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Why Normalise by $h$ in the Partition Function for Classical Harmonic Oscillator?

In classical continuous statistical mechanics, we still introduce a notion of a microstate. The following answer is extracted from Wikipedia: A microstate occupies an extended region in the $2n$-...
Qmechanic's user avatar
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Physical Interpretation of BEC Formation

What I get from Einsteins Original work (https://www.uni-muenster.de/imperia/md/content/physik_ap/demokritov/mbecfornonphysicists/einstein_1924_1925.pdf) is that he found, that the numbers of atom n ...
kai90's user avatar
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Spontaneous symmetry breaking in phase transitions

Now suppose we are at high T. The spins now can have random directions in the ground state and they are not correlated among each other. But anyway I would say that even now if I do a rotation I could ...
MadMax's user avatar
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Why Normalise by $h$ in the Partition Function for Classical Harmonic Oscillator?

The $N$ is not really the Planck's constant $h$. It is denoted as such because that was the convention. This has to do with the history of the subject. Statistical mechanics, in its classical form was ...
Soham Mitra's user avatar
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Why is the number of microstates corresponding to a macrostate even finite for an ideal gas in a box?

For continuous systems, the definition slightly changes. I will quote the following definitions from some lecture notes on soft condensed matter (they are not online sadly): Let $\Gamma=(\mathbf r^N,\...
AccidentalTaylorExpansion's user avatar
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What is the resolution of the Gibbs paradox?

Daan Frenkel discusses this problem very nicely in Why colloidal systems can be described by statistical mechanics: some not very original comments on the Gibbs paradox (preprint). (Colloidal ...
Wrzlprmft's user avatar
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How can the Maxwell-Boltzmann distribution be reconciled with the Boltzmann distribution?

The simplest way to address this is to note that you seem to be missing a piece of the full Boltzmann distribution! The Boltzmann distribution for the canonical ensemble has the form, \begin{gather*} ...
Matt Hanson's user avatar
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Chemical potential of ideal solution in grand canonical ensemble

Even though I follow a bit different way compared to what you have, I will provide an answer that it end up with the same equation. First step: Derive the chemical potential formula from the Grand ...
Theodoros's user avatar
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What are the effects of intramolecular on intermolecular forces?

While I certainly have no idea what you mean by "edgy" physics, I will attempt to allay some of your qualms. First of all, we need to clarify what would constitute an effect from ...
Matt Hanson's user avatar
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Distinguishability in Maxwell-Boltzmann statistics

For the record, the three regimes agree in the dilute limit, i.e. when $N_i\to0$ when fugacity $z = e^{\beta\mu}$ vanishes (not when $z\to+\infty$ as you claimed, this is perhaps due to your sign ...
LPZ's user avatar
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Fermi-Dirac Distribution for Multiple Species

The sum, $n_1 + n_2$, of two Fermi probabilities, as you call it, can exceed $1$ and therefore cannot be considered the correct probability. Instead, for the case of two independent ideal Fermi gases ...
Gec's user avatar
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Correlation functions zero for repeated creation or annihilation operators

Okey, the zero has to come from computing the trace $$ {\rm Tr}(R_0 b_j^\dagger b_k^\dagger)=0, $$ but I don't see easily why this is the case. Here $R_0$ is the density matrix of the bath composed ...
hft's user avatar
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2d Ising model with a longitudinal field -- do the low and high temperature expansions converge for all nonzero $T$?

The following is a partial answer. I realized shortly after posting my question that there is a good reason to expect that the high temperature expansion will have finite radius of convergence in $T$ ...
user196574's user avatar
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Can a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry generated by $\prod_{i} \sigma^x_i$ and $\prod_{i} \sigma^z_i$ be broken in a spin-$1/2$ chain?

