Okay, so the trailer for the new Spider Man movie is out and appearently our friendly physicist from the neightborhood came up with something. However I can't find out what this is.

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$$\begin{align}\frac{\text d \log\Phi}{\text d t} &= \alpha\Biggl({1 - g{{\biggl( {\frac{\Phi }{K}} \biggr)}^\beta }} \Biggr) \\ \Phi &= K\sum_i\prod_j\exp \Biggl(\frac{g^j(1 - E_a)^{j - 1}}{(j - 1)!\bigl(1 - ( {1 - E_a)^j\bigr)}}\Biggr) \log\bigl( \cdots\end{align}$$

(the last part is hidden)

I don't think that it's just jubberish and while my first guess was statistical physics or network theory, it could as well go in the biology direction.

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    $\begingroup$ The text around the box suggests biology(cytokine, cells, etc), but that may be unrelated to the boxed content.. $\endgroup$ – Manishearth Feb 7 '12 at 18:47
  • $\begingroup$ A summation over a PI exp(stuff much greater than one) . nope this is not within the physical universe anymore than the ackermann. Assuming j is real. $\endgroup$ – Captain Giraffe Feb 7 '12 at 19:26
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    $\begingroup$ Partition function for haemoglobin excitations in humans with radioactive blood ? $\endgroup$ – twistor59 Feb 7 '12 at 19:55
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    $\begingroup$ @CaptainGiraffe: that "stuff" doesn't need to be that much greater than one. $(1-E_a)$ is certainly in $[0,1]$, what remains is $\approx \tfrac{g^j}{j!}$ which goes to 0 quickly for large $j$, and the factorial implies that $j$ is a natural. The product can be pulled in the exponential, if it's infitine the sum there gives another exp. Sure $\exp(e^g)$ grows fast for large $g$, but we know nothing about that nor about the $i$-summation. The fact that $i$ doesn't appear in the visible part seems the most suspicious to me. $\endgroup$ – leftaroundabout Feb 7 '12 at 21:12
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    $\begingroup$ I think maybe some of the symbols that look like 1 are i's. Also, there somewhere is the function $\frac{x}{1−x}$, which feels familiar. Lastly, I don't see time in the final expression. If the right hand side of the second expression really is time independent, then the right hand side of the first equation normalizes the sum. So I think maybe an evolution of a probability distribution. $\endgroup$ – Nikolaj-K Feb 9 '12 at 8:16

The two equations are unrelated.

First equation

The first equation is a simple modification of the logistic differential equation, although it is somewhat disguised. The usual logistic equation is

$$ {dx\over dt} = x(1-x) $$

or in terms of the derivative of $\log(x)$, it's the equation Peter Parker writes, with $\beta=1$. The factor of K in conjunction with g sets the scale for $\Phi$, and it is irrelevant, the qualitative behavior for different $\beta$ at long times is not modified, since the equilibrium position is at

$$\Phi_\mathrm{eq}={K\over g^{1\over\beta}}$$

Values above this go down and values below this go up. Further there is a nonzero first derivative of $\Phi^\beta$ for all reasonable idea of what $\beta$ is supposed to be, so this is describing a quantity $\Phi$ which wants to go up exponentially, but is suppressed by competitive effects.

The exponent $\beta$ describes the competitive effects. The logistic equation describes (say) bacteria (or white blood cells) replicating where two bacteria compete for the same limited resources. In this case, the competition is $\beta$-fold, the bacteria crowd each other out worse than quadraticaly (or less worse, depending on whether $\beta<1$ or $\beta>1$).

This equation is consistent with a biological interpretation that $\Phi$ is the concentration of some replicating crowding out agent, like a disease model.

Second equation

The second equation is writing down $\Phi$ in a way that depends on g but not on t. It has a K in it, but there is an unrelated expansion of $\Phi$ in terms of the $E_n$'s, so it isn't the expression for the equilibrium value or the relaxation to this equilibrium value.

Further, you can massage the form by exponentiating, expanding the denominator in a power-series, and performing the sum on j, to produce a second infinite series, but only if you assume the hidden log part does not depend on j, but only on the variable "i" which has so far not been used.

${\Phi\over K} = \exp(g (\sum_{k=1}^{\infty} \Delta^k e^{g\Delta^k})) \sum_i log(...)$$

Where $\Delta=(1-E_k)$, and from the form, I will assume $0<\Delta<1$, so that $0<E_k<1$. The $\log$ part makes no sense as a time development either, this isn't the development of the logistic equation, or any reasonable asymptotic of this, (although the symbol that is partly obscured is probably an $\alpha$ which can only appear multiplied by t on dimensional grounds, so you can assume that it's $\log(\alpha t ...)$, so one can only assume that the movie-makers chose a second equation to look impressive from an unrelated system.

