Program
Module 1: Symmetry, duality, and topology in lattice gauge theories
Jasper van Wezel (Amsterdam)
Lectures and exercises: Feb 3, 10, 17, 24
Exam: March 3
Location: Science Park G3.10
Abstract:
Lattice gauge theories are prevalent both in high and low energy physics, either as discrete approximations to a continuous field theory, or as a direct implementation of lattice models. Besides their inherent interest, lattice gauge theories also provide a particularly nice background for illustrating the correspondence between quantum dynamics and classical equilibrium descriptions, the presence or absence of symmetry breaking phase transitions, the use of duality transformations, and the role of topological defects in mediating phase transitions.
In this lecture series, we will follow one of the classic texts on lattice gauge theory and discover how all these aspects emerge from very simple building blocks. We will start from the famous Kramers-Wannier duality in the Ising model and, time permitting, end up with accessible lattice descriptions of confinement and the Kosterlitz-Thouless phase transition.
Recommended prior knowledge:
-- [Necessary] Working knowledge of basic Quantum Field Theory.
-- [Useful] Some familiarity with spin systems in condensed matter theory.
Module 2: Density Functional Theory
Matthieu Verstraeten (Utrecht)
Lectures and exercises: Mar 10, 17, 24, 31
Exam: Apr 7
Location: TBA (Utrecht)
Abstract:
TBA
Module 3: Active Matter
Yann-Edwin Keta (Leiden)
Lectures and exercises: Apr 14, May 12, 19, 26
Exam: June 2
(No class: April 21 is Easter holiday, April 28 is UvA holiday, May 5 is Liberation day)
Location: TBA (Leiden)
Abstract:
Active matter has emerged as an important class of materials composed of active particles.
These particles are self-driven units, individually capable of using available energy to generate forces.
This broad definition applies to a wide array of synthetic and living elements at all scales, from subcellular elements, to self-driven colloids, to birds and humans.
Due to their continual generation of forces, these elements escape the rules of equilibrium statistical mechanics, and display a wealth of surprising phenomena which challenge our conceptions of equilibrium phases and dynamics [1].
This lecture introduces analytical and numerical tools, rooted in nonequilibrium statistical mechanics, to describe and understand the structural and dynamical features of these systems.
We will first study stochastic differential equations as the natural framework to describe systems subject to fluctuations, with a specific focus on Langevin-like equations, and their representation as Fokker-Planck equations [2].
This will enable to introduce active Brownian particles (ABPs) which is a widely used model to represent crawling living cells and self-propelled synthetic colloids.
We will first explore its properties numerically by integrating the stochastic dynamics of this model using molecular dynamics tools [3].
We will focus on two emergent features: velocity correlations and spontaneous motility-induced phase separation (MIPS).
We will rationalise the former feature with a theory based on linear elasticity [4], and the latter with a mean-field active field theory [5].
Finally we will study large deviation theory and how this provides a natural framework for both equilibrium and nonequilibrium statistical mechanics [6].
Within this framework we will study the concept of dynamical phase transitions in biased ensembles of trajectories, and will characterise one of these in ABPs [7].
[1] Active Matter and Nonequilibrium Statistical Physics. (Oxford University Press, 2022).
[2] Gardiner, C. W. Handbook of Stochastic Methods: For Physics, Chemistry, and the Natural Sciences. (Springer, 2004).
[3] Mannella, R. Integration of stochastic differential equations on a computer. Int. J. Mod. Phys. C 13, 1177 (2002).
[4] Henkes, S., Kostanjevec, K., Collinson, J. M., Sknepnek, R. & Bertin, E. Dense active matter model of motion patterns in confluent cell monolayers. Nat. Commun. 11, 1405 (2020).
[5] Cates, M. E. & Tailleur, J. Motility-Induced Phase Separation. Annu. Rev. Condens. Matter Phys. 6, 219 (2015).
[6] Touchette, H. The large deviation approach to statistical mechanics. Physics Reports 478, 1 (2009).
[7] Keta, Y.-E., Fodor, É., van Wijland, F., Cates, M. E. & Jack, R. L. Collective motion in large deviations of active particles. Phys. Rev. E 103, 022603 (2021).
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