Questions tagged [statistical-mechanics]

The study of large, complicated systems employing statistics and probability theory to extract average properties and to provide a connection between mechanics and thermodynamics.

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Electromagnetic entropy maximum in Planck's black-body radiation law

I am reading Planck's work on black-body radiation. In the paper on the page 19 it is said that the expression $$R_\nu=\frac{\nu^2}{c^2}U\tag1$$ where $R_\nu$ is the intensity of a linearly polarised ...
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Definition of injection spectrum of an electron beam

Referring to Plasma Astrophysics, Part 1, Eq 4.32, The number density of electrons in the beam at a distance $z$ from the injection plane at $z = 0$ is defined as, $$ n_b(z) = \int_{0}^{\infty} \int_{...
Refrigerator's user avatar
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Average number of particles in a Fermi gas

I want look (and plot) how the average number of fermions depends on temperature and the chemical potential. According to Eq.(13.12) in https://itp.uni-frankfurt.de/~gros/Vorlesungen/TD/...
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Applicabilty of the definition of thermodynamic temperature

I have a question about the definition temperature, given by $\frac{\partial S}{\partial E}(E,V,N) = \frac{1}{T}$ Is this valid only for isolated systems (and not applicable, for instance, to a (...
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Is average total energy of two objects is the sum of their individual average energies?

In the context of Boltzmann's distribution, Schroeder states that an average is defined as $$\bar{x}=\frac1Z\sum_sx(s)e^{-\beta E(s)}$$ Where $\beta=1/kT$ and E(s) is the energy corresponding to the ...
GedankenExperimentalist's user avatar
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Are all ergodic classical systems chaotic? [duplicate]

Suppose that we have a classical system with an arbitrary number of particles and a Hamiltonian $H(\vec{x},\vec{p})$. For few particle systems, the KAM theorem tells us that if the Hamiltonian is ...
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Use of Binder Cumulant for Determining Critical Temperature

I am completing a computational project where I am simulating the Ising model using Monte Carlo methods, namely the Metropolis-Hastings algorithm, and the Wolff algorithm. For the Metropolis-Hastings ...
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Interpretation of a probability that does not normalize to one in stat mech?

I am trying to understand the meaning of the "n-particle distribution function" as defined by the three references below([1][2][3]), primarily those by Claudio Zannoni. Setup: For a system ...
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Is there an equipartition theorem for diatomic gases at transitional temperatures?

Context If you have a gas, you can insert a bit of energy $E$ and measure the resulting increase $K$ in the average kinetic energy in your favourite direction. For monatonic gases, $K=E/3$, as the ...
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Hermiticity of Kubo Transform

I want to see whether the Kubo transform($\int_{0}^{1} \rho^{x} O \rho^{1-x} dx$) is hermitian or not. Taking the adjoint of $K_\rho$, we have $K_\rho^\dagger[O] = \left(\int_{0}^{1} \rho^{x} O \rho^{...
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Discrepancy in the Derivation of Maxwell-Boltzmann Distribution

I am reading Maxwell-Boltzmann distribution for describing the velocities of ideal gas molecules. I went through the PSE question Derivation of the Maxwell-Boltzmann speed distribution and the ...
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Derivation of Maxwell Boltzman velocity distribution [closed]

We use the distribution function for the fraction of molecules having velocity $V=(v_x,v_y,v_z)$ as $f(v_x)f(v_y)f(v_z)dv_xdv_ydv_z=F(V)d³V$. Now Zemansky says $dF=0$. Which i don't know how? Second, ...
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The mean kinetic energy of a gas particle

I'm in undergraduate stat mech/thermo. In the context of the Maxwell-Boltzmann distribution, the mean kinetic energy of a gas particle is $\langle KE \rangle = \frac{1}{2}m \langle v^2 \rangle$. I do ...
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Expressions for Entropy in the Canonical Ensemble

In the microcanonical ensemble, we have the standard Boltzmann expression for entropy: \begin{equation}\label{1} S = k_B\ln \Omega \end{equation} where $\Omega$ is the number of elements of the ...
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What is the action of fermionic Hamiltonian $\mu_1 n_1 + \mu_2 n_2 + U n_1 n_2$

Problem Consider a Hamiltonian \begin{equation} H(c^\dagger, c) = \mu_1 c_1^\dagger c_1 + \mu_2 c_2^\dagger c_2 + U c_1^\dagger c_1 c_2^\dagger c_2\,, \end{equation} where $c_i$ are fermionic ...
Michał Jan's user avatar
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Intertial finite-size effects in fluid simulations

