The study of large systems through coarse graining microscopic descriptions, providing a more detailed understanding of thermodynamics.

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Explanation Needed: Thermodynamics of a hot/cold water jet machine

I didn't know where to begin with this problem. I eventually found a solution online, which is why I'm reposting this question with an answer. I was wondering if anyone can explain the one question I ...
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1answer
565 views

Why is velocity normally distributed in a gas, but not energy?

If one looks at a cubic box of gaseous atoms all initially flying in the same direction at the same speed (but flying at an angle to the walls, so as not to reflect up-and-down against the box walls ...
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237 views

Derivation of Pressure/Kinetic Therory problem involving hole in box

A box of volume $V_0$ has a small hole of area $A_0$. The box initially has one mole of an ideal gas at $t = 0$, which is at an initial temperature $T (t = 0)$. Find the rate of energy flow through ...
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3answers
146 views

What are correlated magnetic moments?

My book has the following sentence and I don't understand what correlation or lack of correlation means: At high temperature the magnetic moments of adjacent atoms are uncorrelated (to maximize ...
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70 views

Infinite quon statistics/Quantum Boltzmann statistics: models and hamiltonians

I learned long ago that there are some exotic classes of statistics. One of them is calleq $q$-on or quon statistics. It is given by $$a_ia^+_j-qa^+_ja_i=\delta_{ij}$$ Infinite statistics (Quantum ...
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371 views

Can the laws of classical mechanics be derived from quantum mechanics? [duplicate]

Can classical mechanics be derived from quantum mechanics as the same way thermodynamics derived from statistical mechanics?
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86 views

Nonequilibrium themal QFT

Wick rotation to thermal of QFT in Minkowski space to thermal QFT, which is after this transformation analogue to statistical mechanics, does only describe equilibrium statistical mechanics. On page ...
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114 views

Is there a systematic way to determine the relevant variables needed to describe a nonequilibrium system?

In strong nonequilirium, the statistical operator describing the system depends on an infinite number of variables (BBGKY-hierarchy), contains information about all the previous states starting from ...
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1answer
140 views

What is the broken-sublattice-symmetry phase in an intermediate temperature of the three-state antiferromagnetic Potts model?

I have just read one paper ( Phys. Rev. E 54, R5885 (1996) ) where it was mentioned that the broken-sublattice-symmetry (BSS) phase was stable in the whole low-temperature region. The BSS phase at ...
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2k views

Why do we expect our theories to be independent of cutoffs?

Final edit: I think I pretty much understand now (touch wood)! But there's one thing I don't get. What's the physical reason for expecting the correlation functions to be independent of the cutoff? ...
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308 views

Derivation of differential scattering cross section - off-center target

This is a followup question to this pretty good answer regarding deriving the Boltzmann equation. What if the center of the target particle is actually not the same with the scattering center (or may ...
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279 views

Peierls Argument for Absence of Long Range Order

I'm really confused about the argument in Cardy's book for why there can't be long range order in 1D for discrete models. Let me just copy it out, and hopefully someone can explain it to me. He ...
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1k views

Why the free energy is called 'free'?

The free energy, $F$ of a thermodynamic system at a given temperature $T$, is defined as, \begin{equation} e^{-\beta F} = \mathcal{Z} = \sum_{\{configuration\}} e^{-\beta E(configuration) } ...
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83 views

Possible abuse of notation in statistical mechanics

I know that it often occurs that we need to take a derivitive with respect to $\beta$ in statistical mechanics. However, I think it is poor style to use equations with both T and $\beta$ in them ...
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2answers
376 views

Drude theory of electrical conductivity

I was just trying to calculate the electrical conductivity for a Fermi-Dirac distribution and a Maxwell-Boltzmann distribution, and I ended up with the same result: $$\sigma=\frac{ne^{2}\tau}{m}$$ ...
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85 views

Where else in physics does one encounter Reynolds averaging?

Reynolds-averaged Navier–Stokes equations (RANS) is one of the approaches to turbulence description. Physical quantities, like for example velocity $u_i$, are represented as a sum of a mean and a ...
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427 views

Black magic “Hartree” approximation

The question is about an unusual looking version of the Hartree or mean field approximation. The context is several papers I've been reading recently about the out of equilibrium dynamics of phase ...
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1answer
124 views

What are the state functions telling me or how are they related to total energy?

