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.

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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 ...
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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 ...
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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 ...
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67 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 $...
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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 ...
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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....
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82 views

Relation between phase transitions and free energy?

I'm a mathematics undergraduate student, and I'm struggling with the concept of phase transition. I know that a (quite general) definition of phase transition is that "a physical system undergoes a ...
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Microstates in phase space

Recently i am studying Statistical mechanics and reading about the Boltzman hypothesis about entropy $$S = k \,ln\,\Omega(E)$$ where it says $\Omega(E)$ is total no. of microstates , the available ...
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1answer
56 views

What is the PDF of relative collision speeds in an ideal gas?

In the Kinetic theory of gases, the speeds of particles follow a Maxwell-Boltzmann distribution. However, what if one is interested in the distribution of relative collision speeds, aggregated over ...
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2-level system of indistinguishable particles

It is a typical introductory problem in classical statistical physics to calculate the entropy of a two-level-system: say we have a N particle system in which particles can have energy E or 0. ...
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1answer
59 views

Volume of state in phase space free particle

I have to how a quantum state of a free particle between 0 and a occupies an area of $h$ in the phase space. What I did was to calculate $\Delta x \Delta p$ and show that it was of order $h$, but I ...
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List of Replica Symmetry results for different models?

Does anyone know of a good source that might have a list of problems or models along with what kind of replica symmetry they are conjectured to have? I am aware of some of the more famous results, e....
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Example of two-dimensional Langevin equation?

I am looking for an example of a langevin-equation with two variables: $\dot{x}=f(x,t)+\omega, x=(x_1,x_2)$. Can anyone suggest one?
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the phase space distribution

I am NOT a physicist and am hoping to get a quick answer to a basic question regarding what Boltzmann said (or meant). I (think I) understand the concept of the phase space distribution $D$ - the ...
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Mermin-Wagner and graphene

I have been told that the Mermin-Wagner theorem disallows the existence of the crystal of graphene. However, I don't have enough knowledge to understand the Mermin-Wagner theorem. If possible can ...
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1answer
83 views

How do big rocks split in half? [closed]

So, recently I was on a trip in which I could observe a lot of huge rocks (1 to 20 meters in diameter), and I found odd how in several occasions rocks would be split in two parts, with a neat division....
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Relation between the canonical partition function and the phase space volume

In Kerson Huang's Statistical Mechanics (2nd ed.), it is claimed that the phase space volume occupied by the canonical ensemble is the partition function: $$ Q_N (V, T) \equiv \int \frac{dp dq}{N! h^{...
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Average pressure on a 3D box

I have a particle inside a box of dimensions $L_{x},L_{y},L_{z}$ the energies are given by $E=\frac{\hbar^2}{2m}\pi^2\sum_{i}\frac{n_{i}^2}{L_{i}^2}$ I calculated the force by $-\frac{\partial E}{\...
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Do these two reactions with photons have the same chemical potential?

The reactions in question are: $$e^+ + e^- \longleftrightarrow \gamma$$ $$e^+ + e^- \longleftrightarrow \gamma + \gamma$$ We have two different systems to compare with each other, both composed of ...
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43 views

Phase space harmonic oscillator area and probability

I want to find the probability of finding an oscillator between $x$ and $x+dx$. I calculated the volume $\frac{8\pi EdE}{\omega^2}$ enclosed in the phase space for the oscillator with energy between $...
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Solution to diffusion equation of a random walk

In my class of statistical physics, we studied the classic problem of random walk for the discrete case. In the end, we made the changes necessary for the master equation to be in the continuous ...
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1answer
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Is the expectation value of creation operator zero?

Let $c^\dagger, c$ be creation and annihilation operators respectively. And we denote expectation value of operator $A$ calculated via Hamiltonian without interaction as $\left< A \right>_0$. In ...
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2answers
124 views

What happens to the electronic movements at absolute 0?

what happens to the motion of electrons in the their respective orbits when a substance is cooled down to zero kelvin? assuming they stop moving then are they gonna stick to the nucleus? if yes what ...
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What is entropy? [duplicate]

Can somebody please explain in detail what is entropy not a definition like its measure of disorderness. A complete introduction from basics to what parameters it depends on as a thermodynamics point ...
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Normalization in the $O(n)$ model

The $O(n)$ model is the model of $n$-dimensional spins $\vec{s}_i$ at each lattice $i$ which are restricted to be on the $n-1$ dimensional sphere of radius $S$ with Hamiltionian $$H(\vec{s}_1,\dots,\...
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How to calculate the chemical potential for a 2D lattice gas with excluded volume?

The chemical potential of a 2D lattice gas is given by $\mu = k_bTln(\frac{c}{(1-c)q}) $ , where c is the concentration of particles on the lattice, T is the temperature, and $k_b $ is the ...
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How to extract quadrupole moment and its error from $\chi^2 < \chi^2 + 1$ surface?

