Consequences of the H Theorem from Nonlinear Fokker-Planck Equations
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A Closure Scheme for Chemical Master Equations
A closure scheme for chemical master equations Patrick Smadbeck and Yiannis N. Kaznessis1 Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455 Edited by Wing Hung Wong, Stanford University, Stanford, CA, and approved July 16, 2013 (received for review April 8, 2013) Probability reigns in biology, with random molecular events dictating equation governs the evolution of the probability, PðX; tÞ,thatthe the fate of individual organisms, and propelling populations of system is at state X at time t: species through evolution. In principle, the master probability À Á equation provides the most complete model of probabilistic behavior ∂P X; t X À Á À Á À Á À Á = T X X′ P X′; t − T X′ X P X; t : [1] in biomolecular networks. In practice, master equations describing ∂t complex reaction networks have remained unsolved for over 70 X′ years. This practical challenge is a reason why master equations, for À Á all their potential, have not inspired biological discovery. Herein, we This is a probability conservation equation, where T X X′ is present a closure scheme that solves the master probability equation the transition propensity from any possible state X′ to state X of networks of chemical or biochemical reactions. We cast the master per unit time. In a network of chemical or biochemical reactions, equation in terms of ordinary differential equations that describe the the transition probabilities are defined by the reaction-rate laws time evolution of probability distribution moments. We postulate as a set of Poisson-distributed reaction propensities; these simply that a finite number of moments capture all of the necessary dictate how many reaction events take place per unit time. -
Fluctuations of Spacetime and Holographic Noise in Atomic Interferometry
General Relativity and Gravitation (2011) DOI 10.1007/s10714-010-1137-7 RESEARCHARTICLE Ertan Gokl¨ u¨ · Claus Lammerzahl¨ Fluctuations of spacetime and holographic noise in atomic interferometry Received: 10 December 2009 / Accepted: 12 December 2010 c Springer Science+Business Media, LLC 2010 Abstract On small scales spacetime can be understood as some kind of space- time foam of fluctuating bubbles or loops which are expected to be an outcome of a theory of quantum gravity. One recently discussed model for this kind of space- time fluctuations is the holographic principle which allows to deduce the structure of these fluctuations. We review and discuss two scenarios which rely on the holo- graphic principle leading to holographic noise. One scenario leads to fluctuations of the spacetime metric affecting the dynamics of quantum systems: (i) an appar- ent violation of the equivalence principle, (ii) a modification of the spreading of wave packets, and (iii) a loss of quantum coherence. All these effects can be tested with cold atoms. Keywords Quantum gravity phenomenology, Holographic principle, Spacetime fluctuations, UFF, Decoherence, Wavepacket dynamics 1 Introduction The unification of quantum mechanics and General Relativity is one of the most outstanding problems of contemporary physics. Though yet there is no theory of quantum gravity. There are a lot of reasons why gravity must be quantized [33]. For example, (i) because of the fact that all matter fields are quantized the interactions, generated by theses fields, must also be quantized. (ii) In quantum mechanics and General Relativity the notion of time is completely different. (iii) Generally, in General Relativity singularities necessarily appear where physical laws are not valid anymore. -
Semi-Classical Lindblad Master Equation for Spin Dynamics
Journal of Physics A: Mathematical and Theoretical J. Phys. A: Math. Theor. 54 (2021) 235201 (13pp) https://doi.org/10.1088/1751-8121/abf79b Semi-classical Lindblad master equation for spin dynamics Jonathan Dubois∗ ,UlfSaalmann and Jan M Rost Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany E-mail: [email protected], [email protected] and [email protected] Received 25 January 2021, revised 18 March 2021 Accepted for publication 13 April 2021 Published 7 May 2021 Abstract We derive the semi-classical Lindblad master equation in phase space for both canonical and non-canonicalPoisson brackets using the Wigner–Moyal formal- ism and the Moyal star-product. The semi-classical limit for canonical dynam- ical variables, i.e. canonical Poisson brackets, is the Fokker–Planck equation, as derived before. We generalize this limit and show that it holds also for non- canonical Poisson brackets. Examples are gyro-Poisson brackets, which occur in spin ensembles, systems of recent interest in atomic physics and quantum optics. We show that the equations of motion for the collective spin variables are given by the Bloch equations of nuclear magnetization with relaxation. The Bloch and relaxation vectors are expressed in terms of the microscopic operators: the Hamiltonian and the Lindblad functions in the Wigner–Moyal formalism. Keywords: Lindblad master equations,non-canonicalvariables,Hamiltonian systems, spin systems, classical and semi-classical limit, thermodynamical limit (Some fgures may appear in colour only in the online journal) 1. Lindblad quantum master equation Since perfect isolation of a system is an idealization [1, 2], open microscopic systems, which interact with an environment, are prevalent in nature as well as in experiments. -
Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems
Entropy 2015, 17, 2853-2861; doi:10.3390/e17052853 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems Constantino Tsallis 1;2 1 Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro - RJ, Brazil; E-Mail: [email protected] 2 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, NM, USA Academic Editor: Antonio M. Scarfone Received: 9 April 2015 / Accepted: 4 May 2015 / Published: 5 May 2015 Abstract: It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional (SBG) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger’s Conceptual inadequacy of the Shannon information in quantum measurements, among many other systems exhibiting various forms of complexity. On the other hand, the Shannon and Khinchin axioms uniquely P mandate the BG form SBG = −k i pi ln pi; the Shore and Johnson axioms follow the same path. Many natural, artificial and social systems have been satisfactorily approached P q 1− i pi with nonadditive entropies such as the Sq = k q−1 one (q 2 R; S1 = SBG), basis of nonextensive statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953 uniqueness theorems have already been generalized in the literature, by Santos 1997 and Abe 2000 respectively, in order to uniquely mandate Sq. We argue here that the same remains to be done with the Shore and Johnson 1980 axioms. -
The Master Equation
The Master Equation Johanne Hizanidis November 2002 Contents 1 Stochastic Processes 1 1.1 Why and how do stochastic processes enter into physics? . 1 1.2 Brownian Motion: a stochastic process . 2 2 Markov processes 2 2.1 The conditional probability . 2 2.2 Markov property . 2 3 The Chapman-Kolmogorov (C-K) equation 3 3.1 The C-K equation for stationary and homogeneous processes . 3 4 The Master equation 4 4.1 Derivation of the Master equation from the C-K equation . 4 4.2 Detailed balance . 5 5 The mean-field equation 5 6 One-step processes: examples 7 6.1 The Poisson process . 8 6.2 The decay process . 8 6.3 A Chemical reaction . 9 A Appendix 11 1 STOCHASTIC PROCESSES 1 Stochastic Processes A stochastic process is the time evolution of a stochastic variable. So if Y is the stochastic variable, the stochastic process is Y (t). A stochastic variable is defined by specifying the set of possible values called 'range' or 'set of states' and the probability distribution over this set. The set can be discrete (e.g. number of molecules of a component in a reacting mixture),continuous (e.g. the velocity of a Brownian particle) or multidimensional. In the latter case, the stochastic variable is a vector (e.g. the three velocity components of a Brownian particle). The figure below helps to get a more intuitive idea of a (discrete) stochastic process. At successive times, the most probable values of Y have been drawn as heavy dots. We may select a most probable trajectory from such a picture. -
Master Equation: Biological Applications and Thermodynamic Description
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by AMS Tesi di Dottorato Alma Mater Studiorum · Universita` di Bologna FACOLTA` DI SCIENZE MATEMATICHE, FISICHE E NATURALI Dottorato in Fisica SETTORE CONCORSUALE: 02/B3-FISICA APPLICATA SETTORE SCIENTIFICO-DISCIPLINARE:FIS/07-FISICA APPLICATA Master Equation: Biological Applications and Thermodynamic Description Thesis Advisor: Presented by: Chiar.mo Prof. LUCIANA RENATA DE GASTONE CASTELLANI OLIVEIRA PhD coordinator: Chiar.mo Prof. FABIO ORTOLANI aa 2013/2014 January 10, 2014 I dedicate this thesis to my parents. Introduction It is well known that many realistic mathematical models of biological systems, such as cell growth, cellular development and differentiation, gene expression, gene regulatory networks, enzyme cascades, synaptic plasticity, aging and population growth need to include stochasticity. These systems are not isolated, but rather subject to intrinsic and extrinsic fluctuations, which leads to a quasi equilibrium state (homeostasis). Any bio-system is the result of a combined action of genetics and environment where the pres- ence of fluctuations and noise cannot be neglected. Dealing with population dynamics of individuals (or cells) of one single species (or of different species) the deterministic description is usually not adequate unless the populations are very large. Indeed the number of individuals varies randomly around a mean value, which obeys deterministic laws, but the relative size of fluctu- ations increases as the size of the population becomes smaller and smaller. As consequence, very large populations can be described by logistic type or chemical kinetics equations but as long as the size is below N = 103 ∼ 104 units a new framework needs to be introduced. -
Nanothermodynamics – a Generic Approach to Material Properties at Nanoscale
