Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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. Reference: Wikipedia.

The classical view of the universe was that its fundamental laws are mechanical in nature, and that all physical systems are therefore governed by mechanical laws at a microscopic level. These laws are precise equations of motion that map any given initial state to a corresponding future state at a later time.

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Number of ways to stack LEGO bricks

One of the most surprising combinatorial formulas I know of counts the number of LEGO towers built from $n$ "$1 \times 2$" blocks subject to four rules: The bricks lie in a single plane. Each brick is offset by 1 stud (as in a brick wall). The…
Peter Kagey
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Statistical Mechanics References

I need good references on the subject of Statistical Mechanics having a mathematically rigorous perspective. Almost all physics books on this subject do not care about definitions/rigour/proofs etc. They differentiate discrete functions without any…
heaven-of-intensity
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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, if we let $S_n$ act on $(\mathbb{C}^N)^{\otimes…
Daniel McLaury
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Derivation of the Boltzmann factor in statistical mechanics

I have seen similar derivation of the Boltzmann factor many times before, (http://micro.stanford.edu/~caiwei/me334/Chap8_Canonical_Ensemble_v04.pdf , just for example), which I think is incomplete. The argument is as follows: Consider the system…
velut luna
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Jones Polynomial from Statistical Mechanics

I've been told that, given a knot projection, there is a way of associating a statistical system in such a way that the partition function of the system corresponds to the Jones polynomial of the original knot. I have a rough understanding of how…
Mark B
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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 knows the partition function $Z$, than one…
Nikolaj-K
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Why are Fokker–Planck equation and Feynman path integral formalisms equivalent?

Feynman path integral is equivalent to Fokker–Planck equation. This is mentioned here, but it's not clear why. This page says Schrodinger equation is also equivalent to Fokker–Planck equation which makes me even more confused. Basically what I…
0x90
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An intuitive explanation of how the mathematical definition of ergodicity implies the layman's interpretation 'all microstates are equally likely'.

I'm self-studying Statistical Mechanics; in it I got Fundamental Postulate of Statistical Mechanics and that took me to ergodic hypothesis. In the most layman's language, it says: In an isolated system in thermal equilibrium, all microstates are…
user142971
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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: $$ S= -K\sum_{i=1}^np_i\log (p_i) $$
Wallace Cheng
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Is the Entropy a Function or a Functional?

As in the title, I was wondering whether the entropy of a system (it can be any entropy, from Boltzmann to Renyi etc, it is of no importance) is a function or a functional and why? Since it is mostly defined as: $$S(p)=\sum_{i}g(p_i) $$ for some $g$…
Bazinga
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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 affects its direction and perhaps its speed so that…
mich
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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 model. The review article is Percolation and number of…
Elias Costa
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Taylor expansion using stirling's approximation

This is from a statistical physics problem, but it is the mathematics behind it that I am stuck on here: Consider a large number $N$ of distinguishable particles distributed among $M$ boxes. We know that the total number of possible microstates…
BLAZE
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Help with integral from Boltzmann equation

I have a function $$ g\left(x,v,t\right) = u\left(x,t\right)\cdot v + \theta\left(x,t\right)\frac{1}{2}\left(\left\lvert v\right\rvert^2 - 5\right) $$ where $g(x,v,t),\theta(x,t)$ are scalars and $u(x,t), v\in \mathbb R^N,N=2,3$. I also have a…
Calvin Khor
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What does $-p \ln p$ mean if p is probability?

In statistical mechanics entropy is defined with the following relation: $$S=-k_B\sum_{i=1}^N p_i\ln p_i,$$ where $p_i$ is probability of occupying $i$th state, and $N$ is number of accessible states. I understand easily what probability is: for a…
Ruslan
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