Tagged Questions
0
votes
0answers
24 views
Existence of Boltzmann Distribution With Constraints [closed]
I have a problem with showing the existence of Boltzmann distribution given some constraints.
Consider $p_1,...,p_n$ a Boltzmann distibution, where $p_i=\frac{\epsilon^{-\beta \cdot E_i}}{\sum_{j}^{} ...
0
votes
0answers
28 views
Maxwell-Boltzmann distribution for transport equations
I have to calculate the transport coefficients for the Maxwell-Boltzmann distribution. But I'm not sure what distribution I have to use.
As far as I know it should not be the MB distribution for ...
1
vote
0answers
30 views
Calculating the change in entropy in a melting process
I have a homework question that I'm completely stumped on and need help solving it.
I have a $50\, \mathrm{g}$ ice cube at $-15\, \mathrm{C}$ that is in a container of $200\, \mathrm{g}$ of water at ...
2
votes
1answer
97 views
Energy density of a quantum mechanical ensemble
How do we determine the energy density of a given system? I have seen that the density operator
$$\rho~=~\frac{\exp(-\beta \hat{H})}{\text{tr}(\exp(-\beta \hat{H}))}.$$
What does this mean exactly ...
1
vote
1answer
64 views
Basic energy calculation for N identical spin system
We have a system that has N identical spins $n_i$, and each spin can be in state 1 or 0. The overall energy for the system is $\epsilon\sum_{i=1}^{N}n_i$.
My understanding: There is only one ...
1
vote
1answer
33 views
How can I find the temperature of this system?
A system was given a small amount of thermal energy dE, and its number of states G grew by 25%. How can I find the system temperature?
The system contains gas particles, I know that $dE << ...
0
votes
0answers
94 views
What is the condition for getting Bose-Einstein condensation? [closed]
Consider an ideal Bose gas in three dimension with energy-momentum relation E proportional to $p^s$ with $s>0$. Find the range of $s$ for which this system may undergo a Bose-Einstein ...
1
vote
1answer
62 views
Maximizing Multiplicity of Einstein Solid == (Temperature = $\infty$)?
If I have a system consisting of 2 Einstein solids (A and B) is it equivalent to say that maximizing the multiplicity of the ...
1
vote
2answers
134 views
Energy of particle in electric field
I'm taking a physics class and the professor teaches us really basic things in lecture and then gives homework way beyond what he taught in lecture. Obviously I need to find some resource other than ...
2
votes
0answers
159 views
Pauli paramagnetism for electrons with external magnetic field
Apparently it is to be shown that for electrons under an external magnetic field, in the limit as $B\to 0 $
$$
\chi = \frac{dM}{dB} \approx \frac{n\,\mu^{*^2}}{k\,T}\,\frac{f_{1/2}(z)}{f_{3/2}(z)}
$$
...
1
vote
0answers
77 views
Rotational Constant and Moment of Inertia of Fluorine gas
I have come across some homework question on thermodynamics which needs me to calculate $B$ of $F_2$
My attempt:
$B= \frac{h}{8\pi^2cI}$ where $I=\mu r^2=\frac{m_1m_2}{m_1+m_2} r^2$
Atomic mass of ...
2
votes
1answer
153 views
Canonical partition of a boson gas
I have a 1D gas made of $N$ particles placed in a harmonic potential well, so the Hamiltonian is:
$$ \mathcal H = \sum_{j=1}^N \left ( \frac{p_j^2}{2m} + \frac{1}{2}m\omega^2 x_j^2 \right )$$
The ...
0
votes
0answers
42 views
Gas of hard-core spheres
Consider a system of hard spheres of diameter $d_0$ at temperature $T$ and $V/N=v$. Discuss briefly the existence of the thermodynamic limit for the Helmholtz free energy on the basis of the ...
0
votes
0answers
71 views
Number of microstates of discretized paths
Let us consider a square grid, which has been rotated by 45deg. On this grid we define a path, the directed polymer, which starts at the origin ($t = 0$) and
extends in the positive $t$-direction ...
1
vote
1answer
220 views
Constant pressure and temperature mixing of 2 different ideal gases - possible work and heat?
A simple question I hope...
Initially, have two separate containers of 2 different ideal gases, 1.) N1, P, T, V1 and 2.) N2, P, T, V2.
After mixing, the pressure and temperature are still P and T, ...
1
vote
3answers
478 views
Partition function for quantum harmonic oscillator
Hi guys I'm currently trying to solve a mock exam for an exam in a few days and am a bit confused by the solutions they gave us for this exercise:
Exercise:
A solid is composed of N atoms which ...
1
vote
1answer
293 views
Calculate temperature of the earth through blackbody radiation
I don't understand the solutions to a problem about blackbody radiation and was wondering if anybody could help me out.
Here is the question:
The sun can be considered as a blackbody radiation ...
