Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [statistical-mechanics]

The study of large, complicated systems by means of statistics and probability theory, in order to extract average properties and to provide a connection between mechanics and thermodynamics.

2
votes
0answers
43 views

Degrees of freedom in M-B distributions used for solving Poisson equation

I am having some trouble to understand the Maxwell-Boltzmann probability functions or more precisely how to use it in the Poisson equation. The problem is the following: consider the case of an ion ...
1
vote
1answer
85 views

Why isn't the chemical potential of photons equal to $\hbar\omega$

I am confused about the chemical of photons. Specifically, I don't understand what's faulty with the following derivation: Consider a black body at equilibrium temperature $T$, and let's focus on a ...
1
vote
0answers
43 views

Length of domain wall in Ising model

A subquestion of a homework for my statistical mechanics class this week asked of the 2d Domain wall Ising model approximation: "Now argue that for the formation of a domain wall separating the ...
1
vote
1answer
16 views

Statistical Mechanics Reaction problem

I am trying to do a problem that I encountered on a test, that I couldn't do. It reads as follows: "Consider a system comprised of two types of molecules, A and B, trapped in a volume V. They can ...
1
vote
2answers
33 views

Evaluating Specific Heat at low and high temperature limit

Solving a problem in statistical mechanics, I obtained the following expression: $$C_v=36Nk_B\frac{T^3}{\theta_D^3}\int_0^{\theta_D/T}\frac{x^3}{e^{x}-1}dx-\frac{9Nk_B\theta_D}{T(e^{\theta_D/T}-1)}$$ ...
0
votes
1answer
32 views

When heat flows from hot to cold, do individual particles share energy or is it just a statistical effect?

I'm struggling to get a clear understanding of why heat flows from hot to cold. I understand that temperature reflects the average kinetic energy of the particles, and that kinetic energy transfers ...
2
votes
0answers
27 views

Fisher Information in Statistical Mechanics

I am studying the canonical ensemble and it seems to me there is an analogy between derivatives of the partition function, which can extract energy momenta for the system and Fisher score /information....
0
votes
2answers
47 views

Evaluating the quality of Monte Carlo simulations for 3D Ising model

Suppose I have developed a new Monte Carlo method, and I plan to test this method on studying the magnetization of a 3D Ising model at some non-zero temperature $T$. The coupling is nearest neighbor, ...
1
vote
2answers
54 views

Fermi Dirac distribution derivation

Does anybody understand how my lecturer is normalising the probability distribution at the end to achieve the Fermi Dirac distribution? I don’t understand how he gets 0x1 or the denominator at all.
0
votes
1answer
38 views

Free Fall Conservation of Momentum

So I looked at the invariance of the Lagrangian under the Gallilei Transformations. So for the free fall we have the Lagrangian $$L = \frac{m}{2}\dot{z}^2 -mgz$$ Then I applied the transformation $$x\...
1
vote
1answer
24 views

Interpretation of change in entropy and how to formulate an expression for how much energy required to mix gases.?

From statistical mechanics one cab derive the change of entropy of the mixing of two ideal gases with the result $$\Delta S_\text{mix} = - n R \Sigma_{i} x_i \ln(x_i)$$ where $n$ is the total amount ...
1
vote
1answer
20 views

Disagreement about solving density of orbital per unit energy of photon gas in a cavity

A conducting 2-D cavity (of area $L^2$) contains a photon gas that satisfies the dispersion relationship: $$\omega^2 =\frac{4\pi^2c^2}{L^2}(n_x^2+n_y^2)$$ and I wish to find the density of the ...
0
votes
1answer
23 views

Calculating Total energy of 2D Debye monoatomic solid

I am trying to find the total energy of a mono-atomic 2D Debye solid. I started with the density of states: $$D(\omega)=\frac{A\omega}{\pi c^2} $$ where A is the area, $\omega$ the frequency and c ...
3
votes
1answer
31 views

Graph dimension and phase transitions

In this question I am mainly concerned with phase transitions associated with symmetry breaking. For phase transitions of this type, there are a number of results describing the relationship between ...
4
votes
0answers
44 views

What happens to topological insulators at finite temperature?

There is a similar question here, but I had a few things I wanted to ask. So basically pretty much all analysis/ theory of topological insulators is for pure wave-functions and conservative ...
1
vote
2answers
52 views

Thermodynamic description of few-body systems

How large should a system be to become thermal? Thermodynamic description is well-established for systems with large numbers (say, of order of $N_A\sim 10^{23}$) of constituents. Is there a "lower ...
0
votes
0answers
37 views

Field Theory: Converting $\int_0^{x_0} d^dx$ to $\int_0^{x_0} dr$, where $r=^{\textrm{def}}\|x\|$

For my Statistical Field Theory class (http://www.damtp.cam.ac.uk/user/tong/sft/sft.pdf), the prof converts integrals over each element of a vector $x$ into a single integral over the magnitude of the ...
1
vote
0answers
18 views

What is the Spectral Form Factor?

