## Topic outline

• ### SCP7081763 - PHYSICS OF COMPLEX SYSTEMS 2020-2021 - PROF. ANTONIO TROVATO

• ### Lecture 30-09-2020 (2h)

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

• ### Lecture 01-10-2020 (2h)

Discussion of course syllabus

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)

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi

• ### Lecture 07-10-2020 (2h)

Langevin equation: ballistic and diffusive regime; Einstein relation as a first basic example of fluctuation-dissipation relation. Langevin equation in the presence of a conservative force (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]

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).

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi

• ### Lecture 08-10-2020 (2h)

Chapman-Kolmogorov equation (continuous case) and the Fokker-Planck equation for the transition probability W: forward and backward Kolmogorov equation (section 1.6.4, first part).

Computation of the average exit time: exact formula (section 1.6.4, second part).

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).

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi

• ### Lecture 14-10-2020 (2h)

Brief mention to the polymer translocation problem. Average translocation time.

Markov chains: basic definitions; Markov property, stationary distribution (section 1.5.1)

Markov chains: irreducibility, periodicity, recurrence (section 1.5.1).

Markov chains: existence of and convergence to a unique stationary distribution (section 1.5.3)

Markov chains: Kac lemma (section 1.5.3)

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi

• ### Lecture 15-10-2020 (2h)

Markov chains: 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

• ### Lecture 21-10-2020 (2h)

Normalization of eigenvectors of a stochastic matrix. Left eigenvector associated to the stationary distribution (appendix B)

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: general introduction (section 2.3)

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi

• ### Lecture 22-10-2020 (2h)

Generalized Brownian motion: 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)

Linear response and fluctuation dissipation relations: non equilibrium averages in the case of instantaneous switch off of a constant perturbation (section 2.4.1 first part)

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi

• ### Lecture 28-10-2020 (2h)

Linear response and fluctuation dissipation relations: response function in the frequency domain and the power spectrum (section 2.4.1 second part).

Kramers-Kronig relations between real and imaginary parts; connection between imaginary part and dissipation (section 2.4.1 second part)

Linear response applications: work done by an external perturbation (section 2.4.2).

Static and dynamic susceptibility; thermodynamic sum rule (section 2.4.3)

Response function for the damped harmonic oscillator in the presence of an external force: properties of the poles (section 2.4.3)

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi

• ### Lecture 29-10-2020 (2h)

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)

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)

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi

• ### Lecture 04-11-2020 (2h)

Nonequilibrium conditions between two reservoirs and  in a continuous system: current densities and affinities (section 2.7.5)

Phenomenological equations: kinetic transport coefficients and Onsager reciprocity relations (section 2.7.2)

Variational principle for the entropy production rate: example with a stationary nonequilibrium state (section 2.7.4)

A first example of couple transport with neutral particles: mechanothermal and thermomechanical effects (section 2.8.1 first part)

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi

• ### Lecture 05-11-2020 (2h)

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, the Thomson-Joule effect (section 2.8.2 up to page 120), Thermomagnetic and galvanometric effects will not be covered in the course)

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
• ### Lecture 11-11-2020 (2h)

Non-equilibrium 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: isotropic percolation; bond and site percolation, percolation threshold, spanning cluster. Directed percolation, stochastic update rules, percolation clusters, directed percolation transition from active to inactive phase (section 3.5)

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
• ### Lecture 12-11-2020 (2h)

Other example of directed percolation (DP) universality class: the Domany-Kinzel cellular automata (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

• ### Lecture 18-11-2020 (2h)

The phase transition in DP-like systems with one absorbing state: contact processes and mean field theory with uniform density. Langevin-like equation for non-uniform density: mean field critical exponents from invariant scaling; upper critical dimension (section 3.6)

Beyond the DP universality class with more absorbing states: contact processes with immunization: the dynamical percolation universality class (section 4.2). Epidemic spreading: the SIR model.

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi

• ### Lecture 19-11-2020 (2h)

General discussion of the role of symmetries and conservation laws in determining the universality class:  comparison of equilibrium vs. out-of-equilibrium phase transitions

Beyond the DP universality class: Compact directed percolation and random walk behavior of domain walls dynamics; more inactive states and the DP2 class; the Parity Conserving universality class (section 4.2)

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi

• ### Lecture 25-11-2020 (2h)

Phase transitions in driven systems: the TASEP model (Totally ASymmetric Exclusion Process): periodic boundary conditions: properties of the stationary state; the mean field solution is exact (section 4.4);

TASEP model: 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)

Reference book: "Non equilibrium statistical physics: a modern perspective" by Livi and Politi
• ### Lecture 26-11-2020 (2h)

