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.

0
votes
0answers
30 views

Modelling liquids like water with BBGKY-hierarchy

The BBGKY hierarchy is a well-known useful possibility to derive kinetic equations for gases and Plasma. The N-particle System is reduced to few-particle Systems by Integration over many Phase space ...
1
vote
1answer
36 views

Taking moments of the Vlasov equation

Given the following term: $\nabla_{v} \cdot \left[ \frac{e}{m_{s}} (\textbf{E} + \textbf{v} \times \textbf{B})f_{s} \right]$ where $\textbf{E}$ and $\textbf{B}$ are the electric and magnetic fields ...
3
votes
0answers
48 views

How do you experimentally measure the chemical potential of a gas?

How does one measure the chemical potential of a substance/ thermodynamic system? I am asking this question for two reasons: (1) The measure on phase space: Textbooks typically state that one should ...
2
votes
1answer
86 views

Partition function in spherical coordinates

Suppose I write the Hamiltonian/energy of my system in spherical coordinates ($r,\theta,\varphi$) with conjugated momentums($p_r,p_\theta,p_\varphi$). How do I calculate the partition function? If ...
0
votes
1answer
41 views

Wiedemann-Franz law derivation book recommendation

Can you recommend a good book with a thorough derivation. I know I'm more likely to find in a condensed matter book or a book on conductors but any recommendation would be appreciated, bonus if it ...
2
votes
0answers
27 views

Correlation length amplitudes in Ising 2D model

I am reading the article about Universal amplitude ratios in the 2D Ising model (https://arxiv.org/abs/hep-th/9710019) by G. Delfino. I have a question about page 3 of the paper. For a magnetic ...
1
vote
1answer
66 views

Continuous Transition of Degrees of Freedom in Thermodynamics With Simple Example

In thermodynamics books I have read, I have often come across statements about how certain degrees of freedom are relevant only at certain temperatures (such as the vibration degrees of freedom of ...
4
votes
1answer
134 views

Interpreting heat using information entropy

In an answer to the post about Microscopic Definition of Heat and Work, Ronan says, $$<dE> = \sum \epsilon_idp_i + p_id\epsilon_i$$ We can see that the change in average energy is partly ...
1
vote
1answer
83 views

What is the relation between chemical potential and the number of particles?

Chemical potential is defined as the change in energy due to change in the number of particles in a system. Let we have a system which is defined by the following Hamiltonian: $$H = -t \sum_i^L c_i^\...
0
votes
1answer
35 views

Integral limits in phase space

If I am calculating the partition function for $H=cp$, ultrarelativistic gas in three dimensions. And by breaking down $d \Gamma$ into $dq$ and $dq$ and further using spherical coordinates I will get $...
3
votes
0answers
41 views

How bad is it if we don't know the distribution of an average?

Let's assume that it takes on average $\langle W\rangle$ work to perform some process. While we do know that the fluctuations i.e. the difference between single realizations of the process $W_1$,$...
2
votes
1answer
46 views

Liouville equation with Dirac delta as probability density

I would lke to know if the probability distribution given by $$\rho(q,p,t)=\delta(q-q(t),p-p(t)) $$ with the initial condition $\rho(t=0)=\delta(q,p), $ where $q(t)$ and $p(t)$ are trajectories ...
1
vote
1answer
52 views

What is the order parameter of 2D generalized $XY$ model?

I'm now studying the phase transition of 2D generalized XY model. This model considered here has a mixture with ferromagnetic and nematic-like interactions, $$\mathcal{H}=-\sum_{\langle i j\rangle}\...
0
votes
1answer
45 views

Why can we say that at zero absolute temperature is there only one accesible state?

While studying the microcanonical ensemble, the entropy definition requires that at T=0 there is only one accesible state so that the entropy, S=0. Why is it true?
1
vote
1answer
83 views

Phase separation in physics

I would look to familiarize myself with the current literature of phase separation. If one can direct me to statistical/thermodynamics theories of phase separation. Has phase separation been ...
0
votes
2answers
38 views

What is role of nucleation centre in the formation of ice?