I have following info: plot of $\chi^2$ minimization of 208-Rn Coulomb excitation data, Surface corresponds to regions $\chi^2 < \chi^2 + 1$ with error bars of 1$\sigma$, mean lifetime is 8 $\pm$ 0....
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1answer
84 views

Partition function in renormalization

When studying statistical mechanics, renormalization is understood from attempts to calculate partition function by simplifying. (For example, David Tong's lecture note) While I understand that ...
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Cavity method pedagogic references

I am looking for pedagogic references (textbook, review/expository articles, lecture notes, etc.) explaining the cavity method in detail. I am talking specifically about this: https://link.springer....
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1answer
44 views

Temperature as a function of energy in a container with ideal gas

Let's consider a large container with insulating walls (not subjected to gravity) and with a large number of noninteracting particles (besides elastic collisions) for which I assume that the ergodic ...
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Confusions regarding macrostates and thermodynamic probability

I often came across completely different definitions of a macrostate, or at least they feel different. One of the definitions were supported by an example which I would like to state here: Suppose N=...
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1answer
43 views

Entropy as a function of internal energy in an arbitrary ensemble

The Boltzmann formula for entropy, $S_B=k\ln \Omega(E)$ holds in the microcanonical ensemble, where $E$ is fixed. In other ensembles, entropy is given by the Gibbs/Shannon formula: $S_G=-k\sum_i P_i \...
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Maxwell-Boltzmann Statistics is equivalent to a uniform distribution among acessible states?

Both Fermi-Dirac and Bose-Einstein statistics reduce to the Maxwell-Boltzmann statistics in the classical limit of very high temperatures or very low concentration. In the case of very high ...
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Energy of band of $d$-dimensional semiconductor when voltage $V$ is applied across

Let's say we have a one-dimensional semiconductor and I apply voltage V across it, I want to calculate the energy of a parabolic band, when a source and drain voltage is applied across it. I expect it ...
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941 views

Entropy as a function of temperature: is temperature well defined?

Considering volume and particle number constant, the internal energy $U$ is a function of the entropy: $U=U(S)$. The temperature is then defined as $T=dU(S)/dS$. From here, the temperature is a ...
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Is thermodynamics only applicable to systems in equilibrium?

So I was going through callen's thermodynamics book and their he says that thermodynamics is only applicable to systems which are in equilibrium and that naturally raised a few questions in my mind ...
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How do normal modes have group velocity?

Considering phonon dispersion curves in crystals, the group velocity is given by the derivative of frequency with respect to wave vector. But these modes on the dispersion represent normal modes, ...
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1answer
31 views

Temperature from the Gibbs entropy

In statistical mechanics, temperature is typically introduced via the Boltzmann entropy as follows. The Boltzmann entropy is $S=k_B \ln W$ where $W$ is the number of microstates. If the system's ...
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1answer
70 views

Choosing Zero Chemical Potential

After asking this question: Grand canonical ensemble and chemical potential $\mu=0$, I'm still confused about chemical potential: when have we the freedom to take $\mu=0$? In other words, by ...
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65 views

Why is $\langle{\dot A}A^* \rangle=0$ for dynamical variable $A$?

In page 179 of Hansen and McDonald's book, Theory of Simple Liquids, 3rd edition, 2006, an identity of correlation functions was deduced. Here $C_{AB}(t)$ is the time correlation function between ...
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Cooling effect caused by evaporation

Evaporation is said to cause cooling effect because it absorbs energy from surroundings to change its phase from that of a liquid to gas. I am in doubt as to why would the surroundings be ready to ...
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35 views

Non-Integer Values in Indistinguishable Particle Combinations Quantum Stat Mech

I am taking a thermodynamics course and we have talked about stat mech and the number of possible combinations of $N$ indistinguishable particles given degeneracy $g$. We stated that for the ...
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Canonical and grand canonical formalism at zero temperature

How can I find the expectation value of any operator at zero temperature in quantum statistical mechanics formalism? Expectation value of any operator $\hat{O}$ is given as $$\langle \hat{O}\rangle = ...
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1answer
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How to interpret phase diagrams?

I find quite difficult to interpret phase diagrams in general, for example I see people discuss them along the following lines: Here we see the coexistence line between liquid-solid phases.. a ...
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BKT transition: nature of topological transition

BKT-transition is one of the most well-known topological transition in $O(2)$ model.But I misunderstand the physical interpratation of this transition. I started from the low-temperature expansion of ...
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2answers
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In a NVE ensemble, can a particle access a state with a higher energy than the constraint?

I should start with saying that I know the answer is obviously no but I am trying to make sense of it mathematically. Let's say, for example, that each particle of a system with total energy $E$ in a ...
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1answer
63 views

Quantum statistical mechanics formalism

How do we solve a Hamiltonian written in second quantization by using quantum statistical formalism? For example, the following Hamiltonian $$H = \sum_{i=1}^L c_{i+1}^\dagger c_i + h.c.$$ I have ...
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Accurate derivation of electron-phonon scattering contribution to metal resistivity

My lecture derived the expression for this contribtuin using the collision integral approach but I missed lot of details. He considers the lowest order correction to distribution function $f=f_0+\...
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1answer
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Dealing with thermodynamic processes in Statistical Mechanics

I have recently started studying Statistical Mechanics, and through my study of Classical Statistical Mechanics, I have studied how do we write distribution functions for equilibrium systems which can ...
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Entropy and temperature of small systems

I've been struggling for a while now in understanding the concept of entropy as a function of the internal energy. Textbooks typically call $g(E)$ the number of microstates compatible with an energy $...