All Questions
35 questions
5
votes
1
answer
383
views
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 ...
- 312
7
votes
0
answers
269
views
Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$
I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by:
$$
Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy
$$
This operator ...
- 109
1
vote
0
answers
48
views
Rigorous analysis of phase transitions and universality in a non-linear model of interacting oscillators
Consider a system of interacting non-linear oscillators governed by the McKean-Vlasov equation:
$$\frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x}\left[\frac{\partial V(x)}{\partial x}...
user530766
2
votes
0
answers
115
views
Mixing for a gas of hard spheres
The gas of hard spheres is a model for a gas in a container, where each particle is a sphere of radius $\epsilon$. The spheres interact with each other and with the container with elastic collisions. ...
- 312
2
votes
1
answer
403
views
Fokker Planck equation in the Stratonovich approach
I'm a physics master student and I have difficulties understanding how to derive the Fokker Planck equation from the Stratonovich SDE.
With the Ito SDE it is simple since the noise is independent of $...
- 21
10
votes
1
answer
1k
views
Proving the Replica Trick works
The replica trick attempts to calculate the expectation of the logarithm $X=\log(Z)$ of a random variable $Z$. The wikipedia article describes the logarithm as the limit
$$
\log(Z) = \lim_{n\to 0}\...
- 385
6
votes
2
answers
481
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 ...
- 213
1
vote
0
answers
164
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 ...
- 435
2
votes
1
answer
232
views
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, ...
- 526
4
votes
2
answers
564
views
A relation between the second moment of a distribution and one of its particular probability
I had recently posted a question here: To prove a relation involving a probability distribution
The relations quoted in the above question are used extensively in fluid mechanics and many other fields,...
user318428
1
vote
0
answers
90
views
Proving that a model exhibits either a first or second order phase transition
Motivating example:
Take the (wired) random cluster model $\phi^1_{p,q}$ with parameter $q$ (see http://arxiv.org/abs/1707.00520 for an introduction).
It is now known on $\mathbb{Z}^2$ that it has a ...
- 1,054
4
votes
1
answer
237
views
What is the role of Gibbs states with free boundary conditions in the theory of Gibbs measure?
This is actually a more elaborated version of a previous question of mine, which is now deleted. First, some quick notations:
(1) $\Omega_{0} := \{-1,1\}$ and $\mathcal{F}_{0} := 2^{\Omega_{0}}$ are, ...
- 1,305
4
votes
0
answers
321
views
Examples of measures that satisfy FKG, but not the FKG lattice condition
Let a percolation measure be a measure on $\{0,1\}^n$. We have a natural partial order on $\{0,1\}^n$ given by comparing all coordinates. An event $A$ is called increasing if for all $ \omega \in A $ ...
- 1,054
6
votes
1
answer
1k
views
A set of questions on continuous Gaussian Free Fields (GFF)
As I said in my previous posts, I'm trying to teach myself some rigorous statistical mechanics/statistical field theory and I'm primarily interested in $\varphi^{4}$, but I know that the absense of ...
- 3,515
6
votes
1
answer
708
views
Reformulation - Construction of thermodynamic limit for GFF
I've posted a question about the thermodynamic limit for Gaussian Free Fields (GFF) a couple days ago and I haven't got any answers yet but I kept thinking about it and I thought it would be better to ...
- 3,515
5
votes
1
answer
439
views
Effective action, partition function and the renormalization group
Mayer expansions and the Hamilton–Jacobi equation by D. Brydges and T. Kennedy begins mentioning that many problems in statistical mechanics and QFT center on the analysis of integrals of the form:
\...
- 3,515
1
vote
0
answers
114
views
Spins in classical statistical mechanics
I'm reading Kupiainen's notes on the renormalization group and also caught my attention. Actually, this is something that often causes my some confusion. On page 43, in the section about Ginzburg-...
- 3,515
8
votes
0
answers
195
views
What are the tempered Gibbs measures of classical $\phi^4$-theory?
I consider classical $\phi^4$-theory on the lattice. The model is defined in finite volume with Hamiltonian
\begin{align*}
H(\phi) = - \sum_{x \sim y} J_{x,y} \phi_x \phi_y
\end{align*}
and a-priori ...
- 1,054
2
votes
1
answer
238
views
Thermodynamic limit and Gaussian measures
Let $\Lambda \subset \mathbb{Z}^{d}$ be finite and fixed and consider $\mathbb{R}^{|\Lambda|}$ be the vector space of all sequences $\varphi = (\varphi_{x})_{x\in \Lambda}$. We equip $\mathbb{R}^{|\...
