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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Mayer expansion: from product over i,j to sum over graph

I am studying the Mayer expansion used in Statistical Physics. We arrive at the following expression: $$ \prod_{i<j=1}^N (1+f_{ij}) $$ and then is find out that this expression is equivalent to the ...
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Set $T^{\mathbb N}x$ dense in $\mathbb S^1$ (Poincaré recurrence theorem)

Let $Ω =\mathbb S^1$ be the unit circle in $\mathbb R^2 = \mathbb C$, and let $T : Ω → Ω$ be multiplication by $e^{i\alpha}$. For $α \notin π\mathbb Q$ and every $x ∈ Ω$, is the set $T^{\mathbb N}x$ ...
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Ergodicity on a finite set

Let $\Omega$ be a finite set ($ \#\Omega = n$), how many dynamical systems on $\Omega$ are ergodic?
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Statistical mechanics partition function from probability distribution

I am curious about the mathematical background of something I came across while working on a problem in statistical mechanics. As an example, I am going to use the classical canonical ensemble, though ...
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Non-linear backward Kolmogorov equation

The backward Kolmogorov equation (BKE): $$\frac{du}{dt} = A(x,t) \cdot \nabla_x u(x,t) -\frac12 \text{Tr}(BB^t(x,t) \text{Hess}_x u(x,t) - f(t,x,u, B,\nabla g), \;\;\; t<T$$ If $f\equiv 0$ then ...
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How to interpret integrals that have conditions written beside them

sorry if this question has been asked before. I tried finding similar questions but couldn't find any. I have very little background in statistical mechanics, but I have been reading some literature, ...
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Angle between two random unit vectors uniformly distributed

Consider $x, y \in S_{1}^{d-1}$ (the unit n-sphere in d dimensions) with $(x \cdot y)^2 = 1/d$. I need to compute the angle $\alpha$ between $x$ and $y$ for $d$ = $3$ and asymptotically for large $d$. ...
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Markov chains why the sum of the incoming probabilities is greater than 1?

I have this Markov chain for a set of states, there are the outgoing and the incoming probabilities, it's an exercise, the teacher told us to check that the outgoing probabilities ($p_m^{out}=\sum_n ...
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Is it the only solution $\sinh(u-\xi+\eta)=\sinh(\eta)\rightarrow \eta=\pm\frac{i\pi}{3} \hspace{0.3cm} \& \hspace{0.3cm} u-\xi = \pm\frac{i\pi}{3}$

I am working on a problem that maps the 6 vertex model in statistical physics to alternating sign matrices. The main idea is that there is a one to one correspondence between alternating sign matrices ...
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Correlation function in the Langevin equation

So the Langevin equation of Brownian motion is a stochastic differential equation defined as $$m {d \textbf{v} \over{dt} } = - \lambda \textbf{v} + \eta(t)$$ where the noise function $\eta(t)$ has ...
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Why is the phase space expressed in terms of the cotangent space of the configurational space?

The following is taken nearly verbatim from section 1.1.2.1 Free Energy Computations: A Mathematical Perspective by Mathias Rousset, Gabriel Stoltz and Tony Lelievre. An excerpt, in which the part I ...
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Splitting a probability distribution function into the sum of another PDF and "something else". What is the formal way to talk about "something else"?

Please tell me if I'm in the wrong place to ask this question. I thought this might be the best place to ask since my question is about the formal properties of some function. If I have some dynamic ...
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Solving the Flory-Huggins counting problem when the polymers and solvent have colors

I am trying to construct a Flory-Huggins type lattice for a polymer and solvent with "colors". Essentially, each monomer segment and solvent segment has a color associated with it, and beads ...
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Saddle point method and radius of convergence

I have a question concerning the convergence of the saddle point method. Applying the method to a function $\rho(x)$, defined as \begin{align} \rho(x) = \frac{1}{2\pi}\int dk\, e^{\kappa f(k,\sigma)}, ...
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Transition probability exponential in distance or difference

Is there a situation, for example in physics or in dynamical systems, where we have a Markov chain where the transition probability between two states satisfies a law such as $$ p(y|x) = C e^{-d(x,y)} ...
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Geometry of mean-field family of distributions

I'm approaching this question with the point of view that we consider the space $P$ of probability distributions on $\mathbb{R}^n$ (say with finite second moment), and consider the subspace $Q$ of ...
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Higher derivatives of the log-partition function

I need higher derivatives of the log-partition function $Z(z)=\log \sum_i \exp(z_i)$, has anyone derived the formula? Looking at concrete values of derivatives up to order 8, evaluated at $z=(1,1,1)$ ...
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Invariant distribution for arbitrary Fokker-Planck equation

