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Introduction to the course
Notion of complexity.
Anderson "more is different" and "emergence".
Goldenfeld and Kadanoff: "structure with variations", "complexity vs. chaos", universality allows choice of "most convenient minimal model"
Newman: survey of complex systems; examples and theories
General papers on complexity and complex systems
Random Walk: a basic model of diffusion (section 1.2.2) [a basic formulation of kinetic theory to derive formulas for the mean free path and the typical collision time is given in 1.2.1]
Brownian Motion: importance of separation between dissipative macroscopic time scale and collision microscopic time scale (first part of section 1.4)
Langevin equation: ballistic and diffusive regime; Einstein relation as a first basic example of fluctuation-dissipation relation (section 1.4.1)
Fokker Planck equation: derivation from the master equation and Gaussian solution (section 1.4.2) [be careful: I changed the notation with respect to the one used in the book: I used R(x,x') in place of W(x,x') to be consistent with the notation used in section 1.6]
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Langevin equation in the presence of a conservative force (end of section 1.4.1).
Fokker-Planck as continuity equation. Probability current (beginning of section 1.6.3: I wrote the equation for the current in the general d-dimensional case)
Stationarity vs. macroscopic equilibrium condition (no current flows). Equilibrium distribution for a conservative force. Einstein relation from connection with Boltzmann distribution at thermodynamic equilibrium (last subsection of section 1.6.3).
Chapman-Kolmogorov equation (continuous case) and the Fokker-Planck equation for the transition probability W: forward and backword Kolmogorov equation (section 1.6.4, first part).
Computation of the average exit time: exact formula (section 1.6.4, second part).
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Arrhenius formula as an approximation for the average exit time from a metastable state in
a double well potential (section 1.6.4, second part).
Markov chains: basic definitions; Markov property, irreducibility, periodicity, recurrence (section 1.5.1).
Markov chains: existence of and convergence to a unique stationary distribution (section 1.5.3)
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Markov chains: Kac lemma and time-scale of convergence to equilibrium (section 1.5.3)
Markov chains: discrete time master equation and detailed balance, Detailed balance and connection to equilibrium. Microscopic reversibility (invariance upon time reversal). (section 1.5.4)
Example of a Markov chain with stationary non equilibrium states in the absence of detailed balance: asymmetric random walk on a ring (second subsection of section 1.5.2).
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Detailed balance holds for a Langevin equation only if the Einstein relation is valid (section 1.6.5)
Introduction to fluctuation-dissipitarion relation and linear response theory (section 2.1)
The Kubo formula for the Brownian particle (section 2.2)
Generalized Brownian motion for thermodynamic observables; thermodynamic force and potential; thermodynamic equilibrium susceptibility (section 2.3)
Equilibrium statistical thermodynamics in the presence of a perturbing field; another equation for static susceptibility (first part of section 2.4)
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Linear response and fluctuation dissipation relations: response function in the frequency domain and Kramers-Kronig relations between real and imaginary parts; connection between imaginary part and dissipation (section 2.4.1)
Linear response applications: work done by an external perturbation and susceptibility; thermodynamic sum rule (section 2.4.3)
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Response function for the damped harmonic oscillator in the presence of an external force (section 2.4.3)
Hydrodynamics (continuity equation for conserved quantities) and the Green-Kubo relation to describe transport phenomena in the linear response regime; generalized diffusion coefficients (section 2.5)
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Generalized response function: Onsager regression relation and time reversal (section 2.6)
Nonequilibrium conditions between macroscopic systems: fluxes, affinities and entropy production (section 2.7.1)
Phenomenological equations: kinetic transport coefficients and Onsager reciprocity relations (section 2.7.2)
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Variational principle for the entropy production rate: example with a stationary nonequilibrium state (section 2.7.4)
Nonequilibrium conditions in a continuous system: current densities and affinities (section 2.7.5)
Coupled transport in linear continuous systems (Section 2.8.1)
Physical applications of Onsager Reciprocity relations: Onsager theorem and transport of charged particles; thermoelectric effects; the Seebeck and Peltier effects (section 2.8.2 up to page 119; the Thomson-Joule effect, thermomagnetic and galvanometric effects will not be covered in the course)
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Non-equilibirum phase transitions: equilibrium states vs. stationary states; necessary and sufficient condition on transition rates to achieve detailed balance (section 3.3)
Phase transitions in systems with absorbing states: percolation and direct percolation; bond and site percolation; percolation clusters (section 3.5)
