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

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1answer
239 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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0answers
18 views

Work done by a gas in an expansion

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 ...
1
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1answer
242 views

Virial theorem and the energy in a gas

I clearly am interpreting the Virial Theorem incorrectly, but I don't know how. In dipole gases, the molecules can exhibit five kinetic modes, while they can only experience 2 potential modes. Doesn't ...
2
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1answer
98 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 ...
2
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1answer
935 views

Physics-based derivation of the formula for entropy

I am looking for a derivation of the formula $$S~=~-\Sigma_ip_i \log (p_i).$$ for entropy, from first principles. I only wish to assume the laws of physics, and without involving concepts in ...
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0answers
9 views

Expression for the heat capacity of an ideal Bose gas

I've been desperately looking to solve this problem through various mathematical tricks but it seems that there's something that I'm missing here. Consider an ideal Bose gas. If one notes the ...
2
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0answers
66 views

Why does decay of correlations imply absence of order?

In a few articles I have read, a two-point correlation function $\langle g(x)g(y) \rangle$ is shown to decay with increasing distance of $x$ and $y$, and this is then taken to imply an absence of the ...
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1answer
37 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
14 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 $\,dS = \,dQ/T$ ...
1
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1answer
44 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 ...
0
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0answers
26 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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0answers
19 views

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

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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3answers
102 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 ...
0
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2answers
91 views

How does temperature relate to the kinetic energy of molecules?

In ideal gas model, temperature is the measure of average kinetic energy of the gas molecules. If by some means the gas particles are accelerated to a very high speed in one direction, KE certainly ...
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2answers
294 views

What is the resolution to Gibb's paradox?

This question is essentially a duplicate of Gibbs Paradox - why should the change in entropy be zero?. The question concerns the following situation: I have some gas of identical particles and they ...
6
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2answers
1k 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, ...
11
votes
3answers
952 views

Is there a way to obtain the classical partition function from the quantum partition function in the limit $h \rightarrow 0$?

One would like to motivate the classical partition function in the following way: in the limit that the spacing between the energies (generally on the order of $h$) becomes small relative to the ...
3
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1answer
102 views

Phase Transition at Zero Temperature (Not QPT)

As is well known the Ising model exhibits a phase transition, except the one dimensional case in which the phase transition occurs strictly at $T=0$. Now I have always thought that this makes the case ...
1
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1answer
387 views

phase-space volumes or cells for N particle system

For N non interacting spinless particles in a volume, we have 3N degrees of freedom and we can divide the phase space into 6N dimensional cells of volume h raised to power 3N. And each cell ...
1
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1answer
57 views

Independent boson model with an arbitrary finite-dimensional impurity

The independent boson model consists of the following Hamiltonian: $$ H_s = E \sigma^z $$ $$ H_b = \sum_k \omega_k b^{\dagger}_kb_k $$ $$H_{sb} = \sigma^z \sum_k (g_k b_k + g_k^{\ast}b^{\dagger}_k).$$ ...
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0answers
21 views

Magnetic moment of a paramagnetic crystal

I've been having some trouble with a paramagnetism problem from my Statistical Mechanics class textbook (F. Mandl, Statistical Physics, 2nd edition, p. 25). The problem is as follows Since the ...
1
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2answers
404 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 ...
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0answers
20 views

The physical applications of anomalous diffusion? [closed]

From Einstein's Great work, Avogadro number was determined from Brownian motion. Diffusion coefficient obtained from MSD can be useful. I wonder what is so great about Brownian motion, is there any ...
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0answers
27 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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3answers
307 views

Distinguishable, Indistinguishable Paramagnetic Ideal Gas

In the canonical ensemble, the partition function for an ideal gas is given by: $$\frac{Z}{N!}$$ The factor $N!$ accounts for the indistinguishability of the particles of the ideal gas. What ...
0
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1answer
25 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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3answers
68 views

Entropy change in an irreversible process between 2 equilibrium state

Calculating entropy change in an irreversible process between 2 states requires computing the change in entropy for any reversible process between the 2 same states, but why? If someone could provide ...
2
votes
1answer
139 views

Reaction coordinate as a function of atomic positions

I'm going over some (molecular dynamics) related literature - specifically the derivation of the Weighted Histogram Analysis Method (WHAM). As a quick backdrop WHAM is a method for stitching ...
5
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2answers
420 views

Renormalization Group and Ising with d=1 and D=1

I have a question about the results of RG on Ising model. I know it's possible to obtain two couple of relations $K'(K)$, $q(K')$ $K(K')$, $q(K)$ between the coupling costants. My problem arise ...
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1answer
77 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 ...
2
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1answer
51 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 ...
2
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1answer
33 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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5answers
4k views

Chemical potential

This is something probably very basic but I was led back to this issue while listening to a recent seminar by Allan Adams on holographic superconductors. He seemed very worried to have a theory at ...
0
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1answer
33 views

Does spin degeneracy affect ideal Fermi gases in any way as T->Infinity?

In other words, given any system comprised of an ideal Fermi gas, in the high-temperature (classical) limit, are there any observable thermodynamic quantities (pressure, volume, energy, density, etc.) ...
0
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1answer
47 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
24 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})= ...
0
votes
1answer
39 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 ...
3
votes
2answers
115 views

What materials are used in non thermal plasma?

While reading about non-thermal plasmas, I came across their ionization potentials(~1%), and other capabilities, such as their non Maxwellian energy distributions. At what temperatures, and pressures ...
0
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1answer
26 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 ...
1
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1answer
32 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 ...
1
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1answer
47 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
23 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
21 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?
0
votes
1answer
41 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
46 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 ...
4
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1answer
74 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 ...
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1answer
91 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 ...
3
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2answers
114 views

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
48 views

How can I compute the average number of collisions of a particle in a spherical container? [closed]

I have to compute the average number of collisions per unit time of a particle in a spherical container. These collisions are diffuse, i.e., after colliding, the particle bounces off the inner walls ...
1
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1answer
46 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 ...