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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14 views

transitions in Ising lattice gauge theories in 3+1 dimensions

What is known about the character of the transition (apart from the self-duality of the model and its self-dual point marking the transition point) in the Z2 lattice gauge theory in 3+1 dimensions?
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What research has been done to develop a theory of ideal fluids?

A prof mentioned that there has been a lot of research to develop a theory of ideal fluids. A subsequent literature search wasn't very illuminating. Could somebody please tell me what has been done on ...
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Entropy as a multiplicative measure of disorder as opposed to an additive one. Probability distribution?

Entropy $S,$ is usually defined as an additive measure because mathematically and physically speaking it's usually easier to work with. I'm wondering how to write entropy as a multiplicative measure......
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Why do Repeat Measurements Result in a Reduced Error?

I'm currently reading "Concepts in Thermal Physics", and in the chapter on independent variables it has the following example: If we have $n$ independent variables $X_i$, each with a mean $\...
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Does scaling appear on a phase boundary/first order phase transitions, or it is only reserved to critical points?

In statistical mechanics people talk about scaling laws and critical phenomenon, and one of the textbook examples brought up is the liquid-gas critical point. But in contexts of Superconductivity and ...
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Normalization of the partition function

I am used to think that multiplying the partition function by a constant is meaningless, and shouldn't change any calculation, as under $Z' = A Z = A \sum_n \exp(-\beta E_n)$ for an arbitrary $A$, the ...
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44 views

Asymmetric Random walk with a pause [closed]

EDIT 1 (giving context): In the non-equilibrium statistical mechanics framework, there are two basic paradigms for defining the dynamics of the system: the Langevin and Fokker-Planck equations for ...
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Boltzmann factor

while deriving an expression to represent Boltzmann factor we generally use the classic example where a system of energy E* is immersed a reservoir having energy E and after taking taylor series ...
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38 views

What is the explanation of Quantum Theory to solve the dip down on the UV spectrum on the black body spectrum and overall explanation of the spectrum?

As classical physics predicted the black body spectrum should be a rational graph where the radiation reaches up unlimited high with the increase of frequency. However, it doesn't meet the ...
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Planning new project on sensitivity analysis of simulated crystals [closed]

Not a while ago, I began to read about negative thermal expansion and I started reading about all the work Linus Pauling had done with ice crystals. I am meant to be doing a 7-month research project ...
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38 views

Classical concept: Derivation of Boltzmann distribution [closed]

Consider and read Classical Concept review 7 I don't understand what is $g_i?$. It is defined as statistical weight of state i? But i don't understand is it microstate or macrostate? I also didn't ...
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54 views

Why is the macroscopic Heat capacity related to the standard deviation of microscopic energy fluctuations?

In one of the problems in my textbook (Schroeder Intro to Thermal Physics), it is shown that within the canonical ensemble, the standard deviation of the energy fluctuations for a single microscopic ...
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41 views

Simulation of Boltzmann distribution

I'm tackling with the problem in Blundell_Concepts in Thermal Physics textbook. Let me introduce the context first. Below figure is just taken from Blundell textbook section 4.6. As you can see in ...
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Functional derivatives in density functional theory

I am studying density functional theory and I am currently dealing with manipulating the intrinsic free energy, $\mathcal{F}$, which is defined as $$\mathcal{F} = F - \int dr \rho ^{(1)}(r)\phi (r) $$...
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154 views

Intuition behind power-law scale invariance

I have seen this notion of a scale-invariant power law curve exhibiting the property that $f(cx) = a(cx)^{-k} = c^{-k}f(x)$, and I am confused about how I should be thinking of this as "scale-...
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What Happens if One Couples two Thermal Harmonic Oscillators at Different Temperatures?

Consider two identical initially uncoupled harmonic oscillators with Hamiltonians $$\hat{H} = \frac{p_1^2}{2m}++\frac{m\omega^2x_1^2}{2},$$ $$\hat{H}_2 = \frac{p_2^2}{2m}+\frac{m\omega^2x_2^2}{2}.$$ ...
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44 views

Understanding mean rate of change in Brownian motion

I found a nice discussion of Brownian motion in the Feynman lectures, reproduced online here: https://www.feynmanlectures.caltech.edu/I_41.html Feynman considers a particle undergoing a Brownian ...
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Does "very non-extensive" quantum matter exist?

