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. (Def: http://en.m.wikipedia.org/wiki/Statistical_mechanics)

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Calculating rate of time consumption based on NBA player impact estimate

I'm building a game. I'm looking for help calculating the rate of consumption based on a player's impact estimate (PIE). PIE measures a player's overall statistical contribution against the total ...
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What are the applications of quantum shuffle multiplication?

I am writing a research paper on quantum shuffle multiplication, and there is just one piece that I'm missing: I want to give one or several examples of its applications, and so far I have not been ...
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What is the field of mathematics that describes the transition into statistical mechanics?

There are interesting changes that occur in a sample of interacting objects, such as gas particles, as you approach a statistically significant sample. The position or velocity of any given particle ...
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100 views

How many ways we can find out the strip of N spins contains m “Parallel Pair” out of which m1 of them are “Up Parallel Pair”?

Let us consider a one-dimensional strip containing 8 spins. Spins can be up or down. And spins can be arranged randomly. So the total number of different microstates possible is $2^8$ (Taking Periodic ...
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Help with integral from Boltzmann equation

I have a function $$ g\left(x,v,t\right) = u\left(x,t\right)\cdot v + \theta\left(x,t\right)\frac{1}{2}\left(\left\lvert v\right\rvert^2 - 5\right) $$ where $g(x,v,t),\theta(x,t)$ are scalars and $u(...
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26 views

Pushforward of Liouville measure on configuration space

Say I have $M_1$ a 3-dimensional compact Riemannian manifold, and $M=M_1^{n}$ the product manifold representing n particles on $M_1$. I can identify $TM$ with $T^*M$ via the metric $g$ and then the ...
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49 views

Properties of transfer matrices and their traces

I'm having difficulties understanding some arguments in my statistical mechanics lecture and would like to make them more rigorous by proving some properties. For the Ising model on a lattice we ...
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1answer
13 views

Continuously go from a lognormal distribution to a power law

Do you know any phenomena that are described by a continuous mappings between a lognormal and a power law distribution? Of course, one could give a simple linear combination of the two distributions;...
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30 views

Good book on combinatorics for beginners in statistical mechanics

Im studying stat mech and i want to have a better understanding on counting microstates. What book in combinatorics do you guys recommend for beginners like me? Thanks in advance
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14 views

Propagation of Uncertainty from Variance

Given, for a Brownian Motion, the mean square displacement (MSD) of the particles in the 3D are: $MSD_x = t \cdot 10 \mu m/s$ $MSD_y = t \cdot 10 \mu m/s$ $MSD_z = t \cdot 2 \mu m/s$ By using the ...
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64 views

Deriving the q-Gaussian PDF

Ok it may sound a bit too simple but I am quite confused here. While studying generalized entropic forms, in my case that of $S_q$ or in another words the Tsallis Entropy, I reach a point where I have ...
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31 views

Stuck in using Stirling's approximation to show and justify an approximation of the number of permutations with and without ordering

This is a problem from my applied mathematics class where we are currently working on using Stirling's approximation which is: $ n! \sim (\frac{n}{e})^n \sqrt{2 \pi n} $ and the context of this ...
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16 views

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 ...
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61 views

Statics. Determine the smallest angle $θ$ The coefficient of friction at A and B is $0.5$

Determine the smallest angle θ at which the uniform triangular plate of weight W can remain at rest. The coefficient of static friction at $ A $ and $B $ is $0.5.$ Based on my Free body diagram which ...
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28 views

How does the evaluation of $\dot q_i$ at $q_1+\mathrm dq_1$ yield $\dot q_i +\dfrac{\partial \dot q_i}{\partial q_1}\mathrm dq_1 \;?$

I've been following Reif's Fundamentals of Statistical and Thermal Physics; there I came before the derivation of Liouville's theorem: There I couldn't understood few things. I could conceive the ...
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41 views

Average number of steps to return to the origin of a random walk on a 2-d lattice.

Suppose I have a random walker on a 2-d square lattice with periodic boundary conditions with equal probability of going in any of the four directions. The walk ends when the walker reaches the point ...
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105 views

What does $-p \ln p$ mean if p is probability?

In statistical mechanics entropy is defined with the following relation: $$S=-k_B\sum_{i=1}^N p_i\ln p_i,$$ where $p_i$ is probability of occupying $i$th state, and $N$ is number of accessible ...
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36 views

Need help with $\int\mathrm{exp}[-C(\frac{1}{(x -\frac{1}{2})^2 + (x + \frac{1}{2})^2} -1)^S ] \mathrm{dx}$ for a statistical mechanics problem

Can someone help solving this integral?: $$\int\limits_{-\frac{1}{2}}^{0} \mathrm{exp}\left[-C \left( \frac{1}{(x -\frac{1}{2})^2 + (x + \frac{1}{2})^2} -1 \right)^S \right] \mathrm{dx} = \int\...
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101 views

An intuitive explanation of how the mathematical definition of ergodicity implies the layman's interpretation 'all microstates are equally likely'.

