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Remlein B, Seifert U. Nonequilibrium fluctuations of chemical reaction networks at criticality: The Schlögl model as paradigmatic case. J Chem Phys 2024; 160:134103. [PMID: 38557838 DOI: 10.1063/5.0203659] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2024] [Accepted: 03/15/2024] [Indexed: 04/04/2024] Open
Abstract
Chemical reaction networks can undergo nonequilibrium phase transitions upon variation in external control parameters, such as the chemical potential of a species. We investigate the flux in the associated chemostats that is proportional to the entropy production and its critical fluctuations within the Schlögl model. Numerical simulations show that the corresponding diffusion coefficient diverges at the critical point as a function of system size. In the vicinity of the critical point, the diffusion coefficient follows a scaling form. We develop an analytical approach based on the chemical Langevin equation and van Kampen's system size expansion that yields the corresponding exponents in the monostable regime. In the bistable regime, we rely on a two-state approximation in order to analytically describe the critical behavior.
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Affiliation(s)
- Benedikt Remlein
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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2
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Bone RA, Green JR. Optimizing dynamical functions for speed with stochastic paths. J Chem Phys 2022; 157:224101. [PMID: 36546817 DOI: 10.1063/5.0125479] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
Abstract
Living systems are built from microscopic components that function dynamically; they generate work with molecular motors, assemble and disassemble structures such as microtubules, keep time with circadian clocks, and catalyze the replication of DNA. How do we implement these functions in synthetic nanostructured materials to execute them before the onset of dissipative losses? Answering this question requires a quantitative understanding of when we can improve performance and speed while minimizing the dissipative losses associated with operating in a fluctuating environment. Here, we show that there are four modalities for optimizing dynamical functions that can guide the design of nanoscale systems. We analyze Markov models that span the design space: a clock, ratchet, replicator, and self-assembling system. Using stochastic thermodynamics and an exact expression for path probabilities, we classify these models of dynamical functions based on the correlation of speed with dissipation and with the chosen performance metric. We also analyze random networks to identify the model features that affect their classification and the optimization of their functionality. Overall, our results show that the possible nonequilibrium paths can determine our ability to optimize the performance of dynamical functions, despite ever-present dissipation, when there is a need for speed.
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Affiliation(s)
- Rebecca A Bone
- Department of Chemistry, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - Jason R Green
- Department of Chemistry, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
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3
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Bone RA, Sharpe DJ, Wales DJ, Green JR. Stochastic paths controlling speed and dissipation. Phys Rev E 2022; 106:054151. [PMID: 36559408 DOI: 10.1103/physreve.106.054151] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/24/2021] [Accepted: 10/28/2022] [Indexed: 11/24/2022]
Abstract
Natural processes occur in a finite amount of time and dissipate energy, entropy, and matter. Near equilibrium, thermodynamic intuition suggests that fast irreversible processes will dissipate more energy and entropy than slow quasistatic processes connecting the same initial and final states. For small systems, recently discovered thermodynamic speed limits suggest that faster processes will dissipate more than slower processes. Here, we test the hypothesis that this relationship between speed and dissipation holds for stochastic paths far from equilibrium. To analyze stochastic paths on finite timescales, we derive an exact expression for the path probabilities of continuous-time Markov chains from the path summation solution to the master equation. We present a minimal model for a driven system in which relative energies of the initial and target states control the speed, and the nonequilibrium currents of a cycle control the dissipation. Although the hypothesis holds near equilibrium, we find that faster processes can dissipate less under far-from-equilibrium conditions because of strong currents. This model serves as a minimal prototype for designing kinetics to sculpt the nonequilibrium path space so that faster paths produce less dissipation.