The general Fokker-Planck equation associated with the SDE $dx_t = f(x_t) dt + \sigma(x_t)dW_t$ reads $\partial_t \rho_t(x) = -\nabla\cdot(f\rho_t) + \frac{1}{2}\nabla\cdot\nabla\cdot(\sigma\sigma^T \...
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Simple / modular expression for $\mathbb E_v[(c^\top v)^2]$, in terms of correlation structure of the random vector $v$

Let $c=(c_1,\ldots,c_n)$ be a fixed vector and let $v=(v_1,\ldots,v_n)$ be a random vector in $\mathbb R^n$. Define $V := vv^\top$, a rank-one $n \times n$ random matrix. I'm interested in the ...
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Fundamental Solution / Greens function for Kramers / Langevin equation nowhere to be found

I am looking for a reference to the fundamental solution\Greens function of the two dimensional Kramers PDE $$\partial_t \rho(t,x,v) = -v\frac{\partial}{dx}\rho(t,x,v)+\gamma\frac{\partial}{\partial ...
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Is the classical definiton of probability correct, in case of probability distributions

Does the probability of success always depend upon the exact total number of favorable outcomes divided by the total number of outcomes, especially in the case of distributions? For example, consider ...
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Surface order large deviation in Ising ferromagnet

Background: A familiar behaviour of independent and identically distributed (i.i.d.) random variables $X_1, X_2,\ldots X_n$ is concentration: the probability that the sum $X_1+X_2+\ldots X_n$ exceeds $...
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Furutsu-Novikov Formula Generalisation

The Furutsu-Novikov formula gives the expectation value of a zero mean Gaussian process $z(t)$ and a functional of that process $R[z]$: $$\langle{z(t') R[z]}\rangle = \int^{t}_0 \mathrm{d}s K_2(t',s) \...
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Which PDEs have the Gibbs distribution as its stationary distribution?

The Fokker-Planck equation, which describes the evolution of the pdf of the position of a diffusing particle, has the Gibbs (or Boltzmann) distribution as its stationary distribution. Are there any ...
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Why $\left(\frac{\partial x}{\partial y}\right)_z=-\left(\frac{\partial x}{\partial z}\right)_y \left(\frac{\partial z}{\partial y}\right)_x$?

I was reading a statistical mechanics book, the author use: $\left(\frac{\partial x}{\partial y}\right)_z=-\left(\frac{\partial x}{\partial z}\right)_y \left(\frac{\partial z}{\partial y}\right)_x$ so ...
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Are all continuum systems limits of microscopic descriptions?

Many PDEs modeling nature are derived by reduced-order models of many-body descriptions, most famously the Navier-Stokes equations from the Chapman-Enskog expansion of the Boltzmann equation as well ...
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Rigorous proof of shifting of center during integration

In statistical mechanics, we see a lot of high-dimensional integrals. When using only a pairwise potential, we tend to "integrate away" one of the dimension by shifting it to the origin. For ...
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What does it expression tells us when we write Shannon entropy in terms of continues distribution?

Shannon information is defines as, \begin{equation} H(p_1, p_2,...p_n) = - \sum_{i=1}^{N} p_i log p_i.\end{equation} For continues distribution, we can write shanon information as, \begin{equation}H(x)...
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Integral kernel of the Legendre transform

First of all, I'm not sure, but I think the Legendre transform can be seen as a linear operator between the functions on a normed space and the functions on its dual. (A functional analysis approach ...
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How to calculate large deviations for functions of vectors on the n-sphere with delta functions?

I am copying the setup from Sec 2.5 of these notes describing large deviations of an overlap matrix: Let $\sigma_1,\cdots,\sigma_k\in\mathbb{R}^n$, with $k$ fixed and $n\rightarrow\infty$, and $\...
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Componentwise stationary distribution implies full stationary distribution?

Consider a $d$-dimensional stochastic process $X_t$ having density $\rho(X_t,t)$ at time $t$. Suppose the drift term $b$ of the process depends on the density $\rho = \rho(X_t,t)$: $$ dX_t = b(X_t,\...
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Gallavotti-Cohen action functional

For a time continuous Markov jump process a path $\left\{\sigma_{s}, 0 \leqslant s \leqslant t\right\}$ is time reversed as $\left\{\sigma_{t-s}, 0 \leqslant s \leqslant t\right\}$. The time reversed ...
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Pure state density-operator can not be expressed as non trivial linear combination

Basic Definitions: Let $\mathcal{H}$ be a Hilbert space. Define a density operator $\rho \in \mathcal{L}(H)$ (continous linear operators from $\mathcal{H}$ into itself) by $\rho$ is self-adjoint $\...
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Constraining many Gamma distribution

I'm working on a statistical model which involves many degrees of freedom $i=1...S$. Each degree of freedom is described by a gamma distribution with its own parameters, which we will assume to be all ...
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ConvNet pooling and RG flow

I have a physics/mathematics background and am learning about NNs. The 'pooling' operation applied to layers of CNNs seems to me to very closely resemble 'blocking' (eg decimation or 'majority rule') ...
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Expected amplitude of $N$ randomly shifted sine waves?

