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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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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27 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 ...
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1answer
33 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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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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40 views

What's the best approach for a red light so you'll pass at maximum speed when it turns green?

So me and a couple of friends were driving back home when we started wondering about this (admittedly silly) question. Say you are driving your car on a straight lane and see a red light up ahead. ...
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39 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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49 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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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 ...
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24 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 ...
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1answer
37 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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1answer
75 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 ...
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36 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?
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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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1answer
32 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 ...
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250 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. ...
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1answer
34 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 ...
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92 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 ...
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31 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 ...
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31 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 ...
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2answers
41 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.
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52 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 ...
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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 ...
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Period ground state 1-dim Ising model

Good morning! I'm at the beginning of my study about the Ising model and it has been proposed to me this problem: Find all periodic ground-state configuration for the following one-dimensional Ising ...
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1answer
65 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 ...
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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 ...
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41 views

$k$-space tensor integral in statistical physics

$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$ Please help me to integrate the above tensor ...
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Approximation of sample spaces

I have a tests cases of $N$ different neurons where each of them have M mutually independent features associated with uncertainity value in the range [0,1]. The number of features and their ...
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37 views

How do you obtain the fluctuation spectrum of a tubular membrane?

I am reading through a paper. A tubular membrane, submitted to tension $\sigma$ acting as a Lagrange multiplier to conserve area, fluctuates around a cylindrical shape of length L and radius R. ...
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32 views

How to numerically solve a complex equation? [closed]

I want to know that if you are given a very complex equation g(x)=A(T). How could you solve for x, which is a function of variable T. To be more specific, I encounter a polylogarithmic function I need ...
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99 views

Who are the most influential cows in a herd of cattle?

You have a herd of cattle moving in different directions. The cows in the herd are more or less always moving, at different direction and in different velocities.When a cow bumps another cow it ...
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109 views

Simplifying a Vector Integral

While reading the book - Theory and Applications of Boltzmann Transport Equation by Cercignani (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , ...
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1answer
161 views

Probability of a trajectory in Markov processes

I need help with a simple formula! (My question is taken from here, pag 26 eq 1.112. ) Consider a Markov Process with associated Master Equation: \begin{equation*} ...
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Mathematical Interpretation of Partition Function and Free Energy

Given that The partition function in statistical mechanics tells us the number of quantum states of a system that are thermally accessible at a given temperature ...
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1answer
54 views

How to calculate the log of a sum

I'm going over some basic statistical physics and I need to compute $$\dfrac{\partial}{\partial \beta}\ln\sum_{i}e^{-\beta E_i}$$ Theres probably some simple trick i'm missing but i'm really ...
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1answer
121 views

When are $\Delta x$, $\delta x$, $dx$, and $\text{đ}x$ exactly the same? When are they approximately the same?

As a follow-up to this related question, I'd like to know under what circumstances, if any, $\Delta x$, $\delta x$ and $dx$ all mean the same thing, and under what circumstances they can all be said ...
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Mutlivariable integral, How to compute it? [duplicate]

Can anybody please tell me, how to evaluate a multivariate integral with a gaussian weight function. $$\mathcal{Z_{n}}=\int_{-\infty}^{\infty} ...
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2answers
54 views

Calculation of the Gallavotti-Cohen fluctuation theorem made by Lebowitz

I have a problem understanding a calculation in this paper (another form of the theorem an be found here at equation 11). For those who want to read the paper, I have difficulties with formula 2.14 in ...
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1answer
41 views

When can we write $f(v)dv=f(E)dE$?

In statistical thermodynamics we write $$f(v)\,dv = f(E)\,dE$$ where $v$ is velocity and $E= \frac12mv^2$ is energy and $f$ refers to the distribution function Can someone explain the logic ...
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64 views

Connection between Boltzmann entropy and Kolmogorov entropy

what is the connectivity between Boltzmann's entropy expression and Shannon's entropy expression? mentions a realtionship between Shannon entropy and Bolltzmann entropy. Is there a ...
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41 views

A very basic question about the Boltzmann distribution

I understand the formula for the Boltzmann distribution to be $P(E_i) = e^{-E_i/(kT)}/Z$ When the energy levels vary continuously illustrations of pdf for either the energy or the velocity at a ...
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1answer
42 views

Standard norm of $\mathbb{R}^3$

I am going through the paper, Energy of a Knot by Jun O'Hara. Let me quote from the Definition 1.1 of Section 1 on the first page: Let $f:S^1 = \mathbb{R}/\mathbb{Z} \to \mathbb{R}^3$ be an embedding ...
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Do the algebraic properties of the exponential and log functions specify them uniquely in probability theory?

I come from a physics background and in classical mechanics, we construct a Hamiltonian function whose partial derivatives generates a vector field, two independent systems are assigned a total ...
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Percolation and number of phases in the 2D Ising model.

Update. As my previous figure had conceptual mistakes I decided to change the picture to another, more instructive After a long time I came back to try to understand an article on the Ising ...
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What is the connectivity between Boltzmann's entropy expression and Shannon's entropy expression?

What is the connection between Boltzmann's entropy expression and Shannon's entropy expression? Shannon's entropy expression: $$ S= -K\sum_{i=1}^np_i\log (p_i) $$
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Please help ! A bit confused…I need to figure out exactlty what the question is asking me to determine.

Suppose that over a certain period prices of stocks in the S&P 500 can be approximately modeled using the normal distribution with $\mu = \$40.35$ and a standard deviation of $\$8.97$. The data ...
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Proof of $\sum_{u \in P} \sum_{v \in P} m_u \cdot m_v |u-v|^2 = 2 \cdot Var(P)$

How can we prove that, with $m_u$ being the mass of point $u$ (with a sum of 1) and $Var(P)$ being the weighted variance of points of $P$ considering these masses $m_u$ : $\sum_{u \in P} \sum_{v ...
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Exponent p-value generated in Excel

Excel gave me a p-value of 1.44909E-09 Notice is does not say .09 but 09 This is confusing me, I am trying to analyze my data but am stuck at this point. If it were E-9 it could be ...