Questions tagged [statistical-mechanics]

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.

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Can a non-analytic function provide analytic solution (i.e., exact solution)? [closed]

What is an analytic function? What is an analytic solution? These two terms have the same meaning? If no, can a non-analytic function provide an analytic solution?
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Maximize $f(\mathbf n)=\dfrac{N!}{\prod_{j=1}^M n_j!}$ subject to $\sum_{j=1}^M n_j=N$ and $\sum_{j=1}^M e_jn_j=E$

The following exercise is a recap on probability and maths for statistical mechanics: Maximize $$f(n_1,n_2,\, ...,\, n_M)=\dfrac{N!}{\prod_{j=1}^M n_j!}$$ subject to $$\sum_{j=1}^M n_j=N\ \text{ and }...
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Unusual use of $\lor$ symbol

I am reading the paper "Entropy and Equilibrium States in Classical Statistical Mechanics" by Oscar E. Lanford III (published in 1973). The image below shows an excerpt where the symbol $\...
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Nonlinear Langevin equation and SDE

Recently, I was exposed to SDEs, though I still did not fully understand the concepts. I want to solve a nonlinear undamped oscillator $$\frac{d^2x}{dt^2}=-\alpha^2 g(x) + f(t)$$ where $g(x)= \sin(x), ...
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When is von Neumann entropy differentiable? (Edited)

Given a smoothly time-dependent density matrix (positive-semidefinite matrix with trace 1) $\rho(t)$, its von Neumann entropy is defined as $$ S(t)=\mathrm{Tr}(f(\rho(t))),\ f:[0,\infty)\ni x\mapsto -...
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Can addition of noise to dynamical system reduce estimation errors

I am using Kalman filter to estimate the states of a stochastic dynamical system which has very very small noise( consider zero ). The filter is not aware that the noise is zero. Implementation of KF ...
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Minimum residual error in estimation of deterministic system using Kalman filter

Let the process equation for a state vector $\mathbf{x}_t$ at time $t$ be: \begin{equation} \bf{x}_{t+1} = \bf{f}\left(\bf{x}_t\right) \end{equation} where, $\mathbf{f}\left(.\right)$ is a nonlinear ...
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What are necessary and sufficient conditions for the Bellman Equation to be solvable?

I am studying Markov Rewaed Processes right now, and I wish to gain a deeper understanding of the Bellman equation's relationship with them. I learned the Bellman equation in the following form: $v = ...
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I dont understand the last step. I’m trying to understand how equation 10 follows, especially the last delta equation [closed]

I dont understand following steps of a solution where I need to find the Normalization constant $A(E,P,N)$ . The normalization is given by: $$ \int \rho(\vec x)d\vec x = 1 $$ where $d \vec x = C_N d^...
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Concentration of Gibbs measures with converging energy functions

Let $H$ be a continuous energy function defined on a compact subset $A\subset \mathbf{R}^n$ and let $Q$ be a fixed probability measure on $A$. For each $\theta>0$, define the probability ...
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Can anyone help to solve this task ?In a multiple-choice test with m options, a student knows the correct answer with probability p,...?

"In a multiple-choice test with m options, a student knows the correct answer with a probability p, and in the absence of knowledge, chooses randomly one of the available options. What is the ...
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Definition of discrete Gaussian Free Field

For $V \subset \mathbb{Z}^d$ finite, consider $(h_x^V)_{x\in \mathbb{Z}^d}$, a stochastic process indexed by $\mathbb{Z}^d$. $h$ is a discrete Gaussian Free Field with zero boundary condition outside $...
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Average probability of sampling all clusters

A system consists of $N$ nodes. The nodes are distributed into $M$ clusters such that each node belongs to a unique cluster. Each cluster $i$ has one unit of weight, $w_i = 1$. Thus the initial weight ...
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Mathematical theory of plasma

I am working on a heavily mathematically project about plasma. In particular, I want to find references that treat the problem from microscopic models that include relativistic and magnetic effects (...
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Why do we use the exponential in the Boltzmann distribution

Low effort question incoming. Given a set of states $x_i,i=1,\dots,n$ with energy $0\leq U(x_i)$, we define the probability of a state $x$ as $$ \pi(x)=\frac{1}{Z_T}e^{-\frac{1}{T} U(x)} $$ where $Z_T$...
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The covariance matrix $\mathbf{C}$: Why does $\ln \text{det} \mathbf{C}=\text{Tr}\ln \mathbf{C}$ hold? [duplicate]

Prof. Max Tegmark first introduced the Fisher information matrix into cosmology in his paper titled Karhunen-Loeve eigenvalue problems in cosmology: How should we tackle large data sets? As I read the ...
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Central Limit Theorem for subsamples

I observe a set/realisation of $n$ i.i.d. $\{X_1, X_2, ..., X_n\}$. Because of the Central Limit Theorem, I know that repeating such an observation enough times, the pdf of the mean of such $n$ ...
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Multiple summation for 1D Ising model

I am trying to perform multiple summation in a Matlab code, but I don't know how to write a code to perform multiple for's . $$\sum_{S_1=\pm1}...\sum_{S_N=\pm1}(\Pi_{i=1}^{N-1}e^{S_iS_{i+1}})=\sum_{...
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given SD and mean find bserved value of the test statistic?