A gradient $\nabla \rho$ in the density field $\rho$ of fluids at thermodynamic equilibrium is suppressed at a rate given by $D \nabla^{2} \rho$, allowing to measure the diffusivity $D$ of the fluid ...
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Decorrelation between a system and a harmonic bath

I'm going through the section on generalized Langevin equation (Chapter 15) in Mark Tuckerman's textbook Statistical Mechanics: Theory and Molecular Simulation, and there is a property that I am ...
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Homogeneity Restrictions on the Distribution of states in Thermodynamic Systems

The expected energy in the canonical ensemble is given by \begin{equation} \begin{split} \langle E \rangle &= \frac{\displaystyle\sum_{i=1} E_i e^{-\beta E_i}}{\displaystyle\sum_{i=1} e^{-\beta ...
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Is there a name for a Heisenberg-like model, but instead of the ZZ operator, we have one that favor only spin-up-spin-up configurations?

I understand that the Quantum Heisenberg XXZ model in 1D has the form: $$\hat H = \frac{1}{2} \sum_{j=1}^{N} (J_x \sigma_j^x \sigma_{j+1}^x + J_y \sigma_j^y \sigma_{j+1}^y + J_z \sigma_j^z \sigma_{j+1}...
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The 2d Ising transfer matrix and the effect of anisotropy on more general transfer matrices

The 2d Ising model has a row-to-row transfer matrix that can be written suggestively as $$T = e^{\tau \sum_i \sigma^z_i \sigma^z_{i+1}} e^{ \lambda \tau \sum_i \sigma^x_i}$$ where $\tau$ and $\lambda$ ...
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Quantum Regression Theorem Assumptions

The Quantum Regression Theorem states that if the time evolution of single-operator expectation values is known, then this determines the time evolution of higher-order correlations.Mathematically we ...
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Temperature Definitions Dependent on the Number of Quantum Particles [closed]

In the realm of quantum mechanics, the notion of temperature loses its direct correspondence to the average kinetic energy of particles. When dealing with a single quantum particle, the concept of ...
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Counting Microstates to get the Entropy Generation in Free Expansion of an Ideal Gas

In Elliot and Lira's Introductory Chemical Engineering Thermodynamics on p.134, the authors derive the entropy generated from an ideal gas expanding from a volume V to 2V, by the removal of a ...
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Relation between hamiltonian perturbation theory (classical) and the Fokker Planck drift and diffusion coefficients?

Suppose I have a hamiltonian of the form $$ H(q,p) = H_0 + \epsilon H_1(q,p) $$ In perturbation theory we approximate the solution to the equations of motion as a power series in $\epsilon$: $$ q(t) = ...
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Intuition for smaller upper critical dimension in quantum statistical physics

In the theory of phase transitions, the upper critical dimension (UCD) is the dimension of space above which the phase transition is well captured by mean-field theory. For instance, it is well known ...
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Equal average energies in translational and rotational degrees of freedom

In, An Introduction to Thermal Physics, Schroeder states It’s not obvious why a rotational degree of freedom should have exactly the same average energy as a translational degree of freedom. However, ...
GedankenExperimentalist's user avatar
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Why is entropy a quantification of the typical fluctuation of internal energy around the expected value?

In different books (one example is Statistical mechanics of learning, by Engel and Van Der Broeck) I stumbled upon an idea which should be elementary, but to me it is not easy to grasp. Entropy can be ...
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Carbon dioxide vibration

Can we induce antisymmetric stretching vibration in carbon dioxide molecules by collision with nitrogen molecules in air at standard temperature and pressure?
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Ferromagnetic Potts models in a field and the endpoint of their first-order lines

The $q=3$, $d=3$ ferromagnetic Potts model has a first-order transition on varying temperature. I recently learned that at small $h>0$, where $h$ is a field favoring one of the three colors, there ...
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What is the saturation property of molecular forces?

