WHY? 1 <latexit - PDF document

15/01/19 ADVANCED TECHNIQUES (MC/MD) A (seemingly) random selection. Daan Frenkel Beyond Newtonian MD 1. Langevin dynamics 2. Brownian dynamics 3. Stokesian dynamics 4. Dissipative particle dynamics 5. Etc. etc. WHY? 1

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sha1_base64="BO9H/pr9J2QBNm895TbOG49lyTc=">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</latexit> 15/01/19 There is a relation between the correlation function of the random force and the friction coefficient: The derivation is straightforward, but beyond the scope of this lecture. The KEY point is that the friction force and the random force ARE RELATED. Limiting case of Langevin dynamics: No inertial effects (m=0) m ˙ → → v ( t ) = � γ v ( t ) � r U + ζ ( t ) Becomes: → v ( t ) � r U + ζ ( t ) 0 = � γ “ Brownian Dynamics ” (But still the friction force and the random force are related) 3

15/01/19 What is missing in Langevin dynamics and Brownian dynamics? 1. Momentum conservation 2. Hydrodynamics (1 and 2 are not independent). Is this serious? Not always: it depends on the time scales. Momentum “ diffuses ” away in a time L 2 / ν . After that time, a “ Brownian ” picture is OK. However: hydrodynamics makes that the friction constant depends on the positions of all particles (and so do the random forces…). 4

15/01/19 Momentum conserving, coarse-grained schemes: • Dissipative particle dynamics • Stochastic Rotation Dynamics These schemes represent the solvent explicitly (i.e. as particles), but in a highly simplified way. ADVANCED MC SAMPLING 5