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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Calculating average of a function of molecule's orientation (Euler angles)

In this paper, orientational average of a function of Euler angles, $f(\phi,\theta,\psi)$, is defined as: $$\langle f\rangle=\frac{1}{8 \pi^2} \int_0^\pi \int_0^{2 \pi} \int_0^{2 \pi} f(\theta, \phi, \...
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Why is thermal noise Gaussian distributed in voltage, but Rayleigh distributed in amplitude?

This is a follow-up to a similar but distinct question I recently asked. Background. Consider an ideal antenna. For ideal polarized thermal noise treated as a sum of random, complex phasors, ...
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Properties of random-walk in infinite and finite two-dimensional space: probability of two particles being in the same location at time t

I have been told that one of the property of the continuous-time random walk in two dimensions is that: $$\int_{Z} \, G(z, t | p_1) \, G(z, t | p_2) \,dz = \,G(p_1,p_2,2t)$$ where ...
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What would be the consequences of asymptotic freedom and color confinement of QCD for neutron stars?

Neutron stars are conjectured to have densities between $10^{14}$ and $10^{17}\ \text{g/cm}^3$. In the latter limit, the neutrons could be so close that the interaction between them would not be the ...
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Thermal noise phasor amplitudes are Rayleigh distributed. How are voltages at the antenna distributed?

Background. Consider an ideal antenna. For ideal polarized thermal noise treated as a random phasor sum, bivariate Gaussian statistics apply to the resultant phasor (call it $\vec{v}$) that is ...
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Problem concerning heat capacity of an ideal bose gas [closed]

I have a problem with problem 18.4 From Huang (Introduction To Statistical Physics). Could anyone help me to understand how the second derivative of lnz with respect to t is computed? At the end of ...
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How does an exothermic reaction release energy? [closed]

When a reaction is exothermic it releases energy often denoted in kJ/mol. This is due to the total enthalpy of the reactant(s) being higher than the enthalpy of the product(s) and thus needs to ...
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Unable to derive Grand Canonical Free energy for single electron transistor (SET)

I am try to derive the expression for Grand canonical free energy for single electron transistor (SET) and started with a simpler case as shown below from Physics of Nanoelectronics by Tero Heikkila: ...
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Derive Canonical Ensemble from Maximum Entropy Principle

Consider a quantum system with Hamiltonian $H$, described by a density operator $\rho$. It is known that the expectation value of this system is $$ \langle H \rangle \equiv tr(\rho H) = E $$ I want to ...
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The probability of a circular region being "invaded" by moving spheres as a function of time

I had uploaded the same problem in maths stack exchange but since it got no answer and because I think that it is a problem that can be seen both as a physical one I considered uploading it both on ...
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Why does adiabatic expansion occur in the carnot process?

(Spoiler: Why adiabatic expansion happens in Carnot cycle doesn't really answer the question for me.) In the Carno cycle, the open system is first brought into contact with the warm reservoir, which ...
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On-shell Poisson brackets and time derivative

In classical statistical mechanics, the information about a given system is given by a distribution of probability over phase space $\rho(p,q,t)$. Let $H(p,q, t)$ be the hamiltonian of the system and $...
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From the non-relativistic particle distribution $f(\vec{r}, \vec{v}, t)$ to the relativistic $dN/d \gamma$

In the plasma community it is common to use the 6+1 D particle distribution function $f(\vec{r}, \vec{v}, t)$, and from this obtain the different-order momenta (density, velocity, and so on). ...
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Do Stochastic Differential Equation models conserve energy?

I have recently started looking into stochastic models of chemical reaction systems, particularly the Chemical Langevin Equation (CLE) SDE model (e.g. here). One thing I'm trying to understand is ...
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Do we need atomic theory to do thermodynamics?

With thermodynamics, systems are studied using macroscopic variables (pressure, temperature, volume, etc.) which do not need a mechanical explanation, which is what statistical mechanics deals with. ...
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(Discrepancy in the) Statement of Eigenstate thermalization hypothesis

I am trying to understand ETH and unfortunately came across a seemingly contradicting definition by the same author (Mark Srednicki). I don't know which definition is correct. At this instance in this ...
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How does the phase space transform like this?

