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

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Derivation of Higher-order correlation functions from definition

I'm trying to understand the definition of the n-th order correlation function. My aim is to translate the math into a numerical implementation in order to compute the correlation function $g^{(n)}$ ...
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16 views

Coarse-graining on a second channel decreases mutual information?

Let $X_1,B_1,X_2,B_2$ and $Y_1,A_1,Y_2,A_2$ and $C_1$ and $C_2$ be binary random variables. Suppose: $I(X_2:B_2|C_2=0)+I(Y_2:A_2|C_2=1) \leq 1$. This can be thought of as a bound on the capacity ...
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222 views

Fluctuations in energy for macro and micro canonical ensembles

I was thinking about fluctuations in energy for a system in thermal equilibrium. I think that the Boltzmann distribution itself has an standard deviation approximately equal to the mean, as it is ...
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154 views

Fluctuation-Dissipation theorems in an infinite quantum system

So for a quantum spin chain, one can easily prove via the partition function that you have a fluctuation-dissipation type relation between the magnetic susceptibility and the variance of the ...
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141 views

Understanding the product of partition functions by making sense of the maths and the physics

I have $N$ distinguishable particles in a 1D harmonic oscillator potential with 'proper' frequency $\omega$. The particles also have internal spin-$\frac12$ degrees of freedom in a magnetic field $B$ ...
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56 views

Condensate fraction and single-particle density matrix

In Bose–Einstein condensation (BEC), how to prove the largest eigenvalue of the single-particle density matrix $$\rho_{ij}=\frac{\langle\Psi|a_i^{\dagger}a_j|\Psi\rangle}{N}$$ is ...
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110 views

Ising model. What is large fluctuations of magnetization?

My background is in mathematics. I have studied the Ising model in $\mathbb{Z}^2$. The main model of statistical mechanics. Yesterday, I was reading the preliminary notes of the book Statistical ...
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88 views

Is it possible to define a notion of temperature in a microcanonical ensemble?

I am thinking of a mircrocanonical ensemble as a finite system for which the number of particles, volume and the total energy is fixed. Is there a more refined view of this? Can I think of ...
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241 views

Boltzmann distribution: derivation from canonical distribution

I'm trying to understand the Maxwell-Boltzman Distribution, and in particular the derivation from the boltzman distribution for energy. I have successfully created an incorrect derivation, but I'm not ...
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60 views

What is conductivity?

I read that if we have spin $\frac{1}{2}$-particle, where a magetic force acts on, then the force is given by a drift speed times a conductivity. This conductivity is determined to be $\frac{kT}{D}$, ...
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164 views

Fugacity of the fermi gas

It can be shown that in the high temperature exploration of the Fermi gas, the Fermi function may be expanded to second order in $e^{\beta \mu}$, where $\beta = 1/kT$ and $\mu$ is the chemical ...
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38 views

Can the short-time dynamics of an open quantum system be approximately unitary?

Considering the physics of an open quantum system described by a Hamiltonian $H=H_S+H_E+H_{SE}$, where the subscript $S$ refers to the system of interest, $E$ to the environment and $SE$ to the ...
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174 views

The Maxwell and the Boltzmann distributions

I am trying to understand where the Boltzmann distribution comes from. I recently learned some interesting things of which my interpretation follows below. Did I interpret correctly? If so, is this ...
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86 views

Langevin equation

A molecule consists of two atoms whose centers are located at $\mathbf{r}_1$ and $\mathbf{r}_2$ respectively. The atoms are connected by a bond that can be approximated by a harmonic spring, so that ...
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105 views

How does the Lennard Jones Potential changes for interaction between particles of different sizes?

