R C Ball, Physics Theory Group and Centre for Complexity Science University of Warwick assoc. member Centre for Scientific Computing Incorporating Hydrodynamics into Monte Carlo Simulations

Outline Monte Carlo simulation and Diffusion Challenge of long range hydrodynamic coupling – important in many soft matter systems. Fourier method: order N^2 per unit time –application to polymer dynamics Wavelet adaptation approaching order N

MC of particle systems

More practical interpretation

Particles in a fluid q q p p a  Some studies just use Cholesky …  Banchio & Brady: long range part slow to change – update less often (JCP 2003)  Present approach: (i) Fourier (ii) Wavelets Micro Hydrodynamics Macromolecules Bacteria Colloids

Fourier Approach

Limitation to timestep

Polymer chain N=1000 monomers Using ‘phantom’ chains so that eq’m initial configurations available

Polymer diffusion monomer motion Centre of Mass motion Monomers rel to CoM N=100 monomer chains CoM

N=400 N=1000

Hydrodynamic scaling N=1000 monomers monomers Centre of Mass abs rel CoM

Wavelet version (untested)

Move limits and costs

Limitations and Prospects Imposing relative motion and strain: applicable to SOFT matter only. Not capturing close-to-contact “lubrication friction” Potential to beat conventional MC at equilibration –hydrodynamics accelerates large scale relaxation E.g. polymer D ~ 1/N → 1/R –motion more concerted Opens up hydrodynamic coupling Background flow (e.g. simple shear) can be added Walls easy to add by Fourier, challenging by Wavelets

Acknowledgements Line of enquiry prompted by collaboration on translocation with D Panja and G Berkema: hydrodynamics crucial. CSC CoW ITS desktop CSC seminar