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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The cavity method of Ising model in infinity dimension and dynamical mean field

In the article "Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions" chapter III.A, when discussing the cavity method of Ising model, the ...
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Regarding calculation the moments of a random variable whose probability distribution obeys the Fokker Planck equation

I was going through Van Kampen's Stochastic Processes in Physics and Chemistry, and I was trying to solve the exercises from Chapter 8 about the Fokker Planck equation (just in case context could help ...
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What are critical dimensions in statistical field theories (SFTs) and quantum field theories (QFTs) and how do they relate to divergences?

My question is the following. Statistical field theories (SFTs) and quantum field theories (QFTs) are usually associated with some upper critical dimension (UCD) and lower critical dimension (LCD). ...
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Metropolis-Hastings and underlying Markov process

I tried to understand the workings of the Metropolis-Hasting algorithm. As far as I can understand, it allows to draw samples from an unknown distribution $T(x)$ as long as a function proportional to ...
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Boltzmann vs Gibbs definition of entropy [duplicate]

I am learning Statistical Mechanics and I have a question regarding different definitions of (statistical) entropy. If we use Boltzmann's definition: $$\sigma \propto\ln(W)$$ Where $\sigma$ is the ...
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Currently self-studying QFT and The Standard Model by Schwartz and I'm stuck at equation 1.5 in Part 1 regarding black-body radiation

So basically the equation is basically a derivation of Planck's radiation law and I can't somehow find any resources as to how he derived it by adding a derivative inside. Planck says that each mode ...
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Proving a relationship involving the chemical potential [closed]

I need to prove the relation $$\left(\frac{\partial \mu}{\partial N}\right)_{T,V}> \left(\frac{\partial \mu}{\partial N}\right)_{T,P}.$$ One may start with the total differential of the internal ...
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Pure state vs mixed state in this example

Consider, I have a quantum state $|\Psi\rangle$, such that : $$|\Psi\rangle=c_1|\psi_1\rangle+c_2|\psi_2\rangle$$ This is defined as a pure state, since I have complete information about the system. ...
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Are photons with different frequency distinguishable?

When i learn statistical mechanic, the teacher told me that photons with different frequency are distinguishable, i confused. And the teacher say also photons with different polarization, direction ...
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Are all Local Observables Measured on Gibbs States Analytic as a Function of Temperature Away from Phase Transitions?

Let $\rho(\beta)=e^{-\beta H}/Z$ be the Gibbs state of a quantum Hamiltonian, and $H$ is some local Hamiltonian on $N$ particles, and $Z(\beta)$ is its partition function. Suppose I measure some local ...
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The microcanonical ensemble surface distribution function

on page 58 of the book "the principles of statistical mechanics" by Richard C. Tolman, there refers to a formula for the surface density of distribution in microcanonical ensemble, which is: ...
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How to derive Shannon Entropy from Clausius Theorem?

I am studying Quantum Information now, and I need to understand the entropy of a quantum system. But before I go there, I need to understand Shannon Entropy which is defined as : $H(X) = -\sum_{i=1}^{...
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Understanding the derivation of the variational principle in classical density functional theory

I am trying to understand the derivation of the variational principle as presented in Bob Evan's 1979 work.[1]. The part that is tripping me up is when he presents the following result. He starts by ...
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Did I understand RG correctly?

I am currently self-studying Renormalization Group (RG) in Condensed matter physics (in preparation for graduate school while I'm in Alternative Military Service). While I'm writing bunch of ...
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In the Statistical Mechanics Mark E. Tuckerman 4.4.7

When deriving the energy fluctuations in the canonical ensemble, a step is made to approximate Cv≈N, why is this?
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Master equation with a coherent bath

When we consider an oscillator $a$ acting with a bath of oscillators $b_i$ with the interaction Hamiltonian reads $$H_{int}=\sum_{i}g_ia b_i^{\dagger}+g_i^*a^{\dagger}b_i,$$ with the free Hamiltonian: ...
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Is it possible for a thermoelectric generator to not have a heat source or heat sink?

