### by A. Gambassi (6 credits)

*Main Topics*

*Main Topics*

**Basics:**

- Stochastic and Markov processes: Wiener, Ornstein-Uhlenbeck and Cauchy processes. Chapman-Kolmogorov equation.
- Continuity and differentiability of sample trajectories. Markov processes in physics: Brownian motion.
- Master equation, non-equilibrium and equilibrium stationary states, detailed balance, time-reversal symmetry and increase of entropy.
- Integrability of the detailed balance condition. Kramers-Moyal expansion and Fokker-Planck equation.
- Examples:
- (i) One-dimensional random walk, master equation, equilibrium and non-equilibrium stationary states.Continuum limit of the master equation and the diffusion equation
- (ii) Branching and decay process in zero dimension, absorbing state and non-equilibrium. Master equation, generating function, and non-equilibrium phase transition into the absorbing state
- Diffusion equation and Wiener measure for trajectories: relationship between the Brownian motion and a free quantum particle in imaginary time. Feynman-Kac theorem in the presence of an external potential.
- Applications: (i) First-passage times and distribution of the maximum of a one dimensional process. (ii) First-passage time from the path-integral: persistence probability and persistence exponent. (iii) Gaussian processes and (perturbative) path-integral calculation of the persistence exponent for a non-Markovian process.
- Langevin equation, microscopic and mesoscopic variables, and separation of time scales. Langevin equation for the Brownian, mean square displacement, and Fick’s law. Generalized Langevin equation for a classical particle coupled to a thermal bath of harmonic oscillators: memory kernel and non-Markovian dynamics. Equilibrium distribution function and generalized Einstein relation.
- Formal solution of the Langevin equation via stochastic integrals: definition of the stochastic integral and comparison with Riemann?s integral, Ito and Stratonovich differential calculus.
- Response function and relation to correlation functions. Time-translational and time-reversal symmetries and the fluctuation-dissipation theorem.
- From the Langevin equation to the Fokker-Planck equation.

**Functional methods (from stochastic equations to field theory)**

- Fluctuations, correlations and field-theoretical methods. The Lotka-Volterra model and its mean-field description.
- Master equation of discrete stochastic interacting particle systems (Doi-Peliti formalism): Fock space (bosonic ladder operators) and the associated Hamiltonian. Reaction and diffusion Hamiltonians: the case of the irreversible binary annihilation A+A-> A. Particles with exclusion and SU(2) algebra. Averages of observables and the projection state.
- Path-integral formulation of the master equation: Coherent states and their properties. Representation of the evolution operator. Expectation value of an observable as a path integral over the fields. Continuum limit for the process of binary annihilation reaction with diffusion and structure of the resulting field theory.
- Langevin equation and its field-theoretical representation (Martin-Siggia-Rose/Janssen-De Dominicis-Peliti formalism): average over the stochastic noise and the response field. Jacobian of the change of variable. Normalization and vanishing of the vacuum diagram.

**Applications: Scaling behavior in a non-equilibrium phase transition**

- Field-theoretical analysis of the simple annihilation process A + A -> 0. Relation with the problem of directed percolation. Phase diagram in the absence of annihilation: critical exponents. Reaction Hamiltonian, mean-field solution and the rate equation.

- The role of fluctuations: modification of phase diagram and of the exponents. Power counting, anisotropic scaling, and the upper critical dimensionality. Relevance of the various interaction terms and rapidity reversal. Propagators and perturbative expansion. One-loop contribution to the propagator in the frequency domain: diagrams and combinatorics. Dimensional regularization and calculation of the relevant integral. Renormalized diffusion coefficient. Dimensional expansion, minimal subtraction and calculation of the renormalization constants. Renormalization-group flow and the Callan-Symanzik equation. Scaling behavior at the fixed point.