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
$$
\{ (s_1, s_2, \ldots, s_N ) \in \mathbb{...
- 5,748
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Properties of "Random Natural Packing Graphs"
This question is motivated by the following AI (Microsoft Copilot) generated image of urns with different proportions of blue and red marbles that I had requested and described in detail:
and also by ...
- 14k
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Could critical exponents $z\sim 2.7$ $z\sim2.7$, $\nu\sim 1.3$ in a non-equilibrium reaction–diffusion field indicate a new universality class?
I’m exploring a reaction–diffusion-type scalar field equation of the form
$$
∂_t K=D\nabla^2 K+SK(1-K)(K-K_*),
$$
where $D>0$, $S>0$, and $0<K_*<1$.
Numerical simulations in 2D produce the ...
2
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1
answer
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Characterization of phase transition using Gibbs states
I am far from being an expert on the theory of Gibbs measures, but I know there is a criteria for phase transitions using uniqueness of infinite-volume Gibbs states. This goes roughly as follows. We ...
- 1,465
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Expectation of a functional in a canonical ensemble of weighted bipartite graphs
Let $Q=(Q_{ij})_{1\le i,j\le N}$ be a nonnegative $N\times N$ matrix (investor $i$ investing a dollar amount in asset $j$).
From a given matrix $Q^\star$ (from a financial dataset), let
$$
r_i^\star=\...
- 670
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votes
1
answer
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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
$$
L(u,a,b,n):=
(u+a+b-n)!\times\sum_{i,k,\ell}\
\frac{(-1)^k\ \ (u+a+b-i)!\ (k+\ell)!\ (a+b-k-\ell)!\ (u+a+b-k-\ell)!}...
- 23.7k
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Markov chain mixing: tools for proof?
I'm trying to understand some things about this theorem which comes from the triangle inequality for the transportation metric $\rho_K$:
Suppose the state space $\mathcal{X}$ of a Markov chain is the ...
3
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0
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101
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Spin system with spin values from a discrete metric space
In this article Koehler and Mossel discuss a spin system with spin values from symmetric group $S_q$ for some $q$. They define the Hamiltonian as
$$
H(\sigma)=\sum_{i\sim j}d_\tau(\sigma_i,\sigma_j)
$$...
- 173
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Non-uniform random energy model with site weights $Z=\sum_i w_i e^{-\beta E_i}$
Disclaimer. I have very limited and superficial knowledge of statistical physics.
Let $N$ and $n=n(N)$ tend to $\infty$ such that $(1/N)\log n \to \alpha$ for some fixed "load" $\alpha>0$...
- 7,043
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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}$...
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1
answer
123
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Continuous paths in the space of probability measures on $\mathbb N$ with non-increasing energy
Let $\mathcal X=(\mathcal P(\mathbb N),d_{TV})$ be the space of all probability distribution on the discrete countable set $\mathbb N$ equipped with the total variation metric.
For $m\in \mathcal X$, ...
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1
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Randomising weighted bipartite networks
Squartini et al. have shown how to randomise weighted networks while preserving the expected value of local properties (i.e. sampling from the canonical ensemble preserving e.g. the strength sequence, ...
- 670
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832
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Gibbs measure as stationary distribution of SDEs
I have been trying to understand how one can mathematically explain some of the results from statistical mechanics, especially regarding certain distributions like the Gibbs distribution. It would be ...
- 1,007
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Ashkin-Teller Model
Consider the two-dimensional Ashkin-Teller model on the square lattice $\mathbb{Z}^2$ with Hamiltonian:
$$ H = - \sum_{\langle i,j \rangle} \left[ K \sigma_i \sigma_j + K \tau_i \tau_j + k \sigma_i \...
- 21
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Rigorous statistical mechanics: difficulty of realistic models
Soft question: I am a mathematician self-learning statistical mechanics. The (mathematical) literature is concentrated on lattice models like the Ising model and the lattice-gas model. I understand ...
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