Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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. Reference: Wikipedia.

The classical view of the universe was that its fundamental laws are mechanical in nature, and that all physical systems are therefore governed by mechanical laws at a microscopic level. These laws are precise equations of motion that map any given initial state to a corresponding future state at a later time.

356 questions
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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 parameter $\beta >0$. There exist $x$ and $y$ in $G$…
3m0o
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Finding Moment of Inertia of ellipse.

Show that the moment if inertia of an elliptic area of mass M and semi-axis a and b about a semi-diameter of length r is $$\frac{Ma^2b^2}{4r^2}$$. My attempt. I know that MI about ox is ${Mb^2 \over 4}$. MI about oy axis is ${Ma^2 \over…
namer khan
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Surface measure

I dont understand the following surface measure, My space is $\Omega_E=\{w\in \Omega | H(w)=E\}$ My lecture notes state that my surface measure is given as $\sigma_E(\Omega_E)=\partial_E(E^N/(N!))$. Can anyone enlighten me please? Edit: The…
bios
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Root mean square speed of the Boltzmann distribution in the kinetic gas model

Why can I simply say that multiplying the Boltzmann distribution function with $v^2$ and integrating from 0 to $\infty$ leads to the mean square speed? $\langle v^2\rangle = \int\limits_{0}^{\infty} v^2 \cdot 4\pi \left( \frac{M}{2\pi…
heisenberg_px
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What is the correct definition of correlation length?

What is the definition of correlation length for discrete stochastic process $\{ X_i \}$? We define variance $\text{var}(X) := E[(X - E[X])^2]$, standard deviation $\text{std}(X) := \sqrt{\text{var}(X)}$, covariance $\text{cov}(X, Y) := E[(X -…
Paalon
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Cumulant equations of motion

I am studying some Langevin type dynamics and have an equation of the following form: $$m\dot v= -\gamma v \ +\xi(t) + F_0\cos(\omega t) \\ \langle\xi_0\xi_t\rangle =2D\ \delta(t)$$ And I am asked to find the evolution for the cumulants. I am…
yankeefan11
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Arrange N number of balls into finite area

This may be more of a mathematics question but it comes about in a physics context. I have a box of area $64cm \times 32cm$ and it is partitioned into two parts using a divider which can slide along one axis if hit by a ball.There are $N_{L}$ balls…
Sam
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Expansions of Bose Functions

To study the thermodynamic behavior of the limit $z\rightarrow$ 1 it is useful to get the expansions of $g_{0}\left( z\right),g_{1}\left( z\right),g_{2}\left( z\right)$ $\alpha =-\ln z$ which is small positive number. From, BE…
esilik
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How do you obtain the fluctuation spectrum of a tubular membrane?

I am reading through a paper. A tubular membrane, submitted to tension $\sigma$ acting as a Lagrange multiplier to conserve area, fluctuates around a cylindrical shape of length L and radius R. Parametrisation is as…
ben_afobe
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Central Limit Theorem for subsamples

I observe a set/realisation of $n$ i.i.d. $\{X_1, X_2, ..., X_n\}$. Because of the Central Limit Theorem, I know that repeating such an observation enough times, the pdf of the mean of such $n$ samples, $\bar{X}_n$, has variance $\sigma^2/n$, with…
EllaSingo
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Expanding an expression for small values of a parameter

I have a probability generating function $$ G(z) = \Bigg(\frac{ 1-2d + \sqrt{1-4d(1-d)z}}{2(1-2d)}\Bigg)^{\frac{1-2d}{d^2}\kappa}\ \Bigg(\frac{1-\sqrt{1-4d(1-d)z}}{2dz}\Bigg)^{\frac{\kappa}{d^2}}$$ and I'd like to expand this for $d\rightarrow 0$.…
kevinkayaks
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can ${e^{ikx}}$ and the heaviside step funtion have similar physics content about the distribution of x

in an online video lecture,(around 38min, where the exactly statement is at 38min28secs.) i got one question, suppose we have a system of $N$ particles, $\left\{ {{{\vec r}_i}(t)} \right\}i = 1, \cdot \cdot \cdot ,N$ are the position vectors of…
FaDA
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Gauss Bell Curve - Non negative numbers - Will curve shape from trigonometry be the same as from empirical data?

Im Johan, new to physics stack exchange, second post. How are you doing:) Would you help me with this Gauss Bell Curve question please? Im just looking for a general way to skew the normal distribution curve when its only positive values. Posting it…
Progrmming is fun
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Identifiability of statistical model

I learn the definition of "identifiability of statistical model" as follows; Let W be a parameter space. If for $w \in W$, a map which maps $w$ to $p(|w)$ is one-to-one, it is called identifiable. My question is the reason why this condition is…
ddd
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How does sphere packing fraction in a long cylinder change with sphere size?

Earlier I had to cut up some materials into little pieces and fit them in a glass tube, and I wondered if it's better to cut the pieces as small as possible, or if it wouldn't matter. If we think about sphere packing in infinite 3D space, then the…
user1704042
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