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

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67 views

Kubo formula for general observables

In the wiki page about Kubo formula, the expectation of some observable under weak time-dependent perturbation is derived. However, from my point of view, some crucial steps are missing. I did the ...
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2answers
65 views

Boltzmann distribution for angles?

Consider a system whose sole degree of freedom is an angle $\theta$ that goes from $0$ to $2\pi$. Let $E(\theta)$ be its energy function. Obviously, $E(\theta)$ is $2\pi$-periodic. What's the general ...
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124 views

Obtaining the canonical distribution from Fokker-Planck equation?

First I will provide a summary of the problem. Subsequently, I will provide more detail regarding the problem. Please note that entropy is in units of the Boltzmann constant. Summary I have a ...
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46 views

About the factorial N! in the partition function

After reading these posts: Why is the partition function divided by $(h^{3N} N!)$? , What is the resolution to Gibb's paradox?, and some of these: http://arxiv.org/abs/1012.4111 , ...
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1answer
37 views

Einstein model for thermal capacity of solids and indistinguishability of the oscillators

Albert Einstein's theory of thermal capacity of a solids makes the assumption that a crystal is made up from oscillators which of course oscillate, in all three directions. Thus, for N atoms of the ...
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120 views

Why is it difficult to mix helium and nitrogen gases?

I recently learned an interesting fact: That it's difficult to mix helium and nitrogen gases in a compressed gas cylinder. Gas suppliers that need to mix the two gases have to rotate the cylinders for ...
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1answer
34 views

Negative absolute pressure with positive absolute temperature

Could I ask if the derivative defining pressure $dU \over dV$ or ${∂S \over ∂V}|_{E,N} $can be negative in processes occuring in system not cosmological but statistical(gases or solids or liquids-I ...
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2answers
213 views

Can temperature be a complex number?

Is it possible for a temperature to be a complex number? I want to say "no" but I can't be so sure. If it is possible I would like to know of an example. I found an interesting article which treats ...
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2answers
81 views

How Statistical Physics?

It's a common fact that in physics, we use statistics (or maybe probabilities ) to describe the behaviour of a system. It was from the statistical analysis of a system where quantum statistics arose ...
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1answer
52 views

How can we be sure the Maxwell speed distribution equation is always a rational number?

The Maxwell speed distribution equation is given as $$f(v) = 4\pi \biggl(\frac{m}{2\pi kT}\biggr)^{3/2}\exp\biggl(-\frac{mv^2}{2kT}\biggr)v^2.$$ The left hand side gives the fraction of molecules ...
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1answer
60 views

Understanding chemical potential in AdS/CFT

I always find it very difficult to understand the notion of chemical potential physically/intuitively unlike pressure and temperature in statistical mechanics. Can some one suggest some nice ...
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0answers
47 views

Local and global detailed balance

I'm taking a course on nonequilibrium statistical mechanics and I encountered the terms local and global detailed balance. I'm a bit confused about what is their exact definition and what is the ...
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1answer
29 views

Bending moment and Shear force

Do bending moment and shear force of a beam depend on it's cross sectional dimentions?? Since all the diagrams which I have draw so far don't involve any cross section details. So I think they do not ...
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28 views

One question about critical exponents for first order transition free energy [closed]

I've got a problem to calculate critical exponents for theory given by Landau free action: $$ \tag 1 L = L_{0} - \frac{1}{2}(\nabla m)^{2} + atm^{2} + dm^{3} + bm^{4} - hm, $$ where $$ -\infty < ...
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2answers
64 views

Definition of an irreversible process

I'm a little bit confused as to why quasi-static process cannot lose energy to friction in order to be reversible. This is how I'm thinking: Suppose you have a container of gas with a piston, and on ...
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0answers
37 views

Non-trivial integral with the Bose-Einstein distribution and Cosine function

When I consider the Casimir interaction between an atom and a perfect conducting slab I find the following non-trivial integral: $$\int\limits_0^\infty {\frac{{\cos \left( mx \right)}}{{x + ...
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1answer
84 views

Is 'Boltzon' an accepted name for particles following Maxwell-Boltzmann (MB) statistics?

In my curriculum during one of my statistical mechanics visiting lecture classes, our teacher was referring comparatively macro particles following MB statistics as "Boltzon". But I have searched ...
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63 views

In the derivation of canonical distribution why does one linearize entropy (and not something else?)

I know that there are (at least) two ways to derive the canonical distribution. I am interested in the one where one considers the entropy of the reservoir (with which the system we are considering ...
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26 views

What is normal fluctuation?

