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

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3answers
932 views

Why the free energy is called 'free'?

The free energy, $F$ of a thermodynamic system at a given temperature $T$, is defined as, \begin{equation} e^{-\beta F} = \mathcal{Z} = \sum_{\{configuration\}} e^{-\beta E(configuration) } ...
0
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1answer
80 views

Possible abuse of notation in statistical mechanics

I know that it often occurs that we need to take a derivitive with respect to $\beta$ in statistical mechanics. However, I think it is poor style to use equations with both T and $\beta$ in them ...
1
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2answers
323 views

Drude theory of electrical conductivity

I was just trying to calculate the electrical conductivity for a Fermi-Dirac distribution and a Maxwell-Boltzmann distribution, and I ended up with the same result: $$\sigma=\frac{ne^{2}\tau}{m}$$ ...
3
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0answers
77 views

Where else in physics does one encounter Reynolds averaging?

Reynolds-averaged Navier–Stokes equations (RANS) is one of the approaches to turbulence description. Physical quantities, like for example velocity $u_i$, are represented as a sum of a mean and a ...
9
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2answers
365 views

Black magic “Hartree” approximation

The question is about an unusual looking version of the Hartree or mean field approximation. The context is several papers I've been reading recently about the out of equilibrium dynamics of phase ...
2
votes
1answer
120 views

What are the state functions telling me or how are they related to total energy?

I am quite new to thermodynamics and statistical mechanics so this might be an easy question: In thermodynamics you get a bunch of thermodynamics potentials, so as for example enthalpy, internal ...
1
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1answer
196 views

Infinite-range 1D Ising model

The Hamiltonian for this system is given by \begin{equation} \mathcal{H} \{S\} = -H\sum_i S_i - \frac{J_0}{2} \sum_{ij} S_i S_j, \end{equation} where $H$ is the external magnetic field and there is no ...
3
votes
1answer
532 views

What is a chemical potential good for?

I read that the definition of the chemical potential is, that it is the partial derivative of the Free energy with respect to the number of particles, $$\mu=\frac{\partial F}{\partial N}.$$ ...
5
votes
1answer
82 views

Meaning of the 'deep lattice limit' and 'shallow lattice limit'?

In condensed matter literature, at many places, the phrase 'deep lattice limit' is used. Please tell what is the deep lattice limit and the shallow lattice limit?
3
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2answers
618 views

1D Ising Model with different boundary conditions

The Hamiltonian for one-dimensional Ising model is given by, \begin{equation} \mathcal{H} = -J\sum_{<ij>} S_iS_j; \quad i,j=1,2,...,N+1 \end{equation} where $<ij>$ denotes that there is ...
2
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0answers
271 views

Phase transitions. Conceptual link of my intuitive notions and definition of Georgii's book in terms of probabilities

In his classic book O. H. Georgii (Gibbs Measures and Phase Transitions) in Chapter 2 p. 28 define the concept of phase transition follows. Definition A potencial $\Phi$ will be said exhibit a ...
0
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1answer
57 views

Bariometric formula derivation

I don't understand the following reasoning that I found in a set of lecture notes from a physics course, it's about Perrin's stimate on $N_{a}$ Avogadro's number via the bariometric formula In order ...
0
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1answer
1k views

Bound states and scattering length

What is the relationship between bound states and scattering length? What is the relationship between scattering states and scattering length? When we say, potential is 'like' repulsive for ...
4
votes
1answer
221 views

Discretization of Hamiltonian using finite difference always justified?

I have this continuum version $$ H_{R}=\int dx\psi^{\dagger}(x)(\frac{p^{2}}{2}+V)\psi(x) $$ with $V$ as constant potential. Is it always justified to go from this to $$ \sum_{i}c_{i}^{ \dagger ...
1
vote
1answer
114 views

2nd order phase transition trouble deriving coefficient in fluctuations analysis

I can't get one of the coefficients in the equation for $T < T_c$ in the bottom, specifically the equation with the factor of two. any help appreciated. Consider an ising type expansion of the ...
1
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0answers
117 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.
1
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1answer
156 views

Why NPT ensemble is used for solid state phase transitions?