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Other example of directed percolation (DP) universality class: the Domany-Kinzel cellular automata and contact processes (section 3.5)
The
phase transition in DP-like systems: order parameters (fraction of active sites and survival probability), correlation length and correlation time; critical
exponents; phenomenological scaling theory and scaling laws (section 3.6)
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
The
phase transition in DP-like systems: mean field theory and mean field critical exponents; upper critical dimension (section 3.6)
Beyond the DP universality class: Compact directed percolation; more inactive states and the DP2 class (section 4.2)
Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Universality classes beyond DP: contact processes with immunization (Dynamical; Percolation universality class); the Parity Conserving universality class; general discussion of the role of symmetries and conservation laws in determining the universality class: comparison of equilibrium vs. out-of-equilibrium phase transitions (section 4.2)
Self-Organized critical models: role of driving and dissipation; time scale separation; the sandpile Bak-Tang-Wiesenfeld model (section 4.3)
Phase transitions in driven systems: the TASEP model (Totally ASymmetric Exclusion Process):
periodic boundary counditions: properties of the stationary state; the mean field solution is exact;
open boundary conditions with injection and removal of particles; phase diagram: characterization of High Density (HD), Low Density (HD)and Maximal Current phases; discontinuous transition between HD and LD (section 4.4)
Phase transitions in driven systems: superbrief mention of the Bridge model and of the possibility of spontaneous symmetry breaking (section 4.5)Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
Introduction to information theory: Shannon entropy, Bernoulli
process, Kullback-Leibler divergence, entropy rate of a sequence of
random variables: correlated variables: conditional entropy and mutual
information (sections 1.1, 1.2, 1.3, 1.4)
Data compression: source coding, codewords, uniquely decodable and instantaneous codes, optimal compression, Kraft's inequality and Shannon code (section 1.5)
Data communication through noisy communication channels: memoryless channel, binary symmetric and binary erasure channels, error-correcting codes through redundancy, channel rate and channel capacity, average block error probability, channel coding theorem (section 1.6)
The Random Energy Model as prototype of systems with glassy behaviour: definition and self-averaging properties (Section 5.1)
Thermodynamic properties of the Random Energy Model: micro-canonical entropy; thermodynamic potentials in the canonical ensembles; phase transition with zero entropy low temperature "glassy" phase (Section 5.2)
Properties of the "glassy" phase: "condensation" of the Boltzmann measure; participation ratio (Section 5.3)
Quenched and annealed averages for disordered systems (Section 5.4).
Channel coding theorem (section 1.6).
Code ensembles: MAP decoding, word MAP and symbol MAP (section 6.1).
The geometry of the Random Code Ensemble: distance enumerator, growth rate, Gilbert-Varshamov distance (section 6.2).
Demonstration of the channel coding theorem (direct statement) for binary symmetric channels using the Random Code Ensemble: word MAP decoding, symbol MAP decoding, finite temperature decoding and "phase diagram" for the random code ensemble (Section 6.3).
Similarities between the Random Code Ensemble in channel coding theory and the statistical physics of disordered systems (Section 6.7).
Reference book: "Information, physics, and computation" by Mezard and Montanari
Fluctuation theorems and Jarzynski equality: general introduction
Reference: ""Nonequilibrium fluctuations in small systems: From physics to biology", 2007, F. Ritort, arXiv:0705.0455v1 (sections 3.2,3.3,1)
Fluctuation theorems and Jarzynski equality:
trajectory probability within Markov chain stochastic dynamics (section 3.2.1);
microscopic reversibility and detailed balance (section 3.2.2);
non-equilibrium equality (Kawasaki identity) and stochastic formulation of second law (section 3.2.3);
general formulaton of fluctuation theorem (section 3.2.4);
example within the canonical ensemble: Crook's fluctuation theorem and Jarzynsky equality (section 3.3.1)
References:
""Non-equilibrium fluctuations in small systems: From physics to biology", 2007, F. Ritort, arXiv:0705.0455v1 for the above references
""The Nonequilibrium Thermodynamics of Small Systems", 2005, C. Bustamante, J Liphardt, F. Ritort, Physics Today 58, 43-48 for an overall overview
Effective free energy and density of states
Markov Chains Monte Carlo to sample Boltzmann distribution and estimate expectation values at equilibrium
Metropolis test; it fails in sampling high (free) energy barriers
Wang-Landau method: estimate of the density of states through an iterative procedure which aims at getting flat energy histograms
Reference:
"Thermodynamics and structure of macromolecules from flat-histogram Monte Carlo simulations", 2016, W. Janke and W, Paul, Soft Matter 12, 642 (1. introduction and 2. Background on simulation methods; I did not cover the multicanonical (MuMC) and the SAMC methods in my lecture)
Experimental tests of fluctuation theorems within single molecule pulling experiments
Reference:
Figures 5,6,7,9,10 in ""Non-equilibrium fluctuations in small systems: From physics to biology", 2007, F. Ritort, arXiv:0705.0455v1 for the above references