I had this word problem where I am trying to see whether new version of something is more precise so I know that the formula for test stat is $$\sum_{i=1}^9 \frac{(yi-\mu)}{x^2}$$$ where $$\sum_{i=...
fashionable's user avatar
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How does the probability density function decay off for a 2D random walk with shrinking step size $f(n) = \frac{1}{n}$

Consider a 2D random walk with the magnitude of the nth step fixed by the function $f(n) = \frac{1}{n}$ and the direction being random. I know that the root mean square comes out to be, \begin{...
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Not following derivation of Curie-Weiss-Potts model

I'm reading an article that derives an expression related to the Curie-Weiss-Potts models. The question pertains to how Equation (7) in the article is derived. Below is my summary of the information ...
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Choosing correct parameters in a model with hamiltonian equation

I am working on the hamiltonian of a system related to the extension of the Potts Model which is Cellular Potts Model. The total hamiltonian of the system is: $$ H = H_1 + H_2 $$ $$ H_1 = - J \sum_{\...
wallevic's user avatar
3 votes
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Equivalence of Ising and Random Cluster Model Partition Functions

It is well documented that the partition function of the Ising model is equivalent to the partition function of the Random Cluster model. However I cannot seem to find a resource that actually shows ...
space_kale's user avatar
1 vote
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Natural candidates for energy function of knots in $S^3$?

Let $K\subseteq S^3$ be a knot. In real-life, knots (like protein chains) $K$ moves around stochastically, and experimentally the lowest energy/highest entropy states are particularly simple from a ...
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Existence of Gibbs state on infinite graph

Consider an infinite graph. In an article by Jonasson and Steif, the Ising model on this graph is defined as a generalization of the standard Ising model. On page 551, they cite a book by Georgii for ...
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Effect of the media on the opinion dynamics in online social networks

I am studying a paper called "Effect of the media on the opinion dynamics in online social networks" and I cannot understand a passage. Let's first define a matrix A (n x n) which elements ...
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Is there any applied exposition of the Kahn-Kalai conjecture?

In the Wikipedia entry associated with the Kahn-Kalai conjecture, there is the following assertion This conjecture concerns the general problem of estimating when phase transitions occur in systems. ...
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Concentration of measure in statistical mechanics

I have just finished a course in advanced probability theory (martingales, Brownian motion, Ito calculus, concentration inequalities, Stein's method) and an undergraduate course in stat mech. I am ...
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Sharpness of the maximum of the weight

What does the sharpness of maximum mean? In Statistical mechanics, when we find the most probable configuration, then we also have to define its sharpness. What is this sharpness?
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All probability densities are Gibbs form.

In this https://djalil.chafai.net/blog/2018/03/09/tutorial-on-large-deviation-principles/ blog post on Large Deviations the author says By the way, let me tell you a secret: all probability ...
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Bivariate Lévy-stable PDF in terms of bivariate Fox H-function

In the reference W.R. Schneider, in: S. Albeverio, G. Casati, D. Merlini (Eds.), Stochastic Processes in Classical and Quantum Systems, Lecture Notes in Physics, Vol. 262, Springer, Berlin, 1986. the ...
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Double integral involving products of Fox H-functions

For physics application, I want to calculate the integral of the form $$I(z)=\int_{0}^{\infty}\int_{0}^{\infty}H^{1,1}_{2,2}(x)H^{1,1}_{2,2}(y)H^{1,1}_{2,2}(z+\lambda x +y)H^{1,1}_{2,2}(z-x-\lambda y) ...
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$n$-dimensional spheres and gaussian scalar fields

I'm currently reviewing some problems in Statistical Mechanics and I have come across a question that I'm struggling to resolve. Specifically, in certain parts of the study of ideal gases, the concept ...
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Figuring out a probability distribution

At the instant $t = 0$ a certain radioactive focus starts emitting particles. The infinitesimal probability that the focus emits a particle in the differential interval is $\lambda dt$. Let $N$ also ...
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When is the Gaussian smoothed version of a function the same as the original function?

To be more precise: If $f(x) = \mathbb{E}_{\delta \sim \mathcal{N}(0,\,\mathbb{I}\sigma^{2})} u(x+\delta)$, what sort of function can $u(x)$ be s.t $f(x) = u(x)$ ? I'm not sure what tags to attach, ...
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A positive gradient 2D ODE with multiple stable positive steady states?