In Monte Carlo simulations of solid state phase transitions, why often Isobaric Isothermal ensemble (NPT) is used ? Why not NVT ? Here, N is number of atoms, P is pressure, T is temperature and V is ...
2
votes
1answer
175 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} ...
1
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1answer
641 views

How is the distribution probability in the canonical ensemble derived?

I'm confused by the derivation of the canonical ensemble, namely the origin of the probability density, that is the Boltzmann factor. Here's what I have: We have a system of particles with ...
7
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0answers
180 views

Diffusion of gases in the atmosphere

Suppose that the atmosphere is composed of 21% $O_2$ and 78% $Kr$ (instead of $N_2$). Since the density of $Kr$ is greater than the density of $O_2$, the lower atmosphere (where we live) should be ...
4
votes
3answers
160 views

Is there a phenomenon where physicists are only interested in the standard deviation of the quantity to be measured?

or a phenomenon where we can only measure the standard deviation ($\sigma_w$) of a variable $w$ and not the mean $\overline{w}$
4
votes
2answers
336 views

Continuous phase transition only hold for infinite systems. Real systems are finite, hence, a paradox

Second-order or continuous transitions are usually identified with non-analyticies within the free energy (which is proportional to the logarithm of the sum of exponentials). Such singularities are ...
4
votes
1answer
788 views

quantum mechanics current operators

How to derive the charge current and the energy current operators in second quantized form in Quantum mechanics ? Also if you could comment in a similar way on the entropy current operator, that will ...
1
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1answer
219 views

Magnetic Susceptibility at Arbitrary Temperature

I'm currently working on an assignment where the questions is: Consider a gas of N noninteracting electrons in a uniform magnetic field B = B$\hat{z}$ in a macroscopic system. Assume that the ...
4
votes
1answer
977 views

Chemical Potential of Ideal Fermi Gas

In Wikipedia's article on Fermi Gases, they have the following equation for the chemical potential: $$\mu = E_0 + E_F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{E_F}\right) ^2 - \frac{\pi^4}{80} ...
2
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0answers
93 views

Phase diagram of SO(5) rotor model

It was originally a problem from Professor Eugene Demler's problem set. Consider an SO(5) rotor model: \begin{align}\mathcal{H}=\frac{1}{\chi} ...
14
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4answers
696 views

Why is it often assumed that particles are found in energy eigenstates?

Energy eigenstates provide a convenient basis for solving quantum mechanics problems, but they are by no means the only allowable states. Yet it seems to me that particles/systems are assumed to be in ...
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1answer
106 views

Does time stand still at a phase transition?

For second order phase transition thermodynamic properties can be described in very general terms by their critical exponents. So at every transition the correlation length $\xi$ should diverge as ...
6
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1answer
3k views

Derivation of differential scattering cross-section

I'm trying to follow the derivation of the Boltzmann equation in my Theory of Heat script, but have a little trouble understanding the following: The cross-section $d\sigma$ is defined as: The amount ...
2
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0answers
116 views

Derivation of impact free Boltzmann equation

When deriving the impact-free boltzmann equation ( $\frac{\partial f}{\partial t} + \vec{v} \cdot\frac{\partial f}{\partial \vec{x}} + \vec{a} \cdot \frac{\partial f}{\partial \vec{v}} = 0$) I have a ...
3
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1answer
83 views

Forward-scattering for a single impurity in an infinite system

I'm slightly confused with the following situation: Suppose you have an electron in a tight-binding model, and let's say we are in one dimension with $N$ lattice sites. Add to this a single ...
5
votes
1answer
330 views

The critical point of Bose-Hubbard model

The Hamiltonian of Bose-Hubbard model reads as $$H=-t\sum\limits_{<i,j>}b_i^{\dagger}b_j+h.c.+\frac{U}{2}\sum\limits_{i}n_i(n_i-1)-\mu\sum\limits_in_i$$. In the limit $t\ll U$, the ground ...
0
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1answer
154 views

Mathematics for Statistical Mechanics

I am studying Statistical Mechanics and Thermodynamics from a book that i am not sure who has written it, because of its cover is not present. There is a section that i can not understand: ...
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2answers
100 views

on Brownian motors

From this review on Brownian motors, there is such a statement without detailed explanation: (I think this statement is general enough so that one does not need to read the article) "Apart from ...
2
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2answers
128 views

