From chemistry-request@server.ccl.net Wed Sep 15 04:34:05 1999 Received: from jivdhan.unipune.ernet.in (jivdhan.unipune.ernet.in [196.1.114.31]) by server.ccl.net (8.8.7/8.8.7) with ESMTP id EAA26740 for ; Wed, 15 Sep 1999 04:33:57 -0400 Received: from chem.unipune.ernet.in (chem.unipune.ernet.in [196.1.114.10]) by jivdhan.unipune.ernet.in (8.9.0/UnipuneMailhubV1.3) with ESMTP id NAA31924 for ; Wed, 15 Sep 1999 13:52:21 +0530 Received: (from tcg@localhost) by chem.unipune.ernet.in (8.8.7/8.8.7) id OAA04956 for CHEMISTRY@www.ccl.net; Wed, 15 Sep 1999 14:15:13 +0530 Date: Wed, 15 Sep 1999 14:15:13 +0530 From: "Students of Dr. S.R. Gadre" Message-Id: <199909150845.OAA04956@chem.unipune.ernet.in> To: CHEMISTRY@server.ccl.net Subject: Formamide.(H2O)n Dear Sirs/Madams: Could someone give me refrences on calculations/ modelling of hydration of formamide? Thanks. ANANT KULKARNI From chemistry-request@server.ccl.net Wed Sep 15 10:23:54 1999 Received: from kasei.materials.ox.ac.uk (kasei.materials.ox.ac.uk [163.1.64.46]) by server.ccl.net (8.8.7/8.8.7) with ESMTP id KAA28892 for ; Wed, 15 Sep 1999 10:23:54 -0400 Received: from localhost (hfield@localhost) by kasei.materials.ox.ac.uk (8.9.3/8.9.3) with ESMTP id PAA13268 for ; Wed, 15 Sep 1999 15:18:41 +0100 X-Authentication-Warning: kasei.materials.ox.ac.uk: hfield owned process doing -bs Date: Wed, 15 Sep 1999 15:18:41 +0100 (BST) From: Andrew Horsfield X-Sender: hfield@kasei.materials.ox.ac.uk Reply-To: Andrew Horsfield To: Computational Chemistry List Subject: Diffusion Equation from Kinetic Lattice Monte Carlo Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Hi, I have a question about kinetic Monte Carlo simulations on a lattice, and how they relate to traditional diffusion and reaction equations. I understand Monte Carlo in the following way: 1) You define a set of allowed configurations which you can index 2) You evaluate by some means a set of rates for transitions between these configurations. Let the rate for a transition from configuration i to configuration j be W_{ji} (note the order of the indices). 3) You start from some initial configuration, and make a transition to a new configuration with a probability proportional to the rate. This process is repeated for a number of transitions. I want to derive continuum equations from this algorithm. To do this I begin by defining the probability that my system is in configuration i to be P_i. I then ask how P_i varies with time. The answer (I think) is: d P_i ----- = \sum_j L_{ij} P_j dt where L_{ij} = W_{ij} i \= j = -\sum_k W_{ki} i = j This all looks fine until one transforms this equation into a corresponding equation for the rate of change of concentration (C): d C(r)_n -------- = \sum_{sm} R_{rn,sm} C(s)_m dt where r and s are indices for the type of atom (C, O, H or whatever), n and m are indices for spatial position (lattice site), and R_{rn,sm} is a concentration independent rate matrix that can be derived from L_{ij}. The feature that worries me about this is that it is *linear* in C (as it has to be, given the starting point). I believe that this cannot be correct if there are chemical reactions taking place, since then we would expect the rate to depend on the *product* of two concentrations (at least). Can anyone tell me where I have gone wrong? Is there a standard reference for this kind of approximation? Cheers, Andrew +----------------------------------------------------+ Andrew Horsfield e-mail: horsfield@fecit.co.uk FECIT, 2 Longwalk Road, Stockley Park, Uxbridge, Middlesex UB11 1AB, United Kingdom. phone: +44-(0)181-606-4653 FAX: +44-(0)181-606-4422 +----------------------------------------------------+ From chemistry-request@server.ccl.net Wed Sep 15 11:00:10 1999 Received: from ccl.net (atlantis.ccl.net [192.148.249.4]) by server.ccl.net (8.8.7/8.8.7) with ESMTP id LAA29036 for ; Wed, 15 Sep 1999 11:00:10 -0400 Received: from hscmg02.hou.ucarb.com ([144.68.4.25] (may be forged)) by ccl.net (8.8.6/8.8.6/OSC 1.1) with SMTP id KAA00146 for ; Wed, 15 Sep 1999 10:54:04 -0400 (EDT) Received: FROM hscmg02.hou.ucarb.com BY hscmg02.hou.ucarb.com ; Wed Sep 15 09:55:49 1999 Received: by hscmg02.hou.ucarb.com with Internet Mail Service (5.5.2448.0) id ; Wed, 15 Sep 1999 09:55:49 -0500 Message-ID: <47F7EAA0389AD011840500805FEAB9C60376E27F@sctms01.sct.ucarb.com> From: "Fulton CR (Charles)" To: "'CCL'" Subject: chemisty on linux Date: Wed, 15 Sep 1999 09:55:42 -0500 MIME-Version: 1.0 X-Mailer: Internet Mail Service (5.5.2448.0) Content-Type: text/plain Anyone know of any sites listing all open source comp. chemistry related programs and code that runs on linux. I'm trying to compile a list. thanks, -charlie. ---------------------------- Charles Fulton Catalyst Skill Center Adv. Information Technician Union Carbide Corporation S. Charleston, WV p:(304-747-3175) From chemistry-request@server.ccl.net Wed Sep 15 21:09:54 1999 Received: from anusf.anu.edu.au (root@anusf.anu.edu.au [150.203.5.2]) by server.ccl.net (8.8.7/8.8.7) with ESMTP id VAA32167 for ; Wed, 15 Sep 1999 21:09:52 -0400 Received: from anu.edu.au (hws900@anusf [150.203.5.2]) by anusf.anu.edu.au (8.8.6/8.8.6) with ESMTP id LAA04423; Thu, 16 Sep 1999 11:03:40 +1000 (EST) Sender: hws900@anusf.anu.edu.au Message-ID: <37E041EB.F42BC433@anu.edu.au> Date: Thu, 16 Sep 1999 11:03:39 +1000 From: Harold W Schranz Organization: ANUSF X-Mailer: Mozilla 4.05 [en] (X11; I; SunOS 5.6 sun4m) MIME-Version: 1.0 To: Computational Chemistry List Subject: CCL: Re: Diffusion Equation from Kinetic Lattice Monte Carlo Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Hi Andrew, A brief answer (ignoring some details of notation and implementation): (1) Time dependence of probability --------------------------------- Andrew Horsfield wrote: > I have a question about kinetic Monte Carlo simulations on a lattice, and > how they relate to traditional diffusion and reaction equations. The relationship between chemical kinetics, diffusion and Monte Carlo is indeed a close one as can be observed from some of the treatments in some physical chemistry textbooks (e.g. some editions of Atkins). > I want to derive continuum equations from this algorithm. To do this I > begin by defining the probability that my system is in configuration i to > be P_i. I then ask how P_i varies with time. The answer (I think) is: > > d P_i > ----- = \sum_j L_{ij} P_j > dt > > where > > L_{ij} = W_{ij} i \= j > = -\sum_k W_{ki} i = j The dP_i/dt equation is linear in the probabilities P_i unless the transition rate matrix L_ij also depends on P_i. At the moment it is quite general as you havent specified what the configurations described by P_i are. If L_ij is independent of P_i or P_j then you have a linear system. To make this more concrete (there are other examples), this set of equations, for the case that L_ij is independent of P_i or P_j, is analogous to what is found in the description of unimolecular reactions by a master equation. If the probability is conserved it time you could be describing a closed isomerization system e.g. A->B, B->A or an energy transfer system e.g. A(E) -> A(E') where the indices either tag different species or energies of each molecule. In the former case, L_ij is related to the unimolecular rate constant for isomerization from configuration j to configuration i. In the latter case, L_ij is related to the transition probabilty matrix for energy transfer from a reactant molecule initially with energy E_j which ends up at a final energy E_i. One could think of similar interpretation in terms of hopping rates on a lattice, L_ij determining the rate of hopping from site j to site i. (2) Time dependence of concentrations ------------------------------------- > This all looks fine until one transforms this equation into a > corresponding equation for the rate of change of concentration (C): > > d C(r)_n > -------- = \sum_{sm} R_{rn,sm} C(s)_m > dt > > where r and s are indices for the type of atom (C, O, H or whatever), n > and m are indices for spatial position (lattice site), and R_{rn,sm} is a > concentration independent rate matrix that can be derived from L_{ij}. The > feature that worries me about this is that it is *linear* in C (as it has > to be, given the starting point). > > I believe that this cannot be correct if there are chemical reactions > taking place, since then we would expect the rate to depend on the > *product* of two concentrations (at least). Not true!!! This misconception seems to be a consequence of people being taught kinetics from the bimolecular viewpoint only (A+B->C+D) and most texts disregard the huge literature on unimolecular and related reactions. (Probably since the mathematical treatment is rather more complex than the trivial