We analyze the kinetics of trapping (A+B-->B) and annihilation (A+B-->0) processes on a one-dimensional substrate with homogeneous distribution of immobile B particles while the A particles are supplied by a localized source. For the imperfect reaction case, we analyze both problems by means of a stochastic model and compare the results with numerical simulations. In addition, we present the exact analytical results of the stochastic model for the case of perfect trapping.
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Affiliation(s)
- A D Sánchez
- Centro Atómico Bariloche (CNEA) and Instituto Balseiro (CNEA and UNC), 8400-San Carlos de Bariloche, Argentina.
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Koza Z, Taitelbaum H. Motion of the reaction front in the A+B-->C reaction-diffusion system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:R1040-R1043. [PMID: 9965318 DOI: 10.1103/physreve.54.r1040] [Citation(s) in RCA: 42] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Barkema GT, Howard MJ, Cardy JL. Reaction-diffusion front for A+B-->. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:R2017-R2020. [PMID: 9964596 DOI: 10.1103/physreve.53.r2017] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Cornell S, Koza Z, Droz M. Dynamic multiscaling of the reaction-diffusion front for mA+nB-->0. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:3500-3505. [PMID: 9963827 DOI: 10.1103/physreve.52.3500] [Citation(s) in RCA: 40] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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