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.
375 questions
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Diffusion equation after one time-step
Consider the equation: $$ \partial_tu=D\partial_{xx}^2u $$ with reflecting boundary condition at $x=0$ and with $u(x,0)=\delta(x)$ as an initial distribution. First question: How should I understand a ...
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1 answer
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Understanding the effect of idempotence on mixing in a Markov chain
I'm trying to understand how to get mixing time results when transitions in a chain are idempotent i.e. $P^2=P$ for transition $P$. Following is a simplified example to illustrate my confusion. Toy ...
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2 votes
1 answer
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Integration by parts from perceptron capacity calculation
I am working through Chapter 6 of the book Statistical Mechanics of Machine Learning by Engel and Van den Broeck. I am stuck on the following integral, going from line 6.13 to line 6.14 of the book. I ...
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Computing $P$ from $H$ in $H = - P \log_2 P$
I am currently working through an example of a smoothing technique for information. I have finished applying the technique to my data and now want to figure out the probability of the item after it’s ...
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2 votes
1 answer
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Asymptotics expression in a long-range field theory
In studying a one-dimensional field theory with long-range interactions, I obtain the following scaling function for the two-point correlation function: $$ \Psi\!\left(\frac{r}{\xi}\right) = \int_0^{\...
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Deriving a Probability Distribution Function for Speed Function for Velocity.
Consider a probability distribution for velocity $P(\vec{v})$ function that depends only on the magnitude $v$ of the vector. I would like to derive a probability distribution for speed. For this, I ...
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References for continuous Thomson problem in disk
Consider problem of minimizing total Coulomb energy of configuration of $N$ particles inside unit disk: $$\sum\limits_{i\not =j}\frac{1}{|x_i-x_j|}\longrightarrow \min\limits_{|x_i|\le 1}$$ I think we ...
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Topologies and sigma algebras in infinite spin lattice systems
I don't know most of the terminology, so I appreciate remarks that make further research for me easier. In statistical mechanics we have defined the configuration space as of a system of spins in the $...
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5 votes
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Invariant $\phi_0$ Solutions of $\int^{\pi}_{0} \sin(\phi)I_0(\nu\sin(\phi_0)\sin(\phi))e^{-\nu\cos(\phi_0)\cos(\phi)}d\phi$
The following integral arises from a physics calculation of the finding the expected relative velocity of a known particle moving through a Maxwell-Boltzman distribution. I have grouped together ...
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$x-y$ symmetry breaking in fully packed dimer model.
Suppose I have a fully packed hardcore dimer config on the square lattice with periodic boundary conditions in both directions. The square lattice is bipartite and ca be divided into $A$ and $B$ ...
2 votes
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Optimizing Returns Under Constraints with Fixed Boundary Self-Avoiding Walks: Proof with SLE Theory
Say you have a field, $100$ m by $100$ m. You stand on one end at point $(0,50)$, and you want to get across to point $(100,50)$. In this field are randomly scattered 1500 pieces of gold. Now, you ...
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How can I solve an equation featuring the average of N sigmoids?
I am using a statistical mechanics framework to solve an evolutionary game theoretical problem where different individuals can be in a +1 state (cooperate) or 0 state (defect). Using an Ising-model-...
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Curie-Weiss partition function seems to be written wrongly in a book
I'm working on Statistical Physics Methods in Optimization and Machine Learning by Krzakala and Zdeborova, which can be accessed here: https://sphinxteam.github.io/EPFLDoctoralLecture2021/Notes.pdf. ...
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7 votes
1 answer
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How to express the 6D phase space volume element in coordinate-free notation using the metric tensor?
I'm studying statistical mechanics and kinetic theory and trying to understand how to express the 6D phase space volume element in a coordinate-free form, especially in curvilinear coordinate systems ...
1 vote
1 answer
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Closed-Form sinh Expression for the Partition Function of a Non-Interacting Spin Chain
I’m trying to compute the closed form partition function (sinh) for non-interacting spins in an external magnetic field, but I get stuck transforming the finite geometric series into a closed sinh ...
