Examples where physical heuristics led to incorrect answers? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T00:35:33Z http://mathoverflow.net/feeds/question/30149 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30149/examples-where-physical-heuristics-led-to-incorrect-answers Examples where physical heuristics led to incorrect answers? Alekk 2010-07-01T07:10:20Z 2010-07-09T13:42:09Z <p>I have always been impressed by the number of results conjectured by physicist, based on mathematically non-rigorous reasoning, then (much) later proved correct by mathematicians. A recent example is the $\sqrt{2+\sqrt{2}}$ connective constant of the honeycomb lattice, derived non rigorously by the physicist B. Nienhuis in 1982 and rigorously <a href="http://www.unige.ch/~smirnov/papers/saw.pdf" rel="nofollow">proved</a> this year by S. Smirnov and H. Duminil-Copin.</p> <p>I would be interested in knowing examples of results conjectured by physicists and later proved wrong by mathematicians. Furthermore it would be interesting to understand <em>why</em> physical heuristics can go wrong, and how wrong they can go (for example, were the physicists simply missing an important technical assumption or was the conjecture unsalvagable).</p> http://mathoverflow.net/questions/30149/examples-where-physical-heuristics-led-to-incorrect-answers/30154#30154 Answer by Victor Protsak for Examples where physical heuristics led to incorrect answers? Victor Protsak 2010-07-01T08:05:44Z 2010-07-01T08:05:44Z <p>This question seems overly broad. I also do not think that one should single out physicists here, because all sorts of heuristic arguments can go awry, not to mention that, like mathematicians, physicists correct themselves and each other all the time. </p> <p>Having said that, very much along the same lines as OP's example, but wrong: <a href="http://en.wikipedia.org/wiki/Kelvin_conjecture#Kelvin_structure" rel="nofollow">Kelvin conjecture</a> about the minimal area honeycombs that was disproved by Weaire and Phellan. Also, <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ijm/1255448337" rel="nofollow">Close-packing and froth</a> by H.S.M. Coxeter offers ample evidence in a closely related context of statistical honeycombs. Coxeter himself gave two different answers for the average number of faces ($13.39$ and $13.56$)! In spite of the <a href="http://www.nature.com/nature/journal/v446/n7139/full/nature05745.html" rel="nofollow">recent work</a> of MacPherson and Srolovitz, the mathematical side of the story is still far from complete.</p> http://mathoverflow.net/questions/30149/examples-where-physical-heuristics-led-to-incorrect-answers/30172#30172 Answer by Wadim Zudilin for Examples where physical heuristics led to incorrect answers? Wadim Zudilin 2010-07-01T12:13:51Z 2010-07-01T12:13:51Z <p>I do not count the example below as failure but a remarkable insight with strong mathematical consequences.</p> <p>One of the most famous examples of explicit constructions in Mirror Symmetry was introduced in the epoch-making paper by the physicists Candelas et al. Starting from the family ${\mathbf M}$ of quintic hypersurfaces in $\mathbb{P}^4(\mathbb{C})$ defined by $\sum_ {k=1}^5x_ k^5-5z \prod_ {k=1}^5x_ k=0$ ($z$ being a complex parameter), Candelas et al. naturally associate another family $\mathbf{W}$ of manifolds (the "mirror of ${\mathbf M}$") which turn out to be Calabi--Yau. To $\mathbf{W}$, one can naturally associate a vector of periods (depending on $z$) which are solutions of the same differential equation (namely, the Picard--Fuchs equation of $\mathbf{W}$). This equation is simply a hypergeometric differential equation satisfied by a hypergeometric function $F(z)$ and its "neighbour" $G(z)+\log(z)F(z)$. Then they observed the non-trivial property that the Taylor coefficients of $q(z)=\exp(G(z)/F(z))$ are integers. Furthermore, let us define the <em>Yukawa coupling</em> $$K(q):= \frac{5}{1-5^5z(q)}\cdot \frac{1}{F(z(q))^2}\cdot \bigg(\frac{qz'(q)}{z(q)}\bigg)^3 \in \mathbb{Q}[[q]],$$ where $z(q)$ is the compositional inverse of $q(z)$, and write it as $$K(q) = 5+ \sum_ {d=1}^{\infty} c_ d \frac{q^d}{1-q^d},$$ which is formally possible. Candelas et al. observed that the <em>instanton number</em> $n_ d=c_ d/d^3$ is an integer for all $d\ge 1$, which is already a non-trivial fact, but that furthermore $n_ d$ seems to be the number of rational curves of degree $d$ lying on the initial quintic ${\mathbf M}$, thereby providing an effective algorithm to compute these numbers. These striking observations generated much interest amongst algebraic geometers, and this culminated in the further math work (of Givental, Lian et al.) where it is proved that, if for a given $d$ the curves of degree $d$ are all rigid, then there are $n_ d$ of them. In fact, the coincidence was proved to be true for $d\le 9$, and the first difference occurs at $d=10$ (see [E. Cotterill, <em>Comm. Algebra</em> <strong>33</strong> (2005) 1833--1872]).