Nanothermodynamics – A generic approach to material properties at nanoscale A. K. Rajagopal (1) , C. S. Pande (1) , and Sumiyoshi Abe (2) (1) Naval Research Laboratory, Washington D.C., 20375, USA (2) Institute of Physics, University of Tsukuba, Ibaraki 305-8571, Japan ABSTRACT Granular and nanoscale materials containing a relatively small number of constituents have been studied to discover how their properties differ from their macroscopic counterparts. These studies are designed to test how far the known macroscopic approaches such as thermodynamics may be applicable in these cases. A brief review of the recent literature on these topics is given as a motivation to introduce a generic approach called “Nanothermodynamics”. An important feature that must be incorporated into the theory is the non-additive property because of the importance of ‘surface’ contributions to the Physics of these systems. This is achieved by incorporating fluctuations into the theory right from the start. There are currently two approaches to incorporate this property: Hill (and further elaborated more recently with Chamberlin) initiated an approach by modifying the thermodynamic relations by taking into account the surface effects; the other generalizes Boltzmann-Gibbs statistical mechanics by relaxing the additivity properties of thermodynamic quantities to include nonextensive features of such systems. An outline of this generalization of the macroscopic thermodynamics to nano-systems will be given here. Invited presentation at the Indo-US Workshop on “Nanoscale -
2. Kinetic Theory
2. Kinetic Theory The purpose of this section is to lay down the foundations of kinetic theory, starting from the Hamiltonian description of 1023 particles, and ending with the Navier-Stokes equation of fluid dynamics. Our main tool in this task will be the Boltzmann equation. This will allow us to provide derivations of the transport properties that we sketched in the previous section, but without the more egregious inconsistencies that crept into our previous derivaion. But, perhaps more importantly, the Boltzmann equation will also shed light on the deep issue of how irreversibility arises from time-reversible classical mechanics. 2.1 From Liouville to BBGKY Our starting point is simply the Hamiltonian dynamics for N identical point particles. Of course, as usual in statistical mechanics, here is N ridiculously large: N ∼ O(1023) or something similar. We will take the Hamiltonian to be of the form 1 N N H = !p 2 + V (!r )+ U(!r − !r ) (2.1) 2m i i i j i=1 i=1 i<j ! ! ! The Hamiltonian contains an external force F! = −∇V that acts equally on all parti- cles. There are also two-body interactions between particles, captured by the potential energy U(!ri − !rj). At some point in our analysis (around Section 2.2.3) we will need to assume that this potential is short-ranged, meaning that U(r) ≈ 0forr % d where, as in the last Section, d is the atomic distance scale. Hamilton’s equations are ∂!p ∂H ∂!r ∂H i = − , i = (2.2) ∂t ∂!ri ∂t ∂!pi Our interest in this section will be in the evolution of a probability distribution, f(!ri, p!i; t)overthe2N dimensional phase space. -
[Cond-Mat.Stat-Mech] 3 Sep 2004 H Ee Om Esalte S Tt E H Specific the Get to It Use Then and Shall Tsallis We for Simple Temperature Form
Generalized Entropies and Statistical Mechanics Fariel Shafee Department of Physics Princeton University Princeton, NJ 08540 USA.∗ We consider the problem of defining free energy and other thermodynamic functions when the entropy is given as a general function of the probability distribution, including that for nonextensive forms. We find that the free energy, which is central to the determination of all other quantities, can be obtained uniquely numerically even when it is the root of a transcendental equation. In particular we study the cases of the Tsallis form and a new form proposed by us recently. We compare the free energy, the internal energy and the specific heat of a simple system of two energy states for each of these forms. PACS numbers: 05.70.-a, 05.90 +m, 89.90.+n Keywords: entropy, nonextensive, probability distribution function, free energy, specific heat I. INTRODUCTION heat in both cases and see how it changes with the change of the parameter at different temperatures. We have recently [1] proposed a new form of nonex- tensive entropy which depends on a parameter similar II. ENTROPY AND PDF to Tsallis entropy [2, 3, 4], and in a similar limit ap- proaches Shannon’s classical extensive entropy. We have shown how the definition for this new form of entropy can The pdf is found by optimizing the function arise naturally in terms of mixing of states in a phase cell when the cell is re-scaled, the parameter being a measure L = S + α(1 − pi)+ β(U − piEi) (1) of the rescaling, and how Shannon’s coding theorem [5] Xi Xi elucidates such an approach. -
Finite Size Analysis of a Two-Dimensional Ising Model Within
