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| '''Multi-particle collision dynamics''' (MPC), also known as stochastic rotation dynamics (SRD),<ref name="mpc_review">[http://dx.doi.org/10.1007/978-3-540-87706-6_1 G. Gompper, T. Ihle, K. Kroll and R. G. Winkler "Multi-Particle Collision Dynamics: A Particle-Based Mesoscale Simulation Approach to the Hydrodynamics of Complex Fluids", Advanced Computer Simulation Approaches for Soft Matter Sciences III, Advances in Polymer Science '''221''' p. 1 (2009)] [arXiv: http://arxiv.org/abs/0808.2157]</ref> is a particle-based mesoscale simulation technique for complex fluids which fully incorporates thermal fluctuations and hydrodynamic interactions.<ref>[http://dx.doi.org/10.1063/1.478857 A. Malevanets and R. Kapral "Mesoscopic model for solvent dynamics", Journal of Chemical Physics '''110''' pp. 8605-8613 (1999)]</ref> Coupling of embedded particles to the coarse-grained solvent is achieved through [[molecular dynamics]].<ref name="colloids_malevanets_kapral">[http://dx.doi.org/10.1063/1.481289 A. Malevanets and R. Kapral "Solute molecular dynamics in a mesoscale solvent", Journal of Chemical Physics '''112''' pp. 7260-7269 (2000)]</ref>
| | CMS provides the best platform to create websites that fulfill all the specifications of SEO. It is very easy to customize plugins according to the needs of a particular business. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. Word - Press also provides protection against spamming, as security is a measure issue. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting. <br><br> |
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| ==Method of simulation==
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| The solvent is modelled as a set of <math>N</math> point particles of mass <math>m</math> with continuous coordinates <math>\vec{r}_{i}</math> and velocities <math>\vec{v}_{i}</math>. The simulation consists of streaming and collision steps.
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| During the streaming step, the coordinates of the particles are updated according to
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| <math>\vec{r}_{i}(t+\delta t_{\mathrm{MPC}}) = \vec{r}_{i}(t) + \vec{v}_{i}(t) \delta t_{\mathrm{MPC}}</math>
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| where <math>\delta t_{\mathrm{MPC}}</math> is a chosen simulation time step which is typically much larger than a molecular dynamics time step.
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| After the streaming step, interactions between the solvent particles are modelled in the collision step. The particles are sorted into collision cells with a lateral size <math>a</math>. Particle velocities within each cell are updated according to the collision rule
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| :<math>\vec{v}_{i} \rightarrow \vec{v}_{\mathrm{CMS}} + \hat{\mathbf{R}} ( \vec{v}_{i} - \vec{v}_{\mathrm{CMS}} )</math>
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| where <math>\vec{v}_{\mathrm{CMS}}</math> is the centre of mass velocity of the particles in the collision cell and <math>\hat{\mathbf{R}}</math> is a [[rotation matrix]]. In two dimensions, <math>\hat{\mathbf{R}}</math> performs a rotation by an angle <math>+\alpha</math> or <math>-\alpha</math> with probability <math>1/2</math>. In three dimensions, the rotation is performed by an angle <math>\alpha</math> around a random rotation axis. The same rotation is applied for all particles within a given collision cell, but the direction (axis) of rotation is statistically independent both between all cells and for a given cell in time.
