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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2
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1answer
28 views

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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3answers
69 views

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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0answers
31 views

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 $\{\Omega_{ij}^{\mu \nu}, C_{ij} \in \mathcal{Z}\}$ ...
2
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1answer
26 views

Curie-Weiss probability and expected value distribution?

I was recently working on the following Tripos problem and hit multiple road bumps: The random variables $S_1,...,S_n$ take on values in $\{\pm1\}$, and follow the probability distribution $$\mathbb{P}...
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1answer
37 views

Relation between kurtosis and squared second moment

"The fourth moment is equal to three times the squared second moment." $$\langle X^4 \rangle = 3\langle X^2 \rangle^2$$ Is this generally true for any distribution? I came across this ...
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20 views

Ratio of collision of rate between distributions?

Background Consider the familiar setting of statistical mechanics, an assembly of $N=4$ distinguishable gas particles. Let, the total energy be $4E_0$. The possible ways energy of sharing energy ...
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1answer
34 views

Large N limit of a particular sum

I'm working through a statistical physics book and one of the problems makes the claim that the quantity: $$H =\frac{1}{N}\sum_{n=1}^N\frac{1}{\frac{a}{N} + \frac{b}{N^{5/3}}(n^{5/3}-(n-1)^{5/3})}$$ ...
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0answers
24 views

derivative of a sum - derivation of Boltzman equation

Let $x_i$ is a position vector (for simplicity in 1D) of an $i$-th particle. $V(x_i,x_j)=\phi(|x_i - x_j|)$ is some function that depdends only on the distance between the two particles. I would like ...
3
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1answer
36 views

Markov process in interacting particle systems book

I have trouble understanding the setup of a Markov process in Liggett's interacting particle systems book. Let $X$ be a compact metric space with measurable structure given by the $\sigma-$algebra of ...
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2answers
55 views

Explanation for behaviour of graph of $y=x^2e^{-x^2}$ (Maxwell-Boltzmann distribution)

Consider the function $$y=x^2e^{-x^2}$$ The graph initially behaves as a parabola then in later part exponential part of it dominates; i.e., the graph looks exponential after maximum of the curve. ...
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14 views

Confusion on Shannon's joint entropy

Given the definition of Shannon's joint entropy $H(X,Y) = -\sum_{x,y}p(x,y)log(p(x,y))$, I am not sure of how to compute it for a distribution of random variables such as $X = p$ and $Y = 1 - p$. It ...
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2answers
115 views

Average velocity of overdamped particles in external field

In short: how to obtain the average velocity from the Fokker-Planck equation in the overdamped regime? (i.e. when the probability density is $P(\mathbf{x},t)$ and not $P(\mathbf{x},\mathbf{v},t)$, ...
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25 views

The Definition of Entropy. Boltzmann/Shannon Entropy. The Heat Equation. Stochastic Processes.

Lets talk about Entropy, specifically the statistical mechanics interpretation of Entropy. Loosely speaking Entropy is described as a quantifier of how chaotic a system is. The more micro-states ...
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18 views

saddle point method with conjugate complex roots

Here I want to use the saddle point to calculate something. My equation is $$f(x,t)=\int_0^\infty\exp \left(\underbrace{a N^{\frac{1}{1-\alpha }}+b \ln (N)-\frac{x^2+A^2N^2-2ANx}{2 N \sigma ^2}-\frac{...
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48 views

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 ...
27
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1answer
652 views

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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1answer
48 views

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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10 views

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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1answer
24 views

What is the reasoning behind the definition of Shannon Entropy?

Shannon entropy is defined as the average content of information. Where information content is defined as $Q=-k\log(P_i)$, $k>0$, $$ S = \langle Q \rangle = \sum_{i=1} Q_i P_i = -k\sum_{i=1} P_i\...
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13 views

Integral with function in the exponent / Gibbs Distribution via MaxEnt

In the context of a maximum entropy procedure, I am trying to evaluate the following definite integrals—but they have an arbitrary function $g(x)$ in the exponent : $\int\limits_0^\infty e^{-\lambda ...
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0answers
12 views

Is interchanging the orders of averaging operation with integral operation allowed?

In the book of Zwanzig, Nonequilibrium statistical physics, at page 6, after explaining Langevin equation Brownian motion, to show that $<v^2> = 3/2 k_B T/m$ consistent with the Langevin ...
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0answers
44 views

Reference request for quenched disorder in stat mech

I asked a similar question for references on rigorous results in statistical mechanics here. One thing that I noticed is that the references people mentioned didn't quite cover the effect of quenched ...
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0answers
25 views

What online resources or eBook explains tensor analysis clearly

I would like recommendations for online resources or ebooks that explain tensor analysis in non-technical terms. Most of the resources I have seen are really technical (and unintuitive in some cases).
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0answers
14 views

Near spin expectation value in the 2D classical ising model

I am looking at McCoy's book about on Ising model because I am looking for the expectation value of two adjacent spins at the critical temperature in the infinite volume limit of the anisotropic Ising ...
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1answer
59 views

Problem with an integral in statistical mechanics

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 constant. Now the integral i want to calculate ...
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0answers
15 views

How can I prove this formula for Hopfield model?

