tag:blogger.com,1999:blog-22973357.post5935636029776516086..comments2018-02-25T20:17:59.326-05:00Comments on Backreaction: Pure Nerd Fun: The Grasshopper ProblemSabine Hossenfelderhttps://plus.google.com/111136225362929878171noreply@blogger.comBlogger18125tag:blogger.com,1999:blog-22973357.post-78598661260436083562018-02-07T16:29:37.379-05:002018-02-07T16:29:37.379-05:00Nonlin: No, Sabine is right. You should also look...Nonlin: No, Sabine is right. You should also look at the results after lemma 3.1. <br />Vincent: Results for non-pointlike grasshoppers are left for future work :-) Adrian Kenthttps://www.blogger.com/profile/13679959438473910598noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-3085076992362777902018-01-27T14:38:52.167-05:002018-01-27T14:38:52.167-05:00Did they take the shape of the grasshopper into ac...Did they take the shape of the grasshopper into account or approximate it as a point mass? ;-)Vincent van der Goeshttps://www.blogger.com/profile/18312166792759529390noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-77820244454613335502018-01-23T01:44:18.445-05:002018-01-23T01:44:18.445-05:00Ralf,
You should think about the boundary conditi...Ralf,<br /><br />You should think about the boundary conditions of your integrals. Sabine Hossenfelderhttps://www.blogger.com/profile/06151209308084588985noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-77116488976348952092018-01-22T22:23:30.116-05:002018-01-22T22:23:30.116-05:00Hmm, if the jump is d = 30 centimeters and area is...Hmm, if the jump is d = 30 centimeters and area is 1 sq meter, then a disk is the best lawn shape. <br /><br />The paper you cite shows Lemma 3.1:<br />The disc of area 1 (radius π−1/2) is not optimal for any d>π−1/2 ...where d is the distance jumped. That means the disk turns into a cog only for d > 56.4 cm Nonlin.orghttps://www.blogger.com/profile/15149807667681107150noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-30727354709843170322018-01-22T13:04:25.225-05:002018-01-22T13:04:25.225-05:00I tried to look at the problem using variational c...I tried to look at the problem using variational calculus with a contraint. Symbols are as in the paper unless I say here otherwise (some abbreviations due to the limitations of Unicode).<br />We have the constraint<br /><br />∫dx μ(x) = 1<br /><br />(all my integrals are over ℝ², i.e. x, y, z are vectors) and want to maximize <br /><br />p(μ)=∫dx ∫dy μ(x) μ(y) D(x-y)<br /><br />where D is the Ralf Muschallhttps://www.blogger.com/profile/04261178237250734174noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-19930267076738334162018-01-18T09:19:13.776-05:002018-01-18T09:19:13.776-05:00Amos,
Thanks for pointing out, I've fixed tha...Amos,<br /><br />Thanks for pointing out, I've fixed that.Sabine Hossenfelderhttps://www.blogger.com/profile/06151209308084588985noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-63749074251976739702018-01-18T08:19:23.316-05:002018-01-18T08:19:23.316-05:00...the above mentioned ratio between the distance ...<i>...the above mentioned ratio between the distance the grasshopper jumps and the square of the lawn-area...</i> Square -> square root? Amoshttps://www.blogger.com/profile/00595591283398023248noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-46804058306856080072018-01-18T03:09:16.052-05:002018-01-18T03:09:16.052-05:00Fun problem.
