# 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.

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### 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 ...
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### 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 $T^{\mathbb N}x$ ...
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### Ergodicity on a finite set

Let $\Omega$ be a finite set ($\#\Omega = n$), how many dynamical systems on $\Omega$ are ergodic?
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $d$. ...
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### Gallavotti-Cohen action functional

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 ...
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