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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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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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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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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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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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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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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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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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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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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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} = ...
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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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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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26 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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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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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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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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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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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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95 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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24 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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50 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{ ...
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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 ...
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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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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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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, ...
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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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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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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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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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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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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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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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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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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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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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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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Grand canonical derivative.

I've been trying to work out how to find the density in the thermodynamic limit of a nearest neighbour magnetic lattice gas in the grand canonical ensemble. I'll with hold the Hamiltonian for the ...
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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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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 ...
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what does it mean that a system is attractive?

What does it mean that a system is attractive in the context of Statistical Mechanics? Is this notion related to the presence of some monotonicity properties?
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Is there an analytic solution for this Fokker-Planck equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial ...
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How to determine values of coefficients for a comparisonx using factorial design?

I think my problem is best answered by answering the following example but the more general the answer and explanation the better: Given 2 factors X and Y, x with 2 levels x1 x2 and Y with three ...
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Compute the specific heat capacity of ideal gas under constant $V$ and $p$

Compute the specific heat capacities at constant volume and constant pressure for air at standard temperature and pressure, assuming it is diatomic ideal gas and a molecular mass of 28u. I have ...
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Derivation of the Boltzmann factor in statistical mechanics

I have seen similar derivation of the Boltzmann factor many times before, (http://micro.stanford.edu/~caiwei/me334/Chap8_Canonical_Ensemble_v04.pdf , just for example), which I think is incomplete. ...
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Find the distance travelled by $P$ before it changes direction. (Mechanics)

A particle $P$ starts at the point $O$ and travels in a straight line. At time $t$ seconds after leaving $O$ the velocity of $P$ is v $m/s$, where $v = 0.75t^2 − 0.0625t^3$. Find (i) the positive ...
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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 ...