# Questions tagged [statistical-physics]

The study of physical systems using probabilistic reasoning, especially relating small-scale classical mechanics to large-scale thermodynamics.

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

Suppose we have a function $f_0:{\mathbb R}^3\rightarrow {\mathbb R}_+$ that satisfies the following property
\begin{equation}
\begin{split}
&\mathbf{v}_1^2 + \mathbf{v}_2^2 = \mathbf{v}_1'^2 + \...

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### Langevin dynamics or stochastic gradient flow for grand canonical ensemble

We know that for a measure exp(-U(X)) (canonical ensemble), we can use the dynamic dX=-DU(X)+ noise to sample the measure as t goes to infinity.
Is there any dynamic corresponding to the grand ...

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

I was reading Mathematical Aspects of Quantum Field Theory by. E. de Faria and W. de Melo, and the following caught my attention.
In (Hamiltonian) mechanics, the states of a system are described by ...

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### Norms of Wigner matrices under power law decay

Suppose $\Sigma=\operatorname{diag}(h)$ where $h=(1^{-p},2^{-p},3^{-p},\ldots,d^{-p})$ and $p> 1$
$X$ is a matrix with $b$ rows sampled independently from $\operatorname{Normal}(0,\Sigma)$
Suppose $...

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### $\min(|\lambda_{\min}(A(c))|)$ for a special matrix $A(c)$ defined over $\{-1,-1\}^N$

For a given constant $E$, is there way to find the lower bound of the following expression?
$\min_{c\in\{-1,+1\}^N, -\sum_{i,j}c_ic_j=E}(|\lambda_{\min}(A(c))|)$ for matrix $A(c)$ defined over $\{-1,-...

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### How to make estimation of probabilities of atypically large fluctuations in random matrix theory rigorous?

Suppose we try to rigorously answer questions like:
Given a random $n\times n$ matrix from GUE (Gaussian Unitary Ensemble), what is the probability $P^{\text{GUE}}_n$ that it is positive semidefinite?...

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### Signs of Kravchuk matrix asymptotically produce a large circular region with hyperbolic sinks. Why?

The Kravchuk matrix of dimension $n+1$ is such that its entries satisfy $$K_{i,j}^{(n+1)}=[x^i](1+x)^{n-j}(1-x)^j\quad\forall0\le i,j\le n.$$ It enjoys properties such as involution and has various ...

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### Connectivity constant for lattices

A celebrated result due to Duminil-Copin and Smirnov states that the connectivity constant for the honeycomb lattice is equal to $\sqrt{2+\sqrt{2}}$.
My question is the following: apart from the ...

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### Relationship between heat kernel and Maxwell-Boltzmann distribution

There appears to be a connection between the heat kernel and Maxwell-Boltzmann distribution, but I have not seen this in the literature before. I'd appreciate any kind comments or corrections/...

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### Convergence in perturbative renormalization

Consider the following:
$$G(\phi,W) = -\log \int d\mu_{C}(\psi)e^{-W(\phi+\psi)} \tag{1}\label{1}$$
which is very common in QFT. Here $d\mu_{C}$ is a Gaussian measure with covariance $C$. I want to ...

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### How to analytically solve this ODEs?

I don't think these ODEs have been explicitly solved before, and I'm wondering if anyone can point me to some papers which might help me start.
Here $n$ is an integer and $S_A,S_B$ can be seen as ...

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### Special function: Pulse peak modified with a power term

PeakFit (Systat, v. 4.12) is a software for fitting experimental peaks obtained in physics or chemical experiments. Under the miscellenous peak functions, it shows the following equations with a name, ...

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### Infinite clusters for loopless percolation

I feel like this is maybe an incredibly trivial problem, and I'm just missing something. I may also be describing a well-known model that I cannot find the name for, so any comment/suggestion is ...

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

A very important theorem in mathematical physics is Poincaré’s recurrence theorem.
As you recall, this theorem states that given a dynamical system $(M , \phi , \mu)$ with $ \mu M < +\infty$, for ...

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

$\newcommand\CAR{\mathit{CAR}}\newcommand\Cl{\mathbb C\mathit l}$This question will be rather long and it will be my attempt to finally clarify many issues concerning CCR, CAR and Clifford algebras ...

