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

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

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

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

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

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

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

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

Branching-Annihilation Rate equation ! Zero energy trajectories !

Extinction time Extinction time

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

Refuge R Lifetime: Instanton solution

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

Critical exponents Hinrichsen 2000

Critical Exponents (cont) Critical Exponents (cont) d=1 d=2 d=3 d> 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?

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

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

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)?

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)

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

First Order Transitions Example:

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