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Affiliation(s)
- Rebecca A Bone
- Department of Chemistry, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - Daniel J Sharpe
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, Cambridge, United Kingdom
| | - David J Wales
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, Cambridge, United Kingdom
| | - Jason R Green
- Department of Chemistry, University of Massachusetts Boston, Boston, Massachusetts 02125, USA.,Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
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4
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Skinner DJ, Dunkel J. Estimating Entropy Production from Waiting Time Distributions. PHYSICAL REVIEW LETTERS 2021; 127:198101. [PMID: 34797138 DOI: 10.1103/physrevlett.127.198101] [Citation(s) in RCA: 21] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/23/2021] [Accepted: 08/17/2021] [Indexed: 06/13/2023]
Abstract
Living systems operate far from thermal equilibrium by converting the chemical potential of ATP into mechanical work to achieve growth, replication, or locomotion. Given time series observations of intra-, inter-, or multicellular processes, a key challenge is to detect nonequilibrium behavior and quantify the rate of free energy consumption. Obtaining reliable bounds on energy consumption and entropy production directly from experimental data remains difficult in practice, as many degrees of freedom typically are hidden to the observer, so that the accessible coarse-grained dynamics may not obviously violate detailed balance. Here, we introduce a novel method for bounding the entropy production of physical and living systems which uses only the waiting time statistics of hidden Markov processes and, hence, can be directly applied to experimental data. By determining a universal limiting curve, we infer entropy production bounds from experimental data for gene regulatory networks, mammalian behavioral dynamics, and numerous other biological processes. Further considering the asymptotic limit of increasingly precise biological timers, we estimate the necessary entropic cost of heartbeat regulation in humans, dogs, and mice.
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Affiliation(s)
- Dominic J Skinner
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA
| | - Jörn Dunkel
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA
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Abstract
Temporal order in living matters reflects the self-organizing nature of dynamical processes driven out of thermodynamic equilibrium. Because of functional reasons, the period of a biochemical oscillation must be tuned to a specific value with precision; however, according to the thermodynamic uncertainty relation (TUR), the precision of the oscillatory period is constrained by the thermodynamic cost of generating it. After reviewing the basics of chemical oscillations using the Brusselator as a model system, we study the glycolytic oscillation generated by octameric phosphofructokinase (PFK), which is known to display a period of several minutes. By exploring the phase space of glycolytic oscillations, we find that the glycolytic oscillation under the cellular condition is realized in a cost-effective manner. Specifically, over the biologically relevant range of parameter values of glycolysis and octameric PFK, the entropy production from the glycolytic oscillation is minimal when the oscillation period is (5-10) min. Furthermore, the glycolytic oscillation is found at work near the phase boundary of limit cycles, suggesting that a moderate increase of glucose injection rate leads to the loss of oscillatory dynamics, which is reminiscent of the loss of pulsatile insulin release resulting from elevated blood glucose level.
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Affiliation(s)
- Pureun Kim
- Korea Institute for Advanced Study, Seoul 02455, Korea
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Gaspard P. Stochastic approach to entropy production in chemical chaos. CHAOS (WOODBURY, N.Y.) 2020; 30:113103. [PMID: 33261359 DOI: 10.1063/5.0025350] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/14/2020] [Accepted: 10/14/2020] [Indexed: 06/12/2023]
Abstract
Methods are presented to evaluate the entropy production rate in stochastic reactive systems. These methods are shown to be consistent with known results from nonequilibrium chemical thermodynamics. Moreover, it is proved that the time average of the entropy production rate can be decomposed into the contributions of the cycles obtained from the stoichiometric matrix in both stochastic processes and deterministic systems. These methods are applied to a complex reaction network constructed on the basis of Rössler's reinjection principle and featuring chemical chaos.
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Affiliation(s)
- Pierre Gaspard
- Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles (U.L.B.), Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium
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7
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Nguyen B, Seifert U. Exponential volume dependence of entropy-current fluctuations at first-order phase transitions in chemical reaction networks. Phys Rev E 2020; 102:022101. [PMID: 32942421 DOI: 10.1103/physreve.102.022101] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2020] [Accepted: 07/07/2020] [Indexed: 12/23/2022]
Abstract
In chemical reaction networks, bistability can only occur far from equilibrium. It is associated with a first-order phase transition where the control parameter is the thermodynamic force. At the bistable point, the entropy production is known to be discontinuous with respect to the thermodynamic force. We show that the fluctuations of the entropy production have an exponential volume-dependence when the system is bistable. At the phase transition, the exponential prefactor is the height of the effective potential barrier between the two fixed-points. Our results obtained for Schlögl's model can be extended to any chemical network.
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Affiliation(s)
- Basile Nguyen
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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8
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Rodrigo G. Ab initio scaling laws between noise and mean of gene expression. Phys Rev E 2020; 100:032415. [PMID: 31640034 DOI: 10.1103/physreve.100.032415] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/09/2019] [Indexed: 12/17/2022]
Abstract
Gene expression is inherently noisy due to fluctuations occurring at the molecular level. From a top-down perspective, noise has been traditionally decomposed into an intrinsic component that scales inversely with the mean expression level and an extrinsic component that is constant in absence of regulatory changes. Here, we adopt a bottom-up approach to reveal that extrinsic noise, by itself, can follow the aforementioned decomposition, which entails that one component of it can be confounded with intrinsic noise. Analytical expressions of the noise-mean relationship were derived for different scenarios, which were in part supported by numerical simulations.
