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Imagine that I have a polymer (approximated as a freely diffusing, freely jointed chain with some number of subunits 'N'), and I place this polymer into a sphere of some volume 'V'. Next, I proceed to add a series of infinitely thin, immobile chords of length 'L' to the inner walls of the sphere.

Because these chords are infinitely thin, they will not change the inside volume of the sphere, but should nevertheless place certain topological and geometric constraints on the behavior of the diffusing polymer.

Can we quantify the change in entropy caused by the addition of these infinitely thin chords? For this calculation, what changes if we replaced the polymer with a monoatomic gas?

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+1 This is an awesome question. –  user346 Jan 18 '11 at 4:16
So we are talking here about the entropy associated with all possible configurations of the polymer in phase space (as contrasted with the configurational entropy associated with all configurations in positional space)? Or do you mean to imply that the polymer has finite thickness, so that there is an excluded volume effect due to the wires? For a monoatomic gas the same distinction should be made. If the atoms are point particles they are not influenced by the wires, but if they have finite size, there will be an excluded volume effect due to the presence of the wires. –  Johannes Jan 18 '11 at 4:18
@Johannes the excluded volume effect will be second more AFAI can tell. More important will be the effect the chord has on the possible number of inequivalent topological conformations of the polymer. This is similar to, though not the same as, the effect introducing partitions in a box of gas has on the entropy of its contents. –  user346 Jan 18 '11 at 5:13
I think You mean a polymer molecule? Right? But what is "freely jointed"? Next: in what medium is the polymer "diffusing"? Next: You introduce those "infinetily thin" strands. This is no real material. of course. If You want someone to answer Your question, You have to define all kinds of interaction those strands will do with that "polymer". –  Georg Jan 18 '11 at 11:03
@space_cadet: not sure what MorningCoffee has in mind here. Insofar excluded volume effects can be ignored, I fail to see which topological configurations would be eliminated. Of course, the number of accessible states in the polymer phase space will be strongly reduced, resulting in an entropy drop. Hence my question. –  Johannes Jan 19 '11 at 0:20

1 Answer 1

All states are accessible

Basic assumption of statistical physics is that all "allowed" configurations are equally probable. And the entropy is related to the number of those configurations. Therefore entropy won't change if all the configurations would be still accessible after introduction of the chords. Which is the case if length of joints of your polymer are much smaller than the characteristic length of/between the chords in the volume.

Of course even in this case the chords will affect the diffusion times. The configuration space will have some complex structure with interconnected "islands", which must have some relation to glasses. But the total number of states and, therefore, the entropy won't change.

Inaccessible states

If length of joints becomes comparable to the size of the sphere and/or characteristic length of/between the chords, then the "chordless" configuration space will be broken in several mutually inaccessible subspaces.

I think one can imagine an example when the number of those subspaces equal to the number of
"stamp foldings". And as far as I know this combinatorial problem is yet unsolved -- not only there is no closed expression, but not even guesses about the asymptotic behaviour (here).

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