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

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109 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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29 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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74 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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54 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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125 views

phase-space volumes or cells for N particle system

For N non interacting spinless particles in a volume, we have 3N degrees of freedom and we can divide the phase space into 6N dimensional cells of volume h raised to power 3N. And each cell ...
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104 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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55 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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76 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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27 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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129 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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66 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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116 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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50 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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162 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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86 views

How to formally write down the Boltzmann equation?

Can someone write down the Boltzmann equation, not neglecting any of the variables of the involved functions and integrals? Specifically, how to concisely capture the "primed" variables in a sensible ...
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34 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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60 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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197 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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106 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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96 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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155 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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132 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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33 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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304 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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74 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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98 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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140 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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147 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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106 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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91 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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120 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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190 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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42 views

Euler Equation Arbitrary Quantities

I have a question about the Euler equation. For some state I can write down: $$ U = TS - pV + \mu N$$ In this equation $T$, $p$, $V$, and $N$ are directly measurable so they have fixed values. ...
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41 views

Retarded thermal Green function

I'm working with finite temperature field theory, but I'm having problems understanding the retarded Green's function in this formalism. I'm reading Niemi and Semenoff's article "Finite Temperature ...
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31 views

How do you obtain the fluctuation spectrum of a tubular membrane?

I am reading through a paper. A tubular membrane, submitted to tension $\sigma$ acting as a Lagrange multiplier to conserve area, fluctuates around a cylindrical shape of length L and radius R. ...
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41 views

Entropy of an oscillator in Einstein's solid

This is a homework problem and I need help with it. A solid's (Einstein's model) oscillators are in the first excited state on average. How much entropy does one oscillator have? What I've tried so ...
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56 views

Thermal fluctuations in metals

My professor said that the $k_BT$ displacement in the energy levels of the band electrons is due to the space-thermal displacement of the potential of the ion host. I think that this displacement is ...
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59 views

probability of sequence for given rate constants

lets consider a copolymer, $C_{r,s}^A$ containing r number of A monomers and s number of B monomers with A at the reactive end of the polymer. The equilibrium constant for A-A, A-B, B-A, and B-B bonds ...
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32 views

How to calculate the partial entropy in a fully connected ising system

I'm trying to reproduce a calculation that should lead to the partial entropy in a fully connected ising model for the high-temperature range ($\beta < 2$) in the thermodynamic Limit ($N ...
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20 views

Molecular field meaning in Liquid Crystal Theory

Given the Frank-De Gennes free energy $F = \int f(\boldsymbol{p},\nabla\boldsymbol{p}, ...)\ d\boldsymbol{x},$ for liquid crystals (see De Gennes-Prost, p. 107, formula 3.21), the vector ...
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33 views

Wolff vs Swendsen Wang Algorithm

Following the orginal paper of Swendsen Wang, their dynamical critical exponent $z$ is about $z=0.35$, whereas the Wolff Algorithm seems to have $z=1.19$. When I calculate the Correlation time though, ...
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20 views

What is difference and linkage between power law of phase transition in physics and Zipf law in linguistics

There are power law of phase transition in physics and Zipf law in linguistics which are similiar to each other ,and some expert think they are in fact just the same.But the diagrams of them base on ...
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37 views

Can statistical mechanics be formulated generally in terms of phase space?

In many statistical mechanics books, notably Landau and Lifschitz' volume in the course on theoretical physics, the quantities central to statistical mechanics such as entropy are defined in terms of ...
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33 views

state occupation rate $n_{i}=\frac{1}{e^{\beta (\varepsilon _{i}-\mu )}+{[1/-1/0]}}$ & density matrix $\rho _{m}=\frac{e^{-\frac{E_{m}}{kT}}}{Z(T)}$

Three kinds of distributions. The states occupation rates: F.D. $n_{i}=\frac{1}{e^{\beta (\varepsilon _{i}-\mu )}+1}$ B.E. $n_{i}=\frac{1}{e^{\beta (\varepsilon _{i}-\mu )}-1}$ Boltzmann ...
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14 views

Rationale behind the 'joint cavity distribution'?

I have a question about equation (17) of this paper: http://arxiv.org/pdf/1009.1635v1.pdf First, I was hoping that someone could explain how it is arrived at. Second, I find the notation to be a bit ...
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20 views

Derivation of a formula concerning overlap between spin states

I am reading through Cavagna's Spin glass theory for pedestrians, but I am stuck at equation (35). I'll try to provide a little context. Given two spin configurations $\sigma$ and $\tau$, we define ...
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39 views

Two-dimensional atomic trap--how to set up the problem?

It is possible to trap neutral atoms between two solid surfaces in a potential of the form $V (x, y, z) = ax^2 + by^2$ where a and b are parameters. The allowed space for the gas extends to ...
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43 views

On the relationship between entropy and chaotic noise

I have few conceptual questions related to application of chaos in communications. Kolmogorov-Sinai Entropy1 , Kolmogorov-Sinai Entropy2 and Kolmogorov-Sinai Entropy3 KS is an entropy metric for ...
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46 views

Landau free energy

I am reading the statistical mechanics by Pathria in Chap 12. I have a question about the Landau free energy. What is the physical reasoning for that the free energy could be a functional of the order ...
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74 views

Questions on degenerate ground states and the thermodynamic limit?

For example, let's consider a $N$ spin-1/2 system on a lattice described by the Hamiltonian $H$. My questions are: (1) If $H$ has either global $SU(2)$ spin-rotation symmetry or time-reversal ...