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2
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0answers
54 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
20 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
21 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
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1answer
95 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
28 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
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1answer
44 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
27 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 ...
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1answer
79 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
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1answer
12 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
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0answers
61 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
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
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1answer
31 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
27 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
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1answer
50 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
32 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
38 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
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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
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1answer
44 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
83 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 ...
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2answers
157 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
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1answer
44 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
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
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0answers
309 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
59 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
158 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
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?
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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
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0answers
117 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
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0answers
50 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
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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
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1answer
135 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
222 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
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0answers
104 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
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0answers
214 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, ...