My intuition suggests that for very l...Fun problem.<br />My intuition suggests that for very large d, the solution should consist of very many small stripes, almost dots. In statphys problems there are generically no ordered phases in 1D, so the solution in 1D - many disconnected points - is the disordered phase. So when the disorder parameter grows large in higher dimensions, the solution should be similar to the 1D case.<br /><br />Thomas Larssonhttps://www.blogger.com/profile/01207766078592840926noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-265886536916877292018-01-18T01:00:42.953-05:002018-01-18T01:00:42.953-05:00Amos,
I changed that to disk. Sorry, in my head a...Amos,<br /><br />I changed that to disk. Sorry, in my head a disk is a lower-dimensional sphere, hence the conflation. Sabine Hossenfelderhttps://www.blogger.com/profile/06151209308084588985noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-6557345173246326182018-01-18T00:10:52.374-05:002018-01-18T00:10:52.374-05:00On first blush, it seems obvious that controlling ...On first blush, it seems obvious that controlling where the grasshopper lands is every bit as important as predicting it's subsequent jump.<br />Simple logic can narrow down the pattern almost as neatly as math.<br /><br />It can also notice the loophole in the problem as literally stated.<br />That leads to the obvious solution of a cylinder.<br /><br />The grasshopper "lands on a lawn&AmericanSoccer Archiveshttps://www.blogger.com/profile/00452540394500134769noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-51316174657001925862018-01-17T20:55:05.594-05:002018-01-17T20:55:05.594-05:00...the above mentioned ratio between the distance ...<i>...the above mentioned ratio between the distance the grasshopper jumps and the square of the lawn-area...</i> Square or square root? <i>...For very small d, the optimal lawn is almost spherical...</i> Spherical or circular?<br />Amoshttps://www.blogger.com/profile/00595591283398023248noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-19719711902700042432018-01-17T13:03:54.727-05:002018-01-17T13:03:54.727-05:00... for d=1 the optimal case would probably be a s...<i>... for d=1 the optimal case would probably be a stretch of points with equal distance d, so that wherever you start you'll land on another point</i><br /><br />The d=1 case won't depend on how the curve is embedded in space, as long as the convention is to measure lengths of the grasshopper jump along the curve.<br /><br />How about the d=2 case? e.g., what happens on a sphere?Arunhttps://www.blogger.com/profile/03451666670728177970noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-696863281525078462018-01-17T12:14:29.136-05:002018-01-17T12:14:29.136-05:00The paper has a section "Generalisations to o...The paper has a section "Generalisations to other dimensions" on page 13 of the arXiv version. They write, "The problem generalises to R^n for any positive integer n. The case n = 1 is easy to solve. Consider the lawn defined by the segments [0,1/N], [d, d+1/N], ..., [(N−1)d, (N−1)d+1/N]. This has total length 1 and gives a probability (N−1)/N of the grasshopper remaining on the Gareth Reeshttps://www.blogger.com/profile/15405124248006286547noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-23260785049460860172018-01-17T12:03:30.341-05:002018-01-17T12:03:30.341-05:00John again,
A friend suggests by email that for d...John again,<br /><br />A friend suggests by email that for d=1 the optimal case would probably be a stretch of points with equal distance d, so that wherever you start you'll land on another point. It has some normalization problem of course, but I guess with a suitable epsilon-delta prescription one could fix that. Sabine Hossenfelderhttps://www.blogger.com/profile/06151209308084588985noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-7178129463458694542018-01-17T11:06:51.549-05:002018-01-17T11:06:51.549-05:00John,
Great question. I don't know, but maybe...John,<br /><br />Great question. I don't know, but maybe I can manage zero dimensions ;) Best,<br /><br />B.Sabine Hossenfelderhttps://www.blogger.com/profile/06151209308084588985noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-66945944057597240182018-01-17T10:35:36.049-05:002018-01-17T10:35:36.049-05:00Physics embraces isotropy (re angular momentum con...Physics embraces isotropy (re angular momentum conservation). Identical random direction events here lack it. There may be critical opalescence regimes wherein minima are only symmetric to themselves, suggesting (aargh!) second order phase transition. Fascinating.Uncle Alhttps://www.blogger.com/profile/05056804084187606211noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-68003578782662477682018-01-17T10:26:17.983-05:002018-01-17T10:26:17.983-05:00This is a great example of spontaneous symmetry br...This is a great example of spontaneous symmetry breaking and how simple-sounding problems can give rise to complex structures! One could also look at this problem in different dimensions, for extra fun. Is the 1-dimensional case easy?John Baezhttps://www.blogger.com/profile/11573268162105600948noreply@blogger.comtag:blogger.com,1999:blog-22973357.post-54669922703409031592018-01-17T10:14:33.218-05:002018-01-17T10:14:33.218-05:00This is super neat.This is super neat.Unknownhttps://www.blogger.com/profile/16411970226629138387noreply@blogger.com