Finite size analysis of a two-dimensional Ising model within a nonextensive approach N. Crokidakis,1, ∗ D.O. Soares-Pinto,2, † M.S. Reis,3 A.M. Souza,4 R.S. Sarthour,2 and I.S. Oliveira2 1Instituto de F´ısica - Universidade Federal Fluminense, Av. Litorˆanea s/n, 24210-340 Niter´oi - RJ, Brazil. 2Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud 150, Urca, 22290-180 Rio de Janeiro - RJ, Brazil. 3CICECO, Universidade de Aveiro, 3810-193 Aveiro, Portugal. 4Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada. (Dated: June 26, 2018) In this work we present a thorough analysis of the phase transitions that occur in a ferromagnetic 2D Ising model, with only nearest-neighbors interactions, in the framework of the Tsallis nonextensive statistics. We performed Monte Carlo simulations on square lattices with linear sizes L ranging from 32 up to 512. The statistical weight of the Metropolis algorithm was changed according to the nonextensive statistics. Discontinuities in the m(T) curve are observed for q ≤ 0.5. However, we have verified only one peak on the energy histograms at the critical temperatures, indicating the occurrence of continuous phase transitions. For the 0.5 < q ≤ 1.0 regime, we have found continuous phase transitions between the ordered and the disordered phases, and determined the critical exponents via finite-size scaling. We verified that the critical exponents α, β and γ depend 2 on the entropic index q in the range 0.5 < q ≤ 1.0 in the form α(q) = (10 q − 33 q + 23)/20, β(q) = (2 q − 1)/8 and γ(q)=(q2 − q + 7)/4. -
A Maximum Entropy Framework for Nonexponential Distributions
A maximum entropy framework for nonexponential distributions Jack Petersona,b, Purushottam D. Dixitc, and Ken A. Dillb,1 aDepartment of Mathematics, Oregon State University, Corvallis, OR 97331; bLaufer Center for Physical and Quantitative Biology, Departments of Physics and Chemistry, State University of New York, Stony Brook, NY 11794; and cDepartment of Systems Biology, Columbia University, New York, NY 10032 Contributed by Ken A. Dill, November 7, 2013 (sent for review June 26, 2013) P Probability distributions having power-law tails are observed in functional S½fpkg = − kpklog pk subject to constraints, such as a broad range of social, economic, and biological systems. We the known value of the average energy hEi. This procedure gives −βE describe here a potentially useful common framework. We derive the exponential (Boltzmann) distribution, pk ∝ e k ,whereβ distribution functions {pk} for situations in which a “joiner particle” is the Lagrange multiplier that enforces the constraint. This var- k pays some form of price to enter a community of size k − 1, where iational principle has been the subject of various historical justifi- costs are subject to economies of scale. Maximizing the Boltzmann– cations. It is now commonly understood as the approach that Gibbs–Shannon entropy subject to this energy-like constraint chooses the least-biased model that is consistent with the known predicts a distribution having a power-law tail; it reduces to the constraint(s) (39). Boltzmann distribution in the absence of economies of scale. We Is there an equally compelling principle that would select fat- show that the predicted function gives excellent fits to 13 different tailed distributions, given limited information? There is a large distribution functions, ranging from friendship links in social net- literature that explores this. -
Field Master Equation Theory of the Self-Excited Hawkes Process
PHYSICAL REVIEW RESEARCH 2, 033442 (2020) Field master equation theory of the self-excited Hawkes process Kiyoshi Kanazawa1 and Didier Sornette2,3,4 1Faculty of Engineering, Information and Systems, University of Tsukuba, Tennodai, Tsukuba, Ibaraki 305-8573, Japan 2ETH Zurich, Department of Management, Technology, and Economics, Zurich 8092, Switzerland 3Tokyo Tech World Research Hub Initiative, Institute of Innovative Research, Tokyo Institute of Technology, Tokyo 152-8550, Japan 4Institute of Risk Analysis, Prediction, and Management, Academy for Advanced Interdisciplinary Studies, Southern University of Science and Technology, Shenzhen 518055, China (Received 9 January 2020; accepted 25 August 2020; published 21 September 2020) A field theoretical framework is developed for the Hawkes self-excited point process with arbitrary memory kernels by embedding the original non-Markovian one-dimensional dynamics onto a Markovian infinite- dimensional one. The corresponding Langevin dynamics of the field variables is given by stochastic partial differential equations that are Markovian. This is in contrast to the Hawkes process, which is non-Markovian (in general) by construction as a result of its (long) memory kernel. We derive the exact solutions of the Lagrange-Charpit equations for the hyperbolic master equations in the Laplace representation in the steady state, close to the critical point of the Hawkes process. The critical condition of the original Hawkes process is found to correspond to a transcritical bifurcation in the Lagrange-Charpit equations. We predict a power law scaling of the probability density function (PDF) of the intensities in an intermediate asymptotic regime, which crosses over to an asymptotic exponential function beyond a characteristic intensity that diverges as the critical condition is approached.