# Questions tagged [statistical-mechanics]

Statistical mechanics is a branch of mathematical physics that studies, using probability theory, the average behaviour of a mechanical system where the state of the system is uncertain.

357 questions
Filter by
Sorted by
Tagged with
27 views

### Can a non-analytic function provide analytic solution (i.e., exact solution)? [closed]

What is an analytic function? What is an analytic solution? These two terms have the same meaning? If no, can a non-analytic function provide an analytic solution?
58 views

• 21
53 views

### Can addition of noise to dynamical system reduce estimation errors

I am using Kalman filter to estimate the states of a stochastic dynamical system which has very very small noise( consider zero ). The filter is not aware that the noise is zero. Implementation of KF ...
• 913
28 views

### Minimum residual error in estimation of deterministic system using Kalman filter

Let the process equation for a state vector $\mathbf{x}_t$ at time $t$ be: $$\bf{x}_{t+1} = \bf{f}\left(\bf{x}_t\right)$$ where, $\mathbf{f}\left(.\right)$ is a nonlinear ...
• 913
24 views

1 vote
23 views

### Concentration of Gibbs measures with converging energy functions

Let $H$ be a continuous energy function defined on a compact subset $A\subset \mathbf{R}^n$ and let $Q$ be a fixed probability measure on $A$. For each $\theta>0$, define the probability ...
• 13.2k
67 views

### Can anyone help to solve this task ?In a multiple-choice test with m options, a student knows the correct answer with probability p,...?

"In a multiple-choice test with m options, a student knows the correct answer with a probability p, and in the absence of knowledge, chooses randomly one of the available options. What is the ...
39 views

• 450
48 views

### Ising model, parity of the loops and sign of a spin.

I have a hard time understanding how to reason with these questions Let $G \subset \mathbb{Z}^2$ be a bounded connected domain with $-$ boundary conditions. Consider the Ising model on $G$ with ...
• 783
1 vote
47 views

### Covariance statistics of p-spin models

I am trying to calculate the co-variance of two non-independent variables. $\sigma$ is a string of length $n$ with bits $\sigma_i$ taking values 1 or -1. One has a p-spin model which is defined by the ...
71 views

### Proving that certain integral is positive

Given a compact set $K\subset \mathbb{R}^3$, we consider $f:K^3\subset\mathbb{R^9}\to \mathbb{R}_0^+$ such that $f(x_1,x_2,x_3)=f(x_{\tau(1)},x_{\tau(2)},x_{\tau(3)})$ for every permutation $\tau$ ...
• 423
60 views

### How many ways there are to cover an $n \times n$ tiling with $2 \times 1$ dominoes?

I came across the famous dimer problem in statistical physics and I'm struggling to come up with a mathematical formula for covering an $n \times n$ tiling with $2\times1$ dominoes? How does a ...
105 views

### Why does the condition $\sum_{x\in\Lambda}f_x=0$ imply the discrete Laplacian is invertible?

I'm studying statistical physics these days and have newly learnt the concept discrete Laplacian. For a finite graph $G$ with vertices $\Lambda\subset\mathbb{Z}^d,$ consider the discrete Laplacian ...
• 187
109 views

### Probability that a random walk in $2d$ has small local time at each vertex

Let $P_{n,k}$ be the probability that a simple random walk of length $n$ in $\mathbb{Z}^2$ is such that each vertex of $\mathbb{Z}^2$ is visited at most $k$ times by the walk. Certainly this ...
35 views

### Can some one explain me Planck/Reileigh Jean Law?

I wonder why every proof of these laws consider the number of of oscillator in the end but disregard it while deriving the mean energy. Let me explain. Considering an oscillator in a heat bath, there ...
30 views

### Can sets of functions form a measure space?

For example, continuous functions from $\mathbb{R}$ to $\mathbb{R}$ with nth derivative = 0. Each function acts like a point in an $\mathbb{R}^n$ dimensional space with its Taylor expansion ...
Consider the following agent-based model: There are $N$ agents Every agent starts with $1 At each time interval (i.e. at each step), every agent gives \$1 to a randomly chosen agent. I want to find ...