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Affiliation(s)
- Guillermo Rodrigo
- Institute for Integrative Systems Biology (I2SysBio), CSIC-University Valencia, 46980 Paterna, Spain
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De Decker Y. Stochastic thermodynamics based on an Einstein-Boltzmann definition of fluctuating entropy. Phys Rev E 2019; 99:032143. [PMID: 30999392 DOI: 10.1103/physreve.99.032143] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2019] [Indexed: 11/07/2022]
Abstract
Stochastic thermodynamics is an extension of classical nonequilibrium thermodynamics to small systems, where fluctuations are expected to play an important role. A central difficulty met when developing such an extension is how to define a nonequilibrium fluctuating entropy. Typically, the expression used is based on Gibbs' formula for entropy at equilibrium. In this work, we show that one can construct an alternative framework for stochastic thermodynamics based on an extension of Einstein's formula connecting the probability of fluctuations and entropy around equilibrium states. We compare the two approaches and discuss, in particular, how they lead to different interpretations of what a stochastic entropy and entropy production represent.
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Affiliation(s)
- Yannick De Decker
- Center for Nonlinear Phenomena and Complex Systems (CENOLI), Université libre de Bruxelles (ULB), Campus Plaine, C.P. 231. B-1050 Brussels, Belgium
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Tomé T, de Oliveira MJ. Stochastic thermodynamics and entropy production of chemical reaction systems. J Chem Phys 2018; 148:224104. [PMID: 29907050 DOI: 10.1063/1.5037045] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We investigate the nonequilibrium stationary states of systems consisting of chemical reactions among molecules of several chemical species. To this end, we introduce and develop a stochastic formulation of nonequilibrium thermodynamics of chemical reaction systems based on a master equation defined on the space of microscopic chemical states and on appropriate definitions of entropy and entropy production. The system is in contact with a heat reservoir and is placed out of equilibrium by the contact with particle reservoirs. In our approach, the fluxes of various types, such as the heat and particle fluxes, play a fundamental role in characterizing the nonequilibrium chemical state. We show that the rate of entropy production in the stationary nonequilibrium state is a bilinear form in the affinities and the fluxes of reaction, which are expressed in terms of rate constants and transition rates, respectively. We also show how the description in terms of microscopic states can be reduced to a description in terms of the numbers of particles of each species, from which follows the chemical master equation. As an example, we calculate the rate of entropy production of the first and second Schlögl reaction models.
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Affiliation(s)
- Tânia Tomé
- Instituto de Física, Universidade de São Paulo, Rua do Matão, 1371, 05508-090 São Paulo, SP, Brazil
| | - Mário J de Oliveira
- Instituto de Física, Universidade de São Paulo, Rua do Matão, 1371, 05508-090 São Paulo, SP, Brazil
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11
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Affiliation(s)
- Tie-jun Xiao
- Guizhou Provincial Key Laboratory of Computational Nano-Material Science, Guizhou Synergetic Innovation Center of Scientific Big Data for Advanced Manufacturing Technology, Guizhou Education University, Guiyang 550018, China
| | - Yun Zhou
- Guizhou Provincial Key Laboratory of Computational Nano-Material Science, Guizhou Synergetic Innovation Center of Scientific Big Data for Advanced Manufacturing Technology, Guizhou Education University, Guiyang 550018, China
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Ceccato A, Frezzato D. Remarks on the chemical Fokker-Planck and Langevin equations: Nonphysical currents at equilibrium. J Chem Phys 2018; 148:064114. [DOI: 10.1063/1.5016158] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023] Open
Affiliation(s)
- Alessandro Ceccato
- Department of Chemical Sciences, University of Padova, Via Marzolo 1, I-35131 Padova, Italy
| | - Diego Frezzato
- Department of Chemical Sciences, University of Padova, Via Marzolo 1, I-35131 Padova, Italy
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13
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Marsland R, England J. Limits of predictions in thermodynamic systems: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2018; 81:016601. [PMID: 28976362 DOI: 10.1088/1361-6633/aa9101] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
The past twenty years have seen a resurgence of interest in nonequilibrium thermodynamics, thanks to advances in the theory of stochastic processes and in their thermodynamic interpretation. Fluctuation theorems provide fundamental constraints on the dynamics of systems arbitrarily far from thermal equilibrium. Thermodynamic uncertainty relations bound the dissipative cost of precision in a wide variety of processes. Concepts of excess work and excess heat provide the basis for a complete thermodynamics of nonequilibrium steady states, including generalized Clausius relations and thermodynamic potentials. But these general results carry their own limitations: fluctuation theorems involve exponential averages that can depend sensitively on unobservably rare trajectories; steady-state thermodynamics makes use of a dual dynamics that lacks any direct physical interpretation. This review aims to present these central results of contemporary nonequilibrium thermodynamics in such a way that the power of each claim for making physical predictions can be clearly assessed, using examples from current topics in soft matter and biophysics.