Studying fluctuations theory in Hopfield model I found this formula for the derivative of an observable: $\frac{d\left<O \right>}{ds} = \beta\sqrt{\alpha} \sum_{ab} \left < O \xi_{a, b} \...
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2answers
90 views

Reference request for Gibbs measure

Understanding the Gibbs measure seems to be essential to fulling understanding statistical mechanics. However, I don't know of any mathematically rigorous textbooks/lecture notes that talk about such ...
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2answers
47 views

Permanence time in a markov chain

For a continuous-time Markov process with, say, two states $1,2$ and transition rates $r_{ik}$, over a time interval of duration $T$, what is the probability $P(t)$ of spending, in total, a duration $...
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0answers
76 views

Fermi-Dirac integral

I'm trying to do some integral calculation for Fermi-Dirac distribution, specifically for: $\int\limits_0^\infty {E^2\over 1+\exp{E-\mu\over k_BT}}dE$ I know that it can be only solved numerically, ...
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0answers
14 views

Solution of a modified Poisson-Boltzmann equation

I'm trying to solve a modified Poisson-Boltzmann equation given by $\frac{d^{2}\phi(z)}{dz^{2}}=2k_{1}\sinh(\phi(z))-k_{2}$, where $k_{1}$ and $k_{2}$ are constants, and I'm not sure of how to solve ...
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0answers
11 views

Selecting the best unknown probability distribution given constrains

I'm not sure how to pose the question, so I'll just try to explain. It's inspired by reading lecture notes on statistical mechanics (section 2 on page 3). Let's say I have an unknown probability ...
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1answer
52 views

Reference for Onsager’s solution of the 2D Ising model

I am interested in Onsager‘s famous paper “Crystal statistics I” where he derives the solution of the 2D Ising model. I am reading the original paper, but I search some supplementary material (blog, ...
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0answers
54 views

Differential equation $(y')^3+y^3-yy'=0?$

How to solve differential equation $$(y')^3+y^3-yy'=0?$$ I tried parametrization $y'=p$, Laplace transformation... After a while, I have solved it by help of WolframAlpha... And idea is to divide ...
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1answer
34 views

Dirac delta integral for evolving networks

I'm reading Dynamical Processes on Complex Networks (link), which makes frequent use of dirac delta integrals to examine evolving networks. I'm trying to get a good sense of how to evalute them and ...
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0answers
65 views

How many crossings are there in an X-ray of an idealized ball of yarn?

This is a second question about statistical geometry. The image https://commons.wikimedia.org/wiki/File:Ball_of_yarn_10.jpg shows a typical ball of yarn. The idealized yarn of the question is assumed ...
4
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1answer
81 views

What is the average length of an idealised ball of yarn?

This is a question about statistical geometry. The image https://commons.wikimedia.org/wiki/File:Ball_of_yarn_10.jpg shows a typical ball of yarn. Such a spherical ball of radius $𝑅$ has a volume $...
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0answers
45 views

Theorem about Boolean function inequalities. Proving a BKKKL's result.

I'm currently studying this article, and I got stuck in one of its initial parts. Here is the problem: Suppose you have a probability space $(\Omega, \mathbb{F}, \mathbb{P}_p)$, in which $\Omega = \{...
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0answers
11 views

Re-writing Action on a different slice of space-time, Quantum Hall Effect

I'm looking at QHE notes D.Tong and wondering how he gets from equation 5.46 to 5.48 ( http://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf ) $S_{CS}=\frac{k}{4\pi}\int d^3 x \epsilon^{\mu \nu \rho} tr(...
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0answers
19 views

QHE: effective action as a 'local functional'

' Finally, if we care only about long distances, the effective action should be a local functional, meaning that we can write is as $S_{eff}[A]=\int d^d x... ' Where does this come from and what does ...
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1answer
18 views

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} ...
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0answers
16 views

How do you prove that nondimensionless and dimension expression of microstate equation are equal?

Hello, how do you prove that the two equations in (5) are equal in the picture. I said they are not equal because on the left side, Z cancels out but on the right side Zs do not cancel out and the ...
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0answers
24 views

Lagrange multiplier, different results depending on when to take the derivatives.

In statistical physics, one finds the probability distribution $\rho[q]$ that maximizes the entropy $S$ $$ S=-k_B \sum_{q\in\mathbb{Q}} \rho[q]\ln \rho[q]\tag{1} $$ under the constraints: $$ \begin{...
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0answers
10 views

Lagrange multiplier methods (and taking the derivative of the constraints to be zero)?

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{...
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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 ...
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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 ...
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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 ...
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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}...
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0answers
20 views

Expanding an expression for small values of a parameter

I have a probability generating function $$ G(z) = \Bigg(\frac{ 1-2d + \sqrt{1-4d(1-d)z}}{2(1-2d)}\Bigg)^{\frac{1-2d}{d^2}\kappa}\ \Bigg(\frac{1-\sqrt{1-4d(1-d)z}}{2dz}\Bigg)^{\frac{\kappa}{d^2}}$$ ...
2
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0answers
60 views

Intuition behind defining the information of a random variable as being additive for independent events

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 ...
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0answers
21 views

Integral with Delta function statistical mechanics

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:...

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