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

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Simple estimation of the critical temperature of water

I'm trying to develop fermi estimation skills and I came up with a question for which I don't even know where to start from. Here goes: Is it possible to estimate the critical temperature (say in ...
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0answers
65 views

Is the stability matrix of a linearised RG flow always diagonalisable?

This is a follow up on "Why are the eigenvalues of a linearized RG transformation real?". My question is simple: Is there some physical (or mathematical) reason for the stability matrix of ...
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1answer
37 views

Classical limit of non-interacting, relativistic quantum gas (Kapusta/Gale p.8)

I want to understand two equations in "Finite temperature field theory" by Kapusta and Gale on page 8. The partition function is $$ \ln Z = V\int \frac{d^3 p}{(2\pi)^3}\;\ln\left(1\pm ...
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0answers
24 views

Wigner function vs. backward kolmogorov

We know that a Wigner function is a quasi probability distribution and a backward Kolmogorov equation for a stochastic differential equation (say quantum Lagenvin equation) gives the probability ...
2
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1answer
45 views

What is an order parameter?

I've seen order parameter used in two different ways. One is to distinguish between an ordered and an unordered phase, like whether the net magnetization is stable or not. The second way is to ...
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1answer
62 views

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
43 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
169 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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3answers
271 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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2answers
43 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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0answers
51 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
68 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
67 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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1answer
20 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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1answer
98 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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1answer
155 views

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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2answers
134 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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4answers
209 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? By Liouville's theorem, the total phase space volume is ...
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0answers
29 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 ...
0
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1answer
39 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
74 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
122 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
135 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
180 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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5answers
908 views

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
66 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
117 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
42 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
99 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
117 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
84 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
55 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
75 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
45 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
73 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
191 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
51 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
33 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
95 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
46 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
120 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
319 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
46 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
57 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
154 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
87 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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52 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
153 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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60 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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65 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 ...