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0
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1answer
20 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
votes
1answer
81 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 ...
-1
votes
1answer
28 views

poisitive definite matrix imply matrix of full rank? [closed]

Is a positive definite matrix always of full rank? while is given that the characteristic roots are pisitive
2
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0answers
67 views

Simplifying a Vector Integral

While reading the book - Cercignani, Theory and Applications of Boltzmann Transport Equation (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , \xi_l$ ...
1
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0answers
32 views

Entropy and Gibbs measures (mathematical formalism) [closed]

Entropy and Gibbs measures are both phrases that are thrown around at probability seminars and other analysis events, and sadly this is not "only" bad because I am confused about it, but also because ...
0
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0answers
24 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.
1
vote
1answer
127 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*} ...
1
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0answers
40 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 ...
1
vote
1answer
46 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 ...
0
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0answers
31 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 ...
1
vote
1answer
85 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 ...
0
votes
1answer
16 views

some knowledge about comare means of two group

I'm studying a bout statistic. I can compute very well. But i don't really know well. I want to know clearly about theory. And i have this problem: ...
4
votes
0answers
62 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
vote
2answers
35 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
34 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
votes
1answer
29 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 ...
1
vote
1answer
54 views

rigorous path to statistical mechanics

Is there a path to thermodynamics which is not governed by intuitive and fuzzy postulates but by lemmas and axioms? I also asked this question on physics.stackexchange in order to get different ...
0
votes
1answer
33 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 ...
0
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0answers
18 views

Torsion energy as a component of physical energy of a knot

I am going through the paper, Recognizing knots Using Simulated Annealing by Ligocki and Sethian. This paper uses simulated annealing to solve the KNOT GENUS problem. It has used the concept of ...
0
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0answers
39 views

Invariance of the physical energy of a knot over Möbius transformations

I am going through the paper, Recognizing knots Using Simulated Annealing by Ligocki and Sethian. This paper uses simulated annealing to solve the KNOT GENUS problem. It has used two different ...
2
votes
1answer
36 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
votes
1answer
45 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 ...
5
votes
0answers
87 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
votes
2answers
167 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: $$ S= -K\sum_{i=1}^np_i\log (p_i) $$
0
votes
1answer
45 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
votes
1answer
50 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
votes
0answers
348 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
votes
1answer
62 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}^{} ...
1
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0answers
162 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 ...
0
votes
1answer
40 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?
1
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0answers
49 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
votes
0answers
118 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 ...
1
vote
0answers
51 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 ...
2
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0answers
17 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 ...
0
votes
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 ...
1
vote
2answers
146 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
votes
1answer
137 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?
3
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0answers
231 views

Why is the partition function able to describe the whole system?

No matter what the real system or subject is, if there is a partition function $Z$, then these kind of identities hold $$\langle X\rangle=\frac{\partial}{\partial Y}\left(-\log Z(Y)\right).$$ If one ...
2
votes
0answers
106 views

Surface measure

I dont understand the following surface measure, My space is $\Omega_E=\{w\in \Omega | H(w)=E\}$ My lecture notes state that my surface measure is given as ...
8
votes
0answers
216 views

Intuition for the Yang-Baxter Equation (was: Giving relations via formal power series)

I'm reading a book (Yangians and Classical Lie Algebras by Molev) which regularly uses (what appear to me to be) clever tricks with formal power series to encapsulate lots of relations. For instance, ...