If you do not want to impose translation symmetry, there are trivial (but valid) examples. If the number of sites is a multiple of 4, $$ H = -\sum_{i \text{ even}} (\sigma^x_i\sigma^x_{i+2} + \sigma^...
Nandagopal Manoj's user avatar
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Distribution of states of H$_2$ gas for a given temperature

The total energy for the vibrations and rotations is just the sum of the vibrational and rotational energies, \begin{gather*} E = E_n + F_{n,J} \end{gather*} where $E_n$ is the vibrational energy and $...
Matt Hanson's user avatar
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2 votes

Calculation of canonical partition function for fermion system with degenerate energy levels

I think LPZ already gave a very good answer, so here I'll just add a comment and a general picture. In some sense, you can say that this kind of problem is exactly one of the reasons you would like to ...
Jun_Gitef17's user avatar
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How to go from probability distribution to transitions probability distribution?

There are many possibilities. If I understand correctly, you are asking given a distribution, how to get a Markov chain (like in your example, or a stochastic process more generally) which leaves it ...
LPZ's user avatar
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Calculation of canonical partition function for fermion system with degenerate energy levels

In general it is easier to compute the canonical partition function from the grand canonical one. For a direct computation of the canonical ensemble, it is also easier to think in terms of orbitals ...
LPZ's user avatar
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Bose-Einstein statistics: What would the average or expectation value of velocity $\langle v \rangle$ be for an ideal Bose gas?

I will consider a massive, non relativistic, Bose gas, in the thermodynamic limit and in dimensionless units so that $m=\hbar=k_B=n=1$ (the final variable is particle density). The parameters are ...
LPZ's user avatar
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Density of states - as a function of electron velocity

Consider electron gas moving in 1D ring with length $L$(in thermodynamic limit the boundary effects are not important), with periodic boundary condition, the single particle state can be written as: \...
Yakumo Ran's user avatar
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First excited states of the Heisenberg model on a bipartite lattice/graph

The other answer gives a good discussion of magnons, but there are excitations much lower in energy: global motion within the degenerate continuous symmetry breaking manifold itself. At finite system ...
mbintz's user avatar
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Marginal characteristic function from a multivariate charateristic function

I think it's simpler: given the characteristic function $$G(k_1, k_2) = \int dx_1 dx_2 e^{ik_1 x_1} e^{ik_2 x_2} p(x_1, x_2) \, ,$$ setting $k_2=0$ gives $$G(k_1, 0) = \int dx_1 e^{ik_1 x_1} \int dx_2 ...
MBolin's user avatar
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Why is the volume of an isolated system conserved?

If you use the Gibbs free energy to determine the equilibrium state of the system (a subsystem of SR), it implies that you are considering a process at constant temperature and pressure. You ...
Tanmia's user avatar
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What is the signal of a spin wave?

There are many thorough references on this topic. Theoretical Tools for Spin Models in Magnetic Systems - Pires Fundamentals of Magnonics- Rezende Statistical Mechanics of Magnetic Excitations - ...
catalogue_number's user avatar
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Relation between thermal expansion coefficient and Poisson ratio

There's no general link, no. The thermal expansion coefficient is $\alpha\equiv\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P$; the Poisson ratio is $\nu\equiv-\frac{\varepsilon_\mathrm{...
Chemomechanics's user avatar
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Information theory in classical mechanics

Chaos really plays an essential role here to obtain statistical mechanics out of Newtonian mechanics. As a matter of fact, if you have an integrable system, it is known that the system does not ...
Jun_Gitef17's user avatar
3 votes

First excited states of the Heisenberg model on a bipartite lattice/graph

At least the square-lattice Heisenberg model has been studied very thoroughly, using many state of the art excat techniques (DMRG, QMC, ...); see this recent work for a thorough review of work on the ...
catalogue_number's user avatar
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Onsager-Machlup functional and the Boltzmann distribution

One can find the extremum trajectory and the quadratic fluctuations around it - by usual procedure of varying action. (If I am not mistaken, this leads directly to Newton equations with damping.) This ...
Roger V.'s user avatar
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Definition of the pressure in statistical mechanics

