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Rare Events and Phase Transition in Reaction–Diffusion Systems

Rare Events and Phase Transition in Reaction–Diffusion Systems. Alex Kamenev,. in collaboration with. Vlad Elgart, Virginia Tech. PRE 70 , 041106 (2004); PRE 74 , 041101 (2006);. Ann Arbor, June, 2007. Binary annihilation. Lotka-Volterra model. Reaction–Diffusion Models.

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Rare Events and Phase Transition in Reaction–Diffusion Systems

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  1. Rare Events and Phase Transition in Reaction–Diffusion Systems Alex Kamenev, in collaboration with Vlad Elgart, Virginia Tech. PRE 70, 041106 (2004); PRE 74, 041101 (2006); Ann Arbor, June, 2007

  2. Binary annihilation Lotka-Volterra model Reaction–Diffusion Models Examples: • Dynamical rules • Discreteness

  3. Outline: • Hamiltonian formulation • Rare events calculus • Phase transitions and their classification

  4. Rate equation: • PDF: • Extinction time Example: Branching-Annihilation Reaction rules:

  5. GF properties: • Generating Function (GF): • Multiply ME by and sum over : extinction probability Master Equation

  6. Imaginary time “Schrodinger” equation: Hamiltonian is non-Hermitian Hamiltonian

  7. Conservation of probability • If no particles are created from the vacuum Hamiltonian For arbitrary reaction:

  8. (rare events !) • Assuming: Hamilton-Jacoby equation • Hamilton equations: • Boundary conditions: Semiclassical (WKB) treatment

  9. Rate equation ! Branching-Annihilation Zero energy trajectories !

  10. Extinction time

  11. Equations of Motion: • Rate Equation: Diffusion “Quantum Mechanics”  “QFT “

  12. R Lifetime: Refuge Instanton solution

  13. Phase Transitions • Thermodynamic limit • Extinction time vs. diffusion time Hinrichsen 2000

  14. Critical exponents Hinrichsen 2000

  15. Critical Exponents (cont) • How to calculate critical exponents analytically? • What other reactions belong to the same universality class? • Are there other universality classes and how to classify them?

  16. V(j) Ising universality class: j critical parameter • Critical dimension • Renormalization group, -expansion Equilibrium Models • Landau Free Energy: (Lagrangian field theory)

  17. q V(j) 1 j p 1 1 critical parameter Reaction-diffusion models • Hamiltonian field theory:

  18. Critical dimension Renormalization group, -expansion cf. in d=3 Directed Percolation • Reggeon field theory Janssen 1981, Grassberger 1982 What are other universality classes (if any)?

  19. k Effective Hamiltonian: • Example: k = 2 Pair Contact Process with Diffusion (PCPD) k-particle processes • `Triangular’ topology is stable! All reactions start from at least k particles

  20. Reactions with additional symmetries • Parity conservation: • Reversibility:

  21. Example: First Order Transitions

  22. Wake up ! • Hamiltonian formulation and and its semiclassical limit. • Rare events as trajectories in the phase space • Classification of the phase transitions according to the phase space topology

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