# 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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### Introductory text to the Boltzmann equation?

I'm searching for a good introductory text to the Boltzmann equation and how it gets applied in the relativistic case as well?
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### I am confused relating Entropy in statistics with thermodynamics [duplicate]

The thing is in thermodynamics we learn entropy as a measure of energy of a system per unit temperature that isn't available for the system to do work. Again, statistically, entropy is a measure of ...
1 vote
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### Do first-order phase transitions necessarily imply hysteresis?

As an example of first-order transition with hysteresis, I am thinking of the magnetization of the subcritical Ising model as a function of the magnetic field. Or density in the liquid-gas transition ...
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### Intuition behind entropy and its differentiation

I was reading the following paper about a better intuition of entropy and how it is connected to heat energy without the use of microstates: The problem is when he assumed that volume is constant and ...
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### Confusion regarding the density operator

I'm confused between two representations of the density operator in quantum statistical mechanics. In the first case, we have : $$\hat{\rho}=\sum_i P_i|\psi_i\rangle\langle \psi_i|$$ In the second ...
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### Energy dissipated by friction and entropy

Can we compute the entropy increase in some simple dissipative systems? Imagine a block sliding on a frictional floor and that its initial kinetic energy is $K$, let's imagine that the ambient ...
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### Hamiltonian of an electron in a magnetic field

Suppose I have an electron in a magnetic field given by: $$\vec{B}=B\hat{z}$$ The potential energy of this system is given by: $$U=-\vec{\mu} \cdot \vec{B}=\frac{g\mu_B}{\hbar}\vec{S} \cdot \vec{B}$$ ...
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### Entropy of mixing formula

Could someone give me a proof for the entropy of mixing formula, $$\Delta S_{{mix}}=-R(x_{1}\ln x_{1}+x_{2}\ln x_{2}),$$ with $$x_i= \frac{N_i}{N} = \frac{V_i}{V} \ \ ?$$
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### What is the matter distribution function for the FLRW metric?

For a non-relativistic gas we have the Maxwell Boltzmann distribution For a relativistic gas we have the Maxwell Juttner distribution What is the phase space distribution function for the FLRW metric (...
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### Fermionic System Trouble

I'm trying to solve a question from a test and i don't understand the reasoning behind the solution The question: what is $\frac{P(occupied)}{P(unoccupied)}$ ? Given a fermionic system with a ...
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### Natural log introduced in microstates derivative with respect to energy in equilibrium equation

In Pathria and Beale's Statistical Mechanics, 3rd ed, Chapter 1.2 (Contact between statistics and thermodynamics: physical significance of the number $Ω(N, V, E)$ ) The equation to maximize $Ω^{(0)}$ ...
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### Can more energy be extracted from a system of minimum free energy?

As far as I know, systems tend to minimize their free energy. The free energy is to my knowledge a measure of "useful" energy of a system. So my question is: Is it possible for a system to ...
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### What is temperature: function of energy or the value of this function in thermal equilibrium?

As far as I remember, many textbooks on statistical physics introduce temperature as a condition of equilibrium of a composite thermodynamic system. E.g., if the system consists of two parts with ...
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### What is an intuitive explanation for $T = \mathrm{const}$ when $\Omega(E) = e^E$?

Temperature is related to number of microstates as follows: $$\frac{1}{k_{\mathrm{B}}T} = \frac{\mathrm{d}\ln{\Omega(E)}}{\mathrm{d}E} \ .$$ Hence, if $\Omega(E) = e^E$, then $T = \mathrm{const}$. ...
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### Boltzmann distribution - why does distinguishability increase likelihood?

I am looking through derivations of the Boltzmann distribution. The method I've seen uses an argument that involves counting distinguishable microstates of a system with fixed energy, and then ...
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### How is concept of Lorentz local field considered in Heisenberg model?

The quantum Hamiltonian of Heisenberg model is usually given in the form: $$\textit{H}_{heisenberg}=-\sum_{i,j}J_{ij}\vec{S}_{i}\cdot\vec{S}_{j}-g\mu_{B}\vec{H}\cdot\sum_{i}\vec{S}_{i}$$ such that ...
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Reading about paramagnetism and ferromagnetism i've seen this formula: $$\mu=g \mu_B J$$ where $g$ is the Landé g-factor $$g=1+\frac {j(j+1)+s(s+1)-l(l+1)}{2j(j+1)}$$ From the answer to this question ...