Last update: Jan. 2020


Stochastic dynamics in Statistical Physics

The outline of the syllabus of the course can be found here.

The somehow more detailed content of the various lectures is as follows:

  1. Stochastic processes, joint and conditional probabilities, stationary and uncorrelated processes. Markov processes and examples. Chapman-Kolmogorov equation. Wiener, Ornstein-Uhlenbeck and Cauchy processes. Transition probabilities and properties of sample trajectories: Lindenberg criterion for continuity.
  2. Markov processes in physics: Brownian motion (brief history). Ensemble and time averages, separation of time scales and mesoscopic variables (general remarks). Master equation (ME) and its properties; stationary states and probability currents. ME for a discrete configuration space and relation with Monte Carlo simulations. Non-equilibrium and equilibrium stationary states, detailed balance (DB), time-reversal symmetry and increase of entropy.
  3. Integrability of the DB condition and Kolmogorov criterion. Equilibrium probability distribution from the transition rates. Kramers-Moyal expansion and Fokker-Planck equation. Pawula’s theorem.
  4. Random walk (RW) in one spatial dimension, master equation, boundary conditions, equilibrium vs. non-equilibrium stationary state for periodic boundary conditions. Equilibrium distribution for reflecting boundaries. Continuum limit of the master equation, diffusion equation and its solution. RW in one spatial dimension: combinatorial approach.
  5. Branching and decay process in 0 spatial dimensions. Absorbing state and non-equilibrium. Master equation, generating function and boundary conditions. Method of characteristic and solution of the ME. Survival probability as an order parameter: non-equilibrium phase transition from the active to the absorbing state. Critical exponents and divergence of the characteristic time.
  6. Diffusion equation and Wiener measure for trajectories. Formal limit and connection to quantum mechanics (QM). Comparison with QM and relationship between Brownian motion and free quantum particle in imaginary time.
  7. Wiener path integral and properties of the Wiener measure. External QM potential as an evaporation rate and non-conservation of probability. Feynman-Kac theorem in the presence of an external potential. Bloch equation (Schrödinger equation in imaginary time) and proof of its validity for the Wiener process with evaporation. Relationship with the partition function of the directed polymer.
  8. First-passage times and examples. Relation between first-passage times and the distribution of the maximum of a one-dimensional process. Reflection principle for the calculation of the distribution of the first-passage time. First-passage time within the path-integral approach. Relation with the QM of a particle in the half-line; method of images. Comments on the probability density of the first crossing time: time scales and divergent mean. Recurrence of the Wiener process. Persistence probability and persistence exponent.
  9. Gaussian processes: general definition and examples. Application of Gaussian processes to physical problems: persistence of the global magnetization of the Ising model. Path-integral (perturbative) calculation of the persistence exponent for a Gaussian non-Markovian process. Expression in terms of the non-Markovian part of the correlation function.
  10. Microscopic and mesoscopic variables, separation of time scales and the Langevin equation. Langevin equation for the Brownian motion and properties of the noise term. Green’s function, correlation function for the velocity. Stationary distribution and matching with Boltzmann distribution in equilibrium. Mean square displacement, Fick’s law and prediction of the diffusion coefficient.
  11. Derivation of the generalized Langevin equation: classical dynamics of a particle coupled to a bath of harmonic oscillators. Equilibrium distribution function for the particle and the counterterm for the potential. Solution of the equation of motion for the bath and effective equation of motion for the particle. Properties of the “noise” and meaning of the average. Time-dependent memory kernel and non-Markovian dynamics. Equilibrium distribution function and generalized Einstein relation. Spectral density of the bath: Ohmic, sub-Ohmic and super-Ohmic cases.
  12. Remarks on the Langevin equation: formal solution involving a stochastic integral. Definition of the stochastic integral and comparison with Riemann’s integral: dependence on the partition. Ito and Stratonovich definitions and rules of differential calculus. Ito’s rule for non-linear change of variables. Ambiguities which might arise in the calculation of correlators and the role of the step function Θ(0). Relation between the value of Θ(0), the stochastic calculus and the definition (backward, forward, symmetric) of the time derivative.
  13. From the (non-linear) Langevin equation to the Fokker-Planck equation: dependence on the stochastic calculus for multiplicative noise and consequences. Direct derivation from the Kramers-Moyal expansion for additive noise. Relaxational dynamics, stationary distribution and Einstein’s relation. Response function and its relation with correlation functions. Correlation with the noise and response. Time-translational, time-reversal symmetries and the fluctuation-dissipation theorem. Dynamics which leads to a prescribed stationary distribution: reversible forces.
