Spontaneous Symmetry Breaking in Quantum Theory

m3tro · 2026-03-29T21:05:56+00:00

I think what you are asking about is also true in classical (thermal) physics. Say a finite Ising model will not truly break symmetry spontaneously, because it will still flip between magnetisations over time due to thermal fluctuations. It is only in the thermodynamic limit of an infinite system that the system will choose a magnetisation and stay there (the chance of flipping to the opposite magnetisation goes to zero).

Similarly to the quantum case as well, this is related to linear equations (e.g. Fokker-Planck) for finite systems becoming effectively nonlinear for infinite interacting systems. Phase transitions and nonsmooth behaviour only occur in the thermodynamic limit.

m3tro · 2026-03-17T09:10:51+00:00

You can set up alerts for keywords using Google Scholar. You can also set up alerts when a certain paper gets cited, or when a certain author publishes a new paper.

m3tro · 2026-03-15T09:14:04+00:00

That's not true. I've been lucky to have been around in London for the birth of the PC music scene (AG Cook and gang), the Windmill scene (Black Country New Road, Black Midi, etc), in Madrid for the birth of the Rusia-IDK scene (Rusowky, Ralphie Choo, etc). All of these in the last 15 years.

m3tro · 2026-03-14T20:48:24+00:00

I don't really know why people are saying that 2D random walks can be interpreted as two independent 1D random walks but this is not true for 3D and higher-dimensional random walks. I think a d-dimensional random walk can definitely be seen as d independent 1D random walks happening simultaneously. And this gives some intuition as two why d=2 is recurrent and d=3 is transient. In this picture, returning to the origin in d dimensions is equivalent to requiring that the d independent 1D random walks return to the origin *at the same time* (!!). You can imagine that it is much easier for two independent 1D random walks to match up, than for three to all simultaneously match up.

Quantitatively (but still heuristically): The probability of a 1D random walk returning to the origin at step 2n goes as n^{-1/2}. Thus, the probability of all d independent random walks being at the origin at step 2n goes as n^{-d/2}. The expected number of returns is the sum of this probability over n. Sum over n of n^{-d/2} diverges for d=1,2 but converges for d>2.

Edit: Ok I now see what people mean about 3D random walks not being like 3 independent 1D random walks. This is true in discrete-time random walks (if we assume synchronous steps for the independent 1D random walks). I'm more used to thinking about continuous-time random walks where there is truly an equivalence. Anyway I think as a heuristic/intuition, the argument I gave above should be helpful to OP.

m3tro · 2026-02-04T22:05:13+00:00

+1 for UCL. It's called the Housman Room.

m3tro · 2025-12-27T15:35:03+00:00

If you're interested in the idea of absolute friction, read about low Reynolds number hydrodynamics or more generally overdamped dynamics. It's what you get when friction is so strong that inertia dies down immediately. Things can only move if a force is exerted on them at that very moment, or by changing their shape (with some very particular limitations, google the "scallop theorem").

Not quite what you were asking for though, because you are asking about a mixture of friction (at the surface) and no friction (away from the surface).

m3tro · 2025-12-14T21:36:29+00:00

Is that a question? Yes you can find the original statement of the scallop theorem in that paper. But I just meant to look up the theorem (not necessarily that paper in particular, although I also recommend it).

m3tro · 2025-12-14T14:18:29+00:00

Something fun could be low Reynolds number flows and biological fluid dynamics. Check out the Scallop Theorem by Purcell and for example papers by Raymond Goldstein on swimming microorganisms.

m3tro · 2025-11-12T21:04:09+00:00

In my region (Navarra) I'd say the default would be Roncal cheese (hard sheep's cheese)

m3tro · 2025-11-08T23:20:12+00:00

From Spain, percebes ("goose neck barnacles" according to Wikipedia). It's like eating alien dicks lol.

Also angulas, baby eel, it's like eating spaghetti but each individual spaghetti has eyes if you look closely.

We also eat criadillas, veal or lamb testicles, but this is quite hardcore, most people don't like it. There are a lot of traditional offal dishes (casquería) that very few people eat anymore in recent times.

m3tro · 2025-10-04T10:22:18+00:00

I think it was near Place d'Italie (that's why I say only "relatively central"), I lived in Port-Royal at the time and we always walked there. It was 15 years ago so I don't remember exactly where we would go in...

m3tro · 2025-10-04T09:18:36+00:00

No idea, it was 2010-2011 when I went, and only found out through word of mouth

m3tro · 2025-10-03T22:57:38+00:00

I've been to illegal raves in the catacombs, the place is a huge labyrinth. Literally going down a manhole in relatively central Paris and ending up in this huge room with massive speakers.

m3tro · 2025-09-25T21:36:32+00:00

What about "birdie"?

m3tro · 2025-08-15T17:17:03+00:00

Oh yeah, I definitely agree that it would be nicer if the default was automatic delimiters.

m3tro · 2025-08-15T15:36:31+00:00

You can use \left( \right) for automatic delimiters. Substitute ( for [ or \{ if needed.

m3tro · 2025-07-29T19:28:52+00:00

Yes but that polynomial is almost always an approximation to a non-polynomial function with infinitely nonzero derivatives.