  • $\begingroup$ What's up with the downvote? This is correct. $\endgroup$ – Ron Maimon May 15 '12 at 16:23
  • $\begingroup$ I wouldn't it call a version of the logistic equation, as it is not equivalent to it. it is a variant of the logistic equation.with a different law for the saturation part subtracted. $\endgroup$ – Arnold Neumaier May 16 '12 at 9:53
  • $\begingroup$ I also get a different exponent in your formula for $\ph/K$. But even then the infinite sum doesn't look natural. $\endgroup$ – Arnold Neumaier May 16 '12 at 10:12
  • $\begingroup$ @ArnoldNeumaier: THe different exponent in eq. 1 and eq. 2 shows they are unrelated, should I have called it a "Logisitic equation with competition exponent?" It's the same idea, with a slightly different suppression, and asymptotically, near the equilibrium, it turns into a logisitics equation. $\endgroup$ – Ron Maimon May 16 '12 at 17:38
  • $\begingroup$ @ArnoldNeumaier: The transformation is by expanding the $(1-\Delta)$ on the bottom in an infinite series and resumming, I am pretty sure I got it right. What did I screw up? The point of the infinite sum is to check the case $\Delta$ near 0, to see if there is any content related to the logistic equation: there isn't. This is not a sensible equation in relation to the first. $\endgroup$ – Ron Maimon May 16 '12 at 17:51

Here is a video of the film's science advisor explaining what the equation is and how he came up with it: http://www.youtube.com/watch?v=WjfT6MqTCqQ

It is based on the Gompertz equation, which is a model of mortality rates, with some added "mathematical glitter."

  • $\begingroup$ While this answer is certainly accurately reflecting the psychology of the consultant +1, the "Gompertz equation" gives a cumulative distribution function that obeys a modified logistics equation. This modified logistics equation is what was written down. The second equation is all "mathematical glitz" (meaning it doesn't make any sense in relation to the first, beyond being vaguely combinatorial). My answer is still correct, and there is no modification required. It really doesn't matter what the dude who made it up says. I have to explain this because I got a downvote on my correct answer. $\endgroup$ – Ron Maimon Jun 30 '12 at 2:09

The facts that

  1. there is a sum over $i$ but the product doesn't involve $i$;
  2. the product is a product of exponentials, which as a major result (boxed and marked "DO NOT ERASE") would typically be written as a single exponential;
  3. given that the first equation is a differential equation, one should expect the second equation that gives $\Phi$ to be either an initial condition (ruled out as the left hand side is not called $\Phi(0)$ or so) or the solution (ruled out as neither $t$ nor $\alpha$ nor $\beta$ appears on the right hand side);
  4. $E_\alpha$ looks like a probability but isn't called $p$ or $q$;
  5. the formulas surrounding the box have diffferent variables;

suggest to me that the formula is made up, a combination of imagination coupled with inspiration from the mathematical biology literature.

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    $\begingroup$ @RonMaimon: I had upvoted your answer but still found it appropriate to give my own formulation of the answer. (It is shorter and puts the emphasis differently.) $\endgroup$ – Arnold Neumaier May 17 '12 at 12:43
  • $\begingroup$ Oh, ok, sorry, comment erased. $\endgroup$ – Ron Maimon May 17 '12 at 16:35
  • $\begingroup$ Seems more like $E_a$ to me, which represents the activation energy of a reaction. $\endgroup$ – mikhailcazi Aug 7 '13 at 12:38

I guess $\Phi$ is something like the (grand?) partition function of statistical mechanics, so the formulas may be related to (non-equilibrium) statistical mechanics, maybe to calculation of (chemical) reaction rates.

  • $\begingroup$ Right, and some sort of (grand) partition function would reasonably have its log taken. $\endgroup$ – Jerry Schirmer Feb 19 '12 at 16:44
  • $\begingroup$ @Jerry: Is your comment meant to be sarcastic? A partition function can be seen as a moment-generating function, and this moment-generating function is naturally related to the cumulant-generating function through a logarithm. $\endgroup$ – Raskolnikov Mar 5 '12 at 9:31
  • $\begingroup$ @Raskolnikov: and thus, it makes sense that there is a log on the chalkboard. $\endgroup$ – Jerry Schirmer Mar 5 '12 at 13:50
  • $\begingroup$ @Jerry: It can be avoided by simply multiplying both sides by a factor $\Phi$. $\endgroup$ – Raskolnikov Mar 5 '12 at 20:43
  • $\begingroup$ @Raskolnikov: and I my comment would have only slightly been modified if the differential equation was for $\frac{1}{\Phi}\frac{d\Phi}{dt}$. The log of something equals something that is the sum of a the product of a bunch of terms screams something (grand)partition function-y. $\endgroup$ – Jerry Schirmer Mar 5 '12 at 22:42

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