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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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Diatomic Partition Function

Given the following Hamiltonian: $H = \frac { 1 } { 2 m } \left( \left| \mathbf { p } _ { 1 } \right| ^ { 2 } + \left| \mathbf { p } _ { 2 } \right| ^ { 2 } \right) + \frac { \kappa } { 2 } \left| \...
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How do we imagine the transition between phonons and rotons?

In the theory of superfluidity, we have both phonon and roton excitations. Phonons are long-range density fluctuations (sound waves) and rotons are short-range (atomic scale) circulations of the atoms ...
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local equilibrium of slow time varying thermal system

I'm trying to differentiate a thermal system in local equilibrium (and slow time varying) v/s a non-equilibrium system. For a thermal system which is slowly time varying, how does one define local ...
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What is the physical interpretation of the action integral, without the stationary action principle?

I'm still wondering about the physical interpretation of the action integral of some mechanical system (classical theory here, to simplify things): \begin{equation}\tag{1} A = \int_{t_1}^{t_2} L(q, \, ...
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Degenerate states, Boltzmann factor and statistical mechanics

The probability of finding a particle with energy $E$ according to Maxwell-Boltzmann distribution is: $$ P(E) =\frac{1}{Z}g(E)e^{\frac{-E}{k_BT}} \qquad eq(1)$$ where g(E) is the degeneracy of ...
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42 views

Stochastic differential equations with null mean and unit variance

I have the following: $ \dot{x} = \frac{dx}{dt}= A\left( x\right) + \sqrt{B\left( x\right)}\eta\left( t\right) $ where $ A\left( x\right)=a_0 - a_1x $ and $ B\left( x\right)=b_0-b_1x+b_2x^2 $. All $ ...
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Model linked with Lee-Yang theorem?

i'm studying Lee-Yang three theorems of statistical mechanics. My professor has spoken about a model with this Q(partition function) $ Q = \frac{(1+y)^v(1-y^v)}{1-y} $ My prof. has called it ...
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Product Rule for Partition Sums $Z_N=(Z_1)^N$

For the 1D Ising model with the Hamiltonian $$H=const.-\mu h' \sum_i S_i^z$$ we can write the canonical partition sum as $$Z_N = \sum_{ \{ S_i^z \}_N } e^{-\beta \mu h \sum_i S^z_i} = \sum_{ \{ S_i^...
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Question about the autocorrelation function of the fluctuating force in the Langevin model for Brownian motion

According to the Langevin model, we have, for the motion of Brownian particles, $$\frac{dv}{dt} = -M\gamma v + \zeta(t)$$ with $\zeta(t)$ the random force acting on the particle due to fluctuations. ...
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Ising Model Error Propagation

If I have the statistical uncertainties of the ensemble average magnetisation and the average energy from a monte carlo simulation of an Ising Model, how do I find the errors on the specific heat ...
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How are the coefficients determined in the high temperature expansion of the 2D Ising model?

I have been studying the 2D Ising model lately and have been looking at high and low temperatures. But I'm having problems when trying to understand the high temperature one. The final expansion looks ...
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Statistics of 1D discrete random walks

I have already asked this question in Math.SE. Let $P(n)$ be a probability distribution on the integers. Suppose a random walker starts off at the origin and, at every positive integer time, takes a ...
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Expand the partition fct. of a simple harmonic oscillator

I come across a expansion of the partition fct. of a simple harmonic oscillator $q$ as: $$q=x^{-1}(1-\frac{x^2}{24}+...) \tag{1}$$ where $x=h\nu/kT$. It’s easy to get $$q=\frac{e^{-x/2}}{1-e^{-x}}=\...
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Why are the diagonals of the pressure tensor non-negative?

