# 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{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 ...
339 views

### 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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### 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 ...
1 vote
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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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1 vote
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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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1 vote
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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,...
1 vote
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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 $\SO(n)$ ...
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 \...