I was reading Statistical Mechanics (second edition) by Kerson Huang. On page 146, after equation 7.14, there is a reference to normal fluctuation. What is it? Here is the relevant part from the ...
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30 views

Problem in deriving the second term in perturbation expansion the quantum ising model

So I'm trying to derive the perturbation expansion for one particle states in the quantum ising model (Sachdev 2011 QPTs which this is derived from ) $$ H_I= - J g \sum_i \sigma_i^x - J ...
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1answer
68 views

Boltzmann equation in cosmology

I have a question about the Boltzmann equation in cosmology. Im trying to understand how this can hold? Where does the logarithmic terms come from? It is explained quite well here ...
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1answer
142 views

Difference in partition function of classical and quantum Ideal gas

First, I have read this question:What is meant by the term "single particle state" There is an analysis going on in my book (Mandle F. Statistical Physics) that has brought me in a ...
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0answers
27 views

Work done by a gas in an expansion [duplicate]

1) Consider a gas expanding quasistatically and reversibly from $V_1$ to $V_2$ at constant temperature. I want to calculate the work done. So by convention work done by a system is a negative quantity ...
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1answer
42 views

Struggling with whether its $\pm p dV $

I am struggling to understand when calculating the work done by a gas whether it is postive or negative p. It my notes and in many other notes sometimes it is $-pdV$ and sometimes it is $pdV$. I ...
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1answer
27 views

How can a reversible adiabatic expansion not increase entropy?

In the second stage of the Carnot cycle, a gas is thermally insulated and allowed to expand and do work on the piston. I understand the reason people give is that because entropy is $\,dS = \,dQ/T$ ...
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1answer
55 views

Changing the zero-point energy

I have the following Hamiltonian $$\mathcal{H}(\{x_i,y_i \})=-l\sqrt{2}\sum_{i=1}^N \mathbf{f}_i \cdot \hat{\mathbf{b}}_i+E_0$$ For calculating things like the partition function it would be ...
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1answer
53 views

Why the heat capacity doesn't diverge in the Kosterlitz-Thouless (KT) phase transition?

The KT transition has a special properties that, during the phase transition the heat capacity stay finite (so the behaviour of the heat capacity cannot reflect any critical behaviours). However, the ...
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27 views

Find the fraction of atoms in specific quantum state in stellar atmosphere [closed]

Consider gas consisting of hydrogen atoms at temperature about $T \sim 5 \cdot 10^6 \text{ K} \approx 431 \text{ eV}$ and concentration $N \sim 10^{11} \text{ cm}^{-3}$. I need to find the fraction of ...
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4answers
176 views

Why is the partition function divided by $(h^{3N} N!)$?

When computing partition functions for classical systems with $N$ with a given Hamiltonian $H$ I've seen some places writing it as $$Z = \dfrac{1}{h^{3N} N!}\int e^{-\beta H(p,q)}dpdq$$ where the ...
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35 views

Thermal Equilibrium of two thin sheets

While reading Gibbs' Elementary Principles in Statistical Mechanics I came across this footnote: The most simple test of the equality of temperature of two bodies is that they remain in ...
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1answer
35 views

Calculating average quantities in kinetic theory

Consider a volume $V$ with $5$ particles each of mass $m$ at positions $\mathbf{q}_i=(x_i,y_i,z_i) \in V$ and with velocities $\mathbf{v}_i=(u_i,v_i,w_i)$. The speeds of the particles are between $0$ ...
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1answer
39 views

Books on Liouville Operator

I am looking for a good book doing classical mechanics and statistical mechanics in terms of the Liouville operator. I have not found a lot on this subject and even books like Mathematical Methods of ...
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1answer
87 views

Statistical Mechanics vs Statistics

Just how much of a representation of statistics do we get in a statistical mechanics curriculum. What are some of the useful facets of stat not in stat mech/quantum mech that physicists should really ...
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1answer
282 views

Mean field theory Weiss Approximation for the Isling Model of a Protein

A model for protein in 2D can be formed by adding bonds of fixed length $l\sqrt{2}$ on a square lattice along the diagonal, ie $\hat{\mathbf{b}}_i=\frac{1}{\sqrt{2}}(\pm \hat{\mathbf{x}}\pm ...
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1answer
89 views

Classifying regions of Van der Waal like gas

Given the equation of state $$p+a\left(\frac{N}{V}\right)=\frac{Nk_BT}{V-bN} \tag 1$$ Taking into account of the fact that a realistic model requires $p \geq 0, V \geq Nb, N>0$ classify the ...
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1answer
29 views

Derivation for the expectation value of particle density for a pairwise interactions between particles

my question is why is $<\hat{n}(\vec{r})>=n$ I have the Hamiltonian $H_N= \sum_{i}^{N} \frac{P_i^2}{2m}+U(\vec{R_1},\vec{R_2},..,\vec{R_N})$ where $U(\vec{R_1},\vec{R_2},..,\vec{R_N})= ...
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1answer
57 views

How do you go from a sum over frequencies to an integral?