  14. Fluctuations, correlations and field-theoretical methods. Spatial degree of freedom and interactions as key ingredients for complex behavior. The Lotka-Volterra model and its mean-field description. Conservation law. Stochastic dynamics of a system of discrete and interacting particles and the Doi-Peliti formalism: construction of the Fock space (bosonic ladder operators) and Hamiltonian associated with the master equation. Analogy with the Schrödinger equation. Reaction Hamiltonian: the case of the irreversible binary annihilation.
  15. Hamiltonian of the simple hopping and of diffusion. The case of particles with exclusion: SU(2) algebra. Calculation of the average of observables and the projection state; analogies and differences with quantum mechanics. Properties of the projection state. Conservation of probability and consequences for the structure of the Hamiltonian. Path-integral formulation of the dynamics: Coherent states and their properties. Representation of the evolution operator as a path-integral.
  16. Path-integral representation of the dynamics (cont’d): Contributions of the initial state and of the projection operator. Bulk term and initial conditions. Doi shift of the field. Discussion of the generalization to multi-time observables. Naive continuum limit for the binary annihilation reaction with diffusion. Structure of the resulting field theory.
  17. Dynamics of the simple annihilation process and its interpretation as resulting from a Langevin equation: difficulties with imaginary noise. A quantum detour: coherent-state path-integral representation for expectation values in quantum mechanics and the need of doubling the contour. Schwinger-Keldysh path-integral and its basic properties.
  18. Alternative "phase-space" representation of the Master Equation: properties of the resulting Hamiltonian and role of the noise. General relationship with the Doi-Peliti Hamiltonian. Path-integral representation of the phase-space propagator and the concrete example of binary annihilation with lethal competition. Stationary-path approximation and relationship with classical mechanics. Analysis of the resulting trajectories, role of the noise, stable and metastable states and the activation trajectory.
  19. Field-theoretical representation of the Langevin equation and the response functional formalism (Martin-Siggia-Rose/Janssen-De Dominicis-Peliti formalism). Average over the stochastic noise and implementation of the Langevin equation as a constraint. Introduction of the response field and the Jacobian of the change of variable. Explicit evaluation of the determinant and contribution to the response functional proportional to the step function Θ(0).
  20. Diagrammatic expansion of the response functional in the presence of a non-linear interaction: Diagrams with a self-contraction of the response propagator and the vertex which effectively generates them. Cancellation of the term proportional to Θ(0) in the generating functional. Normalization to unity of the partition function of the response functional: vanishing of vacuum diagrams. Interpretation of the auxiliary field as the response field: linear response function and correlation functions with a response field. Interpretation of the field theory for the stochastic dynamics of the binary annihilation process as a Langevin equation with multiplicative noise: rate equation, qualitative features and problems with imaginary noise. [The Jacobian of the change of variable and the functional integral over Grassmann variables. Supersymmetry and the fluctuation-dissipation theorem.]
  21. Field-theoretical analysis of the stochastic dynamics: the case of the simple annihilation process with diffusion. Relation with the problem of directed percolation, reaction Hamiltonian, determination of the minimum of the action: mean-field solution and the rate equation. The role of fluctuations: modification of phase diagram and of the exponents. Basics of Wilsonian renormalization group (RG) for the construction of the RG flow: separation of fast components, integration over momentum shell, and (anisotropic) rescaling. effective action for the slow modes.
  22. The Gaussian fixed point and the upper critical dimensionality. Relevance of the various interaction terms. Rapidity reversal as an emerging symmetry. Janssen-Grassberger conjecture. Explicit calculation of the propagators in the frequency and time domains. Diagrammatic representation. Interaction vertices. Determination of the effective action at the lowest-order in perturbation theory and modification of the propagator and of the vertices. General consequences of causality for the structure of the effective action and specific consequences of the rapidity reversal. Definition of the effective parameters from the effective action and their flow.
  23. Modification of the vertex and effective coupling constants. Derivation of the RG flow equation: resulting structure, fixed points and their stability depending on the dimensionality. Determination of the anomalous dimension η and of the dynamical exponent z. Linearization of the flow and determination of the correlation-length exponent. Emerging (strongly anisotropic) scaling at the fixed point and consequences for the response function. Interpretation of the response propagator as the local response function of the number of particles.

