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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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Problem while calculating entropy using Gibb's relation

the question homework question from Concepts in thermal physics 2nd edition my attempt how I approached the problem I am stuck at the marked step, specifically what do I do of Pi multiplied by ln(Z). ...
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About derivative of delta function - chain rule for delta function containing a function

I have a problem that relates to derivative of a delta function. The problem originates from a paper I was reading https://aip.scitation.org/doi/full/10.1063/1.2938860 In the paper, it is said that ...
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Non existence of limit of Gibbs distribution

We consider an N-particle system given by the gradient dynamics $dX(t)= -N\nabla H_N (X(t)) dt + \sigma d\beta(t)$ in $(\mathbb{R}^d)^N$, where $\sigma$ is a positive constant. We assume that for $x=...
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Wsin ($\theta$) =W'sin ($\theta'$)

Two heavy particles of weight w and w' are connected by a light inextensible string and placed over a fixed smooth circular cylinder of radius a, the axis of which is horizontal. If $\theta$ and $\...
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43 views

Sum of series $\sum_{n=0}^{+\infty}\frac{1}{e^{\beta n}+1}$ with $\beta >0$

I'm wondering if there exists a closed form or at least some asymptotic expansion in the limit $\beta \rightarrow \infty$ for the sum of the series $$\sum_{n=0}^{+\infty}\frac{1}{e^{\beta n}+1}$$ ...
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Fourier transform of a random walk

I was just reading Statistical Field Theory of Itzykson and Drouffe and saw that they wrote the inverse Fourier transform $$P(\vec{x},t,\vec{x}_0,t_0)=\int_{[-\pi,\pi]^d}\frac{\text{d}^d\vec{k}}{(2\pi)...
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1answer
35 views

are integration limit to find expected value from a pdf inclusive (or exclusive)?

This is my probability density function (pdf) $$pdf = e^{-\frac{r}{\lambda}} \frac{1}{\lambda}$$ I want to find the expected value (EV) from $0<r \leq r_0$, $r$ is the random variable. My first ...
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Phase transitions in voting systems

Please, anyone could direct me a first-approach text in phase transitions of voting systems? I do not have any preliminary knowledge in statistical physics, Ising model and etc., I am only interested ...
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Integration of Poisson brackets by integration by parts

I have to show that the following integral is zero: $$\int \sum_{i=1}^{3N}(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}...
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Examples of functions summables in $\mathbb{Z}^d$.

I know that $f_0(x)={ 1 \over \vert \vert x\vert \vert^\alpha}$, $f_0(0)=1$ , $\alpha>d$, $d\in \mathbb{N}$ is summable in $\mathbb{Z}^d$, i.e. $$ \sum_{x \in \mathbb{Z}^d} f(x)<\infty. $$ I ...
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41 views

Taking moments of a fluid equation

I understand that this question is maybe better placed in the physics stack exchange but thought maybe I would find some help here as well. Given the following term that is taken from the Vlasov ...
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Convolution of molifiers with measure

I would like your opinion on a computation i found in a statistical mechanics paper : Let $\nu$ a prob measure on $\mathbb{R}^{d}$, $V:\mathbb{R}^{d} \rightarrow \mathbb{R}$ continuous, belongs to $L^...
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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 ...
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How to take ensemble average of a given function?

I am going through the calculation by Rawson et al. [J. Opt. Soc. AM. Vol. 70, No. 8, August 1980] and ran into seemingly simple issue with the derivation. I wanted to get some help on solving the ...
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How can I derive the waiting time distribution of gaussian processes?

Given two successive Gaussian events, $X_1 \sim N(\mu,\sigma^2),\qquad X_2\sim N(\mu,\sigma^2)\qquad$ with $X_2 = X_1+\epsilon$,$\quad\epsilon\in {\rm I\!R},\quad\epsilon>0$ how can I derive the ...
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1answer
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Percolation Textbook Recommendation

I was wondering if someone could recommend a Percolation textbook for undergraduates. I have looked at Percolation by Grimmett and it seems quite dense. I was looking for a book that I could self-...
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1answer
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Stochastic differential equations with null mean and unit variance

I have the following: $ \dot{x} = \frac{dx}{dt}= A\left( x\right) + \sqrt{B\left( x\right)}\eta\left( t\right) $ where $ A\left( x\right)=a_0 - a_1x $ and $ B\left( x\right)=b_0-b_1x+b_2x^2 $. All $ ...
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1answer
36 views

