# 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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### Is there a meaning to $\mathrm{e}^{H(p_{i})}$ or $2^{H(p_{i})}$?

In my research I find an equation featuring the "exponential entropy" term $\mathrm{e}^{H(p_{i})}$ and I wonder if it has a specific meaning. I have only found rare references to that term (...
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### Are the remaining samples still independent, identically distributed (IID) after removing the maximum value of the IID samples?

$X_1, X_2, \cdots, X_N$ are independent identically distributed (IID) random variables and $Y_1$, $Y_2, \cdots, Y_{N-1}$ are the remaining after removing the maximum value of $X_k, k=1, \cdots, N .$ ...
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### Estimation of the standard deviation of a zero mean distribution with partial information about it

I'm trying to estimate the standard deviation of a zero-mean distribution of values. Let me start with this brief introduction to the problem: Let $\{\Omega_{ij}^{\mu \nu}, C_{ij} \in \mathcal{Z}\}$ ...
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### Expected value of product of dependent random variables with tightly concentrated $O(\sqrt{N})$ fluctuations.

Let $N_a$ and $N_b$ be two statistically dependent random variables. A textbook I am reading mentions that, if $N_a$ and $N_b$ are tightly concentrated with $O\left(\sqrt{N}\right)$ fluctuations, then ...
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### Number of ways to stack LEGO bricks

One of the most surprising combinatorial formulas I know of counts the number of LEGO towers built from $n$ "$1 \times 2$" blocks subject to four rules: The bricks lie in a single plane. Each brick ...
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### Soft Question - Book Recommendations (Diff Geo, Bose-Einstein stats etc.)

I apologise immediately for the soft question but I will still ask it. I feel there may be a lot of people in the same boat so it may be relevant to a large number of others. With context of this ...
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### Obtaining Logistic Functions from Gibbs Distribution

I am trying to understand how to derive both the binary and multinomial logistic functions from the general Gibbs(Boltzmann) distribution—but I'm a little lost with certain details. Gibbs ...
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In statistical physics, one finds the probability distribution $\rho[q]$ that maximizes the entropy $S$ under the constraints: \begin{align} \overline{E}&=\sum_{q\in\mathbb{Q}}E[q]\rho[q] \tag{... 2answers 92 views ### How to determine the critical exponent of the function f(x)=Ax^{1/2}+Bx^{1/4}+Cx? In the book Statistical Mechanics of Phase Transitions written by J.M. Yeomans there is a set of exercises*, where the objective is to find the critical exponent of specific functions. The critical ... 0answers 44 views ### Properties of the log-normal distribution/ Galton Distribution Context In Statistical Physics, in the heterogeneous problems, we have pebbles, powders in a continuous field. Recently, we started studying the continuous descriptions of the physics phenomenon and ... 1answer 43 views ### 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 ... 1answer 61 views ### Is {e^{ikx}} \to {x^2} when k \to 0 I have a question regarding some information from an online video lecture (around 36min, where the exactly statement is at 36min33secs). Suppose we have a system of N particles, \left\{ {{{\vec r}... 0answers 20 views ### 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}} ...
I'm recently studying the concepts of entropy and I've a fundamental question regarding the conceptual formulation of information content of a random variable $X$, or equivalently, the uncertaintly of ...
I have problems with this integral: There are N identical particles contained in a circle with radius R with Hamiltonian $H=\frac{p^2}{2m} -Aq^2$, A is costant. Now the integral i want to calculate is:...