level required for simple bimolecular reactions!) Unimolecular reactions have a linear rate law: d C(t) / dt = - k C(t) Of course in most cases, the unimolecular rate constant k depends on a lot of factors, which may include concentrations of other species which might be in excess. > Can anyone tell me where I have gone wrong? Is there a standard reference > for this kind of approximation? You havent gone wrong except in interpretation. If you are indeed describing bimolecular reactions on a lattice (or whatever) then your matrices L_ij and R_rn,sm will have to have a dependence on concentration as well. For example: Basic chemical kinetics: if A + B -> C + D is an elementary reaction it has a rate law: Rate = -dC_A/dt = -dC_B/dt = dC_C/dt = dC_D/dt = k C_A C_B which of course be written in a linear form: Rate = -d C_A/dt = k' C_A where k' = k C_B but that is likely to only be useful if C_A is constant or slowly varying in time. I hope the above deliberations help. Regards, Dr. Harold W. Schranz Visiting Fellow, RSC, ANU Fujitsu Area 3 Project, ANUSF -- ------------------------------------------------------------------------ Dr. Harold W. Schranz, Office Ph.: +61 (02) 6249 5988 Computational Chemist, Dept Ph.: +61 (02) 6249 3437 ANU Supercomputer Facility, Fax: +61 (02) 6279 8199 Australian National University, Email: Harold.Schranz@anu.edu.au Canberra, ACT 0200, AUSTRALIA. WWW: http://anusf.anu.edu.au/~hws900 ------------------------------------------------------------------------ From chemistry-request@server.ccl.net Wed Sep 15 22:47:39 1999 Received: from e3.ny.us.ibm.com (e3.ny.us.ibm.com [32.97.182.103]) by server.ccl.net (8.8.7/8.8.7) with ESMTP id WAA32610 for ; Wed, 15 Sep 1999 22:47:39 -0400 From: whuber@almaden.ibm.com Received: from westrelay02.boulder.ibm.com (westrelay02.boulder.ibm.com [9.99.132.205]) by e3.ny.us.ibm.com (8.9.3/8.9.3) with ESMTP id WAA85038 for ; Wed, 15 Sep 1999 22:41:04 -0400 Received: from d53mta03h.boulder.ibm.com (d53mta03h.boulder.ibm.com [9.99.142.3]) by westrelay02.boulder.ibm.com (8.8.8m2/NCO v2.05) with SMTP id UAA231388 for ; Wed, 15 Sep 1999 20:41:31 -0600 Received: by d53mta03h.boulder.ibm.com(Lotus SMTP MTA v4.6.5 (863.2 5-20-1999)) id 872567EE.000EC8AD ; Wed, 15 Sep 1999 20:41:28 -0600 X-Lotus-FromDomain: IBMUS To: chemistry@ccl.net Message-ID: <872567EE.000EC859.00@d53mta03h.boulder.ibm.com> Date: Wed, 15 Sep 1999 19:41:25 -0700 Subject: CCL: Re: Diffusion Equation from Kinetic Lattice Monte Carlo Mime-Version: 1.0 Content-type: text/plain; charset=us-ascii Content-Disposition: inline Andrew Horsfield wrote: > I have a question about kinetic Monte Carlo simulations on a lattice, and > how they relate to traditional diffusion and reaction equations. > ... > Can anyone tell me where I have gone wrong? Is there a standard reference > for this kind of approximation? Andrew, One standard reference, which I found extremely useful, is "Stochastic processes in physics and chemistry", by N. G. van Kampen, Elsevier Science Publishers B.V., 2n ed. 1992. He extensively treats the problem of how to derive macroscopic rate equations from master equations, the method is a systematic expansion being called "Omega-expansion". Higher-order terms in the expansion can even be related to Langevin-like fluctuations. In particular, a master equation is always a linear equation for the probability density P, whereas the resulting equation for the concentrations C can of course be non-linear. In van Kampen's book, spatial dependence and diffusion is not explicitely treated, but this is straightforward. Harold Schranz wrote: > ..If you are indeed describing bimolecular reactions on a lattice > (or whatever) then your matrices L_ij and R_rn,sm will have to have > a dependence on concentration as well. I'm afraid this is not correct. See the above reference. The crucial point is that the probability density in the master equation refers to the many-particle picture, i.e. you are looking at P(m, n1, n2, ...), where m is the index of the lattice site (or spatial cell), and n1, n2,... are the occupation numbers of atoms of type 1, 2,.. It is also good to note that there are master equations which do have a rate equation, the (formal) reason being that the Omega-expansion does not converge. I could try to dig out more references if you're interested. Wolfgang Huber IBM Almaden Research Center