Gibbs Sampling Procedure

let the parameter vector be partitioned into blocks 

Let 0-j be the set of all the other parameters not in block j. The proportional form of Bayes theorem, 

Gibbs sampling requires that we know the full conditional distribution of each block of parameters 0j, given all the other parameters 6-j and the data y = (l/i ) 1/n)- Let the full conditional distribution of block Oj be denoted 

At time n = 0 start from an arbitrary point in the parameter space 0° = 

---

For liquid-vapor interfaces, the correlation length in the bulk is of t he order of atomic distance unless one is close to the critical point Hence the concept of local equilibrium is well justified in most practical circumstances For. solid surfaces above the roughening temperature, the concept also makes sense. Since the surface is rough adding (or removing) an atom to a particular part of the surface docs not disturb the local equilibrium state very much, and this sampling procedure can be used to determine the local chemical potential. This is the essence of the Gibbs-Thomson relation (1). 

Schuller, R., Gibb, J. P., and Griffin, R. (1981). Recommended sampling procedures for monitoring wells. Ground Water Monit. Rev. 1(2), 42—46. 

Both have the advantage of giving a sample volume (and therefore a location of the dividing surface) which is, by definition, perfectly reproducible from one adsorption bulb to another and from one laboratory to another. Even if not always realistic, it is a sound convention, if the aim is to obtain reproducible measurements and calculations and is consistent with the spirit of the Gibbs representation. It is, for these reasons, certainly well suited for the study of reference materials. Of course, this approach would replace Step 3 in the procedure described above, whereas Steps 1 and 2 would remain necessary. 

In the past, the most common method used to obtain a representative aquifer sample was to flush the well-bore by pumping a specified number of well-bore volumes of water. This procedure is considered outdated, and is not advised. The following excerpt is a compilation of the information compiled by Gibbs and others outlining recommendations for the collection of ground-water samples... 

Larionov and co-workers (Institute of Physical Chemistry, the U.S.S.R. Academy of Sciences, Moscow) (308-312) carried out systematic theoretical and experimental investigations of the adsorption from liquid solutions of nonelectrolytes on silica adsorbents. They studied the adsorption of individual substances and binary liquid solutions (benzene/carbon tetrachloride, carbon tetrachloride/isooctane, benzene/isooctane, etc.) on Si02 samples with different degrees of porosity but identical surface chemical properties. The experimental results were compared with the theoretical calculations carried out by the Gibbs method. This procedure made it possible to calculate the dependence of the enthalpy, entropy, and free energy of wetting on the concentration and to obtain expressions describ-... 

Differences in the volumetrically and gravimetrically measured Gibbs excess masses depicted in Fig. 1.24 are partly due to different activation procedures of the zeolite samples used in the experiments. This of course should be avoided, but as a matter of fact happened in our experiments and therefore is brought to reader s attention. 

---