I am quite new to thermodynamics and statistical mechanics so this might be an easy question: In thermodynamics you get a bunch of thermodynamics potentials, so as for example enthalpy, internal ...
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1answer
217 views

Infinite-range 1D Ising model

The Hamiltonian for this system is given by \begin{equation} \mathcal{H} \{S\} = -H\sum_i S_i - \frac{J_0}{2} \sum_{ij} S_i S_j, \end{equation} where $H$ is the external magnetic field and there is no ...
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1answer
632 views

What is a chemical potential good for?

I read that the definition of the chemical potential is, that it is the partial derivative of the Free energy with respect to the number of particles, $$\mu=\frac{\partial F}{\partial N}.$$ ...
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97 views

Meaning of the 'deep lattice limit' and 'shallow lattice limit'?

In condensed matter literature, at many places, the phrase 'deep lattice limit' is used. Please tell what is the deep lattice limit and the shallow lattice limit?
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2answers
791 views

1D Ising Model with different boundary conditions

The Hamiltonian for one-dimensional Ising model is given by, \begin{equation} \mathcal{H} = -J\sum_{<ij>} S_iS_j; \quad i,j=1,2,...,N+1 \end{equation} where $<ij>$ denotes that there is ...
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287 views

Phase transitions. Conceptual link of my intuitive notions and definition of Georgii's book in terms of probabilities

In his classic book O. H. Georgii (Gibbs Measures and Phase Transitions) in Chapter 2 p. 28 define the concept of phase transition follows. Definition A potencial $\Phi$ will be said exhibit a ...
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58 views

Bariometric formula derivation

I don't understand the following reasoning that I found in a set of lecture notes from a physics course, it's about Perrin's stimate on $N_{a}$ Avogadro's number via the bariometric formula In order ...
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1answer
1k views

Bound states and scattering length

What is the relationship between bound states and scattering length? What is the relationship between scattering states and scattering length? When we say, potential is 'like' repulsive for ...
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262 views

Discretization of Hamiltonian using finite difference always justified?

I have this continuum version $$ H_{R}=\int dx\psi^{\dagger}(x)(\frac{p^{2}}{2}+V)\psi(x) $$ with $V$ as constant potential. Is it always justified to go from this to $$ \sum_{i}c_{i}^{ \dagger ...
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120 views

2nd order phase transition trouble deriving coefficient in fluctuations analysis

I can't get one of the coefficients in the equation for $T < T_c$ in the bottom, specifically the equation with the factor of two. any help appreciated. Consider an ising type expansion of the ...
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127 views

Traditional Transfer Matrix on the Potts model — how it grows for strip lattices?

What is the transfer matrix size for a strip lattice of width $n$ vertices, with arbitrary $q$?? I am not sure if it is $q^n$ x $q^n$ or something else. Any reference is also welcome.
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1answer
181 views

Why NPT ensemble is used for solid state phase transitions?

In Monte Carlo simulations of solid state phase transitions, why often Isobaric Isothermal ensemble (NPT) is used ? Why not NVT ? Here, N is number of atoms, P is pressure, T is temperature and V is ...
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1answer
198 views

Thermal radiation in the Unruh Effect

The following formula has been given in 't Hooft's black holes notes ($|\Omega \rangle$ is the vacuum state of Minkowski space, O is a operator): $$\langle \Omega| O|\Omega \rangle = \sum_{n \ge 0} ...
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1answer
795 views

How is the distribution probability in the canonical ensemble derived?

