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15 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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0answers
10 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 ...
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0answers
10 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 ...
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1answer
17 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
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1answer
175 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
23 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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2answers
49 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
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0answers
29 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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0answers
26 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
38 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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1answer
46 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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2answers
132 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 ...
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0answers
17 views

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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0answers
29 views

Need a simple derivation for Stat Mech

I actually know the formula for this which is $\frac{N!}{n_{1}!n_{2}!}$ ,but need some help to derive this. Find the number of distinct ways of arranging N particles in two groups such that one group ...
2
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1answer
54 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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0answers
16 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 ...
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1answer
39 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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0answers
11 views

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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0answers
36 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. ...
0
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1answer
27 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 ...
7
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1answer
92 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 ...
2
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0answers
107 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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0answers
31 views

good primer or intro on quenched stochastic processes

I was hoping that someone could recommend a good intro to "quenched" stochastic processes. I am using quenching in the sense of condensed matter physics where systems can present quenched disorder.
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1answer
153 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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0answers
71 views

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
51 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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0answers
35 views

The function $\sum_{i=1}^\infty i \exp(-a i^2)$

Consider the function $\phi(a)=\sum_{i=1}^\infty i \exp(-a i^2)$ with $a>0$ (see below for the physical motivation). I was very surprised that Mathematica didn't have an expression for a function ...
2
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1answer
118 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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0answers
64 views

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} ...
1
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2answers
51 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 ...
2
votes
1answer
37 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 ...
0
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1answer
53 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 ...
0
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1answer
35 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 ...
2
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1answer
41 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 ...
3
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1answer
47 views

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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0answers
110 views

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 ...
6
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2answers
246 views

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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1answer
49 views

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 ...
0
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1answer
52 views

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 ...
2
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0answers
881 views

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 ...
0
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1answer
68 views

Boltzmann Distribution With Constraints

I have a problem with showing the existence of Boltzmann distribution given some constraints. Consider $p_1,...,p_n$ a Boltzmann distibution, where $p_i=\frac{\epsilon^{-\beta \cdot E_i}}{\sum_{j}^{} ...
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0answers
218 views

Maclaurin series expansion of an expression that involves a fraction

In the context of statistical mechanics the "classical trace" is defined as $Tr(A e^{-\beta H}) = \int dr^N dp^N A e^{-\beta H}$ where $r^N$ and $p^N$ are phase space variables. So if $\Delta H$ is a ...
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1answer
51 views

A good reference to learn the concept of partition function

I am looking for a good reference and easy to learn the concept of partition function in mathematics. Can anyone help me?
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0answers
51 views

Hamiltonian of one and two unknots

Recently I calculated the Ising Hamiltonian of a Hopf link. First, I colored the Hopf link in a checker board pattern and drew the Seifert surface from it. Considering the shaded regions as vertices ...
0
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0answers
126 views

Direct correlation function

The direct evaluation of the configurational partition function of a liquid is very difficult because of the complicated dependence of the potential energy in the liquid.For this reason a formalism of ...
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0answers
53 views

What does $\Delta P_{t+1}(n)$ mean?

In Physics, there is a chapter about "Statistical Mechanics" where I do not really understand a notation: Let there be four coins, each of which can be either heads or tails. Set them in a given ...
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0answers
18 views

Edwards-Anderson Hamiltonian of a Hopf link

I was calculating the Edwards-Anderson Hamiltonian of a Hopf link. A hopf link is like attachment 1. I have drawn the Seifert surface of that link. The surface is shown in attachment 2. It also ...
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0answers
42 views

determining the entropy of a system

Consider a protein A, which has N monomeric units each of length a. For simplicity, treat the chain as one-dimensional: each unit can make a displacement +a (to the right) or -a (to the left) in the x ...
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2answers
150 views

Having trouble understanding the concept of “mixing” in dynamical systems.

I'm trying to understand the concept of mixing in dynamical systems theory, especially when the system in question has a measure-preserving flow. Here's how the condition is expressed mathematically: ...
4
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1answer
151 views

Maximal Ergodic Theorem

Does the maximal ergodic theorem have any dynamical or qualitative interpretations, or is it just a custom-made theorem to leave the demonstration of the Birkhoff ergodic theorem more elegant?