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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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Escaping metallic solids under extreme pressure

Suppose you have metallic solids inside an indestructible tube, with a very powerful and indestructible piston - the piston gets a tiny hole on the piston. What would happen if you compress the solid ...
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why for two level system we consider both energy level while finding number of bosons in ground state?

Let suppose we have N number of particles in two level system. .Effective number of cobosons in ground state is $ <n_0>$ that can be written as $<\hat{n}_0>= Tr [\hat{n_0}\rho ]$ where $\...
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Partition function and probability in finite harmonic trap

Let suppose we have $N$ number of cobosons in 3D harmonic trap. The effective number of cobosons in some state m is $\langle n_m\rangle$ that can be written as $\langle\hat{n}_m\rangle= \mathrm{Tr} [\...
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Kosterlitz-Thouless transition and correlation function

I’m studying Kosterlitz transition on this book: https://tinymachines.weebly.com/uploads/5/1/8/8/51885267/kardar._statistical_physics_of_fields__2007_.pdf#page173 . At page 165 it says:” The gradient ...
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What is meant by finite harmonic oscillator?

What does it mean to take finite harmonic oscillator, In research article "http://iopscience.iop.org/article/10.1088/1367-2630/17/11/113015 ", we were finding effective number of cobosons in ...
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Can we say that density matrix gives probability?

In statistical physics Boltzmann probability is given by $$P= \frac{\exp(-\beta E_i)}{\sum_i\exp(-\beta E_i)}$$ whereas we can also write it $$\rho= \frac{\exp(-\beta H_0)}{\sum_i\exp(-\beta H_0)}.$$...
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Spin spin correlation function in topological phase transition?

during my vacation i have decided to study Kosterlitz and thouless phase transition (i have already posted 2-3 questions about that). I don't know quantum field so I did not expect to understand ...
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What is bose enhancement factor?

I am studying about composite bosons ,when we write N state for composite bosons in terms of its constituent particles, we add factor of normalization constant \chi in denominator.This normalization ...
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Integrating Carnahan-Starling Pressure

Given the Carnahan-Starling equation of state for a solution of hard-spheres, $$ Z = \frac{P}{\rho k_BT} = \frac{1 + \eta + \eta^2 - \eta^3}{(1-\eta)^3}$$ where $\rho = N/V$ is the number density and ...
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Laplace Transform Density of States & Partition function

I am currently going through Pathria's Statistical Mechanics text , and under the Canonical Ensemble description, the author stresses that the partition function of a continuous system is the Laplace ...
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Computing the average energy $\overline{E}$ in thermodynamic equilibrium for a paramagnet

I am having trouble with the following physics exercise: In the case of a paramagnet in a magnetic field $H$, show that from the requirement that $F(E)$ (the Helmholtz free energy) be minimal in ...
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Canonical partition function for different systems

As a homework exercise for Advanced Statistical Mechanics I need to derive the canonical partition functions for the following systems: Single component ideal gas on a square lattice Single component ...
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What is the link between statistical and QFT correlation functions?

I'm studying statistical mechanics in particular correlation function: https://en.wikipedia.org/wiki/Correlation_function_(statistical_mechanics) and I have understood it. Now searching on internet ...
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Computing the Helmholtz free energy for a harmonic oscillator

I would like to calculate the helmholtz free energy for a harmonic oscillator, which has energies given by the equation $\epsilon_{n} = n\hbar\omega$, where $n$ is a natural number greater than or ...
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27 views

Venturi effect on molecular level

Is there a good explanation why the pressure becomes lower on the narrow part of the venturi pipe? I'm not interested in the newton explanation or pressure differences explanation. I want to ...
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In Boltzmann distribution, why is the system at the same temperature as the reservoir?

Consider a boltzmann distribution where the total energy of the reservoir and the system is $E$. The energy of the system can be $\epsilon_i$ and the energy of the reservoir is $E-\epsilon_i$. Now ...
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From emission probability to black body spectrum. (Hawking radiation as tunnelling.)

In the 1999 paper by Wilczek & Parikh (and in many subsequent papers) the "emission rate" (really the probability of tunneling) of a particle with energy $\omega$ from a static black hole (BH) is ...
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Would it be appropriate to say that in presence of Non-Conservative Forces, the Entropy of System Always Increases?

