In this work it is shown how the immersed boundary method of (Peskin2002) for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations. A stochastic numerical method is proposed which deals with stiffness in th
e system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps. An important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved, partially resolved, or fully resolved while retaining a good level of accuracy. Error estimates in each of these regimes are given for the method. A number of theoretical and numerical checks are furthermore performed to assess its physical fidelity. For a conservative force, the method is found to simulate particles with the correct Boltzmann equilibrium statistics. It is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters. The method is also shown to reproduce a well-known hydrodynamic effect of a Brownian particle in which the velocity autocorrelation function exhibits an algebraic tau^(-3/2) decay for long times. A few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method.
The basic ingredients of osmotic pressure are a solvent fluid with a soluble molecular species which is restricted to a chamber by a boundary which is permeable to the solvent fluid but impermeable to the solute molecules. For macroscopic systems at
equilibrium, the osmotic pressure is given by the classical vant Hoff Law, which states that the pressure is proportional to the product of the temperature and the difference of the solute concentrations inside and outside the chamber. For microscopic systems the diameter of the chamber may be comparable to the length-scale associated with the solute-wall interactions or solute molecular interactions. In each of these cases, the assumptions underlying the classical vant Hoff Law may no longer hold. In this paper we develop a general theoretical framework which captures corrections to the classical theory for the osmotic pressure under more general relationships between the size of the chamber and the interaction length scales. We also show that notions of osmotic pressure based on the hydrostatic pressure of the fluid and the mechanical pressure on the bounding walls of the chamber must be distinguished for microscopic systems. To demonstrate how the theoretical framework can be applied, numerical results are presented for the osmotic pressure associated with a polymer of N monomers confined in a spherical chamber as the bond strength is varied.
