All Questions
Tagged with mathematical-physics statistical-mechanics
87
questions
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Correlation length in a 3d Ising slab with one dimension much smaller than the other two
Suppose I have a 3d Ising model on a cubic lattice, but one of its dimensions is much smaller than the other two. That is, I have an $L$ by $L$ by $L'$ slab with $L' << L$; in particular, $L'$ ...
- 2,292
2
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1
answer
34
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Limit for big system size of Fokker-Planck eigenfunctions
I am learning how to use diagonalization methods applied to Fokker-Planck equations with Gardiner's book and these notes. The idea is to find the probability density, $ P[X_t\in[x,x+dx]]=\rho_t \, dx$,...
- 1,091
2
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1
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121
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Thermal ground state?
Consider a system of $N$ fermions in a periodic box $\Lambda \subset \mathbb{R}^{d}$, described by the Hamiltonian
$$H_{N} = \sum_{k=1}^{N}(-\Delta_{x_{k}}-\mu) + \lambda \sum_{i< j}V(x_{i}-x_{j}) \...
- 1,131
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69
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Zero temperature Green function as limit of finite temperature Green function
Consider a system of $N$ fermions in a periodic box $\Lambda \subset \mathbb{R}^{d}$. The Hamiltonian of the system is:
$$H_{N} = \sum_{k=1}^{N}(-\Delta_{x_{k}}-\mu) + \lambda \sum_{i< j}V(x_{i}-x_{...
- 1,131
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1
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62
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Time evolution of mixed state?
Suppose I have a quantum statistical mechanics system in the grand-canonical ensemble. It is given by some Hamiltonian $H = H_{0} + V$, where $H_{0}$ is the free part and $V$ an interaction. The state ...
- 1,131
3
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answers
123
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Mathematical objects on crystal meltings and their relation to particle physics
I am a mathematician interested in analytic number theory, and I found the paper Dimers and Amoebae
, which shows how many mathematical objects like the Mahler measure, the Ronkin function and the ...
- 155
2
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45
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How to show random cluster models with non-integer $q$ have no local description?
It is known that the random cluster model with $q = 1$ corresponds to bond percolation, and $q = 2, 3, ... $ corresponds to the $q$-state Potts model. Both of these have a local description.
What ...
- 49
1
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1
answer
90
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Formulating the variational principle in grand canonical ensemble
After a very nice discussion in my previous question, I decided to move on and try to formulate the variational principle for the grand canonical ensemble. I tried following the references cited in ...
- 585
1
vote
0
answers
36
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Is the Lennard-Jones system in Newtonian mechanics ergodic?
A large part of the computational physics literature relies on solving Newtons equations for deriving phase diagrams and related properties of the LJ system.
In many cases this approach does not ...
- 895
1
vote
1
answer
114
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What is the energy of a knot?
Physicists, mathematicians, people who study protein folding, etc., are all in theory interested in knots moving in $\mathbf{R}^3$:
To try to understand them physically, the first question is:
...
- 394
1
vote
1
answer
57
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Is the random current model tight (in the sense of probability)?
Let us consider the random current model (of the classical Ising model) on $\mathbb{Z}^d$. More specifically, we have probability measures $\mathbb{P}_L$ on the product space $\mathbb{N}^{E_L}$ where $...
- 2,123
2
votes
1
answer
52
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Is the random cluster model ergodic/mixing?
Consider the random-cluster model $\mathbb{P}_G^i$ on a finite graph $G$ with parameters $p\in [0,1]$ and $q \in \mathbb{N}_+$ and boundary condition $i=0,1$ (free and wired). The main 2 references ...
- 2,123
1
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0
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59
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Holley and FKG Lattice Conditions
There's an interesting exercise (page 13, Exercise 11) in Hugo Duminil-Copin's Lectures on the Ising and Potts models on the hypercubic lattice, which states that the following 2 statements are ...
- 2,123
4
votes
1
answer
93
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Infrared bound on Ising model
I'm currently trying to understand aspects of Hugo Duminil-Copin's Lectures on the Ising and Potts models on the hypercubic lattice. In section 4.3, he claims that for the Ising model in $\mathbb{Z}^d$...
- 2,123
3
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58
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Relations between different definitions of critical temperatures
I have noticed the following definitions of critical temperature $T_c$ being used in different subject areas:
MAG: The temperature below which some order parameter (e.g. magnetization or self-overlap)...
- 713