Context I'm studying statistical mechanics using Huang's Statistical Mechanics [1]. Huang writes, Empricially $U$ is an extensive quantity. This follows from the saturation property of molecular ...
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Derivation of kinetic equation for fast particles in a plasma

Referring to Plasma Astrophysics, Part 1, author: Boris V Somov, 4.1.1, We have the kinetic equation with Landau collisional integral, $$ \frac{\partial f}{\partial t} + v_{\alpha} \frac{\partial f}{\...
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A Particular Interaction Matrix in a Fully Connected Ising Model

I’m considering a fully connected Ising model with a Hamiltonian of the form: $$ H = \sum_{i<j} J_{ij} S_i S_j $$ where $S_i = \pm 1$ represents the spin of each particle, and the sum is over all ...
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Is there a galactic "goldilocks" region in the galaxy

I'm wondering if there's a region where the star density in the galaxy create the conditions in outer space where the galactic temperature is between 0 and 200°F. This may cause a ring shaped where ...
Matt Staab's user avatar
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Prove that the second derivative of enthalpy with respect to entropy is always positive [closed]

Is there a way to prove that the second derivative of enthalpy with respect to entropy is always positive? If we are using the specific heat capacity at constant pressure do show how that is positive ...
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Understanding Exceptional Points

Exceptional points occur generically in eigenvalue problems that depend on a parameter. By variation of such parameter (usually into the complex plane) one can generically find points where ...
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Does infrared radiation emitted by an object happen only from its surface, or is emission also from the centre of the object?

Textbook answer of how radiation is emitted is from the surface. Does the inside of a object also emit infrared?
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Calculating the Second Virial Coefficient

I have seen many different derivations of the second virial coefficient, but none of them explains how they turn a double integral into a single integral. The second virial coefficient is: $$ B_2=-\...
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Is it a known fact that quantum theory is the stationary point of (information theoretic) entropy?

Quantum theory is the stationary point of (information theoretic) entropy. Or equivalently a quantum system is (at least at the superficial level) a canonical ensemble in the reservoir of action, ...
y-watarino's user avatar
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1 answer
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Does there exist a 2d continuous phase transition that is first-order in mean-field theory and satisfies the Harris criterion $\nu>1$?

I am looking for an example of a clean, local $d=2$ classical model that undergoes a continuous phase transition on varying temperature that satisfies two properties: The phase transition is ...
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Could compression make gas colder not hotter?

In classical thermodynamics compression always makes gas hotter because of the mechanical work it inputs. However, if the particle density is too high, particles will become degenerate and obey the ...
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Heat Capacity in Mean Field Theory

I have been very confused with calculating the heat capacity when dealing with a Mean Field Hamiltonian. The Hamiltonian I am working with describes a spin lattice of fermions in 2D. I only count the ...
Roger's user avatar
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Regarding the relation between statistical equilibrium and thermal equilibrium

I've seen a similar set of questions on the topic in SE but none seem to satisfy me.So,the question is as follows:1)What is the relation between thermodynamic equilibrium and statistical equilibrium ...
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Chemical equilibrium for distinguishable reactant particles

Denote the partition function of a single particle by $\zeta$. The particles of one type of reactant in a chemical reaction are normally indistinguishable. The partition function of $n$ number of a ...
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Definition of temperature as variance of the momentum

I have always taken the definition of temperature to be the kinetic energy Statistical mechanics definition of temperature as the average kinetic energy. However I have been reading a paper where the ...
Daniel Adams's user avatar
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Relation between Mean Squared Displacement and Velocity Autocorrelation Function [closed]

I'm trying to understand the relation between the Mean Squared Displacement in a sample of moving atoms (for example) and the autocorrelation function of the velocity. So I read in "Understanding ...
chewingram's user avatar
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"By saddle-node" meaning?

On page 2 of this preprint, the authors make a derivation "by saddle point". $$Z \propto \prod_{\sigma=1}^{q} dx_\sigma \, \exp{\left(-N \left[\sum_{\sigma} \frac{\beta Jx^2_{\sigma}}{2} -...
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Justification for extending Gibbs entropy to non-equilibrium states

I am taking a course in statistical mechanics and have seen the Gibbs, Boltzmann and Clausius formulations for entropy in equilibrium thermodynamic states. And I have seen that it can be shown that ...
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Are all microstates always equally probable? [duplicate]

The Gibbs entropy is given by: $S_G = -k_B \sum P_i log(P_i)$, where the summation runs over all possible microstates. I've seen one of the assumptions for statistical mechanics was that all ...
David's user avatar
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Recover non-relativistic density of state

The density of state of a non-relativistic particle ($E = \hbar^2k^2/2m$) in 3D is: $$\rho_{class}(E) = \dfrac{V}{4\pi^2}\left(\dfrac{2m}{\hbar^2}\right)^{3/2}E^{1/2}.$$ The density of state of an ...
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Relation of entropy given in terms of phase space volume vs. multiplicity

I find in the liuterature (e.g. Landau & Lifshitz [1]) that the entropy in a microcanonical ensemble is given as: $S = k_B \log(\Omega),$ where $\Omega$ is the mutiplicity of microstates (Landau ...
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