Corso: SCP7081763 - PHYSICS OF COMPLEX SYSTEMS 2020-2021

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

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Pagina dell'offerta Formativa URL
Physics of Complex Systems lectures - academic year 2020-21 Riunione Zoom
This link will be used for the streaming of lectures and receptions in the whole course. NOTICE FOR STUDENTSIt is forbidden to use downloaded course material for anything else than exam preparation. Any use of course material breaking the above terms (for example, communication, distribution, sharing, even partial, on social networks or websites, either free of charge or against payment) will be prosecuted according to law. NOTICE FOR STUDENTSLectures will be recorded. Therefore, the possible voluntary participation by students (via either chat or by enabling microphone/webcam) will result in the collection of personal data (such as, for example, the picture, the voice and the content of the possible contribution). Lecture recording will remain available on the University website to allow watching by the students who could not participate live. Further information, also related to the rights recognized in privacy legislation, is available at https://www.unipd.it/privacy (in Italian).
exam guideline + exam project list File
*** NEW *** - UPDATED LIST OF AVAILABLE EXAM PROJECTS
The list is updated regularly to keep track of the projects already chosen
A general guideline for exams is also present
Exam_projects
ANOMALOUS_DIFFUSION
Sokolov_PRX17.pdf
FLUCTUATION_THEOREMS
QUANTUM_FT
Morillo_arxiv08.pdf
Sagawa_arxiv18.pdf
Wojcik_arxiv04.pdf
Ritort_0705_rev.pdf
INFORMATION_THERMODYNAMICS
2019_Book_EnergyLimitsInComputation.pdf
KPZ
Corwin_RMTA12.pdf
LARGE_DEVIATIONS
Gherardini_PhysA19.pdf
Touchette_PhysA18.pdf
MAXIMUM_ENTROPY_PROTEINS
Weigt_RPP18.pdf
NON_EQUIL_STAT_PHYS
Attard_PCCP06.pdf
VandenBroeck_JStatMech10.pdf
PERCOLATION
Hinrichsen_AdvPhys00.pdf
Sabari_PhysRep15.pdf
REACTION_DIFFUSION
Frey_PTRSBBS18.pdf
Galic_PTRSBBS18.pdf
LutheySchulten_RPP18.pdf
Voit_PLoSCB19.pdf
STOCHASTIC_THERMODYNAMICS
Seifert_PhysA18.pdf
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Oral exams - Winter session Riunione Zoom
THE EXAMS IN THE WINTER SESSION (JANUARY, 26th AND FEBRUARY, 15TH) WILL TAKE PLACE ONLINE. USE THE ABOVE LINK FOR ZOOM MEETING (ONLY WITH UNIPD ACCOUNT)

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

General papers on complexity and complex systems
Complexity_papers
Anderson_complexity72.pdf
Goldenfeld_Kadanoff_Science99.pdf
Newman_archiv11.pdf
Lecture 30-09-2020 File
Lecture 30-09-2020 part 1 Kaltura Video Resource
Lecture 30-09-2020 part 2 Kaltura Video Resource

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 28-10-2020 File
**NEW** corrected version uploaded November 1st
fixing wrong signs at pag. 7 in the equation for the second pole \omega_2
Lecture 28-10-2020 part 1 Kaltura Video Resource
Lecture 28-10-2020 part 2 Kaltura Video Resource
Lecture 28-10-2020 part 3 Kaltura Video Resource

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 17-12-2020 File
color additions to provide some math details
Lecture 17-12-2020 part 1 Kaltura Video Resource
Lecture 17-12-2020 part 2 Kaltura Video Resource
Lecture 17-12-2020 part 3 Kaltura Video Resource

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 07-01-2021 File
Lecture 07-01-2021 part 1 Kaltura Video Resource
Lecture 07-01-2021 part 2 Kaltura Video Resource
Lecture 07-01-2021 part 3 Kaltura Video Resource
Appendix 07-01-2021 File
Large deviation theory: Sanov theorem for the "empirical vector" (section 4.2) and Jaynes maximum entropy principle for independent particles (example 5.1)

Reference review: "The large deviation approach to statistical mechanics" by H. Touchette (Phys. Rep. 2009)
Appendix 07-01-2021 Kaltura Video Resource

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

Short (Phys. Today) and long review on Fluctuation Theorems
Fluctuation theorems
Ritort_0705_rev.pdf
Ritort_PhysToday05.pdf
Lecture 13-01-2021 File
Lecture 13-01-2021 part 1 Kaltura Video Resource
Lecture 13-01-2021 part 2 Kaltura Video Resource
Lecture 13-01-2021 part 3 Kaltura Video Resource
Lecture 13-01-2021 part 4 Kaltura Video Resource

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

Lecture 14-01-2021 File
It includes the Appendix part as well
Lecture 14-01-2021 part 1 Kaltura Video Resource
Lecture 14-01-2021 part 2 Kaltura Video Resource
Lecture 14-01-2021 part 3 Kaltura Video Resource
Appendix 14-01-2021 Kaltura Video Resource

Indice degli argomenti

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

Lecture 30-09-2020 (2h)

Lecture 01-10-2020 (2h)

Lecture 07-10-2020 (2h)

Lecture 08-10-2020 (2h)

Lecture 14-10-2020 (2h)

Lecture 15-10-2020 (2h)

Lecture 21-10-2020 (2h)

Lecture 22-10-2020 (2h)

Lecture 28-10-2020 (2h)

Lecture 29-10-2020 (2h)

Lecture 04-11-2020 (2h)

Lecture 05-11-2020 (2h)

Lecture 11-11-2020 (2h)

Lecture 12-11-2020 (2h)

Lecture 18-11-2020 (2h)

Lecture 19-11-2020 (2h)

Lecture 25-11-2020 (2h)

Lecture 26-11-2020 (2h)

Lecture 02-12-2020 (2h)

Lecture 03-12-2020 (2h)

Lecture 09-12-2020 (2h)

Lecture 10-12-2020 (2h)

Lecture 16-12-2020 (2h)

Lecture 17-12-2020 (2h)

Lecture 07-01-2021 (2h)

Lecture 13-01-2021 (2h)

Lecture 14-01-2021 (2h)