So I was going through these notes (Exercise $3.2$) and saw the phase space $ d^2 \mathcal{V} $ can be expressed as: $$ d^2 \mathcal{V} = (d \mathcal{V}_x\frac{ p_o}{m})(dp_0 d p_x dp_y dp_z) $$ ...
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Einstein solid multiplicity

I have a question about the multiplicity of an Einstein solid and the probability. I have the book An introduction to thermal physics by Schroeder. It says all accessible microstates are equally ...
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Explaining internal energy from a macroscopic perspective

This question stems from the accepted answer to this question Classical thermodynamics deals with and relates macroscopic quantities, like pressure, temperature, mass, volume, etc. These are the ...
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Can you derive the boltzmann factor from $$ \frac{ \partial P(\epsilon_i)}{\partial\epsilon_i} = -P(\epsilon_i) $$

If you solve the following simple differential equation: $$ \frac{ \partial P(\epsilon_i)}{\partial\epsilon_i} = -P(\epsilon_i) $$ You get: $$ P(\epsilon_i) = c_1 e^{-\epsilon_i } $$ where we ...
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Why is electronic specific heat divided by $V$ rather than $M$?

In Ashcroft and Mermin Chapter 1, just above equation (1.50) and in the context of a classical ideal electron gas, it is said that the electronic specific heat at constant volume $c_v$ is defined by $$...
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Integration by Parts in Liouville's Theorem

I am looking at a proof of Liouville's Theorem, which states that for $F, G \in C_0^\infty$ and a Hamiltonian $H$, the operator $$D_H = \sum_{i=1}^n\Big(\frac{\partial H}{\partial p_i} \frac{\partial}{...
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What quantity can a microstate have?

I confused whether a microstate's chemistry potential is defined. And how about temperature, pressure, entropy? And what is a microstate? A ensemble contain a set of microstates. The microstate is a ...
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What actually are microscopic and macroscopic viewpoints in thermodynamics?

The microscopic viewpoint of studying a system in thermodynamics is the one in which we consider the system on a molecular/atomic/sub-atomic level. (is that even right?) The macroscopic viewpoint is ...
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Unitary evolution and von Neumann entropy

In chapter 5 of the book "Statistical Mechanics" by Pathria it says Since the density matrix evolves in a unitary manner, the von Neumann entropy is time-independent Where the von Neumann ...
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Central limit theorem and fluctuations

Considering a statistical model away from its criticality, the system can be essentially viewed as a collection of subsystems of size $\xi^d$ (with $\xi$ the correlation length) with no mutual ...
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Confusion about fundamental assumption of statistical mechanics

I am confused about the fundamental assumption of statistical mechanics. It says, over a long time scale, that all microstates are equally accessible. I get it so far. But for microstate, there are ...
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What is the theoretical value of this phase space invariant?

So I wanted know how to theoretically calculate this phase space invariant (equation $3.31a$ )$R$ in our universe (FLRW metric) during the cosmological nucleosysthesis: $$R = \int_{p} \frac{\mathcal{...
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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 ...
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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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Partition function for $H=-\vec \mu \cdot \vec B$

The interaction energy between a magnetic dipole $\mu$ and a fixed magnetic field $B$ is $E(\theta)=-B \mu cos\theta $ where $\theta$ in the angle between $B$ and $\mu$. The partition function is $Z=\...
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Proof of pressure of ideal gas from first principles

Here's a question aimed at a deeper intuitive understanding of statistical physics and the theory of ideal gases, which has bothered me for quite a while. Assume a billiard table 2D. The table has ...
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Probability of collision between two particles (Statistical Mechanics)

I'm pretty new to statistical mechanics. While reading an introductory book ("Fisica - Meccanica e termodinamica", translated "Physics - Mechanics and Thermodynamics" by C. ...
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Energy of Quantum Magnetic Dipole

When Amperian loop model is adopted, the energy of magnetic dipole under external field $\vec{B}_{ext}=B\hat{z}$ constitutes of 3 part: $-\vec{m}\cdot\vec{B}$ from work against mechanical force $\...
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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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Linear response theory, project ideas [closed]

As a part of a second course in statistical mechanics I have to present a project related to the subject of linear response theory. We have not talked about linear response theory in the course, so I ...
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Hamiltonian eigenstates in Weiss-Heisenberg model

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 ...
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