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| If the structure of the collision grid defined by the positions of the collision cells is fixed, [[Galilean invariance]] is violated. It is restored with the introduction of a random shift of the collision grid.<ref>[http://dx.doi.org/10.1103/PhysRevE.67.066705 T. Ihle and D. Kroll "Stochastic rotation dynamics. I. Formalism, Galilean invariance, and Green-Kubo relations", Physical Review E '''67''' 066705 (2003)]</ref>
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| Explicit expressions for the [[diffusion coefficient]] and [[viscosity]] derived based on [[Green-Kubo relations]] are in excellent agreement with simulations.<ref>[http://dx.doi.org/10.1103/PhysRevE.70.035701 T. Ihle, E. Tüzel and D. M. Kroll "Resummed Green-Kubo relations for a fluctuating fluid-particle model", Physical Review E '''70''' 035701 (2004)]</ref><ref>[http://dx.doi.org/10.1103/PhysRevE.72.046707 T. Ihle, E. Tüzel and D. M. Kroll "Equilibrium calculation of transport coefficients for a fluid-particle model", Physical Review E '''72''' 046707 (2005)]</ref>
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| ==Simulation parameters==
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| The set of parameters for the simulation of the solvent are: | |
| * solvent particle mass <math>m</math>
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| * average number of solvent particles per collision box <math>n_{s}</math>
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| * lateral collision box size <math>a</math>
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| * stochastic rotation angle <math>\alpha</math>
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| * [[kT (energy)]]
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| * time step <math>\delta t_{\mathrm{MPC}}</math>
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| The simulation parameters define the solvent properties,<ref name="mpc_review" /> such as
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| * [[mean free path]] <math>\lambda = \delta t_{\mathrm{MPC}} \sqrt{kT/m}</math>
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| * [[diffusion coefficient]] <math>D = \frac{kT\delta t_{\mathrm MPC}}{2m} \Bigg[ \frac{d n_{s}} {(1-\cos(\alpha))(n_{s}-1+\exp^{-n_{s}})}-1 \Bigg]</math>
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| * [[shear viscosity]] <math>\nu</math>
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| * [[thermal diffusivity]] <math>D_{T}</math>
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| where <math>d</math> is the dimensionality of the system.
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| A typical choice for normalisation is <math>a=1,\; kT=1,\;m=1</math>. To reproduce fluid-like behaviour, the remaining parameters may be fixed as <math>\alpha = 130^{o},\; n_{s} = 10,\; \delta t_{\mathrm{MPC}} \in [0.01;0.1]</math>.<ref name="elgeti_phd">[http://kups.ub.uni-koeln.de/volltexte/2007/2007/pdf/elgeti.pdf J. Elgeti "Sperm and Cilia Dynamics" PhD thesis, Universität zu Köln (2006)]</ref>
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| ==Applications==
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| MPC has become a notable tool in the simulations of many soft-matter systems, including
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| * [[colloid]] dynamics<ref name="colloids_malevanets_kapral" /><ref>[http://dx.doi.org/10.1103/PhysRevLett.93.220601 J. T. Padding and A. A. Louis "Hydrodynamic and Brownian Fluctuations in Sedimenting Suspensions", Physical Review Letters '''93''' 220601 (2004)]</ref><ref>[http://dx.doi.org/10.1103/PhysRevE.74.021403 M. Hecht, J. Harting, M. Bier, J. Reinshagen and H. J. Herrmann "Shear viscosity of claylike colloids in computer simulations and experiments", Physical Review E '''74''' 021403 (2006)]</ref>
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| * [[polymer]] dynamics<ref>[http://dx.doi.org/10.1063/1.2041527 K. Mussawisade, M. Ripoll, R. G. Winkler, and G. Gompper "Dynamics of polymers in a particle-based mesoscopic solvent", Journal of Chemical Physics '''123''' 144905 (2005)]</ref><ref>[http://dx.doi.org/10.1140/epje/i2006-10220-0 M. Ripoll, R. G. Winkler and G. Gompper "Hydrodynamic screening of star polymers in shear flow", The European Physics Journal E '''23''' pp. 349-354 (2007)]</ref>
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| * [[Vesicle (biology and chemistry)|vesicles]]<ref>[http://dx.doi.org/10.1103/PhysRevE.72.011901 H. Noguchi and G. Gompper "Dynamics of fluid vesicles in shear flow: Effect of membrane viscosity and thermal fluctuations", Physical Review E '''72''' 011901 (2005)]</ref>
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| * active systems<ref name="elgeti_phd" />
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| ==References==
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| {{reflist}}
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| [[Category:Computational physics]]
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| [[Category:Computational fluid dynamics]]
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ve labored so hard to publish and put up on their website. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. You can allow visitors to post comments, or you can even allow your visitors to register and create their own personal blogs. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration.
The primary differences are in the plugins that I install, as all sites don't need all the normal plugins. High Quality Services: These companies help you in creating high quality Word - Press websites. One of the great features of Wordpress is its ability to integrate SEO into your site. It's now become a great place to sell it thanks to Woo - Commerce. Make sure you have the latest versions of all your plugins are updated.
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