Specifically, the type of thermoelectric generator I am suggesting makes use of the ExB drift effect. When electric and magnetic fields are perpendicular to one another, with infinite mobility of the ...
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Quantum versus Classical Partition Function

I am confused as to which partition functions are classical and which are quantum. I am interested in the canonical ensemble. Several places I have seen the "classical" partition function as:...
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What does length dependent entropy means?

I've this problem from homework sheet, which asks to write down the L-dependent part of the entropy. I've no idea what does it mean? The Hamiltonian of the system is defined as $$ \mathcal{H}(p_{i}, ...
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Wick's Theorem and Functional Derivative

In the Quantum Field Theory An Integrated Approach, Fradkin, the author derived the partition functional for a free scalar field (after analytic continuation to imaginary time ) as $$Z_{E}[J]=Z_{E}[0] ...
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Expected energy in micro-canonical and canonical distribution

Which relation $E(β)$ is required to ensure that he micro-canonical distribution and the canonical distribution have the same expected energy?
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Degeneracy of Photons

The density of states for a photon gas is defined by, $$D(\epsilon)=\frac{g}{2\pi^2}\frac{\epsilon^2}{(\hbar c)^2} $$ where g is the number of independent internal states for a photon. The question is ...
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Partition Function Question

I've been looking at calculating the Internal energy of a non-isothermic van de Waals gas, and in doing so have been researching the free energy and hence the partition function necessary to calculate ...
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Equilibrium carrier distribution appropriate to local temperature

In Chapter 13 of Ashcroft and Mermin there is a general discussion about the nonequilibrium distribution function under the relaxation time/semiclassical transport assumptions. One of the key axioms ...
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Momentum and position representation of BCS Hamiltonian

I am struggling to relate the BCS Hamiltonians in momentum and position representations. The free part of the BCS Hamiltonian is often written $$ H_{\text{pos}} = \int d^3 r ~ \psi^\dagger(r) \left(\...
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What do Maxwell's equations tell us about the ultraviolet catastrophe?

I have found it that in elementary discussions of black-body radiation, other than justifying electromagnetic waves should exist from Maxwell's equations, those equations are not used for anything ...
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Net force exerted by pressure

Why do the net force $K$ exerted by the pressure of a hard sphere gas according to the Maxwellian distribution on the walls of the container $A \subset R^3$ vanishes? (In the absence of external ...
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Does statistical mechanics imply that the universe is probabilistic?

Statistical mechanics says that a system will evolve to a state of higher entropy (i.e. states with higher number of microstates) simply because it is overwhelmingly more probable than evolving to a ...
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What is the source of isothermal work?

The usual explanation of isothermal work, especially in the context of the Carnot cycle, is that the "heat" absorbed in the isothermal leg is converted into "work" done on the ...
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Can the water remain in the vapor gas form at zero Kelvin (0K) temperature? (when the pressure is low enough or under other conditions)

Naively, when the water cools down to low temperature, the water goes to the ice solid phase. (Like below 0 celsius at 1 ATM pressure.) Can water remain in the gas form at zero Kelvin (0K) temperature,...
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Why aren't Runge-Kutta methods used for molecular dynamics simulations?

One of the most used schemes for solving ordinary differential equations numerically is the fourth-order Runge-Kutta method. Why isn't it used to integrate the equation of motion of particles in ...
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An exercise on the adiabatic initial condition from Daniel Baumann's new cosmology book

Additional info for those not familiar with cosmology: The $\delta$ s are the density contrast whose subscripts indicate species. It is defined as: $\delta_{a}=\frac{\delta\rho_{a}}{\bar{\rho}_{a}}$ ....
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The entropy given by stefan Boltzmann's law looks remarkably similar to the volume of the sphere; $S(T)=\frac{4}{3}\sigma T^3$

If I am not mistaken the entropy for a blackbody per unit area is given by: $$S(T)=\frac{4}{3}\sigma T^3.$$ The volume of a sphere is given by: $$ V(r) =\frac{4}{3}\pi r^3. $$ Is this coincidental? I ...
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Phonon, photon have chemical potential equal to zero

Hi I know the chemical potential of the phonon and photon is equal to zero. And I know the reason is the number of these particle isn't conserved. But I don't know why the number of the phonon isn't ...
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Motivation for density of states [duplicate]

I have not found a good book on statistical mechanics that explains the quantity density of states well. The books I have read so far make the continuum limit approximation, which does not make much ...
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Is Bogoliubov transformation simply a basis transformation?