After some thought, I think why $S$ should be kept constant can be understood as follows. Microscopically, the pressure exerted by an eigenstate of the system on the wall is $$ P_n = - \frac{\partial ...
poisson's user avatar
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1 vote

Determining free Fermi/Bose gas chemical potentials, given temperature, number density, and baryon density

Your expression for the chemical potential as a function of density is the usual text book result for a non-interacting classical gas (although $n_{max}$ is just terrible notation). I have been ...
Thomas's user avatar
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2 votes

A problem to find the the chemical potential from the free energy (Ginzburg-Landau/Cahn-Hillard)

The free energy is $F[c] = \int d^n x \frac{1}{4}{\left(c^2-1 \right)^2 + \frac{\gamma}{2} \lvert \nabla c \rvert^2}$. In the grand canonical ensemble, the chemical potential is fixed, and the density ...
Archisman Panigrahi's user avatar
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A problem to find the the chemical potential from the free energy (Ginzburg-Landau/Cahn-Hillard)

By following the suggestion of @ArchismanPanigrah. (2) follows straightforwardly from (1) by using the relation of functional derivative (provided in ref. 2) and by noticing that $c$ and $\nabla c$ ...
math-int's user avatar
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Definition of the pressure in statistical mechanics

While you can choose any two independent variables as your basis function to express any quantity, you need to be careful in the choice of your variables when defining new quantities as derivatives. ...
LPZ's user avatar
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3 votes

Is it possible to have an anisotropic temperature to a Brownian motion?

Absolutely Not! But don't let that stop you from musing: For an ideal gas, the temperature is based on the average energy of an ideal gas, via: $$ \frac 1 2 m\bar v^2 = \frac {3}{ 2} kT $$ so $$ T = \...
JEB's user avatar
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Callen's Statistical and Thermo approach

By "fundamental theory," Callan means that when you do statistical mechanics, you appeal to the "underlying physics" in a direct sense. If you're studying the statmech of a gas, ...
Rokas Veitas's user avatar
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Statistical independence of $x,y,z$ dimensions for Maxwell velocity distribution function

the assertion that $v_x = v$ doesn't mean much the other velocities, other than they are unlikely to be large. There is no preferred direction and physics doesn't care about your coordinates. Try to ...
JEB's user avatar
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2 votes

Tight binding of monoatomic chain

If you increase the size of your unit cell, you decrease the size of the unit cell in reciprocal space. In one dimension, the length of the unit cell in reciprocal space is $2\pi/a$, where $a$ is the ...
march's user avatar
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The partition function of a particle in a magnetic field diverges. Why?

Making everything dimensionless so that $\hbar=1$: $$ H = \frac12(p-A)^2\\ A = \frac\omega2(-y,x) $$ with $\omega$ the cyclotron frequency, the usual theory of Landau level gives the spectrum of the ...
LPZ's user avatar
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1 vote

The partition function of a particle in a magnetic field diverges. Why?

I may have solved my issue (thanks to @By Symmetry for the hint). Indeed, the divergence appears to be related to the infinite area covered by the quantum system in 2D. We could consider the similar ...
Cham's user avatar
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1 vote
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Can energy be any energy in canonical ensembles?

When we consider the canonical ensemble (so constant temperature $T$) we have that given that the system consists of a set of energies ${E_i}$, the probability of finding the system in energy $E_j$ is ...
OonyXx's user avatar
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How to derive Shannon Entropy from Clausius Theorem?

You can derive the Gibbs entropy, which is simply the Shannon entropy multiplied by the Boltzmann constant, from the Boltzmann entropy and a multinomial distribution. This video proceeds with the ...
Leonardo Castro's user avatar
1 vote

Proof that Bose-Einstein condensation (BEC) discontinuity happens at the thermodynamic limit

The issue is that the formulas you use are asymptotic expansions in the thermodynamic limit. However, this limit is in some sense pathological due to the emerging phase transition (not uniform). You’...
LPZ's user avatar
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