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### Cluster expansion, Mayer expansion and perturbative renormalization group

This is a second part of my previous question, which I decided to split into two parts not to mix up different topics at one giant question.
Again, according to V. Rivasseau (section 1.5 of ...

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### Width of the critical window in a random graph

In an Erdős–Rényi random graph $G(n,p)$, the giant component emerges with thresholding function $p(n) = c/n$, where $c>1$.
When $c=1$, and $\lambda \in \mathbb{R}$, we can write or "...

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### What is the exact definition of a sharp transition?

In "Sharp threshold phenomena in statistical physics", H. Duminil-Copin, Japanese J. of Math. 14, 2019, a sharp transition of a boolean function is defined as follows:
A sequence of ...

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### Limits of conditional distributions of Mult(k), outside of the range of local limit theorems

Let $X$ be a random vector with uniform distribution on vectors $\{0,e_1,\ldots,e_n\}$ in $\mathbb{R}^n$. Let $X_1, X_2\ldots$ be a sequence of i.i.d. random vectors with the same distribution as $X$ ...

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### physical interpretation of ruelle probablity cascades (SK model)

Background: the Parisi formula gives an exact expression for the free energy of the SK model. The formula (at least the upper-bound) can be derived by looking at the free energy, and then replacing ...

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### How is the quasipotential in Freidlin-Wentzell theory of large deviations affected by $C^1$ transformations?

I have a 3D differential equation I'm interested in studying the potential landscape using quasipotentials, described in depth in this paper. I need to calculate the potential landscape several times, ...

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### What is the justification for using Wiener integrals to integrate over a space of differentiable functions?

In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...

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### Harish-Chandra–Itzykson–Zuber integral with two terms

We know
$$
\int \mathcal{D}U \exp(\mathrm{Tr}(AUBU^*))
$$
can be computed by Harish-Chandra–Itzykson–Zuber(HCIZ) integral. I am wondering whether it is possible to compute
$$
I=\int \mathcal{D}U \exp(\...

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### Placing pins on a Galton board to approximate an arbitrary distribution

Inspired by this reddit post: https://old.reddit.com/r/math/comments/tv3cbg/how_do_you_unbell_curve_a_galtonplinko_board/
The Nth Galton Board, G(N), is a triangular lattice of pegs of height N-1.
...

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

In modern treatments of statistical mechanics, the natural base is conventionally used for the Gibbs and Boltzmann entropy without careful justification. While I am aware that the properties of the ...

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### Binary cellular automata: How slowly can an eroder remove $1$'s?

Consider some deterministic, monotonic, eroding binary cellular automata on some lattice $\mathbb{Z}^d$, and consider the set of initial states $I(L)$ in which all of the vertices are $0$ except for ...

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### Interesting and surprising applications of the Ising Model

One of the most famous models in physics is the Ising model, invented by Wilhelm Lenz as a PhD problem to his student Ernst Ising. The one-dimensional version of it was solved in Ising's thesis in ...

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### Generalising results on superfluid Kubo formulas

In a 2014 article by Chapman, Hoyos and Oz, the authors study non-equilibrium fluid dynamics and describe a method for deriving Kubo formulas for thermal transport coefficients of superfluids (the ...

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### Roots of a family of polynomials forming shapes

Let $f$ be a smooth and strictly concave function on $[0,1]$, where $f(0)=f(1)=0$.
Let $F_n(x)=\underset{k=0}{\overset{n} \sum } \exp(nf(\frac kn))x^k$.
The roots of $F_n$ seems to form "shapes&...

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### From the conceptual idea of the RG to its actual implementation

Everytime I want to understand a little more about the ideas behind Renormalization Group techniques, I get troubled by a gap between the general picture one usually presents (e.g. in books or ...

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### An inequality for a "generalised random energy model"

Let, for all $i, j$, $Z_{i,j}$ be a standard normal, chosen iid. For each $n\geq 1, k\geq2$, define the Hamiltonian $H_{n,k}: [k]^n \to \mathbb{R}$ by
$$(j_1,j_2,\ldots,j_n) \mapsto \sum_{i=1}^n Z_{i, ...

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### Is Toom's rule robust under local but non-on-site noise?

Toom's rule is a 2-dimensional cellular automaton which is known to have two distinct stationary measures in the thermodynamic limit, even after small perturbations to a probabilistic cellular ...

