Information thermodynamics of turing pattern
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This document summarizes a research paper on the thermodynamics of Turing patterns. The paper presents a theoretical framework for analyzing Turing patterns as nonequilibrium phase transitions. Key results include: 1) Deriving reaction-diffusion equations to model Turing pattern formation driven by nonlinear feedback and diffusion. 2) Developing expressions for the Gibbs free energy and entropy production of the system based on these equations. 3) Revealing that Turing patterns represent nonequilibrium phase transitions through analyzing the Jacobian and showing the system can extremize energy.
Information thermodynamics of turing pattern
- 1. intro theory result Information Thermodynamics of Turing Patterns ~Gianmaria Falasco, Riccardo Rao, and Massimiliano Esposito (Complex Systems and Statistical Mechanics, University of Luxembourg) DOI: 10.1103/PhysRevLett.121.108301 I. INTRODUCTION II. MOTIVATION III. THEORY IV. TURING PATTERN V. CONCLUSIONS CONTENTS Nonlinear feedback effects destabilized by diffusion Reaction-diffusion systems searching for general extremum principles Research focus : Continual influx of chemicals and energy for dissipative structures complete framework is still lacking- necessary for technological applications Work needed to manipulate a Turing pattern Efficiency of exchanging information through traveling waves Turing patterns as thermodynamic nonequilibrium phase transitionsBy PREMASHIS KUMAR
- 2. intro 𝒖 𝒗 = 𝒂 𝟏𝟏 𝒂 𝟏𝟐 𝒂 𝟐𝟏 𝒂 𝟐𝟐 𝒖 𝒗 𝝀 𝟐 − 𝐓𝐫 𝛌 + 𝒅𝒆𝒕 = 𝟎 Tr=𝝀 𝟏 + 𝝀 𝟐 < 𝟎 𝒅𝒆𝒕 = 𝝀 𝟏 𝝀 𝟐>0 theor y resul t 𝜕𝑍 𝜎 𝜕t =-⩢.𝐽 𝜎 + ρ 𝑆𝜌 𝜎 jρ + 𝐼 𝜎 Reaction-Diffusion equation: −𝑫 𝝈 ⩢ 𝒁 𝝈 Fick’s diffusion current : Stoichiometry Matrix 'Canonical'Gibbs free energy 𝑮 = 𝑽 𝒅𝒓 𝝈(𝝁 𝝈 𝒁 𝝈 − 𝒁 𝝈) 𝑮 = 𝑮 𝒆𝒒 + ℒ(𝒁 𝝈||𝒁 𝝈 𝒆𝒒 ) ℒ(𝒁 𝝈||𝒁 𝝈 𝒆𝒒 )= 𝑽 𝒅𝒓 𝝈(𝒁 𝝈 𝒍𝒏 𝒁 𝝈 𝒁 𝝈 𝒆𝒒 − (𝒁 𝝈 − 𝒁 𝝈 𝒆𝒒 ) Conservative laws : 𝒙 𝒍 𝒙 𝝀 𝑺 𝝆 𝒙 = 𝟎 𝒙 𝒍 𝒙 𝝀 𝒖 𝑺 𝝆 𝒙 = 𝟎, 𝒙 𝒍 𝒙 𝝀 𝒃 𝑺 𝝆 𝒙 ≠ 𝟎 (For open system) 𝓛 = 𝑮= 𝒅 − 𝒓=− ≤ 𝟎 EPR 𝒙 𝒔, 𝒚 𝒔 Jacobian 𝕁 𝑫 = 𝒂 𝟏𝟏− 𝑫 𝟏𝟏 𝑲 𝟐 𝒂 𝟏𝟐 𝒂 𝟐𝟏 𝒂 𝟐𝟐 − 𝑫 𝟐𝟐 𝑲 𝟐
- 3. intro theor y Det resul t D 𝑲 𝟐 𝑫 𝑲 = 𝟎; 𝒅𝑫 𝒅𝑲 𝟐 = 𝟎; 𝐷 = 𝐷11 𝐷22 𝐾4 − 𝐷22 𝑎11 − 𝐷11 𝑎22 𝐾2 + 𝑑𝑒𝑡 Conclusion: Reveal the existence of nonequilibrium phase transition. Framework to quantify the energy cost of pattern manipulations