**Stochastic Dynamics out of Equilibrium**

- Paolo Dai Pra (Universita degli Studi di Padova) and Pierre Degond (Imperial College London)

*Stochastic mean-field dynamics and applications to life sciences* (pdf) Collective dynamics in life sciences, Mini-cours 4, Part. 2 * (lecture 1) (lecture 2) (lecture 3)*

*Part 1 by Paolo Dai Pra*

Lecture 1. Mean field stochastic dynamics and propagation of chaos.

Lecture 2. Continuous models: Kuramoto model; networks of Fitzhugh-Nagumo neurons.

Lecture 3. Models with jumps: Moran and Write-Fischer model; networks of Hawkes processes.

References:

Sznitman, A.-S. Topics in propagation of chaos. In Ecole d’Ete’ de Probabilites de Saint-Flour XIX—1989, vol. 1464 of Lecture Notes in Math. Springer, Berlin, 1991, 165–251.

Graham, C. (1992). McKean-Vlasov Itô-Skorohod equations, and nonlinear diffusions with discrete jump sets. Stochastic processes and their applications, 40(1), 69-82.

Giacomin, G., & Poquet, C. (2015). Noise, interaction, nonlinear dynamics and the origin of rhythmic behaviors. Brazilian Journal of Probability and Statistics, 29(2), 460-493.

Baladron, J., Fasoli, D., Faugeras, O., & Touboul, J. (2011). Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons. arXiv preprint arXiv:1110.4294.

Collet, F., Dai Pra, P., & Formentin, M. (2015). Collective periodicity in mean-field models of cooperative behavior. Nonlinear Differential Equations and Applications NoDEA, 22(5), 1461-1482.

Cerf, R. (2015). Critical population and error threshold on the sharp peak landscape for the Wright–Fisher model. The Annals of Applied Probability, 25(4), 1936-1992.*Part 2 by Pierre Degond*

Lecture 1. Collective dynamics and self-organization in biological systems : challenges and some examples.

Lecture 2. The Vicsek model as a paradigm for self-organization : from particles to fluid via kinetic descriptions

Lecture 3. Phase transitions in the Vicsek model : mathematical analyses in the kinetic framework.

References:

T. Vicsek, A. Zafeiris, Collective motion, Phys. Rep., 517 (2012) 71-140.

F. Bolley, J. A. Cañizo, J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Letters 25 (2012) 339-343.

P. Degond, S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 Suppl. (2008) 1193-1215

P. Degond, A. Frouvelle, J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Archive for Rational Mechanics and Analysis, 216 (2015) 63-115.

A. Figalli, M-J Kang, J. Morales, Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flow. arXiv:1509.02599

N. Jiang, L. Xiong, T-F Zhang, Hydrodynamic limits of the kinetic self-organized models. arXiv:1508.04640

- Claudio Landim (Universite de Rouen and IMPA, Rio de Janeiro)

Hydrodynamic limit of Interacting particle systems (pdf)

Lectures 1 Interacting particle systems: the exclusion processes

Lecture 2. Introduction to hydrodynamic limits; the hydrodynamic limit of the symmetric simple exclusion process.

Lecture 3. General tools from the theory of Markov chains: entropy, entropy production, Feynam-Kac formula.

Lecture 4. The hydrodynamic limit of gradient models, the replacement lemma.

Lecture 5. Large deviations.

Lecture 6. Non-gradient models.

Reference:

Kipnis C., Landim C.: *Scaling Limits of Interacting Particle Systems*, Springer 1999.

- Ben Leimkuhler (University of Edinburgh)

Molecular and Particle Dynamics Simulation (pdf)

Lecture 1. Molecular dynamics models goals and purposes, numerical methods by splitting, analysis and examples.

Lecture 2. Ensembles for molecular simulation (microcanonical, canonical), stochastic differential equations (Brownian/Langevin dynamics), examples.

Lecture 3. Numerical methods for Langevin dynamics: splitting algorithms, error analysis, superconvergence, examples

Lecture 4. Constraints in molecular dynamics, SHAKE and RATTLE discretization, geodesic integration, isokinetic MD, examples.

Lecture 5. Application of thermostat methods in fluids; examples such as shear flows and vortex dynamics.

Lecture 6. More general statistical sampling from molecular dynamics, applications to Bayesian inference.

Lectures based largely on the book:

B. Leimkuhler and C. Matthews, ‘Molecular Dynamics’, Springer, 2015. __http://www.springer.com/us/book/9783319163741__ (available via Springer-Link and intended for MSc or starting PhD students).

Supplemented by the following references:

B. Leimkuhler, C. Matthews, G. Stoltz, The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics, IMA J Num. Anal., 36 (1): 13-79, 2016. doi: 10.1093/imanum/dru056, Arxiv: __http://arxiv.org/abs/1308.5814__

B. Leimkuhler and C. Matthews, Efficient molecular dynamics using geodesic integration and solvent-solute splitting, Proc. Roy Soc A, 472, 2016, doi: 10.1098/rspa.2016.0138

B. Leimkuhler and X. Shang, Pairwise adaptive thermostats for improved accuracy and stability in dissipative particle dynamics, preprint, 2016.

X. Shang, Z. Zhu, B. Leimkuhler and A. Storkey, Covariance-Controlled Adaptive Langevin Thermostat for Large-Scale Bayesian Sampling, NIPS 2015; C. Matthews, J. Weare and B. Leimkuhler, Ensemble preconditioning for Markov Chain Monte Carlo simulation, preprint, 2016.

- David Mukamel (Physics, Weizmann Institute, Israel) and Gunter Schütz (Institute of Complex Systems, Julich)

*Part 1 by Gunter Schütz, « Driven diffusive systems »* (see here).*Part 2 by David Mukamel, « Steady states and long range correlations in driven systems » *

In these three lectures steady states and dynamical properties of nonequilibrium systems will be discussed.

Systems driven out of thermal equilibrium often reach a steady state which under generic conditions exhibits long-range correlations. This is very different from systems in thermal equilibrium where long-range correlations develop only at phase transition points. In some cases these correlations even lead to long-range order in d=1 dimension, of the type occurring in traffic jams. Simple examples of such correlations induced in the steady state of driven systems will be presented and discussed. Close correspondence of these nonequilibrium steady states to electrostatic potentials induces by charge distribution will be pointed out.

Another class which will be discussed is that of systems with boundary drive, such as in heat conduction problems, where anomalous heat conduction takes place i

n low dimensions. In addition some similarities between driven systems and equilibrium systems with long-range interactions will be elucidated.

References:

B. Derrida, « Non equilibrium steady states: fluctuations and large deviations of the density and of the current » JSTAT P07023 (2007)

T. Sadhu, S. N. Majumdar, D. Mukamel, « Long-range steady state density profiles induced by localized drive » Phys. Rev. E 84, 051136 (2011); « Long-range correlations in locally driven exclusion process » Phys. Rev. E 90, 012109 (2014)

O. Cohen, D. Mukamel » Nonequilibrium ensemble inequivalence and large deviations of the density in the ABC model » Phys. Rev. E90, 102109 (2014)

A. Dhar, « Heat Transport in Low Dimensional systems » Advances in Physics, 57, 457 (2008).

« Thermal Transport in Low Dimensions » S. Lepri, editor, Lecture Notes in Physics 921 (Springer, 2016)

« Physics of Long-Range Interacting Systems » A, Campa, T. Dauxois, D. Fanelli, S. Ruffo (Oxford University Press, 2014)