Water must be impure so that the impurities can act as nucleation centre for ice to form. What is role of nucleation centre? Why can not ice form with some nucleation centre? About existing (and ...
0
votes
1answer
35 views

Statistical averages - integration in phase space

I apologize for my horrible math skills. But how exactly is one supposed to get ensemble averages in the 6N dimensional phase space? Suppose I have two particles in 1D. The phase space element is ...
0
votes
1answer
53 views

1d Ising model: Transfer matrices

we came across a peculiarity when calculating the partition function of $N$ spins $s_i=\pm1$ with Hamiltonian $$H=-J\sum_{i=1}^Ns_is_{i+1}$$ where we impose periodic boundary conditions such that $s_{...
0
votes
0answers
34 views

Partition Factor for $n!$ degeneracy

What would be the partition function $Z$ if we have an $n!$ degeneracy factor such that $\sum n!e^{-nE_o/kT}$. Assuming its shown that this diverges for infinite $n$, but lets define an $n_{max} = ...
0
votes
2answers
47 views

In Wang Landau method, how can I avoid large entropy difference?

Summary: In Wang Landau method, you have to calculate the probability $\exp (- \Delta S)$ but $\Delta S$ is generally large when the size of the system is not small. How can I make $\Delta S$ small? ...
2
votes
3answers
103 views

If I leave a glass of water out, why do only the surface molecules vaporize?

If I leave a glass of water out on the counter, some of the water turns into vapor. I've read that this is because the water molecules crash into each other like billiard balls and eventually some of ...
1
vote
0answers
20 views

Probability density of reactions coordinates (collective variables)

I'm reading Chapter 8 of Statistical Mechanics: Theory and Molecular Simulations by M.E. Tuckerman. I have some difficulties understanding one formula (8.6.4). ... If these reaction coordinates are ...
0
votes
1answer
38 views

Can spontaneous fluctuations cause instantaneous non-equilibrium?

The fluctuation-dissipation theorem says that the linear response of a given system to an external perturbation is expressed as the fluctuation properties of the system in thermal equilibrium. Does ...
0
votes
1answer
58 views

Path integral and Out-of-time-ordered (OTOC) correlator

A simple observation that any insertions within the path integral are classical variables (Not operators) and hence, objects inside the path integral "commute" (is symmetric under exchange). Hence, ...
0
votes
1answer
32 views

Deriving the second law of thermodynamics from an irreversible carnot process

I have studied the ideal carnot cycle extensively where we assume that $$\Delta S_{\mathrm{total}}=\sum_i \frac{Q_i}{T_i} =0$$ Now I was wondering whether it is possible to derive basic properties ...
0
votes
1answer
42 views

Eigenvalues of the thermal state density operator

We define the thermal density operator as $$\tau(\beta) = \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$ where $H$ is the systems Hamiltonian. Today I was told that the eigenvalues of the ...
1
vote
1answer
20 views

Deflection Curve of Fixed-Fixed Beam Under Central Point Load

I have been trying to find the deflection curve of a fixed-fixed (clamped-clamped) beam under a point force at the center... but I could not find it. Any help would be appreciated.
0
votes
0answers
50 views

Self-averaging quantities in physics

This question is about self-averaging quantities in physics. For a definition see: https://en.wikipedia.org/wiki/Self-averaging. For concreteness I give an example below. Example Consider a ...
0
votes
0answers
37 views

Order of phase transition in random walks

If we consider a random walk with step size distribution $P(s)\sim s^{-\gamma}$, we know the order of $\langle s^2\rangle$ changes at $\gamma=3$, while the order of $\langle s\rangle$ changes at $\...
3
votes
1answer
68 views

Does Liouville’s theorem hold for any ensemble?

I’m having some difficulties with my Statistical Mechanics course. In particular, I have some doubts about Liouville’s theorem and the various ensembles. Consider, for instance, the Canonical ...
2
votes
0answers
28 views

Boltzmann transport equation for granular gases

I am researching about granular gases and their collisions, and have come across this Boltzmann transport equation: $$\frac { \partial f ( v ) } { \partial t } = \iint d u _ { 1 } d u _ { 2 } f \left( ...
0
votes
0answers
127 views

The classical limit of QM as a Hamilton-Jacobi equation?

I'am having difficulties to understand the so-called classical limit in quantum mechanics. There is a popular method to transform the Schrödinger equation into two coupled equations that are the ...
0
votes
1answer
38 views

Violation of Virial theorem as indication to ergodicity breaking

Under which conditions the break of virial theorem implies break of ergodicity? I've seen this question, but it is very limited and not sufficient. To constrain the discussion I'm interested in 1D ...
0
votes
2answers
47 views

Deriving pressure from particle velocity: missing 1/3 factor

Let us assume we have a perfect gas, isotropic, homogeneous, inside a cube of side a. We want to calculate the pressure as a function of the velocities of the particles inside. We must find the ...
0
votes
2answers
71 views