</p> http://mathoverflow.net/questions/30149/examples-where-physical-heuristics-led-to-incorrect-answers/30181#30181 Answer by Roland Bacher for Examples where physical heuristics led to incorrect answers? Roland Bacher 2010-07-01T13:18:29Z 2010-07-01T13:18:29Z <p>I believe knot theory got started by 19th-century physicists (Lord Kelvin was one of the initiators) based on the assumption that atoms have to be knots in the ether. They started tabulating knots in the hope to shed light on the periodic table. </p> <p>The physics was rubbish but not the maths.</p> http://mathoverflow.net/questions/30149/examples-where-physical-heuristics-led-to-incorrect-answers/30190#30190 Answer by Gerald Edgar for Examples where physical heuristics led to incorrect answers? Gerald Edgar 2010-07-01T14:37:56Z 2010-07-01T22:47:53Z <p>Physicists originating statistical mechanics asserted that the path in phase space of the state of the system will visit every point on the surface of constant energy. <strong>Bolzmann's ergodic hypothesis</strong>, as formulated by Ehrenfest, 1911, states exactly that. Later mathematicians found it to be in error for topological reasons: a differentiable curve cannot cover a surface of dimension higher than 1. You can then go on to study this...the curve is dense in the surface, and there are "ergodic" conditions much stronger than that which are (presumably) obeyed by physical systems.</p> http://mathoverflow.net/questions/30149/examples-where-physical-heuristics-led-to-incorrect-answers/30197#30197 Answer by jc for Examples where physical heuristics led to incorrect answers? jc 2010-07-01T15:41:00Z 2010-07-01T15:41:00Z <p>I'll describe below a controversy in statistical mechanics in the 1980's: <em>the case of the <a href="http://en.wikipedia.org/wiki/Critical_dimension" rel="nofollow">lower critical dimension</a> of the <a href="http://en.wikipedia.org/wiki/Ising_model" rel="nofollow">Ising model</a> with an applied random magnetic field</em>. </p> <h2>Background</h2> <p>Let me give a little background, though you might want to read <a href="http://terrytao.wordpress.com/2007/08/20/math-doesnt-suck-and-the-chayes-mckellar-winn-theorem/" rel="nofollow">Terry Tao's discussion of basic statistical mechanics</a> instead. The Ising model is a statistical mechanical model of "spins" on a hypercubic lattice. The energy functional is: $E=\sum_{\langle ij\rangle}\frac{1}{2}(1-S_iS_j)-\sum h_iS_i$ where the first sum is taken over nearest neighbor pairs on the lattice and the second is taken over all sites, and $S_i$ is a $\pm1$ valued variable on each site called the spin and $h_i$ is the real-valued "externally applied magnetic field" applied to each site. Each possible configuration of spins on the lattice is assigned a probability proportional to its Boltzmann weight $e^{-\beta E}$ where $\beta>0$ is a parameter that is interpreted physically as the inverse temperature $T$.</p> <p>Given such a model, one question is to determine the "phase behavior", or what are the typical properties of the ensemble of configuration at a given $\beta$, and how does this change with $\beta$. </p> <p>Considering at the moment just the Ising model with $h_i=0$, one might expect that for large $\beta$, the typical configuration will tend to have lower energy, and hence have all its spins aligned to either all $+1$ or all $-1$. At small $\beta$, all the Boltzmann factors tend to 1 and the typical configuration will have random spins. This rough argument is just meant to guide the intuition that there might be a phase transition between "mostly aligned" configurations to "mostly random" configurations at some special value of $\beta$. </p> <p>As it turns out, what happens is highly dependent on the dimensionality of the lattice.</p> <p>The lower critical dimension $d_L$ of a model is the dimension below which no phase transitions can occur because even as $\beta\rightarrow\infty$, there is not enough of an energy gain from ordering to create a phase with long-range correlations. In the ordinary Ising model (with all $h_i=0$), the lower critical dimension is 1, and hence at any finite $\beta$, the average $\langle S_iS_{j}\rangle$ over configurations weighted with the Boltzmann distribution will approach zero (exponentially fast, even) as the distance between sites $i$ and $j$ approaches $\infty$. For two dimensions and above, it can be shown that above a certain $\beta_c$ (depending on dimension) this average will be finite in that long-distance limit.