I understand that the pressure tensor is simply the momentum flux which makes sense to me (pressure is force per unit area which is momentum change per unit time per unit area). From this, a simple ...
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Questions about physics in a C language code about the potential of Lennard Jones [closed]

The code (C language) is about molecular dynamics using the potential of Lennard Jones. The code is 100% working but wanted to understand certain details: 1) why on line 67 appears 1.0e-6? 2) Why add ...
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Compare the level of degeneracy of particles following Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein Statistics

Compare the level of degeneracy of the particles following Maxwell-Boltzmann statistics, Fermi-Dirac Statistics and Bose-Einstein Statistics respectively. I was pondering over this question.. And I ...
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Partition function of an interacting particles

The Hamiltonian of (𝑁+2) interacting classical particles, that are enclosed in a cube of volume 𝑉 at temperature 𝑇, is given by: $H = \sum_{i=0}^{N+1} \frac{|\vec{P_{i}}|^2}{2m} +\frac{1}{2}mw^2 \...
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Superconductor specific heat capacity [migrated]

I would like to obtain expression for heat capacity jump of superconductor. During calculation, I can not deal with the followiwng integral: $$\int_{0}^{\infty}\frac{dx}{x^2}\left(\frac{1}{\cosh^2 x}-\...
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38 views

One-dimensional Ising Model in a three spin chain

I have a system of three aligned spins with $S=\frac{1}{2}$. There are interactions between nearest neighbors, and each spin has a magnetic moment. The Hamiltonian of the system is: $$H=J[S_z(1)S_z(2) ...
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Using probability of camera flash interval to get the probability density equation in Griffith's Quantum Mechanics book

In Griffith's QM, example 1 chapter 1, what is the intuition behind using the probability of camera flash interval to get the probability density equation in terms of "dx". Griffith says that ...
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Probability at temperature in system has energy

Salutations, I'm starting in statistical mechanics and reviewing some related studying cases I would like to understand what occurs in small systems with normal modes of vibration, for example, a ...
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1answer
26 views

Quantum statistics for interacting systems

What is the distribution statistics for the mean occupancy of a many-body state? How can I show that this reduces to the single-particle Bose-Einstein or Fermi-Dirac ones when interactions tend to 0?...
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Can the entropy of mixing be negative? [on hold]

There is a general notion that the entropy of mixing should always be positive (or zero if we are mixing exactly the same stuff). However, I have a seeming counterexample at hand. Consider a box ...
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Bose condensate in 4d

Could a boson gas condensate in a hypervolume $V$ in 4D? How can I find its critical temperature and the heat capacity? In the books it just said volume $V$, it does not specify the dimension. My ...
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Building temperature into the Hamiltonian

Given a quantum Hamiltonian $H$ (e.g. the quantum Ising Hamiltonian $H= -h\sum_{i}X_i-\sum_{\langle i,j\rangle}Z_iZ_j$): we know that at temperature $T$, the system is in the state: $$\rho(T) = e^{-H/...
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Quantum Nuclear Corrections

I am interested in calculating quantum nuclear corrections to energy profiles or free energy profiles. For example, say I perform a DFT simulation for some molecules, and I the energetic profile in ...
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2answers
63 views

Negative probabilities with Wigner quasi-probability distributions

I was toying with Wigner corrections to thermodynamic equilibrium. The semiclassical correction for the position probability density to second order in $\hbar$ is: $$P(x)= \text{e}^{-\beta V(x)}\left(...
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1answer
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Reconciling various expressions of flux and flow in kinetic theory

There are a few expressions I have been shown to describe the flux $\vec{\Phi}_A$ of some quantity $A$ described in kinetic theory. I am having trouble understanding how they are related or if they ...
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77 views

Energy of harmonic oscillators

I've calculated the energy of a classical harmonic oscillator (HO) as: \begin{align*} \overline E = \overline{E_K} + \overline{E_P} = \frac{\overline{p^2}}{2m} + \frac{k\overline{x^2}}{2} = \frac{...
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Multiplicity of statistical weight of macrostates when combining systems in thermodynamics (Boltzmann entropy)

In my lecture notes, the Boltzmann entropy form was motivated as a valid form for entropy partly because it was extensive. However this hinges on the assumption that the statistical weight for a ...
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Confusion on real space renormalization group for Ising model on lattice

For the Ising model with only nearst neighbor interaction on square lattice, if we do the RG by integrating out half degree of freedom, then we would get a new Ising model with many kind of ...
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Introductory to statistical mechanics text query

In the text "Introduction to modern statistical mechanics" by Chandler the following is stated in the first chapter: Entropy obeys several other important properties as well. To derive them, ...
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Regarding this interpretation of the pair correlation function

