# 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.

308 questions
Filter by
Sorted by
Tagged with
12 views

### Mayer expansion: from product over i,j to sum over graph

I am studying the Mayer expansion used in Statistical Physics. We arrive at the following expression: $$\prod_{i<j=1}^N (1+f_{ij})$$ and then is find out that this expression is equivalent to the ...
1 vote
30 views

### Set $T^{\mathbb N}x$ dense in $\mathbb S^1$ (Poincaré recurrence theorem)

Let $Ω =\mathbb S^1$ be the unit circle in $\mathbb R^2 = \mathbb C$, and let $T : Ω → Ω$ be multiplication by $e^{i\alpha}$. For $α \notin π\mathbb Q$ and every $x ∈ Ω$, is the set $T^{\mathbb N}x$ ...
46 views

### Ergodicity on a finite set

Let $\Omega$ be a finite set ($\#\Omega = n$), how many dynamical systems on $\Omega$ are ergodic?
20 views

### Statistical mechanics partition function from probability distribution

I am curious about the mathematical background of something I came across while working on a problem in statistical mechanics. As an example, I am going to use the classical canonical ensemble, though ...
1 vote
32 views

### Non-linear backward Kolmogorov equation

The backward Kolmogorov equation (BKE): $$\frac{du}{dt} = A(x,t) \cdot \nabla_x u(x,t) -\frac12 \text{Tr}(BB^t(x,t) \text{Hess}_x u(x,t) - f(t,x,u, B,\nabla g), \;\;\; t<T$$ If $f\equiv 0$ then ...
33 views

### How to interpret integrals that have conditions written beside them

sorry if this question has been asked before. I tried finding similar questions but couldn't find any. I have very little background in statistical mechanics, but I have been reading some literature, ...
38 views

### Angle between two random unit vectors uniformly distributed

Consider $x, y \in S_{1}^{d-1}$ (the unit n-sphere in d dimensions) with $(x \cdot y)^2 = 1/d$. I need to compute the angle $\alpha$ between $x$ and $y$ for $d$ = $3$ and asymptotically for large $d$. ...
35 views

26 views

The Furutsu-Novikov formula gives the expectation value of a zero mean Gaussian process $z(t)$ and a functional of that process $R[z]$: $$\langle{z(t') R[z]}\rangle = \int^{t}_0 \mathrm{d}s K_2(t',s) \... 0 votes 0 answers 29 views ### Which PDEs have the Gibbs distribution as its stationary distribution? The Fokker-Planck equation, which describes the evolution of the pdf of the position of a diffusing particle, has the Gibbs (or Boltzmann) distribution as its stationary distribution. Are there any ... 2 votes 1 answer 97 views ### Why \left(\frac{\partial x}{\partial y}\right)_z=-\left(\frac{\partial x}{\partial z}\right)_y \left(\frac{\partial z}{\partial y}\right)_x? I was reading a statistical mechanics book, the author use: \left(\frac{\partial x}{\partial y}\right)_z=-\left(\frac{\partial x}{\partial z}\right)_y \left(\frac{\partial z}{\partial y}\right)_x so ... 2 votes 1 answer 65 views ### Are all continuum systems limits of microscopic descriptions? Many PDEs modeling nature are derived by reduced-order models of many-body descriptions, most famously the Navier-Stokes equations from the Chapman-Enskog expansion of the Boltzmann equation as well ... 0 votes 1 answer 30 views ### Rigorous proof of shifting of center during integration In statistical mechanics, we see a lot of high-dimensional integrals. When using only a pairwise potential, we tend to "integrate away" one of the dimension by shifting it to the origin. For ... 3 votes 2 answers 72 views ### What does it expression tells us when we write Shannon entropy in terms of continues distribution? Shannon information is defines as, \begin{equation} H(p_1, p_2,...p_n) = - \sum_{i=1}^{N} p_i log p_i.\end{equation} For continues distribution, we can write shanon information as, \begin{equation}H(x)... 1 vote 1 answer 88 views ### Integral kernel of the Legendre transform First of all, I'm not sure, but I think the Legendre transform can be seen as a linear operator between the functions on a normed space and the functions on its dual. (A functional analysis approach ... 2 votes 0 answers 26 views ### How to calculate large deviations for functions of vectors on the n-sphere with delta functions? I am copying the setup from Sec 2.5 of these notes describing large deviations of an overlap matrix: Let \sigma_1,\cdots,\sigma_k\in\mathbb{R}^n, with k fixed and n\rightarrow\infty, and \... 2 votes 0 answers 40 views ### Componentwise stationary distribution implies full stationary distribution? Consider a d-dimensional stochastic process X_t having density \rho(X_t,t) at time t. Suppose the drift term b of the process depends on the density \rho = \rho(X_t,t):$$ dX_t = b(X_t,\...
1 vote
For a time continuous Markov jump process a path $\left\{\sigma_{s}, 0 \leqslant s \leqslant t\right\}$ is time reversed as $\left\{\sigma_{t-s}, 0 \leqslant s \leqslant t\right\}$. The time reversed ...