I am interested in incorporating a Lennard-Jones potential in a simulation. When the interaction only involves the same type of particle, with same characteristics, we can use reduced units, scaling ...
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250 views

Free Energy of N Spin 3/2 Particles

This question is from the book "Introductory Statistical Mechanics" by Bowley and Sanchez. The question is as follows: Calculate the free energy of a system with N particles, each with spin 3/2 with ...
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59 views

Find out ground sates for large 2D classical spin model

Reaching the ground state of a large 2D classical spin model (e.g. classical Heisenberg model) might be a relatively difficult task while using conventional "flip/reject" Monte Carlo method. The ...
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178 views

Average number of spin up particles

In a paramagnetic system, where $N = N_\uparrow + N_\downarrow$ is fixed, how does one calculate the average number of spin-up particles $\langle N_\uparrow \rangle$? You can assume we have the ...
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40 views

Ergodicity Breaking in Supercooled Liquids

What is a ergodic system? What is Onset temperature of ergodicity breaking in super cooled liquids when we go towards lower temperature?
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91 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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248 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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119 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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206 views

Maxwell-Boltzmann distribution

The short story is, that I have to calculate some transport coefficients, but using the the MB distribution as my distribution function. What I currently need to solve is: ${{\mathcal{L}}^{\,\left( ...
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205 views

Maxwell-Boltzmann distribution for transport equations

I have to calculate the transport coefficients for the Maxwell-Boltzmann distribution. But I'm not sure what distribution I have to use. As far as I know it should not be the MB distribution for ...
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35 views

Is there anything to prevent paired-up neutrons from a complete overlap

The reason "neutrons don't overlap", as DarenW explained it, has to do with intricate forces at play that take into account the spins, iso-spins and symmetry of the wavefunctions. However, assume I ...
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477 views

Rotational Constant and Moment of Inertia of Fluorine gas

I have come across some homework question on thermodynamics which needs me to calculate $B$ of $F_2$ My attempt: $B= \frac{h}{8\pi^2cI}$ where $I=\mu r^2=\frac{m_1m_2}{m_1+m_2} r^2$ Atomic mass of ...
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76 views

Partition function for multidimensional scaling energy

Let $D_{ij}$ a random matrix with i.i.d positive coefficients. One can take for instance $D_{ij}$ uniformly distributed in [0,1]. We consider the following energy function $H(x)$ defined for ...
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109 views

Statistical Mechanic

One can define entropy as $$S=k\log{\omega(E)},$$ where $\omega(E)$ is the numbers of states with energy equal $E$; and the canonical partition function for a set of N particles is defined ...
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191 views

helmholtz free energy of a polymer

You have a polymer chain of $N$ units, which is represented by $N$ independent springs in series. The springs are Hookean, with spring constant $L$, and the end to end vector is $\mathbf r$. So the ...
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161 views

Fock picture of bosonification in condensates

I want to understand how bosonification in a condensate must be interpreted in the Fock states picture Say i have uncoupled fermions in a set of states $E_1$, $E_2$ ... over the vacuum $E_0$. They ...
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110 views

What is meant by correlation propagation?

What is meant by correlation propagation in physics? I have an intuitive understanding but are there any introductory notes ( more mathematical oriented) and with some physical examples?
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94 views

usage of partition function in some number of particles in one-dimensional axis

I just learned some introductory quantum meachnics, but not statistical mechanics, so I am curious how partition functions would be used in the following case: Suppose there are three particles in ...
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187 views

Does the spin glass corresponding to a restricted Boltzmann machine have a characteristic timescale?

From what I gather, a Boltzmann machine can be identified with a spin glass. Though I don't know the details yet (and would welcome any references within the last 5 years--not, e.g. MacKay, etc.), I ...
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216 views

What does it mean for a phase space trajectory to be “long” and “stable”?

What does it mean for a phase space trajectory to be "long" and "stable"? I understand the concept of a trajectory in phase space but not how these adjectives can be applied to one. Thanks.
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27 views

Difficulty in understanding Maxwell Boltzmann distribution in case on ions in a field

I learned that the velocity of molecules obey Maxwell Boltzmann (MB) distribution at a Temperature T. If I have ions of mass 'M' accelerated to 2eV in a specific region. As the ions are not ...
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11 views

Is there a way to get the Bethe Roots, that belong to a given eigenvalue of the transfer matrix?

