1. Rare Events and Phase Transition in Reaction–Diffusion Systems Vlad Elgart, Virginia Tech. Alex Kamenev, in collaboration with Cambridge, Dec. 2008 Michael Assaf, Jerusalem Baruch Meerson, Jerusalem

2. Reaction–Diffusion Models SIR: susceptible- infected-recovered Examples: Binary annihilation Dynamical rules Discreteness

3. Outline: Outline: Hamiltonian formulation Rare events calculus ( Freidlin-Wentzell (?) ) Phase transitions and their classification

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

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

6. Hamiltonian Imaginary time “Schrodinger” equation: Hamiltonian is normally ordered, but non-Hermitian

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

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

9. Branching-Annihilation Rate equation ! Long times: zero energy trajectories !

10. Extinction time Extinction time

11. Time Dependent Rates (e.g. a Catastrophe) Time Dependent Rates (e.g. a Catastrophe) Temporary drop in the reproduction rate  p q 1 1 t  A A B B

12. Susceptible (S) – Infected (I) model Susceptible (S) – Infected (I) model

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

14. Refuge R Lifetime: Instanton solution

15. Phase Transitions Phase Transitions Thermodynamic limit Extinction time vs. diffusion time Hinrichsen 2000

16. Critical exponents Hinrichsen 2000

17. Critical Exponents (cont) Critical Exponents (cont) d=1 d=2 d=3 d 4 0.276 0.5840.811 1 1.734 1.2961.106 1 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? > Hinrichsen 2000

18. Equilibrium Models Landau Free Energy:  V  Ising universality class: critical parameter (Lagrangian field theory) Critical dimension Renormalization group, -expansion

19. Reaction-diffusion models Reaction-diffusion models Hamiltonian field theory: p q 1 1 1  V  critical parameter

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

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

22. Reactions with additional symmetries Reactions with additional symmetries Parity conservation: Reversibility: Cardy, Tauber, 1995

23. First Order Transitions Example:

24. Wake up ! 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