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# Collective Motion in Active Systems

**Lecture course in winter semester 2015 (2 credit points)**

**Lecturer:Dr. Igor Aronson**, Argonne National Laboratory, USA

**Dates:**

November 2 - 26, 2015

**Time:**

Monday, 10.15 - 11.45 h

Thursday, 10.15 - 11.45 h

Exception: No lecture on Monday, November 23

Instead extra lecture on Thursday November 26 from 14:15 - 15:45

Exception: No lecture on Monday, November 23

Instead extra lecture on Thursday November 26 from 14:15 - 15:45

**Place:**

ER 164 (TU Berlin, ER-building)

**First lecture:**

Monday, November 2, 2015

## Course Outline

Active Matter is a new field of soft matter physics, which focuses on the properties of assemblages of interacting self-propelled particles, from molecular motors and synthetic swimmers to living bacteria, motile cells, and large animal swarms. Assemblies of self-propelled organisms are often termed “active fluids” and the study of their collective dynamics has attracted enormous attention in the last decade. Despite simplicity of interactions between the constituents, large assemblies of self-propelled particles exhibit fascinating collective motion. This motion is manifested by spontaneous formation of vortices and jets with the characteristic scale significantly exceeding the size of individual particle. In this course I’ll systematically overview the most recent progress in the field of Active Matter, and will discuss various approaches to the description of collective motion in this class of out-of-equilibrium systems. I will review various types of microscopic models based on the motion of individual particles, then proceed to the mesoscale kinetic description for the probability distribution functions, and then will derive coarse-grained hydrodynamic description of the collective behavior.

**Overview of main experimental studies of collective behavior in synthetic and living systems**

- Patterns in granular systems
*In vitro*cytoskeletal networks- Suspensions of microswimmers

**Pattern-forming instabilities in out-of-equilibrium systems**

- Main types of spatio-temporal instabilities
- Universal equations associated with the instabilities: complex Ginzburg-Landau equation and Swift-Hohenberg equation
- Derivation of the complex Ginzburg-Landau Equation
- Plane wave solution and its stability, absolute vs convective instability
- Sinks and sources, counting arguments
- One–dimensional dynamics: structural instability of the Nozaki-Bekki hole solution and spatio-temporal chaos
- Two-dimensional dynamics: spiral waves and shock solutions, frequency selection, interaction of spiral waves, symmetry breaking, core instability, vortex glass and defect-mediated turbulence
- Three-Dimensional Dynamics, stretching instability of vortex lines, supercoiling, vortex line turbulence

**Self-Organization of active polar rods: self-assembly of microtubules and molecular motors**

- Brief survey of main experimental observations
- Simple micromechanical calculations: inelastic collisions
- Maxwell model for polar rods: analogy with granular systems
- Orientational instability and Landau expansion
- Spatial localization of interaction and the Boltzmann equation
- Ginzburg-Landau expansion of the Boltzmann equation
- Stationary solutions: Asters and Vortices
- Phase diagram, effects of crosslinkers

**Collective motion of interacting self-propelled particles**

- Vicsek model for polar self-propelled particles, phase transition to collective behavior, order parameter
- Phenomenological model of Toner and Tu, giant number fluctuations
- Probabilistic Boltzmann description of the Vicsek model
- Ginzburg-Landau expansion, continuum description, soliton solutions
- Vicsek model for particles with apolar (nematic interaction), band solution and spatio-temporal chaos

**Recommended Literature**

- I. S. Aranson and L. Kramer, The World of the Complex Ginzburg-Landau Equation, Review of Modern Physics, v74, 99 (2002).
- I. S. Aranson and L. S. Tsimring, Patterns and Collective Behavior in Granular Media: Theoretical Concepts, Review of Modern Physics, v78, 641 (2006).
- I. S. Aranson and L. S. Tsimring, Granular Patterns, Oxford University Press, (2009).
- S. Ramaswamy, The mechanics and statistics of active matter. Ann Rev Condens Matter Phys
**1**(1):323–345 (2010). - A. Peshkov, I. S. Aranson, E. Bertin, H. Chate, and F. Gineli, Nonlinear Field Equations for Aligning Self-Propelled Rods, Physical Review Letters, v109, 268701 (2012).