Fluctuation Dissipation in Quantum Gravity

In statistical mechanics we are mostly concerned with the statistical averages of various physical quantities when the system is in equilibrium.

Fluctuation is a common phenomenon in nature. Fluctuation means how much a quantity deviates from its average value. The average value of the thermodynamic observables and the size of their fluctuation about their equilibrium values can be predicted by equilibrium statistical mechanics.

Speeding up Wordpress

One common problem with wordpress sites is that they can seem sluggish compared to, lets say, sites built with static or single-file CMSs such as grav. There are several online tools which can help you check how fast your site is and how to optimize its performance. Two of these are pingdom and Google's pagespeed insights.

Putting my site through these two tools yields the following results:

[caption id="attachment_158" align="alignnone" width="69"] Result from tools.pingdom.com[/caption]

[caption id="attachment_159" align="alignnone" width="207"] Results from Google Pagespeed Insights.[/caption]

A more comprehensive list of online tools for testing your wordpress site speed can be found here.

Nb: The fullscreen browser images were captured using the very useful Firefox plugin Fireshot.

Spacetime Geometry as Information Geometry

In an entry to the 2013 FQXi essay contest and in an accompanying paper, Jonathan Heckman, a postdoc at Harvard, put forward a scintillating new idea - that one can derive the theory of strings and of gravity starting from nothing more but a Bayesian statistical inference model in which a collective of $N$ agents (representing by points on a $d$-dimensional grid) sample a probability distribution in order to obtain the best fits to a set of parameters $\{y_1,\ldots,y_M\}$. In investigating the statistical mechanics of such a collective, he finds that their dynamics can be described by an effective field theory, which happens to be the non-linear sigma model. Further requiring that the "judgements" of the collective be stable under perturbations implies that the dimension $d$ of the manifold in which the agents are embedded must be equal to two. Furthermore, he notes that conformal invariance of the resulting two dimensional "agent space", leads us to Einstein's theory of gravity and that the effective dynamics of the collective is described by a theory of strings.

At first glance his line of reasoning appears to be impeccable, and it is only the profound nature of his conclusions that might lead one to question whether his approach has any fatal flaws. Disregarding that possibility for the time being, let us proceed towards further interpreting this ground-breaking result.

The collective lives on a two-dimensional manifold which one can naturally identify with the worldsheet swept out by a string moving in an $M$-dimensional spacetime. Moreover, the space of parameters to which the collective performs a fit must also naturally be identified with the background geometry the string is embedded in. This leads to ask, whether it makes sense to identify the points of a spacetime geometry with statistical parameters and, if so, how can one then relate our usual geometrical notions of distance, angles, etc. to information based concepts.

Cosmological Rulers

To begin, let us switch to a simpler setting - that of our usual flat Minkowski $3+1$ dimensional spacetime, within which are embedded at random locations a set of agents which resemble the wireless routers commonly used in homes and offices. Each agent transmits a single tone at fixed time intervals indicating its presence to all the other agents in its vicinity. Each agent also listens for the tone broadcast by other agents, and by accumulating many such events performs an estimate of its distance to each of the other agents. (illustration) These agents have no scales and no way to measure distances and areas. How can a distance scale arise solely from exchanging signals between agents?

First, let us consider the situation when we do have a way to measure distances. Each agent transmits a signal, say in the form of an em wave, which propagates outwards isotropically from the location of the agent. Now, conservation of energy implies that the total flux $\Phi$ through any closed surface enclosing the agent should stay the same (illustration). In particular given two spherical surfaces $S_1$ and $S_2$ of radii $r_1$ and $r_2$ (with $ r_2 > r_1 $), the flux per unit area $I$:

$$ I(r) = \frac{\Phi}{4\pi r^2} $$

is smaller the greater the distance from the agent: $ I_2 (r_2) < I_1 (r_1) $. So, when an agent $A_1$ emits a signal containing $n$-bits, another agent $A_2$ situated a distance $r_{12}$ from the first one can receive at most:

$$ m = n \frac{a}{4 \pi r_{12}^2} $$

bits of the original signal. Here $a$ is a unit of area, which characterizes the size of the "aperture" using which an agent captures signals. Alternatively, we can state that $\mathcal{A}_2$ receives a fraction of the total flux emitted by $\mathcal{A}_1$, given by:

$$ \Phi' = \Phi \frac{a}{4 \pi r_{12}^2} $$

Since all agents are identical - emit identical signals and have apertures of the same area - $\mathcal{A}_2$ can use the value of the received flux to determine the distance from $\mathcal{A}_1$ as:

$$ r_{12} = \sqrt{ \frac{a}{4 \pi} \frac{\Phi}{\Phi'} } = \sqrt{ \frac{\Phi_0}{\Phi_{12}} } $$

where, for WTLOG (without loss of generality), we have set the area of the aperture $ a = 4\pi$. $\Phi_0 = \Phi$ is the flux emitted by .$\mathcal{A}_1$ and since we are assuming all agents are identical, this can be set to a universal value $\Phi_0$. Finally, $\Phi_{12} = \Phi'$ is the flux received by $\mathcal{A}_2$ from $\mathcal{A}_1$

Let us note that since, a priori, we do not have access to any "rulers" we can only measure ratios of distances. This, in fact, is exactly what is done in most modern cosmological observations. There is no way to determine absolute distances to stars and galaxies, without reference to some celestial objects which are used as "standard candles". We observe a given standard candle - say a type Ia supernova - in some distant galaxy and determine the amount $z$ its light is red-shifted by the time it reaches us. Using some other methods we determine the physical distance that $z$ corresponds to. In this way, we map out the large scale structure of our Universe (or at least of our local neighborhood) by comparing the spectra received from various objects with each other.

In this manner, each agent $\mathcal{A}_i$ can determine its distance to any other agent $\mathcal{A}_j$ as:

$$ r_{ij} = \sqrt{ \frac{\Phi_0}{\Phi_{ij}} } $$

And since, we are in a flat background without any dissipation it is safe to assume that $\Phi_{ij} = \Phi_{ji}$ and therefore $r_{ij} = r_{ji}$.

There are two other details. First, how does a given agent $\mathcal{A}_i$ distinguish between the flux received from two different agents $\mathcal{A}_j$ and $\mathcal{A}_k$, which lie at equal distances from $\mathcal{A}_i$? Second, even with the ability to measure distances to other agents, how does any one agent reconstruct the geometry in its neighborhood? Without some sense of direction, distances alone are not sufficient to allow an agent to distinguish between two equally distant neighbors.

The first problem can be addressed by equipping each agent with a random number generator. The procedure followed by any agent is then as follows:

1. When its first turned on, the agent generates a random number, its unique ID, and transmits that embedded with its default signal.


2. As it receives signals from other agents, it compares the numbers it reads from their signals with its own. If it receives a signal with a number identical to its own, it generates another random number and sets that as its new ID.


3. This process is continued until every ID the agent receives is different from its own, for some minimum specified duration.


4. Once this equilibrium state is reached, the agent uses the measured value of incoming fluxes to associate a distance to each one of its neighbors.


5. In the event of an ID conflict - agent receives flux signal with ID identical to its own - the system resets and starts from step 2.

The second problem, that of being able to distinguish neighbors which are equidistant from a given agent, but not coincident, can be addressed in several ways.  One possible method is

Area and Information Density

 

 

Conclusion

Thermodynamics and Planetary Motion

In an earlier post, I had discussed a possible derivation of Kepler's Second Law of Planetary Motion within a quantum gravity setting. Kepler's Second Law states that for a planet in orbit around another the Sun under the influence of Newtonian gravity:

the line connecting the planet to the Sun sweeps out equal areas in equal time

The basic idea involves applying Haggard and Rovelli's result about the rate of evolution of a quantum mechanical system, according to which a system in equilibrium evolves in such a way as to pass through an equal number of states in equal time intervals. Quantum Gravity (of the Loop variety) tells us that the fundamental observable in a quantum theory of geometry is the area of a surface embedded within a given spacetime.