A box with cooler and heater on opposite faces

Suppose there's a box with one face cold, and the opposite face hot. So when the air molecules hit the cooler face, it will transfer its momentum and energy to the wall, bouncing back with less ...
5
votes
2answers
284 views

Ising model observables

Is there a formula or equation relating $\langle E\rangle$ and $\langle M\rangle$ (average spin per site) and $\langle E^2\rangle$ to temperature $T$ for the square lattice Ising model at zero ...
3
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0answers
147 views

Impact of the noise distribution on Geometric Brownian motion

I have a problem which includes geometric Brownian motion, with either normally distributed or power-law-distributed noise, and I'm asking for some explanations and if possible references to read in ...
6
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2answers
2k views

Gibbs Paradox - why should the change in entropy be zero?

The Gibbs paradox deals with the fact that for an ideal gas with $N$ molecules in a volume $V$ seperated by a diaphragm into two subvolumes $V_1,V_2$ with $N_1,N_2$ particles in each subvolume, ...
9
votes
2answers
930 views

(Canonical) Partition function - what assumption is at work here?

The canonical partition function is defined as $$Z=\sum_{s}e^{-\beta E_s}$$ with the sum being over all states of the system. The way I saw this derived was by assuming that for each state, the ...
2
votes
1answer
85 views

Bose gas with $T = 0$ and $\mu < 0$

Is it possible to have a Bose gas with $T = 0$ and $\mu < 0$ ? I think that there is a problem, because all the states $k$ are such as $$\langle n_k \rangle = \dfrac{1}{e^{\beta \{\epsilon_k - ...
0
votes
1answer
270 views

what is the combined partition function of two similar but independent systems?

i was reading Runnels' paper on cayley tree where he has squared the partition function of a cyley tree to get that of two exactly similar trees. why square? why not add the two partition functions to ...
4
votes
4answers
1k views

Chemical potential of a Bose gas

In my course, there is this fact : In a Bose gas, the chemical potential $\mu$ must always be lower than the smaller level of energy $\epsilon_0$. I find this strange, because if we put a Bose ...
4
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0answers
116 views

Cauchy Problem for Boltzmann Equations

One of the first profound analysis about the solutions of the Boltzmann Equation was given by DiPerna and Lions in the late 1980s. You can find one of their main papers here: ...
8
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1answer
564 views

The Bhatnagar-Gross-Krook (BGK) approximation of the collision integral

Bhatnagar, Gross and Krook (BGK) proposed a relaxation term for the collision integral $ Q$ as follows $$J = \frac{1}{\tau} (f^{eq} - f)$$ where $f^{eq}$ is the distribution at equilibrium. $Q$ has ...
0
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0answers
615 views

Density of states of a photon gas in volume V and temperature T

I have a question on the density of states for a photon gas: Suppose I have a photon gas in a box of volume $V$ at temperature $T$. If I enumerate the total number of states accessible to the system ...
2
votes
2answers
662 views

1 dimensional Ising model

How to solve the Ising model in 1D by low temperature, and high temperature expansion, and by change of variable method? Can you please give me some reference links?
6
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1answer
1k views

Clear up confusion about the meaning of entropy

So I though, and was told, that entropy is the amount of disorder in a system. Specifically the example of heat flow and it flows to maximize entropy. To me this seemed odd. This seemed more ordered ...
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1answer
134 views

Usefulness of SUSY models when it cannot exist at any non-zero temperature

Unlike other symmetries (like electroweak symmetry), SUSY is spontaneously broken at any non-zero temperature due to some variation of the fact that the boundary conditions on bosons and fermions in ...
0
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1answer
121 views

Second law of thermodynamics

I think this is a simple question. If I have that $E(L)=\tau L$ and we are told that $\tau=BTL$ would this mean that $E=BTL^2$ implies $dE=(2BTL)dL$ or should I sub $\tau$ straight into the second law ...
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3answers
444 views

Fermi-Dirac Statistics

In Fermi-Dirac statistics the probability of being in a certain energy state is $$f(E) = \left[1 + \exp\left(\frac{E-E_F}{k T}\right)\right]^{-1}$$ In the area that I'm looking at the texts always ...