Questions tagged [statistical-physics]
The study of physical systems using probabilistic reasoning, especially relating small-scale classical mechanics to large-scale thermodynamics.
240 questions
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Gibbs distribution via rigorous counting? [closed]
Consider the function $Dist: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ (natural numbers include $0$), by defining $Dist(E,N)$ to be the size of the set
$$
\{ (s_1, s_2, \ldots, s_N ) \in \mathbb{...
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Properties of "Random Natural Packing Graphs"
This question is motivated by the following AI (Microsoft Copilot) generated image of urns with different proportions of blue and red marbles that I had requested and described in detail:
and also by ...
- 293
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C^1 fractals in statistical mechanics
It is well-known--even famous--that the Schramm-Loewner curves appear as domain boundaries between phases at second-order critical points like the critical Ising model or percolation in two dimensions....
- 135
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Could critical exponents $z\sim 2.7$ $z\sim2.7$, $\nu\sim 1.3$ in a non-equilibrium reaction–diffusion field indicate a new universality class?
I’m exploring a reaction–diffusion-type scalar field equation of the form
$$
∂_t K=D\nabla^2 K+SK(1-K)(K-K_*),
$$
where $D>0$, $S>0$, and $0<K_*<1$.
Numerical simulations in 2D produce the ...
- 6,830
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Characterization of phase transition using Gibbs states
I am far from being an expert on the theory of Gibbs measures, but I know there is a criteria for phase transitions using uniqueness of infinite-volume Gibbs states. This goes roughly as follows. We ...
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60
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Expectation of a functional in a canonical ensemble of weighted bipartite graphs
Let $Q=(Q_{ij})_{1\le i,j\le N}$ be a nonnegative $N\times N$ matrix (investor $i$ investing a dollar amount in asset $j$).
From a given matrix $Q^\star$ (from a financial dataset), let
$$
r_i^\star=\...
- 670
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Nonnegativity of an alternating combinatorial sum
Let $u,a,b,n$ be nonnegative integers such that $n\le a+b$.
Define the quantity
$$
L(u,a,b,n):=
(u+a+b-n)!\times\sum_{i,k,\ell}\
\frac{(-1)^k\ \ (u+a+b-i)!\ (k+\ell)!\ (a+b-k-\ell)!\ (u+a+b-k-\ell)!}...
- 23.7k
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votes
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Roots of a family of polynomials forming shapes
Let $f$ be a smooth and strictly concave function on $[0,1]$, where $f(0)=f(1)=0$.
Let $F_n(x)=\underset{k=0}{\overset{n} \sum } \exp(nf(\frac kn))x^k$.
The roots of $F_n$ seems to form "shapes&...
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Markov chain mixing: tools for proof?
I'm trying to understand some things about this theorem which comes from the triangle inequality for the transportation metric $\rho_K$:
Suppose the state space $\mathcal{X}$ of a Markov chain is the ...
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Spin system with spin values from a discrete metric space
In this article Koehler and Mossel discuss a spin system with spin values from symmetric group $S_q$ for some $q$. They define the Hamiltonian as
$$
H(\sigma)=\sum_{i\sim j}d_\tau(\sigma_i,\sigma_j)
$$...
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795
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What's the current state of cluster expansions?
A classic reference on cluster expansions in mathematical physics (specially statistical mechanics) is these lecture notes by professor Brydges for a les Houches course in 1984 on the mentioned topic. ...
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How to calculate S-matrix of critical 3-state Pott's model without considering W-algebra?
Let's consider the critical 3-state Potts model. According to conformal field theory, it corresponds to a CFT with a central charge $c=\frac{4}{5}$. However, there are 10 characters for $c=\frac{4}{5}$...
- 67.4k
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Explanation for why an ideal fluid doesn't have increasing entropy?
The equations of motion for a very simple ideal fluid (specifically a calorically perfect, monatomic, ideal gas) are \begin{align*}\dot{\rho}+\nabla \cdot (\rho u)=0 \;&\text{(mass conservation)} \...
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What is the justification for using Wiener integrals to integrate over a space of differentiable functions?
In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
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Should water at the scale of a cell feel more like tar?
The Navier-Stokes equations are as follows,
$$\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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