Understanding Ising Model in Statistical Mechanics

A section on the Ising model in the text "Introduction to modern statistical mechanics" by Chandler states the following: "We consider a system of $N$ spins arranged on a lattice. In the ...
4
votes
0answers
89 views

van der Waals equation of state plot limitations

When I plot the van der Waals equation of state in terms of Pressure (bar) versus density (mol/L) for propane at 400 K, $$P=\frac{RT}{\big(V_m-b\big)}-\frac{a}{V_m^2}$$ in terms of density, $$P=\...
5
votes
0answers
50 views

Bose Condensation; interacting vs. non-interacting

I have some problems unifying, the two way I learned how a Bose condensate appears. The main problem is that the observables seem to be quite different. In statistical physics lecture one starts with ...
0
votes
1answer
79 views

Obtaining an expression for spontaneous magnetization in 1D Ising model with $H=0$ from the beginning

The usual trick to find the spontaneous magnetization for the 1D Ising model is to calculate the partition function $Z$ with the Hamiltonian $$\mathscr{H}=-J\sum\limits_{i}S_iS_{i+1}-H\sum\limits_{i}...
0
votes
1answer
27 views

Problem in counting bonding pairs (elementary mean-field theory on the Ising Model) [duplicate]

When the Ising model Hamiltonian $$H=-J\sum _{<ij>} \sigma _i\sigma _j-H\sum _i \sigma _i$$ is assumed ($\sum _{<ij>}$ is the summation over all the bonds or adjacent pairs of sites, $\...
0
votes
0answers
54 views

partition function of distinguishable/indistinguishable particle where any number/at most one of them can occupy a state

Say we have a system with 2 particles with energy levels $E_1=0,E_2=E,E_3=2E,E_4=2E$. I want to figure out the partition function of the system depending on whether i) the particles are ...
3
votes
1answer
52 views

Can we always find a Quasi-Probability distribution representation for density operator of any physical system?

For a pure, quantum optical system, the QPD formalism is straightforward and possible because of the existence of coherent states. The question is whether we can find coherent states for any quantum ...
2
votes
1answer
35 views

What is this secondary transition in the simulation of the Ising model?

Here, the horizontal axis is the strength of the ambient magnetic field. The Hamiltonian I used is $$H = -h\sum_i \sigma_i - J\sum_{\langle i \, j \rangle}\sigma_i\sigma_j.$$ The horizontal axis is $h$...
10
votes
2answers
142 views

What is the intuition behind the statement that non-equilibrium systems with static disorder are self-averaging?

In this paper(1) by C. De Dominics, he makes the argument that in a dynamical statistical mechanics system, one doesn't need to apply the replica trick and can directly disorder average the generating ...
0
votes
1answer
46 views

How was Maxwell able to derive velocity distribution? [closed]

Did he assume that their velocities follow some kind of statistical distribution and fit the parameters accordingly. Or did he mathematically prove it?
1
vote
1answer
45 views

Molecular model of liquids

The macroscopic behavior of gasses and solids follow very intuitively from the description of matter as quasi-spherical molecules interacting with each other with attractive and repulsive forces. ...
0
votes
0answers
11 views

Statistical model for local polymer concentration

I'm trying to describe the local constraining of two points located on a 2-part DNA polymer (double and single stranded), with respect to one fixed end. See the schematic below: Here I would like to ...
2
votes
2answers
127 views

Thermodynamic Entropy seems to be contradictory

For an ideal gas the entropy change with energy is inversely proportional to temperature: This must yield: $$S=\frac 3 2 k_B \ N ln(T)$$ For various reasons, this equation is hard to find. ...
3
votes
2answers
85 views

Question on the temperature dependence of the partition function

Let's just say we're looking at the classical continuous canonical ensemble of a harmonic oscillator, where: $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ and the partition function (...
1
vote
0answers
20 views

$p$-spin spherical spin glass

Consider the $p$-spin spherical spin glass model with Hamiltonian $$H_{N,p}(\sigma)=\frac{1}{{N}^{\frac{(p-1)}{2}}} \sum \limits_{i_1,...i_p} J_{i_1,...i_p} \sigma_{i_1} \sigma_{i_2} .. \sigma_{i_p} $$...
1
vote
1answer
31 views

What is the sign of chemical potential of a noninteracting classical ideal gas obeying MB distribution?

The chemical potential of a noninteracting Bose gas can never be negative while that of a noninteracting Fermi gas can be both positive or negative. What can be said about the chemical potential of ...