This is a very well-developed field (non-equilibrium dissipative dynamics); I suggest the classic book "Nonequilibrium Statistical Mechanics" (2001) by Robert Zwanzig. Also some of his seminal papers are particularly illuminating, such as "Hydrodynamic Theory of the Velocity Correlation Function* published in Phys. Rev. A (1970), http://dx.doi.org/10.1103/PhysRevA.2.2005

Others have given good foundational places to look.In terms of current work, you might be interested in a paper about to be published by my colleague Mykyta Chubynsky. In it he talks about modelling "anomalous, yet Brownian" motion.

https://journals.aps.org/prl/accepted/44071Yb7Ccb1444d30ca4c4754825b0caaf72507a

Thank you very much for all your clear answers and papers. Actually I have found recently in a Master thesis that a delta Dirac function was used to simulate the random forces that are applied to a nanoparticle in blood flow. The student used a boundary integral equation to simulate the motion of this nanoscopic particle in the Stokes regime. I would like to know to what extent one could use this formulation in the investigation of Random motion of these particles in the case where there is not a large time scale separation? Otherwise, what is the mathematical form of the random force that could be applied to the particle in order to simulate Brownian motion? Thanks!

I do not think that the topic of the anomalous diffusion is relevant for colloids. Colloidal particles interact with each other, and this interaction affects the laws of diffusion of individual particles (renormalization of these laws). As a result, the effective Brownian particle dynamics may look abnormally while the dynamics of a free particle can be Gaussian.

New fast (and stable) method of simulation of Brownian dynamics for interacting colloidal particles is described here:

https://www.researchgate.net/publication/233959731_Dynamic_self-assembly_of_photo-switchable_nanoparticles?ev=prf_pub

Generalization of the method for systems with mixing is given here:

https://www.researchgate.net/publication/233959738_Kinetic_Monte_Carlo_simulations_of_flow-assisted_polymerization?ev=prf_pub

Article Dynamic self-assembly of photo-switchable nanoparticles

Article Kinetic Monte Carlo Simulations of Flow-Assisted Polymerization

Other simulation tools for describing Brownian motion of colloidal particles in the diffusive and ballistic regimes include Smoothed Dissipative Particle Dynamics methods for simple solvents https://www.researchgate.net/publication/258077112_Multiscale_modeling_of_particle_in_suspension_with_smoothed_dissipative_particle_dynamics?ev=prf_pub          

and complex viscoelastic solvents   

https://www.researchgate.net/publication/257504292_A_SPH-based_particle_model_for_computational_microrheology?ev=prf_pub

Article Multiscale modeling of particle in suspension with smoothed ...

Article A SPH-based particle model for computational microrheology

If asking what do you mean by the temperature in your study, then all the Brownian troubles would be secondary.   

Dear Sir,

it depends.

Basic theory is well clear but You have a lot of possible mathematical 'translations' of brownian behaviours ,depending not only on the space-time scale You compute on.

Do You want a strictly physical point of view (which takes care about simplecticity of the model) about your question or it can be enough to benefit of a more constricted engineering one (more poor about physical consequences but result-oriented, which means sometime ill-posed)? What about the field of your specific applications and the numerical approximation you need for such a computation?

Please, be more precise. Your question is too much general.

Regards, Michele Romeo

Dear Michele,

Thank you for your answer. I would like indeed a strictly physical modeling applied in blood flow. I am using a boundary integral method in order to investigate the motion of rigid particles subject to Brownian motion in blood flow. It is well known that the steady linearized Navier-Stokes equation (Stokes equation) doesn't count for thermal fluctuations. Here I would like to include these fluctuations using a suitable model. Thank you.

a

Dear Sir,

I'm sorry for delay in replying You but I have a lot of stuff to satisfy.

Coming to your question, let's focus on your point.

First - You write "I would like indeed a strictly physical modeling applied in blood flow" but You don't write 'where'.

You have many distinct flow dynamics, depending on your scale-level and body district You investigate,and the weight of Brownian contributions can be as relevant as not.

So, first - please, provide me informations about where you need to model such flows: heart? blood vessels? human body?

Regards,

Michele Romeo

Dear Michele,

Thank you for your answer. Regarding the scale level on which I would like to apply this model is micro-circulation. I would like to investigate the diffusion of nano-particles (e.g. drag delivery agents) in micro-sized blood vessels. The method I use for my simulations is BIM (boundary integral method) which it doesn't account for thermal fluctuations. That's the reason why I would like to accurately model the Brownian motion of these particles which are strongly effected by thermal fluctuations. Your help would be highly appreciated. Thanks!

a

Dear Sir,

aside possible further approximations, your point is clear.

Please, let me have a look to and re-order ideas about your problem.

I'll be back soon from You.

Regards,

Michele Romeo

Dear SIr,

I'm sorry for your waiting. Time is never enough, as You know.

As promised, I've checked different sources to provide You a reasonable help for your case and I convince myself that a good approach to your problem is given inside lectures and relevant publications of Prof. Mark E. Tuckerman, which consist all of a good and accessible physical theory of molecular dynamics in numerical simulation, pointing out the unavoidable constraints to say the model being 'well-posed' (this problem is related to the symplecticity of the equations of motion in case you have a non-hamiltonian mechanics), so that this can be your 'starting point'

http://cims.nyu.edu/~donev/Teaching/CoarseGraining-Fall2013/Lectures.html

where You can find the most important part about your main interests which points out the stochastic motion schemes in molecular dynamics

http://link.springer.com/chapter/10.1007%2F978-1-4419-1002-8_3

and the more useful chapter to your aims about the 'overdamped Langevin schemes' (including metropolized processes and other 'tasteful' features...)

http://cims.nyu.edu/~donev/Teaching/CoarseGraining-Fall2013/Lectures/OverdampedLangevin.pdf

You have to check deep inside for the best managing of your problem, of course. Forget to solve your problem in few steps, without a 'brutal' approach to the mathematics inside the stochastic modelling.

I can ensure You tha all these sources are in respect of the main statistical physics constraints for non-hamiltonian problems and give a solid background to establish a good engineering approach as well.

Furthermore, Mark E. Tuckerman was been one of my best reference guides for every molecular simulation environment that You can imagine.

On the other side, you can take part in the following discussion on Linkedin in which I think You can be implied, following your target, because the presence of a different point of view about the approach, forgetting the brownian question

http://lnkd.in/dUnS6JG

Make the best choice You can.

Regards,

Michele Romeo

Dear Michele,

Thank you for your detailed answer. I will start digging into it and then come out with a feedback for the discussion.

Best

a

There is a general answer,the Mori-Zwanzig equation, but applying it is challenging. In addition to colored thermal noise, one also has a model due to Tateishi, et al, in which there are two thermal forces each with its corresponding memory kernel. Such a model resembles in a very idealized way, e.g., a polymer solution, in which there are relaxations on several different time scales.  My recent paper in Soft Matter* gives examples.  Qualitatively, there are several questions, for example 'does the mean-square displacement increase linearly in time?' and 'is the likelihood of finding a displacement x a Gaussian in x?' The answers to these two questions are independent from each other and may each be YES or NO.

*G. D. J. Phillies, ``The Gaussian Diffusion Approximation is Generally Invalid in Complex Fluids'', Soft Matter, 11, 580-586 (2015). [Not the same paper as my arxiv.org paper of similar title.]