This is a two-fold question: I have read that the entropy of a black hole is proportional to its surface area. It left me wondering if it is possible to create some (non-trivial) quantum matter with ...
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Gas particle collision rate dependence on particle velocity/ Hard sphere simulation

From a basic kinetic theory perspective the A-B collision rate $\theta_{AB}$ is given as a function of the mean speed and the number density.$$\theta_{AB} \propto \bar{C}_A n_B$$ If we are considering ...
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Technique for diagonalising this free spinless fermionic Hamiltonian?

How does one diagonalise the following Hamiltonian? $$ H = \sum_n \epsilon_n c^\dagger_n c_n + g \sum_n (c^\dagger_n c^\dagger_{-n} + c_{-n}c_n), $$ where $c_n$ is a spineless fermionic op. Clearly we ...
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31 views

Thermal state of free fermions in contact with a reservoir at temperature $T$?

Without loss of generality and for simplicity, consider a two fermion Hamiltonian $$ H = \lambda (c_1^\dagger c_2 + c_2^\dagger c_1), $$ where $c_i$ are fermionic ops, i.e. a hopping Hamiltonian. We ...
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Can this relation given in Pathria's Statistical Mechanics book be used in this sytem?

Relation in Pathria's book: (relevant quotes from the text that put this equation and my doubts in context is given below) $$V^{\frac{2}{3}}E=constant$$ The system: $N$ number of classical ...
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Finding the thermal state for a finite dimensional Hamiltonian

Given a finite dimensional Hamiltonian $H$ (a $N\times N$ matrix), I am interested in computing numerically it's associated thermal/Gibbs state $$ \left|\psi_\beta \right> \equiv \frac{1}{Z}\sum_{...
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37 views

Entropy of perfect gas function of internal energy and number of particles [closed]

In a statistical physics lecture, I found an equation describing the entropy $S$ of an ideal gas and it is said that equation can be obtained thanks to elementary thermodynamics formulae. The equation ...
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81 views

How do you derive the Maxwell-Boltzmann distribution?

I have searched for a reasonable derivation online, but so far have been unable to find one which doesn't skip steps or presume prior knowledge. I found a derivation on this Wikipedia article which ...
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1answer
22 views

Why is the expectation value calculated for the Einstein calculation of specific heat the same as the quantum expectation value of energy?

In 29:22 here, the professor says that the expectation value calculated using the partition function when finding the specific heat of an Einstein solid is both a quantum mechanical and a statistical ...
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65 views

Hermitian and non-Hermitian jump operators in Lindblad master equation

Is there a way of rotating non-Hermitian jump operators for a Lindblad master equation (LME) to a basis where they are Hermitian? In other words, I have a (diagonal) LME: $$ \dot{\rho} = -i [\mathcal{...
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23 views

The second law and the advantage of measurement?

Background To the best of my knowledge, almost all living systems make use of information obtained from sensory organs. If this result is a reflection of natural selection, then this fact suggests ...
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82 views

How can free expansion be truly irreversible if particles have a small chance of returning to their original state?

According to Halliday-Resnick, a free expansion of a gas is an irreversible process. However, the text continues that in a system of particles in a box, it is possible (though very unlikely) for a ...
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34 views

Fundamental Relations of Thermodynamic Potentials

One question that I haven't really been able to get a good answer to is that of the fundamental relationships between thermodynamic potentials ($U,F,G,H$) and concepts in statistical mechanics. We ...
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How to find the partition function of the $1$D Ising model?

I am trying to solve a problem that requires finding a partition function. Question: Consider a one-dimensional Ising model with $N$ spins at very low temperature. Let there be $r$ spin flips with ...
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How to derive pressure equation from grand canonical ensemble

The pressure equation is given by $$\frac{\beta P}{\rho} = 1-\frac{\beta \rho}{6}\int dr \cdot r \left[ \frac{dU}{dr}\right]g(r)$$ I am trying to derive it from the thermodynamic definition of ...
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What happens when the PE equals to zero in the potential energy vs intermolecular distance graph? [closed]

In the potential energy versus inter molecular distance graph, we know that atoms/molecules/particles want to be at optimum distance from each other ie $r_0$ and to the left of this position in the ...
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Relative entropy and Lagrange multipliers

I'm reading Barnett's 'Quantum Information Theory' and at some point, he uses Lagrange multipliers to show that the relative entropy is nonnegative (everywhere I looked the proof ends up using Jensen'...
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32 views

Fermi level of a semiconductor

When calculating the career concentrations in the conduction band of a intrinsic semiconductor we consider the integral $\int_{E_c}^\infty g_C(E)f_{FD}(E,T)dE$ where $g_c$ is the density of states in ...
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How do the 2 and 4 point correlation functions depend on volume?

