# MA999 2018-19: From Equilibrium to Extreme Events and Life

**Term 2, Tues 10-11 & Thurs 1-3 in D1.07 **(per Central Timetabling 10/9/18). The ordering of lecturers and lectures within this is subject to change.

Masters students for whom this is not advertised as a usual option (e.g. MASDOC, MASt) are welcome to contact the module leader R.C.Ball@warwick.ac.uk if interested in taking it. All students see MA999 (link) for overarching module structure and assessment patterns.

Robin Ball on **Dynamics of Equilibrium & beyond**. Jan 8, Jan 10 class, Jan 15, Jan 24+class, Jan 29

Thermal Equilibrium is a dynamic property, and we will explore relationships between how you get there and fluctuations, and applications beyond. We will discuss the Fluctuation-Dissipation Theorem, and the associated development of Langevin Equations. This leads us into some intriguing fresh aspects of time reversal (relating to Fixman's Law), and how faster than equilibrium dynamics can still tell you equilibrium information (Jarzynski relations). We will also look at applications beyond conventional equilibrium such as Granular Temperature, and to optimisation challenges in Operational Research, such has how you optimise under uncertainty.

Key bibliography: (probably drop at least one; sorry I still need to check accurate references):

- Classic FDT paper (Kubo)
- Time reversal in Langevin Equations (RCB+OD ms to be submitted soon)
- Jarzynski paper
- Optimisation under Uncertainty (RCB et al PRL)
- Optimal Stochastic Annealing (RCB et al recent paper)
- Granular Temperature (classic PRL)

Matthew Turner on **Physics of cells**. Jan 10, Jan 17+class, Jan 22, Jan 31+class

The living cell is a non-equilibrium physical machine. Its typical building-blocks are 1D polymers or 2D membranes. Function is usually manifested through a non-equilibrium or "active" process that controls the morphology or dynamics of these structures, e.g. transmembrane ion pumps. These lectures will start with a brief introduction to the mechanics of the cell. We will then introduce appropriate continuum descriptions of continuous surfaces constrained by symmetry arguments and use these to build a physical model for intracellular membranes. We will learn how to analyse linear and non-linear deformations, explain the role of topology in determining membrane energy and show how this might be extended to multi-component systems and membrane hydrodynamics. We will conclude with an open question - how does the cell regulate different membrane topologies?

Key bibliography (subject to change):

Background texts (multiple copies held in the library)

▪ B. Alberts et al., Molecular Biology of the Cell, 6th Ed., Norton 2014. (Selected chapters on membrane structure and transport *only*)

▪ S. A. Safran, Statistical thermodynamics of surfaces interfaces and membranes, Perseus, 1994.

Journal articles (available here)

▪ The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell, P. B. Canham, J. Theo. Biol. **26**, 61-81, 1970.

▪ The conformation of membranes, R. Lipowsky, Nature **349**, 475, 1991.

▪ Mechanics of microtubule-based membrane extension, D. Fygensen et al., Phys. Rev. Lett. **79**, 4497, 1997.

▪ Configurations of fluid membranes and vesicles, U. Seifert, Adv. Phys., 46, 13-137, 1997.

▪ Mobility Measurements Probe Conformational Changes in Membrane ￼Proteins due to Tension, R. G. Morris & M. S. Turner, Phys. Rev. Lett., 115, 198101, 2015.

▪ Membrane Composition Variation and Underdamped Mechanics near Transmembrane Proteins and Coats, S. A. Rautu, G. Rowlands and M. S. Turner, Phys. Rev. Lett, 114, 098101, 2015.

Tobias Grafke on **Extreme Events**. Feb 7+class, Feb 12, Feb 19+class, Feb 26

Rare but extreme events can have dramatic influence on the statistics of stochastic systems, but are notoriously hard to handle both analytically and numerically. For example, rogue waves in the sea, extreme weather events induced by climate change, or extreme fluctuations leading to pattern changes in biofilms, are examples of events hard to observe but important to predict. Large deviation theory (LDT) poses an alternative approach to these events by precisely quantifying the probability and evolution of extreme events. From the point of view of statistical mechanics, LDT generalizes the concepts of free energy and entropy to non-equilibrium systems.

We will discuss the basics of extreme events for equilibrium and non-equilibrium systems and the most common methods to treat rare events, including importance sampling for Monte-Carlo, cloning methods, and action minimization methods, to obtain sample pathways realizing extreme events.

