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

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Understanding the mean square displacement in molecular dynamics

In a Molecular Dynamics (MD) simulation, the mean square displacement $\text{MSD}$ is given by $$\text{MSD}(\delta t) = \left\langle\left|\vec{r}(\delta t)-\vec{r}(0)\right|^2\right\rangle,$$ where ...
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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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4answers
162 views

Question on entropy

All of my textbooks mention, that entropy-change of all spontaneous physical, and chemical processes is positive, and that such processes need another condition to fulfill- decrease in the net ...
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7answers
1k views

Mathematically possible vs physically probable outcomes

A good buddy of mine and I have had a friendly debate about the origins of the current state of our universe (namely; Earth and life on Earth) and have fundamentally disagreed in our stances with ...
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23 views

Effusion of particles from one box to another - pressure calculation

Suppose we have a container divided into equal halves. Right half is fixed at temperature $T$, volume $\frac{V}{2}$. Initially it has pressure $P_0$, a hole of area $A$ is opened between them. I ...
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34 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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82 views

Connection between String theory and Statistical Physics

I would like to think via standard transitivity arguments that there should be a deep connection between String theory and Statistical Physics. Why? Statistical Physics $\rightarrow$ QFT 2d QFT ...
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64 views

Mean-field theory : variational approach versus self-consistency

I have a general question concerning mean-field approaches for condensed matter classical of quantum statistical mechanic systems. Does determining the mean-field by a variational approach always ...
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1answer
38 views

state on quantum statistics. 3 particles according to 3 distributions [closed]

consider a system of three identical particles, A B ,and C. Assume that each particle can be in one of three possible quantum states, 1,2 and 3. For the following statistics listed below, enumerate ...
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3answers
119 views

What would be non-ergodic physics processes?

As the title says, what would be non-ergodic processes that occur in statistical physics? Many textbooks do not really cover ergodicity really well so I ask this question. I can't suddenly remember ...
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2answers
48 views

Nontrivial critical exponents in exactly solvable models?

Are there any exactly solvable models in statistical mechanics that are known to have critical exponents different from those in mean-field theory, apart from the two-dimensional Ising model? I wonder ...
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84 views

Wilson's Renormalization Group and Lie's Third Theorem

If you think of a one-parameter group of transformations along a curve in the plane as a (Lie) group, and the tangent vector to the curve as a generator of the curve we can intuitively understand ...
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0answers
39 views

Calculating heat capacity from the equation of state

It is known that within thermodynamics alone, given the equation of the state of a system, one cannot explicitly determine the heat capacity. What is the mathematical reason for this? Intuitively, it ...
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0answers
50 views

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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1answer
64 views

Calculation of the differential of the entropy

In this review (for those who wants a precise reference see page 8 eq 21), the Author says that: \begin{equation*} S=-\sum_{i}P\left(i\right)\ln P\left(i\right) \end{equation*} and using the ...
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0answers
13 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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1answer
77 views

Entropy is constant. How to express this equation in terms of pressure and density?

In hydrodynamics of an ideal, non-compressive flow we use 5 variables: pressure $p$, density $\rho$ and velocity field $\mathbf{v}$. So we need 5 equations. Landau's "Hydrodynamics" states that the ...
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1answer
58 views

Equation of state not linear in energy or system size

Which of these two equations of state are valid? $$S_1 = L_0 \gamma (\theta E/L_0)^{1/2} - L_0\gamma\left[\frac{1}{2} \left(\frac{L}{L_0}\right)^2 + \frac{L_0}{L} - \frac{3}{2}\right]$$ $$S_2 = L_0 ...
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1answer
34 views

Speed distribution in 1 dimension

In 3D, the maxwell velocity distribution is: $$f = \left(\frac{\alpha}{\pi} \right)^{\frac{3}{2}} e^{-\alpha v^2} d^3 \vec v$$ To get the speed distribution in 3D, we simply expand $d^3\vec v = 4\pi ...
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1answer
46 views

What's wrong with this simple derivation of energy flux in a photon gas?

In a photon gas, we know that pressure, $P$, and energy density, $u$, are related by: $$P=\frac{u}{3}$$ We also know from relativity that the momentum of a photon is $$p=\frac{E}{c}$$ Finally, the ...
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0answers
71 views

Reference for stochastic processes which helps moving from a basic level to a measure theory one

I'm looking for a reference (books, notes, lectures) which helps a physicist to understand the language of measure theory in the context of stochastic processes (in particular markov chains). I've ...
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32 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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8answers
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Why is absolute zero considered to be asymptotical? Wouldn't regions such as massive gaps between galaxy clusters have temperatures of absolute zero?

Why is absolute zero considered to be asymptotical? Wouldn't regions such as massive gaps between galaxy clusters have temperatures of absolute zero? I just do not see why our model must work the way ...
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32 views

How to load Bose-Einstein Condensates into an optical lattice?

