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)

learn more… | top users | synonyms

0
votes
0answers
19 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?: $$\int\limits_{-\frac{1}{2}}^{0} \mathrm{exp}\left[-C \left( \frac{1}{(x -\frac{1}{2})^2 + (x + \frac{1}{2})^2} -1 \right)^S \right] \mathrm{dx}$$ The ...
-1
votes
0answers
14 views

Horicontal tension in catenary [on hold]

How is the tension at vertex equal to $T^o = wc$. Where $c$ is parameter and $w$ is weight per unit length.
5
votes
1answer
49 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 ...
0
votes
1answer
19 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 ...
0
votes
0answers
21 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 ...
1
vote
0answers
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 $A$ ...
1
vote
1answer
27 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 $k$?
1
vote
0answers
18 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 ...
0
votes
1answer
10 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 ...
0
votes
0answers
10 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 ...
2
votes
1answer
29 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?
2
votes
1answer
87 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 $d=1$ ...
0
votes
0answers
21 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 ...
0
votes
0answers
46 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{ ...
0
votes
1answer
21 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 ...
0
votes
0answers
15 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 $h=(h_i)_{i=1}^N$ ...
0
votes
1answer
10 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 ...
1
vote
0answers
41 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 ...
0
votes
0answers
10 views

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, ...
1
vote
1answer
22 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 ...
0
votes
1answer
18 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 ...
0
votes
1answer
50 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 ...
0
votes
0answers
20 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 ...
1
vote
0answers
55 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 ...
0
votes
0answers
24 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 ...
1
vote
0answers
20 views

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 ...
0
votes
0answers
37 views

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 ...
0
votes
1answer
37 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 ...
0
votes
0answers
43 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 ...
0
votes
0answers
39 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 $T1$ $\frac{-1i+2j+6k}{6.40}$ $T2$ ...
1
vote
2answers
65 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 ...
1
vote
2answers
323 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 ...
0
votes
0answers
21 views

Symmetrical Probability distribution [closed]

The probability distribution of the velocity component along the direction of X-axis, of molecules of an ideal gas, is symmetrical about zero velocity. The integral under the curve from $- \infty$ to ...
0
votes
0answers
31 views

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 ...
3
votes
1answer
40 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 ...
3
votes
1answer
141 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 ...
0
votes
0answers
43 views

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?
4
votes
0answers
84 views

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 ...
1
vote
0answers
11 views

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 ...
0
votes
1answer
37 views

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 ...
6
votes
1answer
492 views

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. ...
0
votes
1answer
36 views

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 ...
0
votes
2answers
177 views

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 ...
2
votes
0answers
32 views

What are conditions for an infinite sum with a complex parameter not to be analyitically extendable?

I'm looking for a sequence $f(n)$, so that $g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$ with $z$ so that this converges classically, defines a function which can not be ...
1
vote
0answers
34 views

Invariants under Hamiltonian mechanics?

I am interested in certain properties of measures evolving according to Hamiltonian mechanics. Say we have a point $z$ in phase space: $z = (p,q)$ where $p$ is a generalized momentum vector and $q$ is ...
0
votes
2answers
49 views

Statistical Mechanics of interacting Particles. Quantized Fields. Solving Integral?

Hi everyone How we can analytically without using a software solve below integral . Chapter 11 of Pathria (edition 1). and x is dimensionless.
0
votes
1answer
81 views

Partial Differentiation in Statistical Mechanics

I am damn struggling with basics in here. I know that $U=U(N,V,T)$ and $z=z(N,V,T)$ so that $N=N(z,V,T)$. Now, I want to do partial differentiation using chain rule involving three variables so that I ...
9
votes
2answers
211 views

Jones Polynomial from Statistical Mechanics

I've been told that, given a knot projection, there is a way of associating a statistical system in such a way that the partition function of the system corresponds to the Jones polynomial of the ...
2
votes
1answer
117 views

General Gaussian distribution relation

I'm trying to solve a question from Pathria's statistical mechanics textbook (10.21) but it is more math oriented. Show that, for a general Gaussian distribution of variables $u_j$ , the average of ...
1
vote
0answers
17 views

k-space tensor integral in statistical mechanics [duplicate]

k is the modulus of the vector k. Please help me to integrate the above tensor expression in the infinite domain of the vector k. I have tried to let u in the direction of kz and then transform the ...