I'm self-studying Statistical Mechanics; in it I got Fundamental Postulate of Statistical Mechanics and that took me to ergodic hypothesis. In the most layman's language, it says: In an isolated ...
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27 views

How to prove that the Statistical Entropy $S_{BG}$ is concave

So I am for the moment studying the properties of the Boltzmann-Gibbs statistical entropy \begin{equation} S_{BG}=-k_B \sum_{i}p_i\ln p_i, \end{equation} where of course $k_B$ is the Boltzmann ...
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29 views

Proof that the q-Gaussian is derived by maximizing Tsallis entropy

So I have to write a report paper for a course and I would like to prove that the q-Gaussian is the distribution which arises once one maximizes the Tsallis entropy. But I face difficulties in ...
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27 views

Number of connected subsets of a finite connected set in $\mathbb{Z}^d$

Let $A$ be a connected subset of $\mathbb{Z}^d$ having finite cardinality $n$ and containing the origin. I need a good upper bound which depends only on $n$ for the number of connected subsets of $A$ ...
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28 views

Number of connected sets of size $k$ containing a given vertex on $\mathbb{Z}^d$

Consider the graph $\mathbb{Z}^d$. Let $C_k$ be the number of connected sets containing the origin and $k-1$ other vertices. How does $C_k$ grow with $k$?
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22 views

Number of a self avoiding cycles in 2D with a given area

Consider a two dimensional square lattice and let $C_n$ be the number of self avoiding cycles (i.e., a self avoiding walk that starts at the origin and terminates at the origin) such that the number ...
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14 views

How are Canonical and Grand Canonical Ensemble related in the framework of Large Deviations Theory?

I have a toy model of $\rho N$ particles in N boxes and Hamiltonian $H = \sum_{i=1}^{N}log(1+ n_i)$ $P(\underline{n})=\frac{1}{Z(T, \rho, N)}e^{-H(\underline{n})/T}$ The canonical partition function ...
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15 views

Statistics of “minimum weight cycle length” on a randomly weighted graph

I've come across a problem in my research that I don't know how to approach. Consider a fully connected directed graph with $N$ nodes. (That is, each node connected to every other node, including ...
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38 views

Do i get the right MLE and 90% confidence interval of normal distribution?

I think i do right in step1 above. But i wonder whether i get the right confidence interval of mu and sigma in step2?
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1answer
98 views

Uniqueness of Gibbs measure for rotator model in one dimension

I am trying to solve a problem in a course of Y. Velenik (models with continuous symmetry, exercice 8.18: http://www.unige.ch/math/folks/velenik/smbook/index.html): Show that in dimension $d=1$ ...
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27 views

Interchange limit and partial derivative of two different variables

I have seen the following related questions: Interchange limit of one variable with partial derivative of another variable and Interchange of partial derivative and limit But my case is a bit ...
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58 views

Wick/Isserlis' Theorem converse

Let $X=(X_1, \ldots X_{2n})$ be a random vector with centered (zero mean) Gaussian distribution. Then, the $2n$-point correlation function: $$E(X_1 X_2 \cdots X_{2n})=\sum_{P \text{ paring}}\prod_{\{...
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1answer
29 views

How to expand a factorial expression with $N$ and $m$

In a statistical physics book, I don't understand how they moved from this expression: $\Big(\frac{N}{2}-m\Big)! \Big(\frac{N}{2}+m\Big)! = \Big(\frac{N}{2}!\Big)^2$ with $N=2k, k,m\in \mathbb{Z^+}$,...
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26 views

Special case of the inverse Ising problem with equal correlations

Let $s_1,\dots,s_N\in \{-1,1\}$ be $N$ binary spins. The problem of finding a symmetric interaction matrix $J=(J_{i,j})_{i,j=1}^N$ with zero diagonal and an external magnetic field $h=(h_i)_{i=1}^N$ ...
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1answer
12 views

definition of conditional probability $(p_i|\pi_k)$ and Tsallis entropy

Let $\Omega$ be a set of $W$ possible outcomes of an experiment with probability assignments $p_i$ and thus $\sum_{i=1}^{W}p_i=1$. Now, let's divide $\Omega$ into $K$ non-intersecting subsets each ...
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53 views