- 3,515
4
votes
1
answer
96
views
Identifications between different phase spaces
I've discovered Adam's lecture notes on statistical mechanics after posting my first question about Minlo's discussion on continuous Gibbs measures. Adam's lecture notes are really good, but there is ...
- 1,305
1
vote
1
answer
184
views
Measure, volume and cardinality on Minlos' book on statistical physics
The following content was based on Minlos' book on statistical physics. Let $\Lambda \subset \mathbb{R}^{d}$ be fixed (Minlos takes $d=3$ but I think the ideas follow without change to $d \ge 1$). We ...
- 1,305
4
votes
2
answers
267
views
Grand-canonical Gibbs measure for continuous systems
Let's consider a bounded (maybe compact) set $\Lambda \subset \mathbb{R}^{d}$ with particles interacting on it. Suppose, for each $N \in \mathbb{N}$, $U_{N}: (\mathbb{R}^{d})^{N} \to \mathbb{R}\cup \{+...
- 3,515
2
votes
2
answers
294
views
Imprecise Definition of a $\sigma$-algebra
I'm reading some works on the hierarchical model in statistical mechanics and I came across an strange definition, which I need to clarify. Consider a finite set $\Lambda \subset \mathbb{Z}^{d}$. The ...
- 3,515
3
votes
1
answer
2k
views
Understanding Finite Size Scaling in Percolation Theory
Fundamental results in percolation theory are all based on the assumption that the system sizes are infinite, as the spanning/percolating cluster is by definition an infinitely sized cluster that ...
- 649
4
votes
0
answers
164
views
List of Replica Symmetry results for different models?
Does anyone know of a good source that might have a list of problems or models along with what kind of replica symmetry they are conjectured to have?
I am aware of some of the more famous results, e....
- 435
7
votes
3
answers
830
views
What is the link between the Domino Tilings and the Ising Model?
Is there a link between the theory of Domino Tilings and the Ising Model? In the global qualitative sense that physicists use, the answer is "yes". The connections could go like this:
The dimer ...
- 22.8k
3
votes
0
answers
112
views
Uniqueness results for lattice spin systems (graphs)
Are there any nice uniqueness results for Gibbs-measures on lattice spin systems (graphs) that does not rely on Dobrushin's method?
- 244
5
votes
1
answer
697
views
Harmonic Crystal using Random Walk
Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here.
Basically, it is a ...
- 3,574
6
votes
2
answers
912
views
References for a physicist migrating to stochastic processes
I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. ...
- 161
1
vote
2
answers
415
views
$\{\phi:\int \phi d\mu=0\}$ for a fixed shift invariant $\mu$
Given a shift invariant probability measure $\mu$ on a mixing subshift of finite type.
What are the Lipschitz functions with zero integral with respect to the measure $\mu?$
Clearly any $\phi\in\{-u+...
- 1,805
1
vote
0
answers
1k
views
What is the characteristic functional for Brownian motion on a sphere?
I'm a physicist, somewhat familiar with stochastic processes, but I'm a little unsure of what follows. What I basically have is a complicated quantity involving a vector that is equivalent to ...
- 111
7
votes
0
answers
627
views
Elementary proof of lack of phase transition in Ising models with external fields
I have a question about the phase transitions in the Ising model in the presence of a (constant) external magnetic field. I will state the question on $\mathbb Z^2$ for simplicity. A definition of the ...
- 23.2k
13
votes
7
answers
1k
views
Probabilistic (and other mathematical) methods of physics without the physics?
Many of the methods of physics are vastly more general than their use in that discipline. For example, information theory overlaps with a lot of statistical mechanics, and the latter actually ...
3
votes
0
answers
229
views
For Ising models on finite graphs, is the gradient of Z (w/r/t coupling and field) easier to compute than Z?
Suppose we have a graph $G$ with $n$ vertices, edgeset $E$, $\mathcal{X}=\{1,-1\}^n$. The partition function of the spin-1/2 Ising model on $G$ is
$$Z(J,h)=\sum_{x\in \mathcal{X}} \exp\left(J \sum_{(...
- 2,252
5
votes
1
answer
623
views
For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ?
Background:
Let $\mathbb Z^d$ denote the $d$-dimensional integer lattice with norm $|x|=\sum_i|x_i|$.
For each $x\in\mathbb Z^d$ we associate a spin variable, $\sigma_x$ taking values on the set
$\{...
- 2,044