Would it be appropriate to say that in Presence of Non Conservative Forces the Entropy of a system would always increase? Can we relate Non-Conservative Forces and Entropy in some way? I intuitively ...
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Why are the Lyapunov and Lindeberg Central Limit Theorem conditions often satisfied in the real world?

Some background for the question. I've been trying to understand why so many things have a Gaussian Distribution. There are a lot of questions about this on StackExchange but none of them were ...
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Why is potential energy zero in this calculation of partition function?

In section 7.2 of Rief's "Fundamentals of Statistical and Thermal physics". While calculating the partition function for ideal gas he writes: $$ \begin{array}{l} \displaystyle{Z' = \int{ \...
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Do there exist phases of matter where the order parameter space is non-orientable?

For example, are there order parameter space that is homeomorphic to a Klein bottle?
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Book(s) fills the gap from introductory thermo to nonequilibrium thermo/stat mech for self-taught student?

I know this is a big question. But as a graduate student, my research is somehow related to nonequilibrium thermodynamics/statistical mechanics. TBH, I hate how some research treat this subject like a ...
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What determines the timescale for fluctuations in the electromagnetic field from a light source?

Let's say you place an electric field meter some distance from a light bulb. As a function of time the output of the meter would be $\mathbf{E}(t)$. I would guess that the electric field will be some ...
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One dimensional paramagnet

I have a lattice model consisting of $N$ spins $s_{j}$ which can take the values $s_{j}=\pm1$. The spins are considered to be non interacting. The probability for a spin spin to be 1 is p and the ...
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122 views

How to find density of states in harmonic oscillator?

Density of state should be number of states per volume .Why weThe take derivative of "number of states " with respect to energy to get density of states ?
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Why does Critical Points have fluctuations on all scales (Infinite correlation length?

I have been studying statistical field theory for a while and I still haven't found a physical explanation for this question. Every answer seems to be kind of circular. Basically something like this: "...
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The average velocity of a particle

The Maxwell distribution of velocities is: $$p (v) = (\frac{m}{2\pi K_b T})^{\frac{3}{2}} e^{\frac{-mv^2}{2 k_b T}}$$ I want to understand how to obtain the average value of the velocity. The ...
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How to handle bra-ket in logarithm?

$\newcommand{\ket}[1]{\left| #1 \right>}$ $\newcommand{\bra}[1]{\left< #1 \right|}$ Say the following two equations: $$ S = - k_B \text{Tr} (\rho \ln \rho) $$ $$ \rho = \sum _\epsilon \ket{\...
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Computing the average energy and specific heat at constant volume

Consider an Einstein solid. Each oscillator has quantized energy $E = n\hbar\omega$, where $n \geq 0$ is an integer. How can I compute the average energy and the specific heat at constant volume of an ...
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63 views

Bose Einstein Condensation in Grand canonical ensemble

Why we develop formalism of Bose Einstein Condensation in framework of grand canonical ensemble ?
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Calculating the mean separation using a canonical ensemble [closed]

I am having trouble with the following physics question. Two atoms, each of mass $m$, interact with each other by a force that can be derived by the mutual potential energy equation given by $$U = ...
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Long Range order in 2D Ising model

We know from the exact solutions for 2D Ising model on square lattice the long range order appears bellow critical temperature, but how does this agree with the Mermin-Wagner theorem, from which we ...
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Pauli Exclusion Principle

In a box which has length $L$, there are 3 distinguishable particles which have same mass $m$ and 1/2 spin. When one is in search for lowest energy level and degeneracies, I personally think one must ...
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60 views

Phase transition 2D Ising model magnetization around critical temperature

I am stuck on a part in the derivation of the critical exponent of the magnetization in case of the 2D Ising model. I know that I should find $M \propto (T_c-T)^{1/8}$ but I am having some trouble ...
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What are enthalpic and entropic forces?