I'm confused by the derivation of the canonical ensemble, namely the origin of the probability density, that is the Boltzmann factor. Here's what I have: We have a system of particles with ...
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215 views

Diffusion of gases in the atmosphere

Suppose that the atmosphere is composed of 21% $O_2$ and 78% $Kr$ (instead of $N_2$). Since the density of $Kr$ is greater than the density of $O_2$, the lower atmosphere (where we live) should be ...
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165 views

Is there a phenomenon where physicists are only interested in the standard deviation of the quantity to be measured?

or a phenomenon where we can only measure the standard deviation ($\sigma_w$) of a variable $w$ and not the mean $\overline{w}$
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425 views

Continuous phase transition only hold for infinite systems. Real systems are finite, hence, a paradox

Second-order or continuous transitions are usually identified with non-analyticies within the free energy (which is proportional to the logarithm of the sum of exponentials). Such singularities are ...
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1answer
1k views

quantum mechanics current operators

How to derive the charge current and the energy current operators in second quantized form in Quantum mechanics ? Also if you could comment in a similar way on the entropy current operator, that will ...
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1answer
263 views

Magnetic Susceptibility at Arbitrary Temperature

I'm currently working on an assignment where the questions is: Consider a gas of N noninteracting electrons in a uniform magnetic field B = B$\hat{z}$ in a macroscopic system. Assume that the ...
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1answer
1k views

Chemical Potential of Ideal Fermi Gas

In Wikipedia's article on Fermi Gases, they have the following equation for the chemical potential: $$\mu = E_0 + E_F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{E_F}\right) ^2 - \frac{\pi^4}{80} ...
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93 views

Phase diagram of SO(5) rotor model

It was originally a problem from Professor Eugene Demler's problem set. Consider an SO(5) rotor model: \begin{align}\mathcal{H}=\frac{1}{\chi} ...
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4answers
807 views

Why is it often assumed that particles are found in energy eigenstates?

Energy eigenstates provide a convenient basis for solving quantum mechanics problems, but they are by no means the only allowable states. Yet it seems to me that particles/systems are assumed to be in ...
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1answer
120 views

Does time stand still at a phase transition?

For second order phase transition thermodynamic properties can be described in very general terms by their critical exponents. So at every transition the correlation length $\xi$ should diverge as ...
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4k views

Derivation of differential scattering cross-section

I'm trying to follow the derivation of the Boltzmann equation in my Theory of Heat script, but have a little trouble understanding the following: The cross-section $d\sigma$ is defined as: The amount ...
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0answers
121 views

Derivation of impact free Boltzmann equation

When deriving the impact-free boltzmann equation ( $\frac{\partial f}{\partial t} + \vec{v} \cdot\frac{\partial f}{\partial \vec{x}} + \vec{a} \cdot \frac{\partial f}{\partial \vec{v}} = 0$) I have a ...
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1answer
85 views

Forward-scattering for a single impurity in an infinite system

I'm slightly confused with the following situation: Suppose you have an electron in a tight-binding model, and let's say we are in one dimension with $N$ lattice sites. Add to this a single ...
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1answer
400 views

The critical point of Bose-Hubbard model

The Hamiltonian of Bose-Hubbard model reads as $$H=-t\sum\limits_{<i,j>}b_i^{\dagger}b_j+h.c.+\frac{U}{2}\sum\limits_{i}n_i(n_i-1)-\mu\sum\limits_in_i$$. In the limit $t\ll U$, the ground ...
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1answer
162 views

Mathematics for Statistical Mechanics

I am studying Statistical Mechanics and Thermodynamics from a book that i am not sure who has written it, because of its cover is not present. There is a section that i can not understand: ...
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105 views

on Brownian motors

From this review on Brownian motors, there is such a statement without detailed explanation: (I think this statement is general enough so that one does not need to read the article) "Apart from ...
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2answers
160 views

A box with cooler and heater on opposite faces

Suppose there's a box with one face cold, and the opposite face hot. So when the air molecules hit the cooler face, it will transfer its momentum and energy to the wall, bouncing back with less ...
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338 views

Ising model observables

Is there a formula or equation relating $\langle E\rangle$ and $\langle M\rangle$ (average spin per site) and $\langle E^2\rangle$ to temperature $T$ for the square lattice Ising model at zero ...
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151 views

Impact of the noise distribution on Geometric Brownian motion

I have a problem which includes geometric Brownian motion, with either normally distributed or power-law-distributed noise, and I'm asking for some explanations and if possible references to read in ...
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2k views

Gibbs Paradox - why should the change in entropy be zero?

The Gibbs paradox deals with the fact that for an ideal gas with $N$ molecules in a volume $V$ seperated by a diaphragm into two subvolumes $V_1,V_2$ with $N_1,N_2$ particles in each subvolume, ...