Key bibliography (subject to change):

- Weinan E, Weiqing Ren, and Eric Vanden-Eijnden.
*String method for the study of rare events.*Physical Review B, 66(5):052301, August 2002. doi:10.1103/PhysRevB.66.052301. [link] - Weinan E, Weiqing Ren, and Eric Vanden-Eijnden.
*Minimum action method for the study of rare events.*Communications on Pure and Applied Mathematics, 57(5):637–656, May 2004. ISSN 1097-0312. doi:10.1002/cpa.20005. [link] - Cristian Giardina, Jorge Kurchan, Vivien Lecomte, and Julien Tailleur.
*Simulating Rare Events in Dynamical Processes.*Journal of Statistical Physics, 145(4):787–811, November 2011. ISSN 0022-4715, 1572-9613. doi:10.1007/s10955-011-0350-4. [link] - Tobias Grafke, Tobias Schäfer, and Eric Vanden-Eijnden.
*Long Term Effects of Small Random Perturbations on Dynamical Systems: Theoretical and Computational Tools.*In Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Fields Institute Communications, pages 17–55. Springer, New York, NY, 2017. doi:10.1007/978-1-4939-6969-2 2. [link] - J. Wouters and F. Bouchet.
*Rare event computation in deterministic chaotic systems using genealogical particle analysis.*Journal of Physics A: Mathematical and Theoretical, 49(37):374002, 2016. ISSN 1751-8121. doi:10.1088/1751-8113/49/37/374002. [link] - Giovanni Dematteis, Tobias Grafke, and Eric Vanden-Eijnden.
*Rogue waves and large deviations in deep sea.*Proceedings of the National Academy of Sciences, 115(5):855–860, January 2018. ISSN 0027-8424, 1091-6490. doi:10.1073/pnas.1710670115. [link]

Gareth Alexander on **Active Matter**. Feb 5, Feb 14+class, Feb 19, Feb 28+class

Active matter is an umbrella term for systems driven out of equilibrium by the consumption of energy at the small scale, by individual particles or organisms. Many examples come from biology — in cell cytoskeleton dynamics, swarming bacteria, biofilms, flocking and crowds — but synthetic examples are also plentiful, for instance self-propelled colloids or shaken granular systems. In addition to stimulating advances in non-equilibrium statistical physics, models of active matter also illuminate natural processes in living systems.

We will introduce the basic theoretical concepts that have arisen from statistical physics and how they are being applied in biological contexts.

Key bibliography (subject to change):

- J. Prost, F. Jülicher, and J-F. Joanny,
, Nature Physics**Active gel physics****11**, 111-117 (2015). DOI: 10.1038/nphys3224

- Amin Doostmohammadi, Jordi Ignés-Mullol, Julia M. Yeomans, and Francesc Sagués,
, Nature Communications**Active nematics****9**, 3246 (2018). DOI: 10.1038/s41467-018-05666-8 - Michael E. Cates and Julien Tailleur,
, Annu. Rev. Condens. Matter Phys.**Motility-Induced Phase Separation****6**, 219-244 (2015). DOI: 10.1146/annurev-conmatphys-031214-014710 - Sriram Ramaswamy,
, Annu. Rev. Condens. Matter Phys.**The Mechanics and Statistics of Active Matter****1**, 323-345 (2010). DOI: 10.1146/annurev-conmatphys-070909-104101 - Tim Sanchez, Daniel T.N. Chen, Stephen J. DeCamp, Michael Heymann, and Zvonomir Dogic,
, Nature**Spontaneous motion in hierarchically assembled active matter****491**, 431-434 (2012). DOI: 10.1038/nature11591 - Felix C. Keber, Etienne Loiseau, Tim Sanchez, Stephen J. DeCamp, Luca Giomi, Mark J. Bowick, M. Cristina Marchetti, Zvonomir Dogic, and Andreas R. Bausch,
, Science**Topology and dynamics of active nematic vesicles****345**, 1135-1139 (2014). DOI: 10.1126/science.1254784 - Thuan Being Saw, Amin Doostmohammadi, Vincent Nier, Leyla Kocgozlu, Sumesh Thampi, Yusuke Toyama, Philippe Marcq, Chwee Teck Lim, Julia M. Yeomans, and Benoit Ladoux,
, Nature**Topological defects in epithelia govern cell death and extrusion****544**, 212-214 (2017). DOI: 10.1038/nature21718