In cold atom experiments, what techniques are used to load Bose-Einstein Condensates into an optical lattice??
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96 views

2d Ising model in CFT and statistical mechanics

When I recently started to read about conformal field theory, one of the basic examples there is the so called Ising model. It is characterized by certain specific collection of fields on the plane ...
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1answer
54 views

Partition function for composite systems

I want to understand the derivation of the partition function for two distinguishable non-interacting particles. Let the energy of particles $1$ and $2$ be $E_1$ and $E_2$ respectively. Setting ...
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0answers
84 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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0answers
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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1answer
60 views

what does “an average over noise” mean in Zwanzig's book

This is a very specific question about Robert Zwanzig's book Nonequilibrium Statistical Mechanics. Specifically, what is he talking about in equation 1.25 on page 10 that he calls "an average over ...
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0answers
15 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 ...
6
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1answer
49 views

Minimum connectivity required for mean field to be a good approximation?

In spin models, it is known that mean field becomes a better approximation as the connectivity increases. My question is: Is there an estimate for the threshold connectivity (as a function of the ...
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2answers
109 views

Quantum entropy in term of density matrix

Why in von Neumann expression of quantum entropy we have trace of density matrix expression? Why don't off diagonal term play a role?
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42 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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1answer
79 views

What is the phenomenological logic behind Fermi liquid theory

I am a super beginner when it comes to Solid State Physics and when wanting to learn more on the subject, I end up reading on Landau's Fermi liquid theory that supposedly justifies the quasi-free ...
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1answer
49 views

How can the microstates be measured with zero energy expenditure?

James P. Sethna. Statistical Mechanics. Exercise 5.2: What prevents a Maxwellian demon from using an atom in an unknown state to extract work? The demon must first measure which side of the ...
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1answer
114 views

Is temperature discrete

Because an object's temperature is inversely proportional to the wavelength of blackbody radiation which it emits, physicists have theorized the existence of Planck temperature at around $1.4×10^{32}$ ...
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1answer
191 views

Flory-Huggins ternary phase diagram with a neutral component

I am searching the literature for the Flory-Huggins phase diagram with the following components : polymer, solvent, and a third component that does not interact with the other components (just entropy ...
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1answer
64 views

Entropy and chemical potential of an ideal gas

I am reading Schroeder's book "Thermal Physics". One calculation in the text was not quite clear to me. The entropy of an ideal gas is given by the Sackur-Tetrode equation: $$ ...
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2answers
75 views

Classical and Semi-classical treatments of the ideal gas

In the semi-classical treatment of the ideal gas, we write the partition function for the system as $$Z = \frac{Z(1)^N}{N!}$$ where $Z(1)$ is the single particle partition function and $N$ is the ...
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2answers
218 views

Grand canonical partition functions for Bose-Einstein statistics vs. Maxwell-Boltzmann statistics

In Bose-Einstein statistics, the grand canonical partition function is $$\mathcal{Z}=1+e^{-\beta(\epsilon-\mu)}+e^{-2\beta(\epsilon-\mu)}+e^{-3\beta(\epsilon-\mu)}+\cdots$$ In Maxwell-Boltzmann ...
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1answer
160 views

Why are there gapless excitations in the anti-ferromagnetic Heisenberg model while the true ground state is a singlet?

The true ground state of the anti ferromagnetic quantum Heisenberg Model (nearest neighbor only)is known to be a singlet (I think this is Liebs theorem.) Since a singlet is invariant under rotations, ...
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1answer
75 views

I'm getting weird autocorrelations when simulating an Ising model below the critical temperature

So I'm simulating an Ising model using Monte Carlo and the Metropolis algorithm. After letting it reach equilibrium, I try to calculate the autocorrelation of the magnetization. As long as the system ...
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0answers
72 views

Intuitively, why does removing solutes cost $k_B T$ of free energy per molecule?

I can calculate that if you want to, for example, desalinate water, you will have to pay a free energy cost of $k_B T$ for each ion you remove. In other words, removing an ion from a volume of water ...
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1answer
119 views

Energy fluctuations in quantum canonical ensemble

How would you go about showing that in the quantum canonical ensemble (that is, in the density matrix and operator formulation), the energy fluctuations, namely $\langle H^2\rangle - \langle ...
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0answers
48 views

Is entropy a dynamical quantity [closed]

I am confused whether Entropy is a dynamical quantity or not. Gibbs entropy, and quantum mechanical Von Neumann entropy.
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7answers
3k views

Homemade salad dressing separates into layers after it sits for a while. Why doesn't this violate the 2nd law of thermodynamics?

The oil, vinegar and other liquids in homemade salad dressing separate into layers after sitting for a while, making the mixture become more organized as time evolves. Why doesn't this violate the ...
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1answer
83 views

The “replica trick” initial formula?

In Spin-glass theory for pedestrians by Castellani and Cavagna, the initial formula used to introduce the replica trick is written as: $$\overline{\log ...
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2answers
206 views

Is there an equation to calculate the average speed of liquid molecules?

I seem to remember from first year physics that we can calculate the RMS speed of a stationary, ideal gas with $v=\sqrt{\frac{3RT}{M}}$. Does a similar equation exist for liquids?
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106 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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2answers
266 views

Chemical potential in Thermodynamics

In many scenarios, on computing the partial derivative of the internal energy (U) with respect to mole number (N) is negative. This implies that adding more moles of the substance decreases the U of ...