Variance of the Wiener Increment

I have a Wiener process $W(t)$, which is a normally distributed random variable with mean $\langle W(t)\rangle = \mu = 0$ and variance $\langle W(t)^2\rangle = \sigma^2 = t$. The angled brackets $\...
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9 views

Superconductor specific heat capacity

I would like to obtain expression for heat capacity jump of superconductor. During calculation, I can not deal with the followiwng integral: $$\int_{0}^{\infty}\frac{dx}{x^2}\left(\frac{1}{\cosh^2 x}-\...
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Contour integral with branch points inside argument of logarithm

This question comes from the context of calculating the grand potential for a simple toy problem (a linear chain of masses connected by springs with a mass defect) using statistical field theory. In ...
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1answer
54 views

How to solve or workaround the integration?

From statistical mechanics I've found $ \left( c,\alpha \in \mathbb{R}^+ \right) $: $ lnZ = c\int_0^\infty dx\ x^2ln\left( 1 + e^{-\alpha x^2}\right) $ what I performed a integration by parts to get:...
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Gaussian concentration inequality for the maximum of gaussians

The problem goes like this: Let $N\leq e^{c/ \epsilon}$ and $0\leq j\leq \log_2 N^2$. Suppose for some $C>0$, we have $$\mathbb{E}(\max_{A\in\mathit{A_j}}h_A)<C j \epsilon 2^j,$$ where $A\subset ...
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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}(...
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$2\sum_{i=1}^{n}(t_{i,n} - t_{i-1,n})^2 \ \leq\ 2\max(t_{i,n} - t_{i-1,n}) \sum_{i=1}^{n}(t_{i,n} - t_{i-1,n})$?

I am reading the book by McCabe and Tremayne "Elements of modern asymptotic theory with statistical applications" and in Chapter 8 about Brownian motion I ran into this inequality: $$2\sum_{i=1}^{n}(...
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1answer
48 views

Help with Difficult Integral - $\dfrac{\partial}{\partial z} f_v (z) = z^{-1} f_{v-1} (z)$

I need a hand with the following integral, I'm not sure how to go about it. I'm trying to verify the following relation, found in Pathria and Beale's Statistical Mechanics, appendix E. $$\dfrac{\...
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33 views

A relation I have never seen before

I was going through the book Information, Physics and Computation by Mezard and Montanari, where in Chapter 8, I found the following relation: \begin{equation} \mathbb{E} \log Z = \lim_{n\rightarrow ...
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1answer
78 views

Relation between waiting time distribution and probability that an event occurs within time $dt$

Waiting time distribution is defined as the distribution of the time interval between two successive events. I'm looking at stochastic processes in discrete space and continuous time with non-...
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1answer
28 views

Understanding how to compute the space volume for an ideal gas

There is a system of N non-interacting particles (Ideal Gas). The Hamiltonian of a system of free particles is given by: $$H = \sum_{i=1}^{N}\frac{p_{i}^2}{2m} + \sum_{i=1}^{N} \psi(q_i)$$ where to ...
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1answer
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Is square of probability $p^2$ is less than probability $p$?

suppose we have $N$ possible states in system. There is a probability $p_n$ that system is in state $|n\rangle$, and the sum of all probabilities is one. Is there any general rule in math or physics ...
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1 dimensional Ising Ring

I have the following Hamiltonian describing a 1 d Isining model with periodic boundary conditions. Hence $\sigma_{N+1}=\sigma_{1}$ with $\sigma=\pm1$ describing a state up or down. $$ H=-J\sum_{i=1}^{...
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1answer
102 views

Solving a partition function in polar coordinates

The Hamiltonian for a particle is given by: $$H_1 = \frac{P^2}{2m} + \frac{p^2_\theta}{2I} + \frac{p^2_\phi}{2Isin^2 \theta}$$ $I$ is the moment of inertia $$I = \frac{mR^2}{4}$$ To get the ...
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1answer
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Conjugate Momenta From Generalized Tensor

I'm stuck on some of the math behind this problem and could use some help working this out. I'm trying to calculate the Lagrangian, find the conjugate momenta, and finally calculate the Hamiltonian ...
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179 views

Solving the diffusion equation with an absorbing boundary

There is a one-dimensional diffusion process in which particles start running at $t = 0$ and from $x_o > 0$. When particles reach x = 0 they are removed from the system, thus the total ...
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1answer
86 views

