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

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

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

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

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

6. Hamiltonian Imaginary time “Schrodinger” equation: Hamiltonian is 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 ! Zero energy trajectories !

10. Extinction time Extinction time

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

12. Refuge R Lifetime: Instanton solution

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

14. Critical exponents Hinrichsen 2000

15. 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?

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

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

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

19. 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)

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

21. First Order Transitions Example:

22. 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