104
votes

Accepted

### Should water at the scale of a cell feel more like tar?

There is a beautiful article (a write-up of a talk, actually), by E.M. Purcell, Life at low Reynolds number, that explains how bacteria swim.
Low Reynolds number is the technical way to phrase the ...

27
votes

### Interesting and surprising applications of the Ising Model

An application of the Ising model in social sciences is to voter models: The dynamics of the Ising model tries to align neighbouring spins, similarly, perhaps, to humans deciding on their political, ...

18
votes

### Should water at the scale of a cell feel more like tar?

You may be interested in Shapere, A., and F. Wilczek. 1987. Self-propulsion at low Reynolds number. Phys. Rev. Lett. 58: 2051–2054 where they use gauge theory to describe micro-swimming. Because the ...

17
votes

Accepted

### Representation theory of $\operatorname{SO}(n)$ for large $n$

In this answer I'll focus on the representation theory of $SO(n)$ as $n \to \infty$ (rather than the group theory). Strictly speaking, I want to discuss $O(n)$ rather than $SO(n)$, but I hope that it'...

14
votes

Accepted

### Good overviews on $\phi^{4}$-field theory?

This reference is a bit older, but it should be a good starting point for items 2 and 3: $\phi^4$ field theory in dimension 4: a modern introduction to its unsolved problems.
Concerning item 1, you ...

13
votes

Accepted

### Is there a systematic theory for Gibbs measures (better if on Hilbert spaces)?

Any probability measure $\mu_1$ absolutely continuous with respect to $\mu_1$ can be written as a Gibbs measure if you allow $G$ to take values $\pm \infty$. If the density is bounded above and below,...

13
votes

Accepted

### The origin of the natural base in statistical mechanics

As Matt F. points out, we could just absorb a change of base of the logarithm into the coefficient. The reason that is not convenient in physics is that we would like the same coefficient $k$ to ...

13
votes

Accepted

### Proving the Replica Trick works

Q: Am I overlooking something important?
I think you are ignoring the role played by the thermodynamic limit.
There are two interplaying limits here, the replica limit $n\rightarrow 0$ and the ...

12
votes

Accepted

### Poincaré recurrence and its implications for statistical physics and the arrow of time

Since the question is about physical implications of Poincaré recurrence one should take both quantum effects and gravitational effects into consideration. Quantum mechanics does not spoil the ...

11
votes

### Interesting and surprising applications of the Ising Model

The Ising model defines a universality class, meaning lots of systems simplify to something that looks basically like a magnet. Renormalisation tells us that lots of systems share universal asymptotic ...

10
votes

Accepted

### Explanation for why an ideal fluid doesn't have increasing entropy?

This is a very important issue, to which an answer must be made in mathematical terms, rather than by waving hands.
Yes, the Euler system (conservation of mass, momentum and energy) is time-reversible....

10
votes

Accepted

### Innovations in number theory leading to breakthroughs in statistical mechanics

The paper “The reasonable and unreasonable effectiveness of number theory in statistical mechanics” by George Andrews comes to mind. It is a nice survey that mentions some of the more striking ...

9
votes

### Chebyshev's other inequality

The source for the inequality is
P.L. Chebyshev, On approximate expressions of some integrals in terms of others, taken within the same limits, Proc. Math. Soc. Kharkov 2, 93–98 (1882).
This article ...

8
votes

### Chromatic number of the plane and phase transitions of Potts models

I'm not an expert, but I believe your question is predicated on something which is not true. In particular, the antiferromagnetic (AF) Potts model and the ferromagnetic Potts model exhibit very ...

8
votes

### Ising model, phase transition

I presume this is for a ferromagnetic interaction with all bonds of strength $J$.
The critical temperatures can be found using Kramers-Wannier duality and a star-triangle transformation. According ...

8
votes

### Is there a systematic theory for Gibbs measures (better if on Hilbert spaces)?

The field of improving convergence of sample averages is known as "enhanced sampling". As Robert pointed out, this is an incredibly hard problem. In my field (theoretical chemistry), we have been ...