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### In percolation on a lattice, how is "infected" status correlated for points in a region around the origin?

Consider independent bond percolation on $\mathbb{Z}^2$, with $p>p_c$ so that the process is supercritical. For any site $x$ let $Y_x$ be the indicator of $x$ belonging to the infinite open cluster....

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### A relation between the second moment of a distribution and one of its particular probability

I had recently posted a question here: To prove a relation involving a probability distribution
The relations quoted in the above question are used extensively in fluid mechanics and many other fields,...

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### To prove a relation involving a probability distribution

I'm reading a book and have encountered a relation which seems to me to be impossible to prove, I would like to be sure if this is the case. The author gives a probability function as
$$p_n = \frac{e^...

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

Might there be a good reference on the interaction of number theory with statistical physics? I am particularly interested in innovations in number theory that have led to breakthroughs in statistical ...

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### Use statistical physics ideas ("replica trick") to compute asymptotic value of $\inf_{\|w\| \le r} (1/n)\|Xw-y\|^2$ for random $X$ and $y$

I'm trying to get my head around the "replica trick" and it's mathematically rigorous formulations (due to Talagrand, Parchenko, etc.). I was wondering to myself that a solution or insight ...

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### Simplification on the estimation on error of the ratio of 2 random variables

Let $Z=\dfrac{X}{Y}$ the ratio of 2 random variables.
Distribution of $Z=\dfrac{X}{Y}$
Consider the case of two independent normal variables $X$ and $Y$ with strictly positive means and variances $\...

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### Bochner-Minlos for moment-generating functions?

It is well-known that the Bochner-Minlos theorem characterises measures on duals of nuclear spaces by their characteristic functions. Is there a similar version for moment-generating functions?
I have ...

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### in Euclidean space defined by multivariate normal distribution, what fraction of points falls within n-ball (centered at origin) tangent to point p?

In a Euclidean space defined by the multivariate normal distribution, what fraction of all points falls within or are tangent to (as opposed to falling outside of) the n-sphere whose center is at the ...

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### Entropy per site of quantum spin chain

It’s fork lore that von Neumann entropy (and free energy) grows linearly with respect to the size of a quantum system. Is there a rigorous demonstration in the toy model of a quantum spin chain with (...

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### Find a way to apply the MLE on Fisher or Covariance matrix to make cross-correlations

I have 2 Fisher matrixes which represent information for the same variables (I mean columns/rows represent the same parameters in the 2 matrixes).
Now I would like to make the cross-correlations ...

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### What's the current state of cluster expansions?

A classic reference on cluster expansions in mathematical physics (specially statistical mechanics) is these lecture notes by professor Brydges for a les Houches course in 1984 on the mentioned topic. ...

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### How are Clifford algebras and spinors used to study the Ising model?

I've heard Clifford algebras and spinors are useful tools to study the Ising model, but I've never find any good discussion on this matter. Also, as far as I know, in the original solutions of the 2-D ...

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### Generalized Ising Model

I am in very trouble with a particular expression. I leave the original pages in order to have everything available and what I am goin to leave are the first pages of nine chapter of Non Perturbative ...

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### Chebyshev's other inequality

It is a simple fact, the granddaddy of correlation inequalities that if $f,g$ are monotone functions on $[0,1]$ then $$\int_0^1 f(x)g(x) dx \ge \int_0^1 f(x) dx \int_0^1 g(x) dx.$$ In their famous ...

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### How can one recover/obtain information from the renormalization group procedure?

I know the basic idea behind the renormalization group approach as it is used in mathematical physics to study both QFT and statistical mechanics. However, I have trouble understanding how can one ...

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

$\DeclareMathOperator{\SO}{\operatorname{SO}}\DeclareMathOperator{\SU}{\operatorname{SU}}$Background: In the quantum field theory literature people commonly consider "expansions" of $\SO(n)$ ...

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### Random sequential adsorption of rectangles with random aspect ratio

Jian-Sheng Wang, in his 1994 article A fast algorithm for random sequential adsorption of discs (arXiv link), considered deposition only on (small) squares which are not fully covered by the excluded ...

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### GKS inequality with boundary condition

I want to know whether the following version of GKS inequality with boundary condition for Ising model hold or not. Consider Ising model on $\mathbb{Z}^d$ and $\varnothing \neq A\subset \Lambda_1 \...