Solving the one particle partition function

We have $N$ oscillators and each of them is described by the Hamiltonian: $$H = \frac{p^2}{2m} + \frac{Kq^4}{4} $$ I have to compute the average total energy $\langle E\rangle$ of the $N$ ...
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1answer
82 views

Examples of graphs that are amenable and non-amenable

The amenable graph $G=(V, E)$ is a graph that satisfies the following $$ \inf\limits_{K \subset V,\, |K|< \infty} \frac{\partial K}{|K|}=0$$ I know for example that $\mathbb{Z}^2$ is amenable ...
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2answers
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How to tell which groups have different means than others in a One-way Anova

The presence of harmful insects in farm fields is detected by erecting boards covered with a sticky material and then examining the insects trapped on the board. To investigate which colors are most ...
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Evaluating Matrix Elements of the Density Operator

I'm taking my first course in graduate statistical mechanics and I'm struggling a bit with the math. I think I understand how to use dirac notation, at least the basic stuff for now, but I want to ...
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2answers
82 views

Calculating Probability Distributions

I've posted a few times about this specific distribution question, but I am confused about what exactly my professor is asking for here. The homework question is: A physical measurement of $x$ ...
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ensemble average of two dimensional deterministic map

I am calculating mean square displacement of the two dimensional map, but i'm confused in ensemble average. Is following true? $$\langle\Delta x(t)^2 + \Delta y(t)^2 \rangle ~=~ \langle \Delta x(t)^2\...
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4answers
91 views

Limit As n goes to infinity of $ \sum_{n=1}^\infty e^{- \alpha n^2 }$.

I suspect the following is exactly true ( for positive $\alpha$ ) \begin{equation} \sum_{n=1}^\infty e^{- \alpha n^2 }= \frac{1}{2} \sqrt { \frac{ \pi}{ \alpha} } \end{equation} If the above is ...
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1answer
103 views

Solving non-linear ODEs

I am trying to solve a differential equation of the form: \begin{equation}x^2y''+2xy'+x^2e^{ay}=0\end{equation} This arises from calculating the electric potential of ions following the Boltzmann ...
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What is rank invariance?

I work in the area of soft matter and recently I have been reading a paper that talks about developing a new optimization algorithm for statistical physics problems. Turning statistical physics ...
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1answer
172 views

Derivative of von Neumann entropy

Similar/related questions to the one I have here that I have looked at are this and this. Problem description I have an N-body density matrix (for all intents and purposes, this is just a square ...
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Fokker Planck forward equation

I am studing Fokker Planck equation, but I am struggling with its derivation. In particular there is a passage with which I have some problems: the last line on the left page says that with some ...
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1answer
40 views

Is the notation $\text{ch}$ standard for some function? What does it mean?

A paper I'm reading has the following formula: $\ln ( 2 (\text{ch}(t') - cos ( \theta))] = t' - \sum_{k \in \mathbb{Z}^*} \frac{ e^{-|k|t'}}{|k|}e^{ik \theta}$. Which they call the Fourier ...
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1answer
79 views

Distinguishable groups of balls into distinguishable boxes with constraints

We have $E$ balls of $N$ colors. Let's call $e_i$ the number of balls of color $i$ (of course across group balls are distinguishable while within a color they are not). We can split these balls among $...
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1answer
110 views

How important is ergodic theory in fundamentally understanding statistical mechanics?

Recently, I realized the physics course in my uni has been lacking mathematical rigor and I have been attempting to compensate by adjoining it with personal mathematical study. For example, learning ...
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40 views

Branching annihilating random walks and mean field theory

I am attempting a project on modelling branching morphogenesis, but am getting very confused looking at the literature. On the one hand, the structure formation itself is clearly best described by a ...
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Characteristic function of the sum of random variables

The problem statement, all variables and given/known data I am trying to understand the very last equality for (let me replace the tilda with a hat) $$\widehat{P_X(K)}=\widehat{P(k_1=k_2=\cdots=k_N=k)...
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Can anyone tell me the name of this function?

So recently I came across this function (it was in the context of Bose-Einstein condensation in Statistical Mechanics): $F_\nu (\xi) = \frac{1}{\Gamma(\nu)}\int_0^\infty{\frac{x^{\nu-1}}{e^x/\xi -1}...