Am I right when thinking of entropic force to be an entropy minimizing mechanism and enthalpic force to be an energy minimizing mechanism (which is basically an entropy maximization mechanism). What'...
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Microcanonical Ensemble of a two state system

I have a system consisting of N distinguishable particles. Each particle has two states, one with energy E and the other with energy 0. The number of particles in the state with energy 0 is $n_{0}$ ...
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Formally from $Z=tr e^{-\beta H}$ to $Z=\sum_{n_1, n_2} e^{-\beta (E_1 + E_2)}$ for two non-interacting quantum harmonic oscillators

Suppose we have a Hamiltonian of two noninteracting quantum harmonic oscillators. Then the Hamiltonian can be written $H = H_1 \otimes I_2 + I_1 \otimes H_2$. When I start with $Z=tr e^{-\beta H}$ I ...
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Distributing distinguishable particles in dinstinguishable boxes and computing canonical partition function

I have n distinguishable particles and m distinguishable boxes. If all particles are in the same box the system has an energy of -$\epsilon$ in all the other cases the energy is 0. Now I want to ...
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How to determine the off-diagonal term of magnetic susceptibility tensor from fluctations?

I have run a Monte Carlo simulation of the classical Heisenberg model (in the future I am planning to add other interaction terms). I would like to extract information about the property of the system ...
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142 views

Average Kinetic Energy from Canonical Partition Function

I want to compute the average kinetic energy of a particle at a certain temperature T given by the Hamiltonian: $$ H = \sum_{i=1}^{N}\frac{\mathbf{p}_i^{2}}{2m}+V(\mathbf{r}_{1},...\mathbf{r}_{N}), $$...
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Feynman tricks to reproduce Onsager's solution of the 2D Ising model

I found the following quote in this paper: Wilson, Kenneth G. "The renormalization group and critical phenomena." Reviews of Modern Physics 55.3 (1983): 583. Later, Jon Mathews explained some of ...
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Are there examples of nondegenerate Fermi gases?

A degenerate Fermi gas is an ensemble of fermions with very low interactions and at temperatures that are low enough (lower than Fermi temperature). Most of the examples in the literature are about ...
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84 views

Derivation of relativistic pressure

As you can find in many cosmology textbooks, the relativistic pressure in quantum statistical mechanics can be witten as below: $$p=g \int \frac{d^3P}{(2 \pi \hbar)^3} \frac{c^2 |\mathbf{P}|^2}{3E(\...
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$dQ=Tds$ and quasistatic process

See Ján Lalinský's commment on Entropy $dQ=TdS$ and Work $dW = -pdV$ conditions? My question are: Is there a way to prove that $dQ=Tds$ is a quasistatic process? What's the funcitonal description ...
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Deriving an entropy expression from two other expressions

I would like to show that an equation for entropy is given by $$S = k\frac{\partial}{\partial T}\left[T \ln Z\right].$$ I am given the equations $$S = -k_{B}\sum_{i = 1}^{N} p(n_{i})\ln(p(n_{i}),$$...
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What does blackbody radiation look like for negative-temperature systems?

Using the thermodynamic definition of temperature, $$\frac{1}{T}=\frac{\partial S}{\partial U}\bigg|_{V,N}$$ negative temperatures are possible, in systems where the entropy decreases when energy is ...
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Does the order of operators in the the hamiltonian in second quantised form matter?

For a particles that not interact (free particles) we can write the Hamiltonian in second quantized form as $$\hat{H} = -\frac{\hbar^2}{2m} \int \psi^{\dagger}(\vec{x}) \nabla^2 \psi(\vec{x}) d^3x \,...
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Physical examples of log CFTs

There are examples of CFTs having correlators with logarithms. What are the examples of physical systems exhibiting such logarithmic behaviour (particularly in $d>2$ dimensions)?
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Finding the partition function for a three-level system

I am having difficulty finding the partition function of a system with two particles, each of which can be in any of three states with energies $0, \epsilon, 3\epsilon$. The system is in contact with ...
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How to prove Fermi Dirac distribution by taking thermal average of particle number operator?

thermal average definition is given as; $$\langle...\rangle_{th}=\frac{1}{Z}\sum_{n'}\langle n'|...|n'\rangle \exp(-\beta En')\tag{1}$$ how ever lets assume I have an non interacting system now we ...