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WHY? 1 <latexit - PDF document

Jul 12, 2023 •279 likes •818 views

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 <latexit

  1. 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

  2. <latexit sha1_base64="olvJEVQ3QL9Th1HTE2VO1GMpk=">ACg3icdZDPbhMxEMadhUIJf5qWIxerEVKq0Gg3QmovlSK4cCwSaYviKJr1ziZWvd6VPdsSVn4X3oTn4AG4wivgJD3QFkay9Pmbn+WZL620chTHP1rRg4dbjx5vP2k/fb8xU5nd+/MlbWVOJalLu1FCg61MjgmRovKotQpBrP08v3q/75FVqnSvOJlhVOC5gblSsJFKxZ53MhspIa4QjkpUXdCKvmCwJry2vfXHnfo4OTQzGHoD/QoE5FAZSDXzcF1+RIDizTjcexOvi90VyI7qjfdH/xhg7ne29sIsi7QkNTg3CSJK5o2YElJjb4taodVmADm2Kw39/x1sDKelzYcQ3zt3uKgcG5ZpIEsgBbubm9lvkGnDP2LmNSUH08bZaqa0MjNd3mtOZV8FSbPlEVJehkESKvCnFwuwIKkEHlbGLyWZQjOZI3I8uyLnwynjdCYU0/kgWpEtVoONO8m/q/L0G8CPvDtkGJyN7P74mw4SOJB8jHpjt6xTW2zV2yf9VjCjtiIfWCnbMwk+85+sl/sd7QV9aNh9HaDRq2bNy/ZrYpO/gAb38fk</latexit> <latexit sha1_base64="nDNUaWsoVdCUNhn/rsrAHf5zbgs=">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</latexit> <latexit sha1_base64="nDNUaWsoVdCUNhn/rsrAHf5zbgs=">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</latexit> <latexit sha1_base64="BO9H/pr9J2QBNm895TbOG49lyTc=">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</latexit> 15/01/19 1. Can be used to simulate molecular motion in a viscous medium, without solving the equations of motion for the solvent particles. 2. Can be used as a thermostat . Friction Conservative force force First, consider motion with friction alone: After a short while, all particles will stop moving, due to the friction.. “ random ” force Better: m ˙ → → v ( t ) = � γ v ( t ) � r U + ζ ( t ) 2

  3. <latexit sha1_base64="QaWk7lyQE8P/BovuWUXTM0eAeaI=">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</latexit> <latexit 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sha1_base64="BO9H/pr9J2QBNm895TbOG49lyTc=">ACg3icdZDPThsxEMadbWlp+ocAx16sRpWCKNFuVAkuSIheqRSA1RxFM16ZxMLr3dlz0JTa5+tz8ED9Epfoc6fQ4F2JEufv/ksz/zSitHcXzbip483Xj2fPNF+Wr12+2Ots756srcShLHVpL1NwqJXBISnSeFlZhCLVeJFefVr0L67ROlWarzSvcFzA1KhcSaBgTrfCpGV5IUjkFcWtRdWTWcE1pY3jb9umh7tHR+IKRQF/DcUMgfCQKqBD/fFDyQIzqTjfvxsvhjkaxFl63rbLd2gmzyLpAQ1KDc6MkrmjswZKSGpu2qB1WYQKYol9u3vD3wcp4XtpwDPGley8HhXPzIg3JAmjmHvYW5gd0ytC/EqOa8qOxV6aqCY1cfZfXmlPJFzB5pixK0vMgQFoV5uRyBhYkBeRtYfBGlgGcybzI8ux7MxqMvdCYU0/kIeVFtVgONO8mzV+XQbMCvNe0A8XkIbPH4nzQT+J+8iXpnpyueW6yt+wd67GEHbIT9pmdsSGT7Cf7xe7Y72gj2o8G0cdVNGqt3+yexUd/wGliMZb</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

  4. 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

  5. 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

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