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

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Ising model on lattices with (vertical side length) $\neq$ (horizontal side length)

Consider the Ising model with nearest neighbours interactions on a rectangular lattice $L\times M$. If $L=M$ (2-dimensional square lattice), it is known (e.g. by Peierls argument or Onsager explicit ...
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1answer
31 views

Do Monte-Carlo updates have a physical significance in stat. mech?

One of the archetypical example to introduce Monte-Carlo methods in stat. mech. is to work out the properties of the 2D square lattice Ising model and compare the obtained results with Onsager's exact ...
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1answer
28 views

What is the difference between reversible and irreversible adiabatic expansion?

What is the difference between reversible and irreversible adiabatic expansion? Is it true that the work done by the gas is the same but the pressure applied externally differ between two process? ...
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107 views

Does entropy have a physical meaning?

Entropy is incredibly useful as a mathematical tool. But what does it actually mean? I understand that the Boltzmann entropy is defined by: $$S=k\ln{\Omega}$$ With $\Omega$ being the multiplicity ...
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1answer
20 views

Exponentially increasing $\Omega(E)$

If I choose the number of microstates for energy $E$ to be $\Omega(E) = e^{aE}$ ($a>0$), its temperature is constant: $$ kT = \left( {d\ln \Omega \over dE} \right)^{-1} = 1/a $$ If I choose ...
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32 views

ensembles and lagrange multipliers

In the derivation of maxwell-boltzmann distributions, the method of Lagrange multiplier is $\sum n_i = N$ $\sum n_i E_i = E$ where $N$ is the total number of particles, and $E$ is the total ...
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1answer
37 views

Does the q-states Potts become the XY model in large q state?

I have met several times in papers, the order of the phase transition of the $q$-state Potts model depends on $q$. E.g., in two dimensions, for $q = 2$ (the Ising model), $3$, $4$ the order-disorder ...
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3answers
44 views

When the low temperture reservoir with negative temperture (Kelvin), such as Ising model, is the efficiency of ideal heat engine larger than 1?

The ideal Carnot engine works between two heat reservoir with two temperatures $t_h$ and $t_l$. Its efficiency is then $1-\frac{t_l}{t_h}$ . If the low temperture reservoir is the Ising model with ...
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15 views

Is it possible to cool magnetic dipoles with a magnetic static field?

Suppose you have a bath of magnetic dipoles, with a common mean rotational kinetic energy. Now you apply a very strong magnetic field so that the dipoles align with the field, thus "losing" their ...
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30 views

Calculation of the partition function for a classical 2D gas lying on the surface of a sphere of constant radius $R$

I'm kind of confused with this system. My first question is. Is the Hamiltonian of one particle of this gas the following? ...
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20 views

Comparison of the classical and quantum distribution functions [closed]

The distribution functions $f_{MB}(E)$, $f_{BE}(E)$ and $f_{FD}(E)$ all have the undetermined constant $A$ (= $\exp(-α)$ ). This may be viewed as a normalization constant, whose value can be found ...
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28 views

Maxwell-Boltzmann speed distribution [closed]

A container has $10^4$ $N_2$ molecules per $\text{cm}^3$. Assuming that the particles are distinguishable and $$A = \frac NV \left( \frac{m}{2 \pi k \, T} \right)^{3/2}$$ , Tabulate the values of ...
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47 views

Phase space volume and correlation dimension

This may seem trivial but I will appreciate help in determining the functional form of the probability density function (pdf) for the following case. Will highly appreciate some guidelines on how to ...
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83 views
+50

Why are the eigenvalues of a linearized RG transformation real?

The RG transformation $R_\ell$ maps a set of coupling constants $[K]$ of a model Hamiltonian to a new set of coupling constants $[K']=R_\ell[K]$ of a coarse-grained model where the length scale is ...
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129 views

Why the self-information is -log(p(m))?

