Questions tagged [statistical-mechanics]
The study of large, complicated systems employing statistics and probability theory to extract average properties and to provide a connection between mechanics and thermodynamics.
7,306 questions
1
vote
1
answer
40
views
Phase transition for first moments in the 2D random-bond Ising model
I have a very basic confusion about the 2D random-bond Ising model on a square lattice with Boltzmann weight
$$\omega(J_{ij},\sigma_j)=\prod_{ij}(1-p)^{\delta_{J_{ij}=1}} p^{\delta_{J_{ij}=-1}} e^{-\...
- 395
0
votes
0
answers
39
views
Maxwell-Juttner distribution and its derivation
I am looking for good notes on the Maxwell-Juttner distribution and its derivation. I stumbled across this post Calculation of Maxwell-Juttner distribution integral but I was unable to find reference ...
- 9
0
votes
1
answer
67
views
What causes the difference in the graphs showing the dependence of a molecule's potential energy on distance?
"The potential curve will have the shape shown in the figure above if the molecules approach each other in plane A along the line connecting their centers".
"If the molecules approach ...
- 27
2
votes
1
answer
120
views
What does grand potential for $2D$ systems mean?
In $3D$ the grand potential $\Omega = -PV$ has a well-defined physical meaning. But for $2D$ systems what would $\Omega$ actually mean physically? Dimensionally speaking it looks like $\Omega = -\...
- 1,726
3
votes
0
answers
68
views
Diffusion equation after one time-step [closed]
Consider the equation:
$$
\partial_tu=D\partial_{xx}^2u
$$
with reflecting boundary condition at $x=0$ and with $u(x,0)=\delta(x)$ as an initial distribution.
First question: How should I understand a ...
- 31
1
vote
0
answers
61
views
Kolmogorov complexity in physics - applications [duplicate]
Has there been application of Kolmogorov complexity to physics? Shannon entropy has an interpretation from thermodynamics. In principle Kolmogorov complexity and Shannon entropy are the same but is ...
- 11
2
votes
1
answer
233
views
Alternate statement for Liouville's theorem: Can you say that the relative velocity between any two representative points is zero?
So Liouville's theorem basically says the local density of representative points stays constant or that the flow of representative points resembles that of an incompressible fluid. Can you then say ...
- 31
3
votes
1
answer
319
views
Reweighting method for Metropolis algorithm
I read the Wiki page about the reweighting procedure as a way to use Monte Carlo methods with the sign problem, but I'd like to know more about how this could be implemented in the Metropolis ...
0
votes
0
answers
18
views
How to choose the initial state of the reverse trajectory when applying fluctuation theorems and calculating entropy production?
In the literature on fluctuation theorems and stochastic entropy production, I often find different choices for the initial state of the reverse trajectory. This choice seems to depend on how the ...
- 80
1
vote
4
answers
183
views
Why can a change in randomness (entropy) be mathematically expressed as $\frac{Q_{rev}}{T}$?
The change in entropy of a system resulting from a thermodynamic process can be calculated as:
$$\Delta S = \frac{Q_{rev}}{T}$$
where $Q_{rev}$ is the heat evolved/absorbed in the process had it ...
- 37
1
vote
0
answers
31
views
Confused about derivation - electron gas with binary collisions and impurities scattering
I am confused about a derivation for the behaviour of electrons in a conductor with binary collisions, and scattering due to static impurities. The derivation begins as follows:
The distribution ...
1
vote
1
answer
115
views
When are $N$ particles in $D$ dimensions the same as 1 particle in $ND$ dimensions?
Here is the Hamiltonian for a particle moving in a 2-dimensional box:
$$
H = \frac{p_x^2}{2m} + \frac{p_y^2}{2m} + V_2(x,y) .
$$
Here is the Hamiltonian for two non-interacting particles moving in a 1-...
- 66.5k
9
votes
4
answers
1k
views
What is the meaning of "knowing all the Green functions implies knowledge of the full theory"?
I have encountered many times the sentence in the title. Either written in books or told by more experienced friends, there seems to be a consensus that "If we know all Green functions of a ...
- 251
1
vote
1
answer
70
views
Stochastic calculus clarification
In an introductory statistical physics class, the overdamped Langevin equation was introduced as: $\frac{dx}{dt} = \frac{1}{\gamma}\xi$, where $\xi$ is the white noise representing the fluctuations. ...
- 310
1
vote
1
answer
63
views
How does one go from the short-time stochastic evolution of a single quantum trajectory to the ensemble master equation?
In the quantum-jump (Monte Carlo) method described in Quantum Optics by Gerry and Knight, consider a single-mode cavity field where photons are lost at rate $\gamma$. For a single trajectory $|\Psi\...
- 288
2
votes
0
answers
96
views
Cumulant expansion in computing generating functional at third order
In Kerson Huang's Quantum Field Theory 2nd, p 278, the author introduces "cumulant expansion"
from
$$
\ln \langle e^x \rangle = \ln \sum_{n=0}^\infty \frac{\langle x^n \rangle}{n!} = \...
- 63
2
votes
0
answers
88
views
Computing the ground state energy using perturbation theory?
I know that, in quantum statistical mechanics, the ground state energy of a system at finite temperature can be computed from the partition function:
$$Z_{\beta} = \sum_{n=0}^{\infty}e^{-\beta E_{n}} =...
- 251
3
votes
1
answer
302
views
What is the average formula for interacting picture at positive temperature?
Although this looks a very simple question, it has been difficult to find an answer on textbooks since many of them develop the theory of interacting many particles for zero temperature and just ...
- 1,367
3
votes
1
answer
219
views
How to derive Fourier-Laplace transform of random walk master equation on 2D lattice (Part 2)?
This question is linked with this question and is related to this paper.
The Fourier-Laplace transform is given by:
$$P(q,r,s)=\sum_{t=0}^{\infty}\sum_{m,n=-\infty}^{\infty}\frac{e^{iqm+irn}}{(1 + s)^{...
- 291
4
votes
1
answer
208
views
Is the second law of thermodynamics only a statistical and emergent law or not?
According to this answer,
According to the Fluctuation Theorem one can derive a specific formulation of the Second Law of Thermodynamics as manifested in the statistical effects over large number of ...
- 149
1
vote
2
answers
148
views
How to derive Fourier-Laplace transform of random walk master equation on 2D lattice (Part 1)?
I have certain gaps in clearly understanding the derivation given in this paper. Suppose a particle moves on a 2D lattice randomly. The probability of going in any one direction outb of four available ...
- 291
3
votes
3
answers
2k
views
Will gases contained inside a box eventually reach zero temperature?
Suppose I have some amount of gas inside an isolated box.
As the gas molecules continue colliding with box wall and each other, there will be a loss of energy due to these collisions.
So, will over ...
- 393
1
vote
1
answer
170
views
Nuclear Magnetic Resonance interaction of individual $μ_i$ with $B_1$
In Nuclear Magnetic Ressonance (MRI purpose) I am trying to understand how the magnetic field perpendicular to $B_0$ with intensity $B_1$ which supposedly is time dependent (the magnetic field is ...
- 247
4
votes
1
answer
209
views
Law of increasing entropy when the temperature is not well defined
In an isolated system with well defined initial and final states, we can argue that $\Delta S\geq{0}$ easily. Consider the system's initial and finial states are denotes by points $A,B$. Then the ...
- 79
0
votes
0
answers
50
views
How to incorporate boundary conditions in mean field descriptions while deriving macroscopic equations from microscopic stochastic processes?
The question is linked to this question.
The microscopic stochastic processes are defined using homogeneous jump probabilities between sites. The assumption will be broken when we have physical ...
- 291