Questions tagged [statistical-physics]
The study of physical systems using probabilistic reasoning, especially relating small-scale classical mechanics to large-scale thermodynamics.
225
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Limiting value of trace of resolvent matrix involving two independent Wishart random matrices
Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that
$$
d/n_k \to \phi_k \in (0,\infty).
$$
Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...
5
votes
2
answers
460
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+100
Algebra/Algebraic geometry in statistical mechanics
This is a soft question. I am currently studying statistical mechanics and I found this one by chance: Algebraic statistical mechanics
And I also found some workshops on interactions between ...
1
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0
answers
55
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Under which condition, such that all second-order critical points satisfy $\sum_j\cos(\theta_i-\theta_j)>0$ for all $i\in[n]$?
Consider the following non-convex function
$$E(\theta):=-\sum_{i,j}A_{ij}\cos(\theta_i-\theta_j)$$
where $A$ is a symmetric, diagonal-free matrix whose non-diagonal element are $\pm 1$. In other words,...
1
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0
answers
134
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Recommendation to understand mean field theorem
I am studying Rodnianski and Schlein - Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics. Everything was clear for me and I reproved everything before inequality (3.22) (except ...
4
votes
4
answers
403
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Why computing $n$-point correlations?
I am trying to find a sufficiently convincing answer to this question, but it has been taking so much of my time and I can't get anywhere. It also follows my previous question on PSE.
In axiomatic QFT,...
4
votes
2
answers
191
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Reference for rigorous interacting many-body quantum mechanics
Are there good references for (both zero and finite time) interacting systems of quantum many-body theory? More precisely, I would be interested in references discussing the following topics:
Second ...
1
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0
answers
85
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Mixing for a gas of hard spheres
The gas of hard spheres is a model for a gas in a container, where each particle is a sphere of radius $\epsilon$. The spheres interact with each other and with the container with elastic collisions. ...
0
votes
1
answer
125
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Overview resources for (rigorous) critical phenomena
I recently came across this overview which discusses some results in the theory of critical phenomena. It is already quite old and I would like to know if there are other (more recent) overviews in ...
11
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1
answer
579
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Progress on Simon's 1984 problem of the proof of Universality
I am writing this post to inquire if any progress has been made in solving problem 8B (Proof of Universality) proposed by Barry Simon in 1984.
The problem goes like this:
Show that the critical ...
5
votes
0
answers
231
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$\log\det$ asymptotics of a skew-circulant matrix with additive diagonal bimodal disorder
I'd like to share a problem that I have been dealing with for a longer time now.
In the framework of quenched disorder in the square-lattice Ising model I want to calculate, for large even $M$, the ...
6
votes
2
answers
592
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Explicit form of this unitary transformation
Disclaimer: This question has its motivation from physics. It is probably not entirely rigorous at the moment. I just want to clarify some steps and try to make the arguments rigorous afterwards, if ...
1
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0
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55
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Limiting value of expectation of trace of truncated Gram matrix
Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, ...
2
votes
1
answer
196
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Fokker Planck equation in the Stratonovich approach
I'm a physics master student and I have difficulties understanding how to derive the Fokker Planck equation from the Stratonovich SDE.
With the Ito SDE it is simple since the noise is independent of $...
1
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0
answers
125
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Large-deviation inequalities for a class of simple random multivariate polynomials
Let $N$ be a large positive integer and let $[N] := \{1,2,\ldots,N\}$. For any $k$, let $K_{N,k}$ denote the collection of $k$-element subsets of $[N]$. Let $x=(x_1,\ldots,x_N)$ be a uniformly random ...
1
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0
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Large deviation inequalities for number of coupon types collected by a coupon collector with fixed budget
In the generalized Coupon Collector's Problem, there are $N$ types of coupon, and for any $i \in [N] := \{1,2,\ldots,N\}$, $p_i \ge 0$ is the probability of obtaining a type-i coupon on any trial. ...
