# Questions tagged [statistical-physics]

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

173
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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 $\...

**4**

votes

**1**answer

239 views

### 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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193 views

### 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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33 views

### 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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**1**answer

155 views

### 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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97 views

### Multivariate gaussian integral

I am looking for the solution of a multivariate gaussian integral over a vector x with an arbitrary vector a as upper limit and minus infinity as lower limit. The dimension of the vectors x and a are ...

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32 views

### Hammersley Welsh decomposition of SAW for Hexagonal lattice

I am stuck in the process of decomposing any self avoiding walk into bridges on the hexagonal lattice. Does anyone know how to make sense of the decomposition in the Hugo-Smirnov paper of "...

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175 views

### 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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472 views

### 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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40 views

### Central limit theorem behavior in a mean-field model with quenched randomness

Let $\mathbf{h} := \{h_i \}_{i=1}^\infty$ be a sequence of i.i.d random variables and $J > 0$. For $n \in \mathbb{N}$ and a realization of $\mathbf{h}$, we define the Hamiltonian $H_n (\cdot, \...

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58 views

### 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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413 views

### 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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214 views

### 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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436 views

### 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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33 views

### 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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136 views

### Maximal eigenvalue of a correlation matrix

We note $C^M_N = \begin{pmatrix} N \\ M \end{pmatrix}$ the binomial coefficients.
Let $(\alpha_i)_{i \in \{1, \dots, N \} }$ be $N$ real numbers, and $M \in \mathbb{N}$, $1 \le M \le N$. Let $S = (s_k)...

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79 views

### 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 \...

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423 views

### What are applications of Jones polynomial on von Neumann algebras?

I have read according list of below papers a basic connection between Jones polynomial and statistical mechanics is that the Kauffman bracket or Kauffman polynomial a polynomial invariant of knots is ...

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33 views

### How to use the mixed normal distribution to construct a proper statistics?

For a random vector $\xi_n \in \mathbb{R}^p$, if $\xi_n \rightarrow_d N(\mu, \Sigma)$, we can construct
\begin{equation*}
\Psi := \xi_n^{\top} \widehat{\Sigma}^{-1} \xi_n
\end{equation*}
for ...

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**1**answer

99 views

### Understanding the wrapping criterion in percolation theory

Context:
When studying percolation in finite sized systems, there exist various definitions and criteria for determining when a given system is percolating, i.e., given a definition for connectivity, ...

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68 views

### Proving that a model exhibits either a first or second order phase transition

Motivating example:
Take the (wired) random cluster model $\phi^1_{p,q}$ with parameter $q$ (see http://arxiv.org/abs/1707.00520 for an introduction).
It is now known on $\mathbb{Z}^2$ that it has a ...

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557 views

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

I am not an expert on either QFT or $C^{*}$-algebras, but I'm trying to learn the basics of QFT. In all books/papers and other materials that I know, QFT is studied mainly using a lot of functional ...

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124 views

### What is the role of Gibbs states with free boundary conditions in the theory of Gibbs measure?

This is actually a more elaborated version of a previous question of mine, which is now deleted. First, some quick notations:
(1) $\Omega_{0} := \{-1,1\}$ and $\mathcal{F}_{0} := 2^{\Omega_{0}}$ are, ...

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334 views

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

The equations of motion for a very simple ideal fluid (specifically a calorically perfect, monatomic, ideal gas) are \begin{align*}\dot{\rho}+\nabla \cdot (\rho u)=0 \;&\text{(mass conservation)} \...

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5k views

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

The Navier-Stokes equations are as follows,
$$\dot{u}+(u\cdot \nabla ) u +\nu \nabla^2 u =\nabla p$$
where $u$ is the velocity field, $\nu$ is the viscosity, and $p$ is the pressure.
Some elementary ...

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157 views

### What is the definition of the thermodynamic limit of a thermodynamic quantity?

Statistical mechanics is all about taking thermodynamic limits and, as far as I know, there are more than one way to define such limits. Consider the following theorem:
Theorem: In the thermodynamic ...

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95 views

### Gibbs distribution as a pushfoward

Let $\Omega_{0}:=\{-1,1\}$ be a single spin space. If $\Lambda \subset \mathbb{Z}^{d}$ is a fixed finite set, take $\mathcal{F}_{0}$ to be the $\sigma$-algebra $2^{\Omega_{0}}$ on $\Omega_{0}$. We ...

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107 views

### Examples of measures that satisfy FKG, but not the FKG lattice condition

Let a percolation measure be a measure on $\{0,1\}^n$. We have a natural partial order on $\{0,1\}^n$ given by comparing all coordinates. An event $A$ is called increasing if for all $ \omega \in A $ ...

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335 views

### A set of questions on continuous Gaussian Free Fields (GFF)

As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...

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486 views

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

I'm looking for nice overviews on $\phi^{4}$-field theory from the mathematical-physics point of view. To be a little more specific, here are some topics I'd like to read about:
(1) What are the ...

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201 views

### Is there a general theory for Wilsonian renormalization?

I know that Wilson's renormalization group is not a theory per se and that there are many ways to implement it in a given system. Also, renormalization group techniques are applied in a large number ...