(Quantum) integrable systems, that belong to solutions to the Yang-Baxter-equation, are often solved by the (algebraic) Bethe Ansatz. Solutions to the Bethe-equations lead to the eigenvalues of the ...
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24 views

Books that cover Mode-Coupling Theory

I am looking for a book that covers the schematic mode-coupling theory and that are not too arcane (i.e., recent book). Basically the only book so far on this I have come across is "nonequilibrium ...
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28 views

about Conservation laws and Correlation function

I'm reading a review paper by Gorden Baym-(http://www.worldscientific.com/doi/abs/10.1142/9789812793812_0002) In the second part, he raised that: According to conservation law $\frac{\partial ...
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44 views

Riemannian generalization of Weierstrass transform

As it has been written on this Wikipedia page, one can define the Weierstrass transform on any Riemannian manifold. Even though, I couldn't find any references guess that the Weierstrass transform on ...
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29 views

Is there a Lagrangian that can lead to the Rayleigh-Jeans law?

Is there a way to derive the Rayleigh-Jean's law using classical statistical mechanics only? On the internet there is a common way to arrive at the equation by using concepts in electrodynamics. This ...
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41 views

What is the physical fundamentals of Pascal's law

Pascal's law or the principle of transmission of fluid-pressure (also Pascal's Principle) is a principle in fluid mechanics that states that pressure exerted anywhere in a confined incompressible ...
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14 views

What does the particle to volume density physically mean for Bose-Eisenstein condensate?

The average number of particles $\langle N\rangle$ for a Bose-Eisenstein condensate in 3D is given as $$ \dfrac{\langle N\rangle}{V} = \dfrac{V^{-1}}{e^{\beta (0-\mu)}-1} + \int_{0}^{\infty} ...
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12 views

Relaxation time approximation in anisotropic potential scattering event

In relaxation time approximation (RTA) of Boltzmann transport theory, the relaxation time is calculated by $\frac{1}{\tau(\mathbf{k})}=\frac{2 \pi}{\hbar V}\sum_{\mathbf{k^{'}}} \delta ...
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24 views

Exact expression for the coefficient in Bloch-Grüneisen (BG) formula?

In most representations of the BG formula, there is a coefficient (usually left vague as an experimental parameter, but sometimes written out "analytically") in front of the integral: $$\rho=\rho_0 +A ...
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29 views

bose einstein phase transition

From Carter's book Thermodynamics and Statistical Mechanics, the partition function of a bose-einstein gas in $d$ dimensions is $$ \ln(Z) = ...
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28 views

Can I measure the volume of a locked room by pumping gas through keyhole and measuring its entropy?

Suppose that I have a locked room and a keyhole in the door and I want to measure the room's volume. Suppose also that I have some "magical" "artificial" inert gas A that doesn't interact with ...
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65 views

Equipartition theorem and virial theorem differences?

The classical virial theorem and the classical equipartition theorem are clearly related. A version of the virial theorem is, \begin{equation} \bigg\langle \sum ^{3N}_{i=1} x_i\frac{\partial ...
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28 views

Mean-field solution of Potts model

The mean-field equation for the three-state Potts model $H= -J∑δσiδσj$ can be derived as follows using this: a) show that $H$ is equivalent to $-J∑Si.Sj$ where, $Si=(1 0) , (-1/2 √3/2 ) , (-1/2 ...
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39 views

Non-trivial integral with the Bose-Einstein distribution and Cosine function

When I consider the Casimir interaction between an atom and a perfect conducting slab I find the following non-trivial integral: $$\int\limits_0^\infty {\frac{{\cos \left( mx \right)}}{{x + ...
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27 views

What is normal fluctuation?

I was reading Statistical Mechanics (second edition) by Kerson Huang. On page 146, after equation 7.14, there is a reference to normal fluctuation. What is it? Here is the relevant part from the ...