0
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96
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Phase-transitions for a property of random bipartite graphs
Let $N_1$, $N_2$, and $k$ be positive integers. Let $V_1$ and $V_2$ be finite sets with $|V_i| = N_i \ge 1$. Consider a bipartite graph $G=(V_1,V_2,E)$ constructed as follows. For every $x \in V_1$, ...
3
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0
answers
105
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Differential entropy of random Gibbs measure
There is a question I have been wondering about for a while, which I have thus far not been able to resolve. The problem revolves around random Gibbs measures. I am not very well-versed in the more ...
1
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0
answers
112
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Mathematical justification for the use of an energy shell in the microcanonical ensemble
I would like to understand an identity used in the deduction of the explicit formula for the probability distribution of the microcanonical ensemble in statistical mechanics.
Consider $\Lambda$ to be ...
9
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1
answer
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Proving the Replica Trick works
The replica trick attempts to calculate the expectation of the logarithm $X=\log(Z)$ of a random variable $Z$. The wikipedia article describes the logarithm as the limit
$$
\log(Z) = \lim_{n\to 0}\...
2
votes
0
answers
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Limiting value of $\dfrac{1}{m}\mathrm{tr}(FAF^\top (FBF^\top)^{-1})$, where $F$ has iide $N(0,1)$ entries and $A,B$ are deterministic
Let $F=F_{m,d}$ be a random $m \times d$ matrix with iid entries from $N(0,1)$. Let $A=A_d$ and $B=B_d$ be deterministic $d \times d$ positive-definite matrices. In case it helps, it may be assumed ...
0
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0
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42
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Asymptotics of a certain trace involving random matrices with general elliptical covariance structure
Let $n,d,m$ be large positive integers that the ratios $d/n$ and $d/m$ are fixed in $(0,\infty)$. Let $G \in \mathbb R^{n \times d}$ and $S \in \mathbb R^{d \times m}$ be independent random matrices ...
10
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2
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597
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Proving the simple form of a function from statistical mechanics
I have discovered a pertinent solution to my problem in the article On the Kinetic Theory of Rarefied Gases by Harold Grad and the book Thermodynamik und Statistik by Arnold Sommerfeld, both of which ...
1
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0
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40
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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 ...
5
votes
1
answer
404
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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 ...
0
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0
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58
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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 $...
1
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0
answers
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$\min(|\lambda_{\min}(A(c))|)$ for a special matrix $A(c)$ defined over $\{-1,-1\}^N$ [closed]
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,-...
6
votes
2
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278
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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.$$ It enjoys properties such as involution and has various ...
0
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0
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Connectivity constant for lattices
A celebrated result due to Duminil-Copin and Smirnov states that the connectivity constant for the honeycomb lattice is equal to $\sqrt{2+\sqrt{2}}$.
My question is the following: apart from the ...
2
votes
1
answer
263
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Relationship between heat kernel and Maxwell-Boltzmann distribution
There appears to be a connection between the heat kernel and Maxwell-Boltzmann distribution, but I have not seen this in the literature before. I'd appreciate any kind comments or corrections/...
3
votes
0
answers
73
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Convergence in perturbative renormalization
Consider the following:
$$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 ...
-1
votes
1
answer
121
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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
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0
answers
184
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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, ...
6
votes
2
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404
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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 ...
6
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2
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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 ...
8
votes
1
answer
451
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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 ...
4
votes
1
answer
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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 ...
1
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0
answers
133
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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 "...
1
vote
1
answer
275
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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 ...
1
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0
answers
146
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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 ...
1
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0
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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, ...
2
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0
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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 ...
2
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0
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180
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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(\...
5
votes
2
answers
306
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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.
...
10
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2
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486
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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 ...
3
votes
1
answer
144
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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 ...
38
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4
answers
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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 ...
0
votes
1
answer
127
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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 ...
8
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0
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168
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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&...
8
votes
1
answer
201
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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 ...
2
votes
1
answer
228
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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, ...