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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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How do I solve this $\mu(T) \approx \mu_f \left( 1 - \frac{\pi^2 k_B^2 T^2}{12 \mu(t)} \right)$? [closed]

I'm trying to solve this equation in the context of the statistical physics and because I have to do a code that is able to return the value of $\mu(T)$ but because $\mu(T)$ is a function of itself I ...
Alejandro's user avatar
1 vote
1 answer
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Probabilistic 2D cellular automata with memory lifetime increasing like $e^{L^2}$

Consider 2-state probabilistic cellular automata on an $L\times L$ torus square lattice which has the all-$0$ and all-$1$ configurations as fixed points, thinking of something similar to Toom's rule ...
Andi Bauer's user avatar
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Stein's Lemma for conditional expectation?

Let $X=(X_1,\ldots,X_d)$ be a standard normal random vector in $\mathbb R^d$, let $m:\mathbb R^d \to \mathbb R$ be a function, and let $E=E_m$ denote the expectation operator conditioned on $m(X) > ...
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Recursive relation to represent the last element of a matrix using determinant [closed]

$J$ is a $N\times N$ matrix, each element of $J$ is sampled from a Gaussian distribution with zero mean and variance $N^{-1}$. The resolvent matrix is defined as $R^{(N)} = [\mathcal{E} \mathbb{I} - J]...
Richard's user avatar
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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 ...
dohmatob's user avatar
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5 votes
2 answers
555 views

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 ...
FFjet's user avatar
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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,...
happyle's user avatar
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1 vote
0 answers
146 views

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 ...
Mr. Proof's user avatar
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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,...
MathMath's user avatar
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2 answers
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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 ...
MathMath's user avatar
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1 vote
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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. ...
Plemath's user avatar
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1 answer
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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 ...
MathMath's user avatar
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11 votes
1 answer
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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 ...
Leibniz's Alien's user avatar
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235 views

$\log\det$ 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 ...
Fred Hucht's user avatar
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6 votes
2 answers
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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 ...
JustWannaKnow's user avatar
1 vote
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55 views

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 $v \in \mathbb R^d$, ...
dohmatob's user avatar
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2 votes
1 answer
256 views

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 $...
user515206's user avatar
1 vote
0 answers
127 views

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 ...
dohmatob's user avatar
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1 vote
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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. ...
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0 votes
0 answers
98 views

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 $x \in V_1$, ...
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3 votes
0 answers
106 views

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 ...
Jesse van Rhijn's user avatar
1 vote
0 answers
117 views

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 ...
MathMath's user avatar
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9 votes
1 answer
1k views

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}\...
Felix B.'s user avatar
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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 ...
dohmatob's user avatar
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0 votes
0 answers
42 views

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 ...
dohmatob's user avatar
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10 votes
2 answers
600 views

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 ...
LuckyJollyMoments's user avatar
1 vote
0 answers
48 views

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 ...
Hausdorff's user avatar
5 votes
1 answer
435 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 ...
JustWannaKnow's user avatar
0 votes
0 answers
59 views

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 $...
Yaroslav Bulatov's user avatar
6 votes
2 answers
279 views

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 ...
TheSimpliFire's user avatar
0 votes
0 answers
89 views

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 ...
Johnny Cage's user avatar
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2 votes
1 answer
275 views

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/...
Aleph1234's user avatar
-1 votes
1 answer
125 views

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 ...
chongwen wang's user avatar
1 vote
0 answers
194 views

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, ...
AChem's user avatar
  • 801
6 votes
2 answers
431 views

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 ...
PeaBrane's user avatar
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6 votes
2 answers
3k views

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 ...
display llvll's user avatar
8 votes
1 answer
502 views

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 ...
truebaran's user avatar
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4 votes
1 answer
239 views

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 ...
MathMath's user avatar
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1 vote
0 answers
139 views

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 "...
apg's user avatar
  • 612
1 vote
1 answer
290 views

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 ...
apg's user avatar
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1 vote
0 answers
151 views

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 ...
DJA's user avatar
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1 vote
0 answers
88 views

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, ...
Rono's user avatar
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2 votes
0 answers
113 views

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 ...
Harmenszoon's user avatar
2 votes
0 answers
184 views

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(\...
user853186's user avatar
5 votes
2 answers
321 views

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. ...
weissguy's user avatar
10 votes
2 answers
498 views

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 ...
Aidan Rocke's user avatar
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3 votes
1 answer
150 views

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 ...
user196574's user avatar
39 votes
4 answers
4k views

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 ...
JustWannaKnow's user avatar
0 votes
1 answer
127 views

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 ...
Hollis Williams's user avatar
8 votes
0 answers
180 views

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&...
LeechLattice's user avatar
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