In many papers in Random Matrix Theory [1-3] related to quantum chaos (and, in particular, to the SYK model) they analytically continuate the partition function of the system $Z(\beta)$ into $Z(\beta +...
0
votes
0answers
25 views

What is a good introductory text on statistical mechanics for math-student [duplicate]

Here a "math-student" is an advanced undergrad or a 2nd-3rd-year grad student but only has taken the first-year undergraduate physics classes. Ice-type models, Yang-Baxter Equations are of particular ...
0
votes
0answers
12 views

Rotational contribution towards heat capacity of an ideal gas made up of arbitrary shaped particles

Consider the particles of an ideal gas to be arbitrarily shaped, but all particles have the same shape. The Hamiltonian has a translation term and a rotational term. The translation contribution ...
0
votes
1answer
28 views

Relation of Corresponding principle and law of large numbers

Is it possible that Corresponding principle can be derived from the law of large numbers? Also is the principle a postulate of Quantum Mechanics?
2
votes
2answers
95 views

Why do electrons revolve around the nucleus? [duplicate]

Why do electrons revolve around the atom's nucleus? Where does it get the energy for the revolution? Do the electrons stop revolving at absolute zero temperatures?
0
votes
0answers
26 views

Reference request: good introduction to landau theory and phase transition

My background is in statistics, so I am not well versed in the details of physics, lattices, and ideal gases, etc.--beyond my high school familiarity. However, I have been looking at some models of ...
1
vote
1answer
83 views

Need verification for the entropy equation in statistical thermodynamics

The relationship between entropy $S$, the total number of particles $N$, the available energy levels $E_j$ and a yet to be defined parameter $\beta$ is: $$S(\beta)=k_B \cdot N \cdot \ln\bigg(\sum_{j=1}...
0
votes
1answer
36 views

Statistical mechanics Calculating sums [closed]

The energy is given by: $$ E(\sigma) = -\sum_{i=1}^{N}\sigma_i + a \sum_{i=1}^{N}\sigma_i^2 $$ where $a$ is constant. I have to calculate the following: $$\left\langle \frac{1}{N}\sum_{i=1}^{N}\...
3
votes
0answers
71 views

Question about a hand-waving estimation done by Landau

I was reading the book written by Landau on statistical mechanics (Statistical Physics I 3rd Ed). In §1 there is a footnote that Landau wants to illustrate how accurate a statistical average can be ...
0
votes
0answers
12 views

How to compare scales in a thermodynamic system

Suppose we have a generic system with hamiltonian \begin{equation} H= JH_1 +\mu H_2 \end{equation} where $J$ and $\mu$ are couplings and $H_1$ and $H_2$ are just two dimensionless parts of the ...
0
votes
1answer
23 views

Fastest way heat water (or some other liquid or material)

If I want to heat lets say, 1 liter of water - would it be faster if I heat half a liter, and then another half a liter? or slower? my guess is that its not equal (lets assume that the time it takes ...
2
votes
4answers
93 views

Approximation of multiplicity when Ideal gas low density is applied $\frac{M !}{(M-N)!} \approx M^{N}$

Our lecturer today mentioned how a piston's head being at equal pressure maximised the multiplicity of states. He mentioned the following: If I have a fixed number of particles $N_A$ on left and $...
1
vote
2answers
48 views

Using Boltzmann distribution, what is the ratio of probabilities of two states?

I got the probability of state $i$ (in terms of Boltzmann distribution) as $$p_{i}=\frac{1}{Z_{i}}e^{-\epsilon _{i}/{kT}},$$ where $Z_{i}$ is the canonical partition function: $$Z_{i}=\sum_{i}e^{-\...
-1
votes
1answer
56 views

How did Carnevale et al. could argue mass in a ballistically aggregating systetm evolves as $t^{2d/(d+2)}$?

I am trying to understand this paper but I am not able to do it. Please help. See the paper here. First of all how could they write collision time $t_0$ as given there?. I am not able to get it even ...
9
votes
2answers
199 views

Spontaneous symmetry breaking: proving the equivalence of two definitions

This question can be posed for both quantum and classical set-ups. For concreteness, let me consider a local, classical Hamiltonian $H$. The expectation values I consider are with respect to the usual ...
1
vote
0answers
26 views

Metropolis method under the Gand Canonical Ensemble

I am writing code to simulate Langmuir adsorption using Metropolis method under the Gand Canoninal Ensemble. Aided by the Wikipedia Article, I derived the corresponding equation, proposed a sample ...
1
vote
0answers
35 views

Scaling limit of the Ising model with nonzero order parameter

I'm interested in simulating the continuum limit of the 2D Ising model $$H=J\sum_{\langle i j\rangle} s_i s_j+ h \sum_i s_i$$ In one dimension I can fix average magnetization $m=\langle s\rangle$ and ...
2
votes
1answer
48 views

Validity of Maxwellian distribution for interacting particles?

