tag:blogger.com,1999:blog-5439168179960787195.post143162640028370246..comments2024-03-28T17:13:01.117+10:00Comments on Condensed concepts: A limit to my understandingRoss H. McKenziehttp://www.blogger.com/profile/09950455939572097456noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-5439168179960787195.post-67691741080030184002012-09-14T09:07:55.463+10:002012-09-14T09:07:55.463+10:00Thanks for the comments.
I agree spontaneous symm...Thanks for the comments.<br /><br />I agree spontaneous symmetry breaking provides an important and puzzling example of how the order of limits matters. Thanks for stressing this.<br /><br />However, my main point was slightly different. For both examples I gave one takes the thermodynamic limit first. Then for the infinite system the order of taking the omega and q to zero limits matters.<br />I found this unsettling. Ross H. McKenziehttps://www.blogger.com/profile/09950455939572097456noreply@blogger.comtag:blogger.com,1999:blog-5439168179960787195.post-3681197153896766772012-09-13T00:06:42.844+10:002012-09-13T00:06:42.844+10:00Ben beat me to it (spontaneously broken symmetry i...Ben beat me to it (spontaneously broken symmetry in e.g. magnets as THE example). I don't agree with the statement that it's distasteful though. My "aesthetic sense" always though it a really nice, deep feature. Due to finite size effects&co you never get the infinite limit, only really large numbers, and the "correct" order of the limits depends on which factor is less infinite in your experiment. <br /><br />The simplest example I can think of [1] is the long term behavior of a ball balanced on the middle hill out of three (i.e. in a double well potential). Here $x_init -> 0$ and $t -> \infty$ are the competing limits.<br /><br />[1] I first encountered it in Wilsons http://prb.aps.org/abstract/PRB/v4/i9/p3174_1Erik Edlundhttps://www.blogger.com/profile/12590749446690386925noreply@blogger.comtag:blogger.com,1999:blog-5439168179960787195.post-15741096674657783842012-09-12T15:58:05.735+10:002012-09-12T15:58:05.735+10:00Isn't spontaneously broken symmetry THE exampl...Isn't spontaneously broken symmetry THE example? If the "field", H->0 before N-> infinity then we don't get SBS, but in the other order of limits we do. <br /><br />I agree that the fact that the order of limits matters is deeply distasteful. Ben Powellhttps://www.blogger.com/profile/04312113344388752854noreply@blogger.comtag:blogger.com,1999:blog-5439168179960787195.post-18290561404187296402012-09-11T03:18:34.152+10:002012-09-11T03:18:34.152+10:00Hi Ross,
I feel honored that you mentioned my pa...Hi Ross, <br /><br />I feel honored that you mentioned my paper with Sriram on your blog...I have been looking at your blog for quite a while now already!<br /><br />Anyway, I agree that it is interesting messing about with order of limits and what this says about the, I guess, non-analytic nature of these functions. The connection above to the electromagnetic response of superconductors is also not something I realized before so I appreciate you pointing it out.<br /><br />As far as S_Kelvin goes I would like to point out that, while it is approximate, S_Kelvin is exact for the fractional quantum Hall effect (FQHE). Nearly the last section of Peterson-Shastry discusses how if the system is dissipationless and the particle velocities are energy independent then S_Kelvin is exact. Previous calculations by Yang and Halperin of the thermopower for the FQHE have yielded S/(q_e N) (S is entropy) while S_Kelvin = (1/q_e)(\partial S/\partial N)_{T,V}. In certain FQHE systems (filling factor 5/2, for example), the entropy goes as S~N to first approximation. Thus, \partial S/\partial N and S/N are the same. However, S_Kelvin is exact and is able to capture higher order corrections that Yang and Halperin have missed!<br /><br />Also, I made a comment on a previous post of yours http://condensedconcepts.blogspot.com.au/2011/11/deconstructing-chemical-potential-of.html regarding a paper by Jaklic and Prelovsek.Anonymoushttps://www.blogger.com/profile/01457912761572923790noreply@blogger.comtag:blogger.com,1999:blog-5439168179960787195.post-69111913536280689852012-09-06T23:24:27.333+10:002012-09-06T23:24:27.333+10:00Hi,
A paper somewhat related to the comments by To...Hi,<br />A paper somewhat related to the comments by Tony is Luttinger's paper on coefficients for thermal tranport<br />http://prola.aps.org/abstract/PR/v135/i6A/pA1505_1<br />I believe it might be enlightening.Anonymoushttps://www.blogger.com/profile/17177464545831711502noreply@blogger.comtag:blogger.com,1999:blog-5439168179960787195.post-66430424438838559962012-09-06T22:50:08.075+10:002012-09-06T22:50:08.075+10:00You might also like the following paper: http://ww...You might also like the following paper: http://www.nature.com/nphys/journal/v4/n6/abs/nphys963.html (or arXiv:0706.0212). The abstract explicitly states that «... the limits of the ramp rate going to zero and the system size going to infinity do not commute ...», which is exactly what you are looking for.Igor Ivanovhttps://www.blogger.com/profile/00567251406214176660noreply@blogger.comtag:blogger.com,1999:blog-5439168179960787195.post-17151657712809485162012-09-06T19:23:04.899+10:002012-09-06T19:23:04.899+10:00Nice title by the wayNice title by the wayTonyhttps://www.blogger.com/profile/08582644751812812675noreply@blogger.comtag:blogger.com,1999:blog-5439168179960787195.post-64065053637700844212012-09-06T19:19:56.831+10:002012-09-06T19:19:56.831+10:00Mahan discusses in his Kubo formulae section (ch 3...Mahan discusses in his Kubo formulae section (ch 3 from memory) that when taking the dc conductivity, you calculate the finite \omega, finite q current-current correlation function, and then you must first take q->0, then \omega->0. <br /><br />Done the wrong way, ie first taking \omega->0 enforces a static electric field. In this case no current will flow. Taking first the q->0, one gets a finite conductivity (when appropriate). <br /><br />It seems that for Kubo formulae, you tend to have this 1/\omega term which when treated properly will often cancel with some linear in \omega numerator. For this reason I wouldn't be surprised if there are other examples.<br /><br />In fact, it's not the same thing, but taking the \omega->0 limit for graphene doesn't work, and one gets a spurious result from linear response for the dc conductivity. I believe this was part of the 'case of the missing pi' controversy.<br /><br />I'm not 100% sure though, I never really got into that field, just remember some comments made in passing. Tonyhttps://www.blogger.com/profile/08582644751812812675noreply@blogger.com