Logic behind Metropolis algorithm

I am using Metropolis algorithm to make a program for Ising model in Statistical Physics. In Ising model, we take a collection of spins with initial energy, say $E_i$, then we randomly flip one of the ...
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find expression for limiting mean/covariance

Let $\mathbf{X}_i\in\mathbb{R}^{m\times m},i=1,2,\cdots,N$ be realizations of a random matrix $\mathfrak{X}$ having mean $\mathbf{X}_0 = \mathbf{0}$ and covariance $\boldsymbol\Sigma$. Let, $\mathbf{X}...
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25 views

Choosing the variance for a Gaussian Distribution for the Langevin Equation

I am trying to numerically integrate the Langevin equation which is given by $$ \ddot x= -\frac{6\pi\eta a}{m} \dot x + \frac{\xi(t)}{m} $$ where $\xi(t)$ is a probability distribution. I'm ...
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22 views

Statics Find the range value of P

Answer is $29.3 N ≤ P ≤ 109.3 N$ I tried solving it for quite some time already which I don't understand why I don't get the values. Can someone help me? $Fv$ Vertical force $Fh$ Horizontal force $...
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54 views

Probability of Observing $N$ particles in a given volume?

I'm having an issue with a probability problem concerning solutions. Assume there is an "observational region" in a dilute solution with a volume $V$, and as solutes move across its boundary, the ...
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21 views

Metropolis Markov Chain and Mixing Time

I have a statistical mechanical system, which I would like to sample with the Gibbs distribution using a Metropolis-Markov chain. I think the following are standard questions, but I am not sure what ...
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87 views

What is the relationship or difference between MLE and EM algorithm?

I am trying to study EM algorithm and Maximum Likelihood Estimation. Somehow, they both sound the same to me but can't really say the difference. Maybe I don't really understand any of them. I have ...
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27 views

Generating pseudo-random numbers around a distribution with an uncertain/chaotic mean

I originally asked this question on cross-validated, but apparently it is too mathematical a problem for that site. I want to simulate data collected by an instrument realistically. The problem is ...
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Covering a family of sets of $\mathbb{Z}^d$ with boxes of a given diameter

Let $diam(A)$ be the graph distance between the two farthest vertices contained in a finite set $A \subset \mathbb{Z}^d$. Is it true that there exists a real $\eta(d)>0$ such that for any finite $...
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Conformal field theories and critical points

I apologize in advance if this question belongs to physics.stackexchange. I've been trying to learn CFT following Zee for QFT background (approximately first and second chapter,) and then Di ...
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1answer
39 views

Integrating $\prod_{i=k+1}^{kN} \left(\int_{-\infty}^{\infty}dp_i\right)\times$ with conditions on $p_i$

I am trying to integrate this expression which came up in a derivation of the momentum distribution function for an ideal gas. $\Theta(x)$ is the Heaviside step function which is $1$ when $x$ is ...
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83 views

Finding trends in data with multiple variables, analysis software?

I have a series of many test results (hundreds). They are vibration test results of electronic equipment that has been built the same standard way (same materials, adhesives, structural shape and ...
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48 views

Find the other forces on the plate that are equivalent to $210k$ $kN$

Determine the three forces acting on the plate that are equivalent to the force $R=210k$ $kN$ First I tried to find the components of $T1$$T2$ $T3$ So for $T1$ $\frac{-1i+2j+6k}{6.40}$ $T2$ $\...
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67 views

Scalar derivative of ${\rm tr}~[A(x)\log A(x)]$ where $A(x)$ is a square matrix

How do i proceed to calculate $$\frac{d}{dx}{\rm tr}\left[{A(x) \log A(x)}\right]$$ where $A(x) \in \mathbb{M}(n)$ and $x \in \mathbb{R}$? The $\log$ function is the one defined by the exponential ...
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615 views

Proof that a log-of-sum-of-exponentials is a convex function

It's well known in statistical mechanics that the following is a convex function of the vector $\theta$: $$ A(\theta) = \log \left( \sum_{i=1}^\infty e^{\theta \cdot f(i)} \right) $$ where $f(i)$ is ...
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41 views

On random rotational fluctuations in $\mathbb{R}^n$

Imagine first a disk that is mostly stationary, except for random ("thermal" if you like) "rotational fluctuations" around its axis (which is fixed). Something a bit like what's shown in the figure ...
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184 views

Regarding “Two Singular Diffusion Problems” by William Feller

I'm currently reading the research paper, Two Singular Diffusion Problems, by William Feller (1950). However, I don't understand how Feller derived the solution $(3.5)$ given equation $(3.4)$ in his ...