The area swept out by a planet during the course of its motion around the Sun is far greater than $ A \gg A_p $(the basic quantum of area, where $A_p = l_p^2$ is the Planck length). However, if classical geometry as described by (classical) general relativity arises from a more fundamental quantum theory, and consistency of any theory of quantum gravity would require this, then it is natural to assume that the macroscopic area $\delta A$ swept out of by a planet in a time $\delta t$ emerges from an ensemble of quanta of Planck areas. If one could argue that planetary motion corresponds to an "equilibrium" configuration of the gravitational field, then Haggard and Rovelli's result can be applied and we obtain Kepler's Second Law as a trivial consequence.

Transcending Bad Sci-Fi

So I was watching "Transcendence", the latest cinematic attempt to generate public hysteria about Artificial Intelligence (AI), or more specifically about Hard AI. I stopped watching around the 42 minute mark. Johnny Depp's remarkably good portrayal of a scientist dying a slow and painful death from Polonium poisoning had kept me watching despite the utter nonsense being thrown around in the name of Sci-Fi. Then I got to the point, where one of a gang of anti-AI extremists says the following whopper:

If that thing [the AI computer armed with Johnny Depp's consciousness] connects to the internet, the first thing it will do is copy itself to every networked computer in the world and there will be no way to stop it.

Take a moment to ponder this sentence. First of all, if your script centered on an AI gone rogue, hinges on whether or not the said AI successfully copies itself to "every networked computer", then you should think of a different profession that scriptwriting. I mean, seriously? THAT'S the best AI gone rogue scenario you could come up with? Second, and more seriously, this sentence suggests that the makers of the film really didn't give a crap about actually understanding the promise and perils of strong AI. Even an AI powered scriptwriting program could have come up with dozens of fantastic script ideas centered around the theme of a strong-AI gone rogue which were not in violent conflict with our basic understanding of the subject.

The problem is with the notion that said AI, which is powered by "the most powerful quantum processors on the planet", could exist independently of the cutting-edge computing hardware on which it was originally conceived. The point is simply that of all the examples of "strong"-AI (or perhaps just I) which exist in Nature all involve some sort fleshy matter built out of neurons and neurotransmitters. I am talking about brains, of course. At least considering mammalian brains as formal working examples of biological machines that exhibit strong-Intelligence, the following observation is crucial:

The software does not have an existence independent of the hardware.

In other words the "consciousness" aspect of the behavior of these strong-I machines cannot be simulated on a computer which is less complex than the brain-matter itself. Sure, one could always (in principle) take an imprint of the consciousness in brain 'X', store it on some digital memory and at some point in the future restore the consciousness by uploading the stored data into some other brain 'Y'. However, and this is crucial, while the imprint represents the individual aspects of the brain 'X's behavior, as long as it is separated from the actual brain itself it cannot exhibit any "conscious" behavior.

Coming back to the film, if the rogue AI copies "itself" (or more properly speaking creates "imprints" of itself) onto other computers of the network, those imprints would simply be fossils of the original consciousness without the advanced hardware required to sustain the computation. In other words, the threat of an AI spreading like a global digital pandemic simply cannot be realized unless and until every networked computer is as advanced as the original hardware. Given that in the film, the original hardware is described as consisting of "the most powerful quantum processors on the planet", connecting the AI to the internet would not pose any harm as long as most of the hardware on the planet was not as sophisticated as Depp's original quantum computers.

The fact that the filmmakers were unable to grasp this fact is what leads to "Transcendence" being at best a campy sci-fi movie with more in common with "Flash Gordon" that with "Blade Runner". But don't let these deep observations stop you from enjoying the rest of the film. After all, how would the AI feel if you gave up on it halfway through the movie?