1
vote
0answers
90 views
helmholtz free energy of a polymer
You have a polymer chain of $N$ units, which is represented by $N$ independent springs in series. The springs are Hookean, with spring constant $L$, and the end to end vector is $\mathbf r$. So the ...
0
votes
0answers
50 views
energy and number of configurations for particular microstate [closed]
there are N molecules on an interface of A sites. if the molecule is perpendicular to the interface, it has an energy of -b, and if it is parallel to the interface it has an energy of -d (b>d>0). Let ...
1
vote
1answer
140 views
Occupied lattice sites, determining number of microstates and energy
A solid consisting of $N$ molecules on a lattice of $N$ sites is isolated from its environment, and has energy $E$. Each molecule is fixed in position and independent of all others. It can be in any ...
5
votes
1answer
184 views
few fermions in a harmonic trap — position density matrix from diagrammatics
I'm trying to calculate the momentum distribution of a 1D system of non-interacting identical fermions in a harmonic trap.
Given Feynman's answer (from his Statistical Mechanics book) for the ...
1
vote
1answer
588 views
Partition function of bosons vs fermions
I have two atoms, both of which are either bosons or fermions, with four allowed energy states: $E_1 = 0$, $E_2 = E$, $E_3 = 2E$, with degeneracies 1, 1, 2 respectively.
What's the difference between ...
0
votes
1answer
212 views
Expressions for canonical partition function and probabilities $p(E_i)$
Given an atom with 4 allowed states corresponding to the energy levels
$E_1 = 0$, $E_2 = E$, and $E_3 = 2E$ with degeneracies 1, 1, and 2 respectively.
How do I find the expressions for the ...
0
votes
1answer
281 views
A problem from Pathria, canonical ensemble, how to calculate $\left\langle \left(\Delta E\right)^{3}\right\rangle $
This is problem 3.18, from the book Statistical Mechanics by Pathria. Show that for a system in the canonical ensemble
$$\left\langle \left(\Delta E\right)^{3}\right\rangle =k^{2}\left\{ ...
2
votes
1answer
208 views
What is non-thermal plasma?
I read about non-thermal plasma, but I still have some questions:
The ions and neutral particles are not in thermal equilibrium with the electron, does that mean that the overall temperature is low ...
2
votes
0answers
69 views
Randomly sampling a “well-mixed” solution of Brownian particles
I place $N$ Brownian particles in $V$ liters of solution, shake until I assume that the particles are "well-mixed", and sample and randomly sample an $S$ liter volume. What is the probability ...
0
votes
1answer
481 views
Ideal gas with two kinds of particles, Grand canonical partition function
Consider an ideal gas contained in a volume V at temperature T. If all particles are identical the Grand canonical partition function can be calculated using
$$Z_g(V,T,z) := \sum_{N=0}^\infty z^N ...
3
votes
2answers
202 views
What fraction of electrons is captured in semiconductor defects?
I'm having trouble with the following exercise:
Some point defects (impurities, holes, etc.) in semiconductors can trap an electron in a localized state with energy $E_{1}$ and spin $-1/2$. A second ...
0
votes
2answers
159 views
How was transformed an integral below?
I know how transform an integral below,
$$
\iint f(\mathbf v_{1})f(\mathbf v_{2})d^3\mathbf v_{1}d^3\mathbf v_{2},
$$
using relative speed coordinates: we just use
$$
m_{1} \mathbf v_{1} + ...
-1
votes
1answer
216 views
Black body balloon in vacuum [closed]
The problem statement, all variables and given/known data
There is a perfectly spherical balloon with surface painted black. It is placed in a perfect vacuum. It is gently inflated with an ideal ...
1
vote
1answer
129 views
Quantum Stat-Mech Proof of an Inequality for the Partition Function
I have the following problem that I was unable to solve for class, but I had a couple first steps that I started with that I am unable to finish. I know I can't get this since it's already been ...
4
votes
1answer
731 views
Infinite-range 1D Ising model + Hubbard-Stratonovich-Transformation
I have a probably quite simple question RE the HST.
After some work, I obtain as the partition function for the infinite range 1D Ising model
$$Z = \int_{-\infty}^\infty \frac{dy}{\sqrt{2\pi / ...
3
votes
2answers
315 views
Paramagnet: Negative specific heat?
for a simple paramagnet ($N$ magnetic moments with values
$-\mu m_i$ and $m_i = -s, ..., s$) in an external magnetic field $B$, I have computed the Gibbs partition function and thus the Gibbs free ...
3
votes
0answers
451 views
How do I derive the critical temperature for bose condensation in two dimensions?
In class we derived the 3D case, but there's a step I don't understand:
$$ N = g \cdot {V \over (2 \pi \hbar)^3} \cdot \int\limits_{0}^{\infty}{1 \over{e^{\left( E_p \over{K_B T}\right)}-1}} d^3 p = ...