Why is self-information given by $-\log(p(m))$? Shannon derived a measure of information content called the self-information or "surprisal" of a message $m$: $$I(m) = \log \left( ...
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1answer
55 views

Liouville's theorem and preservation of topology

What might be a simple proof showing that the time evolution of the phase space volume can't lead to splitting off of the phase space volume? I don't know much about topology and stuff.
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27 views

What is the fluctuations of the energy of a simple harmonic oscillator? [closed]

$$\begin{align} \varepsilon&=\frac{\vec{p}^{\,2}}{2m}+\frac{K}{m}\vec{q}^{\,2}\\ \rho(q,p)&=\biggl(\frac{\omega}{2\pi k_BT}\biggr)^3e^{-\frac{\varepsilon}{k_bT}} \end{align}$$ where ...
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32 views

compressibility of cold atoms in optical lattices

The compressibility of cold bosons in an optical lattice is defined as $\kappa = \frac{\partial \langle n\rangle}{\partial \mu}$, where $\langle n\rangle$ is the density and $\mu$ is the chemical ...
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1answer
41 views

MIcrocanonical and Canonical - The thermodynamic limit

Considering a two level system with energies $ 0 $ and $ \epsilon$, we write out the single particle partition function with ease to be, also N-particle partition function for non-interacting ...
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1answer
92 views

How does statistical mechanics predict that hot air rises?

Does hot air rise -- from a statistical-mechanical viewpoint Question #6329 asks whether and why hot air rises. The consensus answer is straightforward: - hot air is less dense than cold air - ...
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1answer
121 views

Why do phase transitions even exist? Why not smooth density change curves?

Why do phase transitions even exist? Why not smooth density change curves? What properties of matter, quantum or otherwise, predicts that matter will undergo phases at different pressures and ...
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1answer
133 views

Strange definition of microcanonical partition function

I always thought that the microcanonical partition function would measure the number of states that correspond to some fixed energy. Despite, I found in this paper (equation 3.4) that we integrate ...
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Is there a classical analog to quantum mechanical tunneling?

In comments to a Phys.SE question, it has been written: 'Tunneling' is perfectly real, even in classical physics. [...] For sufficiently large temperatures this can put the system above a hump in ...
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1answer
50 views

How is free energy built into a Metropolis Monte Carlo simulation of an Ising model?

In the Metropolis algorithm, the change in the energy given by the hamiltonian is compared for flipping a spin. This is not the free energy, but for systems above absolute zero you are trying to ...
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1answer
57 views

H-theorem and Boltzmann equation applied to Boltzmann distribution

Using the Boltzmann equation: $$ \frac{dH}{dt} = \int_0^{\infty} dr \int_0^{\infty} ds W(r,s)[p_r - p_s][\ln{p_r} - \ln{p_s}]$$ and assuming $p_r = e^{-\beta r}$, the equation looks like $$ ...
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1answer
39 views

How can entropic effects be prevalent at low temperatures?

I read in a book that at low temperature the hydrophobic effect (for example) is entropic but at high temperatures it is enthalpic. I thought that entropy should decrease at very low temperatures. ...
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1answer
58 views

Helmholtz free energy from a relation for entropy

The Legendre transformation defines the helmholtz free energy (at least according to my lectures) as: $F(T,V,N)=E-TS$ It also says to start with $E(S,V,N)$ and $T=\frac{\partial{E}}{\partial{S}}$ ...
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3answers
109 views

Why does the Metropolis algorithm allow changes even for ∆E > 0?

In the Metropolis Monte Carlo algorithm, why can you accept changes even for ∆E > 0 (provided that a random number is less than a given probability ratio, e.g. exp(-β∆E))?
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1answer
68 views

Definition of entropy in nonequilibrium states

Thermodynamical definition of entropy $$S(p)=-\int p\ln p~dx$$ is defined only on equilibrium system. But why can't we use it for non-equilibrium system? Is there a well-accepted definition for it?
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1answer
31 views

Spontaneity / Free Energy of Non-Isothermal Process

I'm trying to determine a lower bound for the work input necessary to make an entropy-reducing process "spontaneous" in the sense that the 2nd law is not violated. For a constant temperature and ...
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1answer
39 views

The BBGKY Hierarchy

The collision term in the Boltzmann equation can be derived from the BBGKY hierarchy. Wikipedia says: In statistical physics, the BBGKY hierarchy [...] is a set of equations describing the ...
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1answer
38 views

Statistical mechanics: What is a “microscopic realization” of a system?