Some videos and material from my courses

Under construction

Past courses

In the past years I gave the following courses in Physics (in English) and in Science Communication (in Italian):


  • 2011-present: Lectures for PhD students in Physics at SISSA - International School for Advanced Studies, Trieste (Italy): Stochastic dynamics in Statistical Physics. (40 h)
  • 2011 - 2014 (October - December): Lectures for PhD students in Theory of Elementary Particle and in Astroparticle Physics at SISSA: Quantum Field Theory. (ca. 30 h)
  • 2010 (February - April): Lectures for PhD students in Statistical Physics at SISSA: Stochastic dynamics in Statistical Physics. (35 h)
  • 2010 (January - March): Lectures for Master Students in Condensed Matter Physics at ICTP - The Abdus Salam International Centre for Theoretical Physics, Trieste (Italy): Stochastic processes and applications. (20 h)
  • 2009 (May - June): Lectures for PhD students in Statistical Physics at SISSA: Stochastic dynamics in Statistical Physics. (30 h)
  • 2008 (Summer Semester): Tutorials for the Diplom and BSc courses at the University of Stuttgart (Germany): Theoretische Physik I, Mechanik.
  • 2007 (November - December): Invited lectures for PhD students in Physics at SISSA - International School for Advanced Studies, Trieste (Italy): Stochastic dynamics in Statistical Physics.
  • 2007 (September - October): Invited lectures for PhD students of the Department of Physics of the University of Milano (Italy): Stochastic dynamics in Statistical Physics.
  • 2006 (Winter Semester): Tutorials for the MSc program in Physics of the University of Stuttgart (Germany): Quantum Mechanics.
  • 2006 (Summer Semester): Tutorials for the MSc program in Physics of the University of Stuttgart (Germany): Thermostatistics.
  • 2005 (Summer Semester): Special lectures (Spezialvorlesungen) at the Department of Physics, University of Stuttgart (Germany): Field Theory in Statistical Physics (together with Dr. M. Oettel).

Science Communication:

  • 2017/18 - 2016/17 (November-May): Coordinator (together with Gianluigi Rozza and the science journalist Daniele Gouthier) of the lectures for the Master in Science Communication "Franco Prattico" at SISSA: Communicating Physics and Mathematics. (ca. 12 h)
  • 2015/16 - 2014/15 - 2013/14 (November-June): Coordinator (together with Luca Heltai and the science journalist Daniele Gouthier) of the lectures for the Master in Science Communication "Franco Prattico" at SISSA: Communicating Physics and Mathematics. (ca. 20 h)
  • 2012/13 - 2011/12 (December - May): Coordinator (together with the science journalist Daniele Gouthier) of the lectures for the master in Digital Science Journalism at SISSA: Physics. (ca. 20 h)