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Hartich D, Seifert U. Optimal inference strategies and their implications for the linear noise approximation. Phys Rev E 2016; 94:042416. [PMID: 27841626 DOI: 10.1103/physreve.94.042416] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2016] [Indexed: 12/11/2022]
Abstract
We study the information loss of a class of inference strategies that is solely based on time averaging. For an array of independent binary sensors (e.g., receptors, single electron transistors) measuring a weak random signal (e.g., ligand concentration, gate voltage) this information loss is up to 0.5 bit per measurement irrespective of the number of sensors. We derive a condition related to the local detailed balance relation that determines whether or not such a loss of information occurs. Specifically, if the free-energy difference arising from the signal is symmetrically distributed among the forward and backward rates, time integration mechanisms will capture the full information about the signal. As an implication, for the linear noise approximation, we can identify the same loss of information, arising from its inherent simplification of the dynamics.
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Affiliation(s)
- David Hartich
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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15
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De Decker Y, Derivaux JF, Nicolis G. Stochastic thermodynamics of reactive systems: An extended local equilibrium approach. Phys Rev E 2016; 93:042127. [PMID: 27176274 DOI: 10.1103/physreve.93.042127] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2015] [Indexed: 06/05/2023]
Abstract
The recently developed extended local equilibrium approach to stochastic thermodynamics is applied to reactive systems. The properties of the fluctuating entropy and entropy production are analyzed for general linear and for prototypical nonlinear kinetic processes. It is shown that nonlinear kinetics typically induces deviations of the mean entropy production from its value in the deterministic (mean-field) limit. The probability distributions around the mean are derived and shown to qualitatively differ in thermodynamic equilibrium, under nonequilibrium conditions and in the vicinity of criticalities associated to the onset of multistability. In each case large deviation-type properties are shown to hold. The results are compared with those of alternative approaches developed in the literature.
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Affiliation(s)
- Yannick De Decker
- Center for Nonlinear Phenomena and Complex Systems (CENOLI), Université libre de Bruxelles (ULB), Campus Plaine, C.P. 231. B-1050 Brussels, Belgium
- Nonlinear Physical Chemistry Unit, Université libre de Bruxelles (ULB), Campus Plaine, C.P. 231. B-1050 Brussels, Belgium
| | - Jean-François Derivaux
- Center for Nonlinear Phenomena and Complex Systems (CENOLI), Université libre de Bruxelles (ULB), Campus Plaine, C.P. 231. B-1050 Brussels, Belgium
- Nonlinear Physical Chemistry Unit, Université libre de Bruxelles (ULB), Campus Plaine, C.P. 231. B-1050 Brussels, Belgium
| | - Grégoire Nicolis
- Center for Nonlinear Phenomena and Complex Systems (CENOLI), Université libre de Bruxelles (ULB), Campus Plaine, C.P. 231. B-1050 Brussels, Belgium
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Polettini M, Wachtel A, Esposito M. Dissipation in noisy chemical networks: The role of deficiency. J Chem Phys 2015; 143:184103. [DOI: 10.1063/1.4935064] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022] Open
Affiliation(s)
- M. Polettini
- Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit, University of Luxembourg, 162a Avenue de la Faïencerie, Luxembourg L-1511, Luxembourg
| | - A. Wachtel
- Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit, University of Luxembourg, 162a Avenue de la Faïencerie, Luxembourg L-1511, Luxembourg
| | - M. Esposito
- Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit, University of Luxembourg, 162a Avenue de la Faïencerie, Luxembourg L-1511, Luxembourg
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