This question is inspired by this physics question, but it is purely mathematical. Let $\{f_i\}$ be a set of sine waves $f_i=a\sin(kx+\phi_i)$ where $i\in\{1,2,\dots,N\}$ and the $\phi_i\in[0,2\pi/k]$ ...
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How do we count the number of ways a loop can go around a torus?

As Cumrun Vafa explains in this video, he was able to calculate the number of micro-states for a black hole by counting the number of ways a loop can go around a torus. Obviously, there are infinite ...
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Fractal dimension of domain wall in Ising model at criticality

Consider the Ising model on an $L \times L$ lattice with periodic boundary conditions in the east/west directions and with spins on the north boundary fixed as $+1$ and the spins on the south boundary ...
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Does KL divergence *ever* satisfy the triangle inequality?

I know that KL divergence does not satisfy the triangle inequality, in that $$ D(q||p) \leq D(q||r) + D(r||p) $$ is not always true. This statement can be readily proven by providing a counterexample ...
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Partition function of an Ideal monoatomic relativistic gas | Solving an integral

I'm interested on finding the partition function $Z(\beta)$ of an ideal monoatomic relativistic gas. The partition function $Z(\beta)$ is given for this case as $$Z(\beta) \equiv \prod\limits_i^{N_A} ...
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Double sum of two delta functions

I am working on a problem in statistical mechanics involving a double sum of two dirac-delta functions. I am not sure how to $$ \text{relate} \quad \sum_{i=1}^{N} \sum_{j=1}^{N} \delta (r - r_{i}) \...
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Overdamped vs Underdamped Langevin

Consider the : $\textbf{Underdamped Langevin}$ \begin{align} dX_t&=V_tdt \\ \frac{m}{\gamma}dV_t&=-V_tdt-\nabla \phi(X_t)dt+\sqrt{2D} W_t. \end{align} I believe $m$ is the mass, $\gamma$ ...
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Functional equations: $f(x^3) = \frac{1}{3}f(x)$ and $g(x/b) = b^{a}g(x)$ (uniqueness)

Here are two functional equations, both of which come up in the theory of 2nd order phase transitions in statistical physics: $$f(x^3) = \frac{1}{3}f(x)$$ and $$g(x/b) = b^{a}g(x)\rm{,}$$ where $b$ is ...
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Gibbs measure is concentrated in the set of global minima

So I was reading Chii-Ruey Hwang's paper called "Laplace's Method Revisited: Weak Convergence of Probability Measures" I will sort of give the basic premise of the paper: Let $Q$ be a fixed ...
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What is $\mu (i, j)$ in the context of an Ising model on a small-world network?

Chapter 5 of Dynamical Processes on Complex Networks includes a discussion of an Ising model on a small-world network. It considers a system of $N$ Ising spins $\sigma_i = \pm 1$, $i = 1, ..., N$ with ...
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When does the correlation length increase and diverge in an Ising model?

I was reading chapter 5 of Dynamical Processes on Complex Networks and encountered the following paragraph: The importance of critical phase transitions lies in the emergence at the critical point of ...
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What does $\sum_{i/k_i = k}$ mean in the equation below?

I was reading chapter 5 of Dynamical Processes on Complex Networks, which discusses the Ising model, where I encountered the following equation for the average magnetization of the class of nodes with ...
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A modified 4-coloring problem

Given a graph $G$ with vertex set $V$ and edge set $E$, the standard coloring problem is to ask the number of ways that the graph can be colored with $k$ colors such that no adjacent vertices have the ...
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Brownian motion with 2 absorbing boundaries

Consider the stochastic process \begin{equation*} X_t = \mu t + \sigma W_t \end{equation*} where $W_t$ is the standard Brownian motion. Suppose that $X_0 = x \in (0,1)$ and that $0$ and $1$ are ...
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Probability of getting more heads than tails

The probability of getting heads in an biased coin is $p<\frac{1}{2}$ . We toss the coin $2n$ times.We win if the number of times heads appear is more than that of tails. We need to determine n (...
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