On some lecture notes of the statistical mechanics of Fermi systems I found the following, specifically regarding spin 1/2 systems: The correlation function $$ g_{\uparrow\uparrow}(s) \equiv \...
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Why is -273.15 °C the low temperature limit for the universe? [closed]

According to Ideal Gas Law the lowest temperature of an ideal gas can be $-273.15 °C$. This temperature is also considered the lowest temperature in the universe. But it is the lowest possible ...
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Derivation of 2nd law of Thermodynamics from ergodicity assumption

In Wikipedia it is claimed that: Assumption of the ergodic hypothesis allows proof that certain types of perpetual motion machines of the second kind are impossible. Since perpetual motion ...
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Blackbody photon variance and number of modes [closed]

In an experiment to measure the photon statistics of thermal light, the radiation from a black- body source is filtered with an interference filter of bandwidth 0.1 nm centered at 500 nm, and allowed ...
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Indistinguishable particles and statistical mechanincs

i'm studying the paragraph 5.5 (page 119) of this book: http://sciold.ui.ac.ir/~sjalali/MSc.Students/statistical.mechanics/pathria.pdf Now at page 121 we have: $$ \sum\limits_{p} \delta_p{u_{k1}}(...
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Chemical Potential and interactions

I'm interested in an model with interactions between different kind's of particles. Each particle species has it's own chemical potential. I want to treat the system in the Matsubara formalism. Here, ...
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1answer
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Quantum statistics from the (anti)commutation relations of the operators?

From a QFT point of view, the difference between bosons and fermions is that their creation/annihilation operators ($a^{\dagger}$, $a$ and $c^{\dagger}$, $c^{\dagger}$ respectively) obey the following ...
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Many-body quantum tunneling: Is quantum tunneling sensitive to decoherence?

If we have a many-particle System that is strongly correlated, the tunneling probability can significantly increase; see this article here: https://www.sciencedaily.com/releases/2014/06/140612142215....
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1answer
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Why the chemical potential of phonon gas in Einstein 's solid model is not zero

In Einstein’s model of solid, each atom in the solid is considered to be an independent three-dimensional quantum harmonic oscillator with characteristic frequency $ω$ that is constant. Each degree of ...
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1answer
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Why do we study the Ising model on $\mathbb{Z}^d$ for $d > 3$?

I'm a beginner in statistical physics and I'm reading some stuff about the Ising model. So this might be a silly question. My question is: why we study the Ising model for high dimension cases, ...
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What are good books covering information theoretic approaches to theoretical physics?

I am about to finish my undergraduate studies and am very interested in going into the applications of information theory to either general relativity, or quantum mechanics. However I have been ...
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How do the fluctuations of the order parameter depends on the dimensionality of the system?

In Landau theory of second order phase transitions, how do the fluctuations of the order parameter depends on the dimensionality of the system? In superconductors, the fluctuations of the order ...
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What is the probability distribution for a subsystem in canonical ensemble?

Suppose we have a 3d Ising Model(NN interaction) in simple cubic lattice, if we define a subsystem of it to be a 2d plane of spins(for example all sites with z = L/2, L being the linear system size) ...
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Phase transtions. Why has Ehrenfest classification been replaced by modern classification?

Why did modern classification replace Ehrenfest classification? What are the advantages of the modern one?
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Shot noise derivation

In the lecture notes I use to study on shot noise I found the following derivation for the Fourier transform of $$\phi_{shot} (t) = \langle (I(t+t_0)-\langle I\rangle)(I(t_0)-\langle I\rangle)\rangle =...
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What is the potential of mean force?

I've come across the term potential of mean force (PMF) in polymer physics, colloidal physics etc., but have not come across a complete definition. As far as I understand, the PMF determines the ...
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Why gas molecules move with different speed at a given tempreture?

As per my understanding we know that molecules of an ideal gas are identical in all aspects (size, shape, mass). Since collisions are elastic in nature, they don't lose their kinetic energy. That ...
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Path integrals vs operator

I have a statement that the path integrals formalism is eqivalent to operator formalism in quantum mechanics. Is it a correct statement? I understand that each of these two formalisms has its ...