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136 views

### Path integrals on statistical mechanics

In (rigorous) statistical mechanics and statistical field theory one is usually concerned in giving meaning to integrals of the form:
\begin{eqnarray}
\langle \mathcal{O}\rangle = \frac{1}{Z}\int D\...

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287 views

### Reformulation - Construction of thermodynamic limit for GFF

I've posted a question about the thermodynamic limit for Gaussian Free Fields (GFF) a couple days ago and I haven't got any answers yet but I kept thinking about it and I thought it would be better to ...

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73 views

### Convergence of probability measures which (asymptotically) concentrate along a submanifold

Let $V : (-1, 1)^d \to \mathbf{R}_+$ be a smooth function, and for $\beta > 0$, define
\begin{align}
P_\beta ( dx ) &= \exp \left( - \beta V ( x ) \right) / z (\beta) \, dx\\
z (\beta) &= \...

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162 views

### Effective action, partition function and the renormalization group

Mayer expansions and the Hamilton–Jacobi equation by D. Brydges and T. Kennedy begins mentioning that many problems in statistical mechanics and QFT center on the analysis of integrals of the form:
\...

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186 views

### Renormalization group strategies

Before introducing block spin transformations in chapter four of Random Walks, Critical Phenomena and Triviality in Quantum Field Theory, the authors state the following:
"In this chapter we sketch ...

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100 views

### Spins in classical statistical mechanics

I'm reading Kupiainen's notes on the renormalization group and also caught my attention. Actually, this is something that often causes my some confusion. On page 43, in the section about Ginzburg-...

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109 views

### What are the tempered Gibbs measures of classical $\phi^4$-theory?

I consider classical $\phi^4$-theory on the lattice. The model is defined in finite volume with Hamiltonian
\begin{align*}
H(\phi) = - \sum_{x \sim y} J_{x,y} \phi_x \phi_y
\end{align*}
and a-priori ...

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155 views

### Thermodynamic limit and Gaussian measures

Let $\Lambda \subset \mathbb{Z}^{d}$ be finite and fixed and consider $\mathbb{R}^{|\Lambda|}$ be the vector space of all sequences $\varphi = (\varphi_{x})_{x\in \Lambda}$. We equip $\mathbb{R}^{|\...

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69 views

### Sine-Gordon transformation and functional integrals

In the past months, I've been trying to understand the so-called Sine-Gordon transformation, so I've posted some questions here about this topic. I also did an extensive research about this subject, ...

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**1**answer

83 views

### Identifications between different phase spaces

I've discovered Adam's lecture notes on statistical mechanics after posting my first question about Minlo's discussion on continuous Gibbs measures. Adam's lecture notes are really good, but there is ...

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101 views

### Measure on a set and its value on $\emptyset$

After my first post here, I have one more doubt which is bothering me. It concerns Minlos's book Introduction to mathematical statistical physics again. To fix the notation, we have $\Lambda \subset \...

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126 views

### Measure, volume and cardinality on Minlos' book on statistical physics

The following content was based on Minlos' book on statistical physics. Let $\Lambda \subset \mathbb{R}^{d}$ be fixed (Minlos takes $d=3$ but I think the ideas follow without change to $d \ge 1$). We ...

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212 views

### Grand-canonical Gibbs measure for continuous systems

Let's consider a bounded (maybe compact) set $\Lambda \subset \mathbb{R}^{d}$ with particles interacting on it. Suppose, for each $N \in \mathbb{N}$, $U_{N}: (\mathbb{R}^{d})^{N} \to \mathbb{R}\cup \{+...

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270 views

### Imprecise Definition of a $\sigma$-algebra

I'm reading some works on the hierarchical model in statistical mechanics and I came across an strange definition, which I need to clarify. Consider a finite set $\Lambda \subset \mathbb{Z}^{d}$. The ...

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182 views

### Mathematical meaning for the (continuous) Sine-Gordon transformation

I've been trying to understand the so-called Sine-Gordon Transformation which occurs in both classical and quantum statistical mechanics. One of the most cited references on this topic seems to be ...

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115 views

### A combinatorial identity on even spanning subgraphs in the Erdös-Renyi random graph with relations to the Ising model

Let $x \in \lbrack 0,1 \rbrack$. Then for any finite graph $G$ consider the Erdös-Renyi random graph where we independently keep each of the edges with probability $x$. Denote the corresponding ...

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219 views

### Examples of particle systems with higher-order collisions

In kinetic theory, one often comes across interacting particle systems with a collisional flavour. I'll currently prefer to think about them as systems of ODEs (or SDEs, Jump Processes, $\ldots$), ...

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640 views

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

During these first months in my PhD, I realized how my computational problems can be drastically reduced to one single problem:
Find an efficient way to sample from a Gibbs measure.
Let me ...

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96 views

### Computing the partition function via one of the three methods

I am trying to compute the averaged partition function for some system (with very large $N$) and I reach this point:
\begin{equation}
\left < Z\right > =\int \prod_i^N \left (\frac{\...