Why is the two point function $\langle\phi(x) \phi(x^\prime) \rangle_c = \frac{\delta^2 F[J]}{\delta J(x)\delta J(x)}$ not extensive, since in the thermodynamic limit, the free energy behaves as $F[J] ...
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29 views

Intrinsic carrier concentration of a semiconductor without Boltzmann approximation [closed]

The carrier concentration of conduction band of a intrinsic semiconductor at a temperature $T$ is $$n_i=\frac{1}{2\pi^2}\left(\frac{ 2m_e^* }{ \hbar^2}\right)^{3/2}\int_{E_c}^{\infty}\sqrt{E-E_c}f_{FD}...
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What's wrong with the Maxwell demon? Seems very trivial consequence of the second law of Thermodynamics...?

Maxwell Demon's Paradom: The entropy production of the system $X$ becomes negative: $$\Delta S(X) + Q_X \leq 0$$ Exorcism of the demon: I believe that this paradox has been resolved by an inequalities ...
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23 views

Gravitationally-driven electrical potential differences in conductors

This question asks Free electrons in a metal are attracted by gravity towards Earth. So why don't they lay down to the bottom of the conduit, like sediment at the bottom of a river? The current ...
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How are these two expressions for the canonical partition function equivalent?

In Equilibrium Statistical Physics by Plischke and Bergersen the canonical partition function is defined (on page 37, eq. (2.33)) as $$Z_C = \int \frac{dE}{\delta E} \Omega(E) \exp\{-\beta E\}\tag{1},$...
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Simplest exactly solved model displaying a phase transition?

The classical example of an exactly solved model which displays a phase transition is the 2D Ising model. However, all the proofs I've seen of this have been very long and complicated. So, I wanted to ...
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Is it possible to attain absolute 0 temperature? [duplicate]

I think it is impossible to attain this temperature in normal circumstances. As energy would be applied from surrounding areas and energy flow from higher to lower potential and hence all the heat ...
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Quantization of the Gibbs distribution

Consider a simple quantum mechanical system, for example, the 1d harmonic oscillator. Given the inverse temperature $\beta$, the classical Gibbs distribution is the following function over the phase ...
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Splitting probability over a saddle point for diffusion

Consider a system with two metastable reactant and product states whose reaction dynamics can be approximated by a multidimensional Smoluchowski equation. If the potential of mean force around the ...
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27 views

Simplifying double integrals of isotropic functions

I am studying statistical mechanics, and I am studying ideas surrounding potential of mean force and n-body density functions. In a derivation, they mention that $$-\left\langle \frac{\partial U}{\...
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109 views

Why is quantum entanglement associated with increase rather then decrease of a system's entropy? [duplicate]

As far as i understand quantum entanglement (and I'm far from being an expert on it), it is a phenomena involving two or more subatomic particles that reflects itself in immediate correlation between ...
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Canonical ensemble Pathria's derivation

I'm reading Pathria's Statistical Mechanics but I have one question regarding the derivation of the Canonical ensemble. The derivation is resumed in this question: Canonical Ensemble and Combinatorics ...
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1answer
57 views

Why do we use Boltzmann distribution rather than Fermi-Dirac distribution in a transistor?

Considering electrons are fermions I would think it is better to use Fermi-Dirac distribution when discussing the physics of the electrons and holes in a transistor. However, I have been told ...
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147 views

Dirac delta-function integral in Kerson-Huang

I found a portion in Kerson Huang $\textbf{Statistical~Mechanics}$ book, in Page No. $\textbf{61}$ where they have performed an integral over a delta function and got the last equation (given at the ...
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159 views

Can every global conservation law be written as following?

Consider a physical quantity $\phi$ that is globally conserved. From Feynman's argument (in his volume 2 I think), which states that local conservation follows from global conservation due to special ...

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