Statistical Physics-II-S3

Statistical Physics II

Contact: Pr. Alberto Verga or Pr. Marco Pettini

 

DYNAMICAL FOUNDATIONS OF STATISTICAL MECHANICS

  • Invariant Measure for the Hamiltonian Dynamics on the constant energy level sets ΣE .
  • The Irreversible Approach to Equilibrium. The Zeroth Law of Thermodynamics.
  • Loschmidt’s paradox, the Poincaré recurrence theorem, the Kac recurrence theorem.   Connection with microscopic chaos.
  • Ergodicity and Mixing: Khinchin ergodic theorem, Birkhoff ergodic theorem, metric transitivity and Hamiltonian chaos.

EQUILIBRIUM PHASE TRANSITIONS

  • Phenomenology, Clausius-Clapeyron equations for first and second order phase transitions.
  • The Braggs-Williams approximation; the Bethe-Peierls approximation.
  • Mermin-Wagner Theorem; Elitzur Theorem
  • Transfer-matrix and the Onsager exact solution of the 2D Ising model
  • Geometric Approach to Chaotic Dynamics and PhaseTransitions
  • The Topological Theory of phase transitions to tackle: transitional phenomena in finite systems (mesoscopic and nanoscopic systems); amorphous and disordered systems; filament-globule transitions in homopolymers; protein folding transitions.

CRITICAL PHENOMENA

  • Correlation function and the correlation-response theorem
  • Critical exponents; scaling hypothesis; scale invariance
  • Goldstone excitations
  • Landau theory; mean-field theory
  • The Gaussian model
  • Ginzburg criterion
  • The Migdal-Kadanoff approach
  • The Renormalization Group: fixed points and scaling fields; renormalization of the partition function; Renormalization Group equation

NONEQUILIBRIUM STATISTICAL MECHANICS

  • From the Liouville equation to the Boltzmann equation, and Navier-Stokes equation via the BBGKY hierarchy
  • Quasi Stationary States and Vlasov equation ; the Lynden-Bell’s statistics
  • Ensemble inequivalence for systems with long-range interactions
  • Non-equilibrium phase transitions in out-of-equilibrium stationary states (laser, Rayleigh-Bénard convective instability)

FLUCTUATIONS

  • Thermodynamic fluctuations
  • Spatial correlations in a fluid
  • Einstein-Smoluchowski theory of Brownian motion
  • Langevin theory of Brownian motion
  • Approach to equilibrium: the Fokker-Planck equation
  • Fluctuation-dissipation theorem

 

References:

  • Chaikin et Lubensky, Principles of Condensed Matter Physics, Cambridge, 1995.
  • Kardar, Statistical Physics, particles (I) and fields (II), Cambridge, 2007.
  • Krapivsky, Redner et Ben-Naim, A Kinetic view of Statistical Physics, Cambridge, 2010.
  • Leach, Molecular Modelling, Prentrice Hall, 2001.
  • Mori et Kuramoto, Dissipative Structures and Chaos, Springer, 1998.
  • Onuki, Phase Transition Dynamics, Cambridge, 2002.
  • Safran, Statistical Thermodynamics of Surfaces, Interfaces and Membranes, Westview, 2003.
  • Sethna, Statistical Mechanics: entropy, order parameters and complexity, Oxford, 2006.