I am trying to figure out how to go, with help of a density of states function $g(\omega)$, from a sum like this $$K=\sum \limits_{j=0}^N f(\omega_j)$$ to an integral over the frequencies for $N \to ...
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1answer
45 views

Meaning of solutions of an equation of state [closed]

Question Let $p \geq 0, V \geq Nb, N > 0$. Now we are given the equation of state $$p+a\left(\frac{N}{V}\right)=\frac{Nk_BT}{V-b} \tag 1$$ Classify the solutions of the equation of ...
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1answer
32 views

Problem with indistinguishability in partition function

Consider an ideal gas of classical particles of mass $m$ in uniform potential $\xi$ in 3d. The gas $N$ molecules, volume $V$ and is at temperature $T$. I believe that the Hamiltonian of this system is ...
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1answer
43 views

Deriving pressure from a given partition function

If the partition function for some system is given as $e^{\text{$\alpha $T}^3V}$; please note note that $\alpha$ is a constant. I have computed $$\left[\frac{\text{$\delta $Z}}{\text{$\delta ...
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1answer
26 views

bridging the connection from the Helmholtz free energy in classical thermo to stat mech

The Helmholtz-free energy from classical thermo is defined as $$\text{F=u-TS}$$ taking the differential and algebraic manipulation, we arrive at $$\text{dF=-pdv-sdT}$$ Observe that: ...
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1answer
50 views

Strange Vector Matrix Operation

I am reading "A Modern Course in Statistical Mechanics" by Linda E. Reichl. Where i encountered this notation: $$\Delta S = \bar g : \vec \alpha \vec \alpha$$ Here $\bar g$ is $$ g_{i,j}=-{ ...
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1answer
32 views

What is the definition of 'relative population' in context of partition function?

In statistical mechanics, what is the definition (or mathematical definition) when authors refer to relative population in the case of a classical particle system?
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1answer
47 views

is it necessarily true that the partition function $Z$ (with degeneracies) $ =1$?

The partition function with degnerate energies is $$\text{Z}=\sum _ig_ie^{{-E_i}/{k_BT}}.$$ Because the partition function Z is defined as the normalisation constant, does Z always = 1?
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1answer
106 views

The energy contribution of a frequency at finite temperature

This is from a paper I'm reading: Since each frequency contributes $\hbar \omega/2$ of energy (or at finite temperature, $\hbar \omega /2 \coth(\hbar\omega/2kT)$), we can find the energies for the ...
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1answer
107 views

How do we find the phase space density from the Hamiltonian?

How do we find the phase space density from the Hamiltonian? For example: Consider a classical gas made of N identical non-interacting particles in 1d. Each molecule is characterised by centre mass ...
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1answer
56 views

The grand partition function of non interacting hamiltonians

In the case of non interacting particles I know we can write the Hamiltonian as $$H(\mathbf{q}_1,\dots,\mathbf{p}_1,\dots)=\sum_{i=1}^N h(\mathbf{q}_i,\mathbf{p}_i)$$ but I am having trouble ...
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46 views

Finding the phase space density of $N$ harmonic oscillators

Consider a system of $N$ identical harmonic oscillators in 1d. The Hamiltonian will be given by $$\mathcal{H}_N=\sum_{i=1}^N \frac{p_i^2}{2m}+\frac{m\omega^2}{2}q_i^2$$ Now supposedly the Hamiltonian ...
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1answer
45 views

Are there any units involved in the partition function for a classical particle system?

Is the output of a partition function dimensionless or are there units involved? The question as it is: $$E_1\text{=0}K_B\text{,g=1}$$ $$E_2\text{=0}K_B\text{,g=3}$$ $$E_3\text{=0}K_B\text{,g=5}$$ ...
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1answer
79 views

What happens to the planck distribution if the temperature is set to zero?

BE Problem I am currently working on modelling the density of states and optical conductivity of graphene utilizing the GW algorithm. In calculating the exchange self energy of the system, the ...