</p> <h2>Controversy</h2> <p>In the 1980's there was a controversy in the physics literature over the value of $d_L$ for the Random Field Ising model, a model where the $h_i$ are independent Gaussian random variables with zero mean and constant variance $\epsilon^2$. </p> <p>I'm not in a position to describe the history accurately, but I believe that there were <a href="http://prl.aps.org/abstract/PRL/v35/i21/p1399_1" rel="nofollow">physical arguments by Imry and Ma</a> originally that $d_L\leq 2$, which were disputed when an amazing connection between random systems in $d$ dimensions and their pure counterparts in $d-2$ dimensions was found, known as the "<a href="http://prl.aps.org/abstract/PRL/v43/i11/p744_1" rel="nofollow">Parisi-Sourlas</a> correspondence". My understanding of Parisi-Sourlas is that it is based on a hidden supersymmetry in some series representation of the model which yields order-by-order agreement in the "epsilon expansions" of the two systems. Their argument was also made rigorous by <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.cmp/1103941404" rel="nofollow">Klein, Landau and Perez</a> (<a href="http://ams.org/mathscinet-getitem?mr=0763747" rel="nofollow">MR</a>). Based on this, since the Ising model has $d_L=1$, the RFIM was argued to have $d_L=3$ by various authors, though this was never a consensus view.</p> <p>This controversy was settled by work of <a href="http://math.virginia.edu/~ji2k/Home/Papers/GroundState.pdf" rel="nofollow">John Imbrie</a> (<a href="http://ams.org/mathscinet-getitem?mr=786570" rel="nofollow">MR</a>) and later work of <a href="http://projecteuclid.org/DPubS?verb=Display&amp;version=1.0&amp;service=UI&amp;handle=euclid.cmp/1104161515&amp;page=record" rel="nofollow">Bricmont and Kupianen</a> (<a href="http://ams.org/mathscinet-getitem?mr=0943702" rel="nofollow">MR</a>) building off his results that proved rigorously that $d_L\leq2$ in this system. Apparently terms like $e^{-1/\epsilon}$ become important and the epsilon expansion breaks down in low dimensions, though I'm not sure if this has been made precise, and even today the RFIM is far from being completely understood.</p> http://mathoverflow.net/questions/30149/examples-where-physical-heuristics-led-to-incorrect-answers/30241#30241 Answer by John Sidles for Examples where physical heuristics led to incorrect answers? John Sidles 2010-07-01T22:15:16Z 2010-07-01T22:15:16Z <p>Many physicists believed that quantum entanglement would (eventually) be understood in terms of some simple-to-compute measure. The surprising mathematical finding that the quantum separability problem is NP-hard proved these expectations to be wrong. </p> <p>See, for example, Lawrence M. Ioannou's review, "Computational complexity of the quantum separability problem", (arXiv:quant-ph/0603199).</p> http://mathoverflow.net/questions/30149/examples-where-physical-heuristics-led-to-incorrect-answers/31189#31189 Answer by Willie Wong for Examples where physical heuristics led to incorrect answers? Willie Wong 2010-07-09T13:42:09Z 2010-07-09T13:42:09Z <p>Somewhat related to the ergodic hypothesis mentioned in another answer is the assumption that generic non-linearities leads to thermalization and equipartition of energy. To be more precise, start with a linear, completely integrable, finite dimensional Hamiltonian system (say de-coupled system of many harmonic oscillators). The system has independent excitation modes that, if the initial data is set to be one of the modes, the evolution will stay on the mode. The assumption from physics is that by addition a non-linear coupling, this would let modes interact and in the long run, the system will settle down to a thermalized state where each mode contributes the same amount to the total energy. </p> <p>This, of course, is now known to be false, in view of the KAM theorem. </p> <p>But an interesting side development is that Fermi, Pasta, and Ulam were convinced that the thermalization should take place (in fact Fermi had published a "proof" to that effect), so they ran a computer simulation (way back when in Los Alamos on one of the first computers built) for a vibrating string, taking in account of the second order effects (the first order effects are just the linear wave equation, which in finite grid approximation is completely integrable ODE), and tried to numerically compute the rate at which thermalization will occur. What they observed, however, is that the system is quasi-periodic. This discovery gave birth to the modern study of solitons. See an account of this in Palais' article in the Bulletin <a href="http://dx.doi.org/10.1090/S0273-0979-97-00732-5" rel="nofollow">http://dx.doi.org/10.1090/S0273-0979-97-00732-5</a></p>