Dear Andrea and Rodrique, Just in case I was obscure. The equations I refer to are eq 2.1-2.9 of the attached paper, [[Schuss, Nadler, and Eisenberg PHYSICAL REVIEW E, VOLUME 64, 036116]] namely the Langevin equations 2.1 and the electrostatic forces that specify the f function in equations 2.1 We write out the f function for channels but you can treat it simply as the sum of Coulombic forces between the ions that are moving as specified in eq. 2.1. Boundary conditions can be dealt with later. The goal is to a) simulate the time series of the number density of 1) positive particles 2) negative particles 3) mass (i.e., sum of positive and negative) 4) charge (i.e., difference of positive and negative0 b) derive a Fokker Planck like equations that describes the number densities 1) through 4) identified above. As ever Bob On Sat, Apr 5, 2008 at 6:26 AM, Bob Eisenberg wrote: Dear Andrea and Rodrique I thought you might a more formal statement of the problem I think is so important in statistical physics before you discuss it with Mitya and Pete. The problem I am interested in is central to the treatment of ions (hard charged spheres, which to begin with are approximated as points) in water and in proteins. This is a central system in biology, chemistry, physics, and stochastics, the latter since it is the system that Brown originally studied and is described by Einstein's treatment of Brownian motion. The problem is best introduced by considering Na (positive hard spheres) and Cl (negative hard spheres with similar but not equal diameters and diffusion coefficients) in a dielectric background. Many many chemists (including my friends Stuart Rice, Doug Henderson, and Jean-Pierre Hansen) have shown that a thermodynamic analysis of this system is remarkably successful (but of course not totally successful) in understanding the properties of ionic solutions over a wide range of compositions. We have shown that a similar approach works surprisingly well in biology, for ionic channels (proteins with a hole down their middle, of enormous importance in health and disease). The problem starts by describing the motion of Na^+ as a Langevin equation (in the high friction limit if you wish) in an electric field. The electric field MUST itself be calculated from the charges present in the system (i.e., the other Na ions, the Cl ions, and dielectric charges, and the boundary conditions). The potential can NOT be independently prescribed because in the physical world there are no sources to maintain that potential fixed (in either space or time or both) as charges move in thermal motion. The Cl^- ion is described the same way. The electric field is described by Poisson's equation ( i.e., for the electric potential) and boundary conditions. The standard problem is to find a description for the number density of Na, the number density of Cl as a partial differential equation in the spirit of a Fokker Planck equation. The problem I pose is to find a similar description for the probability of the 1) number density of charge Q defined as the difference of the density of Na and Cl and 2) for the total number density of both ions C C defined as the sum of the density of Na and Cl. The charge fluctuates, the 'concentration' C fluctuate. A simulation can clearly keep track of the charge and the total number density. The question is can we write equations for these. The reason this issue is so central is that it is an experimental fact (verified by electrochemists since around 1890) that in many domains the system of Na and Cl in water behaves according to "Ohm's law" (i.e., one need be concerned only with charge, to the first order, except perhaps one scaling constant) and in other domains the system of Na and Cl in water behaves according to "Fick's law" ( i.e., one need only be concerned with the concentration, to first order). Those domains are the domains that determine a large fraction of biological behavior of nerve cells, muscle cells, etc etc. The mathematical question is can we derive these equations and the domains in which they are valid. There are then many things that could be done. If one develops a mean field description, changing variables from number density of each ion, to number density of charge, and of all ions, gives large simplifications. I suggest you write the time dependent version of PNP and then change its variables to p+n (mass) and p-n (charge0 and look at the enormous simplification. I imagine that the equivalent stochastic change of variable (which is what I seek) would be a least as successful. There is more to this than I write here but I am writing to propose the key problem. As ever Bob -- ======================== Return Address for email: beisenbe@rush.edu Bob aka RS Eisenberg Bard Professor and Chairman Dept of Molecular Biophysics & Physiology Rush Medical Center 1653 West Congress Parkway Chicago IL 60612 USA Office Location: Room 1291 of Jelke Building at 1750 West Harrison Email: beisenbe@rush.edu Voice: +312-942-6467 FAX: +312-942-8711 FAX to Email: +801-504-8665 Department WebSite: http://www.rushu.rush.edu/molbio/ Personal WebSite: http://www2.phys.rush.edu/RSEisenberg/physioeis.html -- ======================== Return Address for email: beisenbe@rush.edu Bob aka RS Eisenberg Bard Professor and Chairman Dept of Molecular Biophysics & Physiology Rush Medical Center 1653 West Congress Parkway Chicago IL 60612 USA Office Location: Room 1291 of Jelke Building at 1750 West Harrison Email: beisenbe@rush.edu Voice: +312-942-6467 FAX: +312-942-8711 FAX to Email: +801-504-8665 Department WebSite: http://www.rushu.rush.edu/molbio/ Personal WebSite: http://www2.phys.rush.edu/RSEisenberg/physioeis.html