Phase transitions in driven systems: Bridge model and 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, Shannon entropy of correlated random variables (sections 1.1, 1.2)

Reference book: "Information, physics, and computation" by Mezard and Montanari
• ### Lecture 02-12-2020 (2h)

Introduction to information theory: entropy rate of a sequence of random variables: correlated variables: conditional entropy and mutual information (sections 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 (section 1.6)

Reference book: "Information, physics, and computation" by Mezard and Montanari

• ### Lecture 03-12-2020 (2h)

Data communication through noisy communication channels: 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,  Canonical partition function: the saddle point method (Section 5.2)

Reference book: "Information, physics, and computation" by Mezard and Montanari
• ### Lecture 09-12-2020 (2h)

The Random Energy Model: 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)

Code ensembles: random Code Ensemble (section 6.1).

The geometry of the Random Code Ensemble: distance enumerator and growth rate (section 6.2).

Reference book: "Information, physics, and computation" by Mezard and Montanari
• ### Lecture 10-12-2020 (2h)

Code ensembles: MAP decoding, word MAP and symbol MAP (section 6.1).

The geometry of the Random Code Ensemble: Gilbert-Varshamov distance (section 6.2)

Demonstration of the channel coding theorem (direct statement) for binary symmetric channels using the Random Code Ensemble: closest distance decoding, word MAP decoding, symbol MAP decoding (Section 6.3).

Reference book: "Information, physics, and computation" by Mezard and Montanari

• ### Lecture 16-12-2020 (2h)

Random code ensemble: finite temperature decoding and "phase diagram" (Section 6.3).

Reference book: "Information, physics, and computation" by Mezard and Montanari

Large deviation principle: general formulation for continuous random variables; exponential decay and rate functions (section 3,1, 3,2)

Examples of direct computation of rate functions: random bits (example 2.1), gaussian sample mean and the law of large numbers (example 2.2), symbol frequence and Kullback-Leibler divergence (example 2.4)

Reference review: "The large deviation approach to statistical mechanics" by H. Touchette (Phys. Rep. 2009)

• ### Lecture 17-12-2020 (2h)

Large deviation theory: calculating rate functions. The Gartner-Ellis theorem and the Legendre-Funchel transform (section 3.3.1); connection with saddle-point (Laplace) approximation (section 3.3.2).

IID examples (Cramer' theorem): Gaussian and exponential sample mean (section 3.4)

Properties of rate functions: convexity and Legendre transform (section 3.5). Connection with central limit theorem (section 3.5.8)

Reference review: "The large deviation approach to statistical mechanics" by H. Touchette (Phys. Rep. 2009)

• ### Lecture 07-01-2021 (2h)

Large deviations: non convex rate functions, convex envelopes and non differentiable generating functions (section 4.4)

Large deviations in equilibrium statistical mechanics: microcanonical entropy density and canonical free energy (section 5.1, 5.2).

Large deviations in the microcanonical ensemble: maximum entropy principle from macrostate "contraction" to energy;  minimum free energy principle in the canonical ensemble (section 5.3)

Reference review: "The large deviation approach to statistical mechanics" by H. Touchette (Phys. Rep. 2009)

• ### Lecture 13-01-2021 (2h)

Fluctuation theorems and Jarzynski equality: general introduction

Dissipated work and total entropy production. Stochastic thermodynamics is relevant for small systems

Trajectory probability within Markov chain stochastic dynamics; average of total entropy production (section 3.2.1);

Non-equilibrium equality (Kawasaki identity) and stochastic formulation of second law (section 3.2.3)

References:

"Non-equilibrium fluctuations in small systems: From physics to biology", 2007, F. Ritort, arXiv:0705.0455v1 for the above references

"The Non-equilibrium Thermodynamics of Small Systems", 2005, C. Bustamante, J Liphardt, F. Ritort, Physics Today 58, 43-48 for an overall overview

• ### Lecture 14-01-2021 (2h)

General formulation of fluctuation theorem (section 3.2.4);

microscopic reversibility and detailed balance (section 3.2.2);

example within the canonical ensemble: Crook's fluctuation theorem and Jarzynsky equality (section 3.3.1)

Experimental tests of fluctuation theorems within single molecule pulling experiments (figures 5,6,7,9,10)

APPENDIX: Fluctuation theorems within large deviation theory (example 6.10 in Touchette review, Phys. Rep. 2009)

References:

"Non-equilibrium fluctuations in small systems: From physics to biology", 2007, F. Ritort, arXiv:0705.0455v1 for the above references

"The Non-equilibrium Thermodynamics of Small Systems", 2005, C. Bustamante, J Liphardt, F. Ritort, Physics Today 58, 43-48 for an overall overview