I am looking for a 2D ODE of the form: $\begin{array}{rl} \dot x&= f(x,y) \\ \dot y &= g(x,y) \end{array}$ that satisfies the following: 1- there exists a function $H(x,y)$ such that $f(x,y)=-\...
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Ising model, parity of the loops and sign of a spin.

I have a hard time understanding how to reason with these questions Let $G \subset \mathbb{Z}^2$ be a bounded connected domain with $-$ boundary conditions. Consider the Ising model on $G$ with ...
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Covariance statistics of p-spin models

I am trying to calculate the co-variance of two non-independent variables. $\sigma$ is a string of length $n$ with bits $\sigma_i$ taking values 1 or -1. One has a p-spin model which is defined by the ...
Ognorelep's user avatar
3 votes
1 answer
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Proving that certain integral is positive

Given a compact set $K\subset \mathbb{R}^3$, we consider $f:K^3\subset\mathbb{R^9}\to \mathbb{R}_0^+$ such that $f(x_1,x_2,x_3)=f(x_{\tau(1)},x_{\tau(2)},x_{\tau(3)})$ for every permutation $\tau$ ...
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How many ways there are to cover an $n \times n$ tiling with $2 \times 1$ dominoes?

I came across the famous dimer problem in statistical physics and I'm struggling to come up with a mathematical formula for covering an $n \times n$ tiling with $2\times1$ dominoes? How does a ...
physics22's user avatar
2 votes
1 answer
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Why does the condition $\sum_{x\in\Lambda}f_x=0$ imply the discrete Laplacian is invertible?

I'm studying statistical physics these days and have newly learnt the concept discrete Laplacian. For a finite graph $G$ with vertices $\Lambda\subset\mathbb{Z}^d,$ consider the discrete Laplacian ...
Chang's user avatar
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5 votes
1 answer
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Probability that a random walk in $2d$ has small local time at each vertex

Let $P_{n,k}$ be the probability that a simple random walk of length $n$ in $\mathbb{Z}^2$ is such that each vertex of $\mathbb{Z}^2$ is visited at most $k$ times by the walk. Certainly this ...
QuantumLogarithm's user avatar
3 votes
0 answers
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Can some one explain me Planck/Reileigh Jean Law?

I wonder why every proof of these laws consider the number of of oscillator in the end but disregard it while deriving the mean energy. Let me explain. Considering an oscillator in a heat bath, there ...
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2 votes
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Can sets of functions form a measure space?

For example, continuous functions from $\mathbb{R}$ to $\mathbb{R}$ with nth derivative = 0. Each function acts like a point in an $\mathbb{R}^n$ dimensional space with its Taylor expansion ...
Peter Hodgson's user avatar
2 votes
1 answer
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Agent-Based Wealth Model: Proving Inequality

Consider the following agent-based model: There are $N$ agents Every agent starts with $1 At each time interval (i.e. at each step), every agent gives \$1 to a randomly chosen agent. I want to find ...
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1 answer
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How to solve this integral with exponent depending on two independent variables?

The question is I was given the Hamiltonian of two classical particles with masses $m$ with which I must calculate Partition function $$H =\frac{P_1^2}{2m} +\frac{P_2^2}{2m} + q_1^2 + q_2^2 +\frac{...
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1 vote
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Bounding integral with quadratic complex exponential

I am reading this book (Euclidean Harmonic Analysis, Benedetto, 1979) and on pages 24-25 Carleson proves a lemma related to the Kolmogorov-Seliverstov-Plessner method. There is one small step in ...
Cloudfire's user avatar
1 vote
1 answer
95 views

histogramming phases between a periodic function and another periodic, quasiperiodic or almost-periodic function with irrational period relationship.

Background and motivation: The Astronomy SE question How often does a full moon happen on the weekend? touches on issues I've always wondered about. The current answer says 2/7 of full moons occur on ...
uhoh's user avatar
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2 votes
1 answer
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When can you substitute $w!$ for $\sqrt{2\pi}(w/e)^w$? [duplicate]

Hi in one of Boltzmann's discussions on thermal equilibrium he performs a minimization which relies on the assumption that $$x! = \sqrt{2\pi}(x/e)^x$$ or as he states it " here $x!$ and $\sqrt{2\...
phntm's user avatar
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2 answers
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What does the "3" in $d^3x_1d^3p_1\ldots d^3x_Nd^3p_N$ mean here?

On page 4 of this online lecture notes, I find the following notation: $$d^3x_1d^3p_1\ldots d^3x_Nd^3p_N$$ What does the raised "3" next to the "d" mean?
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