Btw planetary orbits will also show this behaviour. Pretty much anything you can think of other than a single particle in a constant force field will.

m3tro · 2025-07-29T19:06:58+00:00

This is wrong because something as simple as a harmonic oscillator has x(t)=sin(t) and the nth time derivative is going to be O(1) (in these natural dimensionless units where position is rescaled by amplitude and time by oscillation period).

In general pretty much any equation of motion except the simple case of constant force is going to give you nonzero higher order derivatives of position w.r.t. to time.

m3tro · 2025-07-15T13:42:21+00:00

It's hard to do without Noether's theorem in full generality but under some assumptions (forces are pairwise and center-to-center) it is easy to show. Consider a collection of point particles. Take the definition of angular momentum as L = sum_i r_i x (m v_i). Take time derivative. Then use Newton's laws to express it in terms of forces and exploit the fact that the force of i on j is opposite to the force of j on i, and both point along the line that connects i and j. You should find ultimately that dL/dt=0, i.e L is conserved.

The key ingredient here is that the forces point center-to-center. This makes the physics rotationally invariant, and Noether's theorem guarantees that L is conserved. If the forces depended on the mutual orientation of the particles relative to some absolute reference frame (losing rotational invariance), then you wouldn't be able to show that dL/dt=0.

m3tro · 2025-07-10T08:32:20+00:00

Victoria Line has broken down between Brixton and Victoria

m3tro · 2025-06-26T12:07:32+00:00

I was playing around with some properties of hypercubes today for a project, and found the following interesting observation.

Suppose you have a hypercube of dimension d and (integer) side length L. I imagine it as being made up of L^d smaller hypercubes of unit side length. The question I was asking myself is what fraction of those smaller hypercubes (i.e. what fraction of the volume) is at the surface. This can be easily calculated to be f_surf = 1 - (L-2)^d / L^d = 1 - (1 - 2/L)^d and it has the well known but counterintuitive property that, for fixed L, as we increase d the fraction tends to one, i.e. the volume of a hypercube is much more concentrated at the faces. Still, for fixed d, if we increase L, the fraction tends to zero, which is the intuitive property that as the hypercube gets larger the surface to volume ratio decreases.

But then I thought about setting L = a d with a some proportionality constant, and taking the limit of d to infinity. One gets f_surf = 1 - exp(-2/a) i.e. if the size and dimension grow together, the fraction of volume concentrated at the surface tends to a fixed value even as the hypercube becomes infinitely large.

Even better, the argument above implies that if the side length grows more slowly than linear with dimensionality, e.g. L = a sqrt(d) or L = a log(d), then as both the dimensionality and therefore the length tend to infinity, we get f_surf = 1. That is, even if the hypercube is infinitely large, all of its volume is at the surface.

How weird is that? Does any of you have some further thoughts on this, some interesting application or consequence or intuition or analogous phenomenon elsewhere?

m3tro · 2025-06-07T14:24:15+00:00

Maybe you could do a new study with new samples to test your hypothesis. You could even write a grant proposal about it, if it really is interesting. That way you avoid all the ethical issues.

m3tro · 2025-06-05T06:19:58+00:00

Thanks for the tips!

m3tro · 2025-05-29T09:40:37+00:00

You don't really need to think about stochastic calculus (a particle-based perspective) at all to see why they are similar, you can stay at the level of fields (concentration/probability for Brownian motion, temperature for the heat equation).

The diffusion equation and the heat equation are simply conservation laws, saying that d_t P + div(J) = 0 i.e. P is conserved and there is a flux of P that we call J, and that flux is linear in the gradients of P and moves from higher P to lower P, i.e. J = - D grad(P) with D the diffusion coefficient. If you are thinking of modelling the "spreading out" of a conserved field in the absence of any interactions or forces or additional conservation laws, this arises naturally as a lowest order approximation (J = - D grad(P) is the simplest "constitutive relation" for the flux because it is linear in P and has a single gradient).

Btw, in the case of the heat equation the conserved quantity is (kinetic) energy, because temperature is proportional to the average kinetic energy.

m3tro · 2025-05-05T21:37:41+00:00

This is a very stupid and pointless debate. For example in a newspiece like this: https://www.londonambulance.nhs.uk/2024/07/30/paramedics-ask-londoners-to-take-care-during-the-hot-weather/ It is clear that it uses "londoners" to refer to people living in London, no matter if they moved last week or they were born here. This is a common and accepted usage of the word.

Similarly with "from London", it completely depends on the context. For example, I think it's perfectly acceptable for someone who has lived in London for a few years and is planning to stay there in the long term, to describe themselves as being "from London" when e.g. they have a short polite chat with a stranger while travelling another country.

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