8
votes

### Is $C^{*}$-algebra the most modern way to study QFT?

My PhD work used C*-algebras quite heavily, so I guess I can claim some expertise there, but I'm not an expert in QFT. That will be the main perspective of my answer.
A good starting point for this ...

8
votes

Accepted

### Proving the simple form of a function from statistical mechanics

We can indeed prove this for reasonable functions, $\log f_0\in C^2$, say.
Let me write $F=\log f_0$. By replacing $F$ by $F(v)-C-d\cdot v$, we can also assume that $F(0),\nabla F(0)=0$.
If $a,v$ are ...

7
votes

Accepted

### map from 6-vertex model to domino tiling

In short: unlike the mapping between alternating-sign matrices and the configurations of the six-vertex model with domain-wall boundary conditions, the mapping between the six-vertex model and domino ...

7
votes

### Poincaré recurrence and its implications for statistical physics and the arrow of time

The Poincare recurrence (or, more general, the ergodic theorem that says that a system will, over time, evolve through essentially all microscopic states that are consistent with the total energy, ...

7
votes

Accepted

### Infinite clusters for loopless percolation

These fascinating questions have been studied recently, e.g. by Bauerschmidt, Crawford, Helmuth and Swan (no percolation on $\mathbb{Z}^2$) and by Bauerschmidt, Crawford and Helmuth (percolation phase ...

7
votes

### Why computing $n$-point correlations?

Quite generally, three-point (and higher order) correlators are used to reveal the non-Gaussian (read: nonclassical) character of the fields, see for example Experimental characterization of a quantum ...

6
votes

Accepted

### A counterexample for the Mean Ergodic Theorem in $L_\infty$

Let $I_n$ be a collection of rapidly-shrinking intervals with $|I_n|=1/n!$, say. Let $f_0=0$.
Inductively define
$$
f_n(x)=\begin{cases}
(-1)^n&\text{if $x\in\bigcup_{i=0}^{n-1}\tau^i(I_n)$;}\\
...

6
votes

Accepted

### Grand-canonical Gibbs measure for continuous systems

Looking at your formula (1), it appears that $\mu$ must be a measure defined on a $\sigma$-algebra $\mathscr F$ over the finite set $\Lambda$. The natural $\sigma$-algebra over the finite set $\Lambda$...

6
votes

Accepted

### CCR vs. CAR vs. Clifford algebras, infinite tensor products and type of the corresponding von Neumann algebra

Wow, that's a lot of questions. For a more in-depth discussion you might look at my book Mathematical Quantization, particularly Section 2.5 and Chapters 4 and 7. But anyway, let me start with the ...

6
votes

### Reference Request for a particular approach of (rigorous) statistical mechanics

You'll want to distinguish equilibrium from non-equilibrium systems. For the former, a classic text is Thermodynamic Formalism
The Mathematical Structure of Equilibrium Statistical Mechanics by David ...

6
votes

### Explicit form of this unitary transformation

Denote by $b_x^*$ and $b_x$ the right-hand sides of (1) and (2). They satisfy the anticommutation relations, and therefore there is an isomorphism from the $C^*$-algebra generated by the $a_x$'s onto ...

5
votes

### What (if anything) happened to Viennot's theory of Heaps of pieces?

Heaps are significant in the theory (especially the combinatorial theory) of minuscule representations of Lie algebras. I guess Stembridge in his 1996 paper "On the fully commutative elements of ...

5
votes

### What is the link between the Domino Tilings and the Ising Model?

For a more modern treatment of the links between dimers and Ising, and exactly where the small print is, I would advise you to read the habilitation memoir of Béatrice de Tilière from 2013, available ...

5
votes

### Single quantum particle entropy

A quantity that might well satisfy the OP is the socalled "entropy of position",
$$S(t)=-\int |\psi(r,t)|^2\log|\psi(r,t)|^2 dr.$$
As derived in The entropy of position and the spreading of wave ...

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