What is a "microscopic realization" of a system? The context is statistical mechanics. The microscopic system consists of many atoms (too many to track individually) with an assigned probability ...
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1answer
64 views

Question about Metropolis Monte Carlo in the case of equal energies

If configuration A is equal to configuration B in a Metropolis Monte Carlo method, do you still do the attempted update?
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2answers
107 views

Statistical mechanics: Meaning of “accessible” in “accessible microstates”

What does "accessibility" mean in statistical mechanics? Is it an equivalent concept to accessibility in mathematical control theory? I'll provide an example: When two systems A and B interact on a ...
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1answer
32 views

Basics of osmosis. What about excluded volume?

I may not understand osmosis very well. Let us suppose two compartments filled with water, separated by a semi-permeable membrane. At equilibrium, both levels are equals. Let us introduce now a given ...
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0answers
29 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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1answer
64 views

What does the behavior of the pair correlation function look like in the vicinity of the critical point?

What does the g(r) look like near the critical point? I know what the pair correlation function (radial distribution function) should look like for a solid, which has regular packing and therefore ...
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1answer
41 views

Can an Ergodic dynamical system approach equilibrium?

An ergodic dynamical system $(\Omega,\phi^t,\mu)$ is such that the time average $\bar{f}$ of every function $f\in L_1(\Omega,\mu)$ equal the space average $\langle f \rangle_\mu$, i.e. the system ...
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1answer
51 views

How to calculate the ground-state energy for the Ising model?

I'm learning about the 2D ferromagnetic Ising model in zero field and trying to verify what I know by calculating the ground-state energy for the state with all 'up' spins in a 3x3 lattice. $$H = ...
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3answers
266 views

Axioms behind entropy!

The concept of entropy is very ubiquitous, we learn about its uses starting from Information Theory (Shannon entropy) up to its basic definition in statistical mechanics in terms of number of ...
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0answers
35 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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0answers
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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2answers
122 views

Who invented the perfume bottle thought experiment?

A common thought experiment used to explain the second law of thermodynamics, the "arrow of time", etc. is perfume escaping from an opened perfume bottle; the perfume is likely to diffuse into the ...
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2answers
70 views

Is a superposition of (anti)symmetric states (anti)symmetric?

Let's say we have the following wavefunction of two identical particles, $A$ and $B$: $$\frac{1}{2}[(\chi(A)\psi(B)\pm\psi(A)\chi(B))+(\phi(A)\eta(B)\pm\eta(A)\phi(B))]$$ Is this properly ...
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2answers
49 views

Average ocupancy in an ideal gas at high-temperature

In David Chandler's 'intro to statistical mechanics' he states that for an ideal gas at high-temperature $$ \langle n_j\rangle=\langle N\rangle\frac{e^{-\beta \epsilon_j}}{\sum e^{-\beta \epsilon_j}} ...
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2answers
143 views

How do you measure numerically the central charge of a system?

Let's say that you are doing some Monte-Carlo simulations of a statistical system on a lattice and you observe scale invariance, meaning that you are at a conformal point. Can you get a numerical ...
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58 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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58 views

Use of Boltzmann over Maxwell distribution

Why is the Boltzmann distribution used over the Maxwell distribution in many cases such as the derivation of Plancks law of thermal radiation, derivation of Einstein A and B coefficients, Langevin ...
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1answer
57 views

Grand partition function of gas of non-interacting spin-1 bosons in magnetic field

Consider a gas of non-interacting spin 1 bosons in a uniform B field, each subject to a Hamiltonian of the form: $ H(\vec{p},s_z) = \frac{p^2}{2m} - \mu_0 s_z B$ where $s_z$ can take the three ...
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2answers
66 views

Dimensionless entropy interpretation

Measuring temperature in joules instead in the artificial units of Kelvin would render entropy as a dimensionless quantity. This is quite appealing since entropy has always been quite a misterious ...