I encounter Bogoliubov transformation when I'm learning 1D transverse field Ising Model.But I think it's just a simple basis transformation, I don't understand why people give it a special name?
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Why an energy constraint gives the canonical ensemble (NVT) when we maximize von Neumann entropy, when energy is not conserved in NVT?

In Sakurai and Napolitano's Modern Quantum Mechanics 2nd Edition, I'm learning Section 3.4, page 188 where they derive the canonical ensemble by maximizing von Neumann entropy with energy constraint. ...
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Equivalent Formulation of Least Squares Method in Terms of Entropy

I have learned that people in the field of machine learning and statistics often use the least squares method. Is there any alternative formulation of the least squares method using entropy? (It might ...
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Radial dependence of temperature for adiabatic expansion of ideal gas

If the solar wind propagates out adiabatically with a constant speed and can be regarded as an ideal gas, how the solar wind temperature depend on radial distance from the sun $r$, i.e. $T$ as a ...
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What is the interpretation of the following version of the first law: $\sum p_i E_i = \sum p_i dE_i + \sum E_i dp_i $

The first law of thermodynamics can be written as: $$ U= <E> = \sum p_i E_i = \sum p_i dE_i + \sum E_i dp_i $$ $$ U= E = \delta Q + \delta W $$ Where $$\delta W = \sum_{i=1}^N p_i\,dE_i$$ $$\...
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Most likely velocity vector of ideal gas?

$$\left( \text{probability of a molecule having velocity }\vec{v}\right) \propto e^{-mv^{2}/2kT}$$ So the most likely velocity vector for a molecule in an ideal gas is zero. Given what we know about ...
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Finding equation of internal energy in adiabatic process

I am doing a review of thermodynamic and I encounter the following question, Show that if a single-component system is such that $PV^k$ is constant in an adiabatic process (k is a positive constant) ...
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Why are globular clusters always denser in the core?

Stars falling towards the core speed up--thus, the amount of time they spend in any given volume goes down. Stars out near the edge move more slowly, thus their residence times in outer volumes should ...
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What is the type of system, if I have an opened container with hot water inside, but no heat input to the system?

I have a school project, where I am trying to generate electricity using TEG modules that are attached to an aluminium container that contains hot water. The container is opened and there is no heat ...
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Why aren't the thermodynamic suceptibilities zero in the thermodynamical limit?

As it is explained in this answer (and nicely so!), the second derivative of a thermodynamic potential to an intensive quantity, for example pressure or magnetic field strength (or temperature) will ...
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Ising Model without periodic boundary conditions (PBC)

I try to calculate the correlation function $<\sigma_i \sigma_j>$ with the method of transfer matrices. I do understand how to use this method with PBC. But how can I do it without PBC? My ...
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What's the definition of temperature ? What's the meaning of reaching a zero temperature and infinity temperature? [duplicate]

I'm taking a course about statistical mechanics and it is completely new to me . We've learned about entropy and how when 2 different sub-systems comes to contact the entropy of the whole system keeps ...
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Why Photon gas's Equation of State Diverges?

We know that, for Photon gas, the equation of state is given by $pV=\frac{1}{3}U$; where $p$ is the Pressure, $V$ is the Volume and $U$ is the Internal Energy of the Photon Gas. (see Equation of State ...
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Why all cumulants of energy $H$ are proportional to the particle number $N$ (Kardar’s book 4.6)?

In the Chapter 4.6 of Kardar ‘s Statistical Physics of Particles. It claims that: In fact $<\cal{H}^{n}>_{c} = (-1)^{n}\frac{\partial^{n} lnZ}{\partial \beta^{n}}$ shows that all cumulants of $\...
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If temperature is the expected value of kinetic energy: $T\propto \text{E[}E_k\text{]}$, what is exactly entropy in statistical similar terms?

If temperature is the expected value of kinetic energy: $T\propto \text{E[}E_k\text{]}$, what is exactly entropy in terms of moments similar to expected value? Is there a relation with moments and ...
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