I have read in a few (relatively credible) sources (e.g. Cambridge Tripos exam) that the Maxwell-Boltzmann speed distribution can be valid for interacting particles, but I have not been able to find a ...
2
votes
1answer
40 views

Clausius paper “ On the motive power of heat & on the laws which can be deduced from it for the theory of heat”

My question is simple. Clausius expesses his fundamental proposition as follows: "In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to ...
1
vote
0answers
29 views

Is the Vlasov equation a good kinetic equation for solids or even liquids?

Suppose we know the intermolecular potential $V(r-r')$ acting between the positions $r,r'$. Then we have the kinetic equation for a one-particle Distribution function $f(r,p,t)$ given by: $\frac{\...
3
votes
1answer
72 views

Regarding the Boltzmann entropy formula, is the Boltzmann constant really arbitrary?

In the top answer to this question (Is the Boltzmann constant really that important?) I read that the Boltzmann constant is just a dummy factor which converts energy to temperature. But that allows ...
0
votes
0answers
33 views

Spinwaves, Mermin-Wagner theorem, Two-point correlation function and Heisenberg model

I was looking at the Mermin-Wagner theorem (as following the previous question) and the Heisenberg model seems to be presented, and they split the Hamiltonian $H$ in the matrix or vector n-components ...
0
votes
1answer
46 views

Chemical Potential of an Intrinsic Semiconductor

I was going through an article called "The chemical potential of an ideal intrinsic semiconductor" and I just cannot understand how the author gets that expression for the chemical potential. I know ...
1
vote
1answer
54 views

Understanding the statistical mechanics of Recombination Epoch (“Cosmology” Weinberg)

I also posted this in Astronomy.stackexchange, but realize it is primarily the physics I am trying to understand, not astronomy. In Steven Weinberg's 'Cosmology' Chapter 2.3 (pg 113), he begins with ...
1
vote
1answer
35 views

Can energy increase during neuron dynamics?

Consider a system of Ising spins, with Hamiltonian: $$H(\mathbf{S})=-\sum_{i, j} J_{i j} S_{i} S_{j} \qquad (1)$$ where $J_{ij}$ are symmetric real couplings ($J_{ij}=J_{ji}$) and $S_i=\pm1$. ...
0
votes
0answers
33 views

Single atom states with energy 0, probability and occupation number fermions and bosons

Two mutually non-interacting atoms are trapped in a double-well potential in equilibrium at a temperature $T$, such that an atom can only occupy two possible single- atom quantum states, $Ψa(x)$ and $...
1
vote
1answer
76 views

Momentum average on phase space for free particle

I'm studying from Greiner statistical mechanics, and he uses an approximation which I don't really understand. On averaging over many phase-space points we have $$\langle\vec{p}^2\rangle=3\langle ...
1
vote
0answers
50 views

Entropy density (entropy per particle)

How I can get this definition for an ideal gas: $s=C v \ln (Pρ^{−\gamma})+ \text{constant} $ ? In this answer it is mentioned that there is a connection with the Sackur-Tetrode relation of the ...
0
votes
1answer
34 views

What is the link between the meaning of excluded volume and its mathematical derivation?

When we examine real polymer chains we have to consider the interactions between single monomers. Therefore we consider a Lennard-Jones-like potential for bringing two monomers together and calculate ...
0
votes
1answer
24 views

Is the isothermal isobaric ensemble suitable to describe systems with inhomogeneous pressures (e.g. the atmosphere)

Let's consider a simplified atmosphere which can be described via a hydrostatic equilibrium: https://en.m.wikipedia.org/wiki/Hydrostatic_equilibrium We therefore have a pressure which depends on the ...
2
votes
1answer
64 views

Partition Function of System of Atoms in a Magnetic Field

Problem: Consider a system of N atoms in a magnetic field $B$ pointing along the z-axis. Each atom has angular momentum J and the Hamitonian of each atom is $$H=-MB=-g\mu_B B\sum_i^N J_z^i$$ where $...
0
votes
0answers
28 views

Chemical potential of a BEC

I know that the chemical potential of a BEC can be calculated with $\mu=\frac{\partial E}{\partial N}$, where $E$ is the energy and $N$ is the number of particles. For the Gross-Pitaevskii equation ...
0
votes
1answer
67 views

Why are statistical approaches used in Physics?

Statistical analysis is the analysis of a very large number of particles / sample space where the general behaviour and trends of these particles is studied. But this also means that its probabilistic....