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

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Which function denotes the energy of thermal motion within a system?

In thermodynamics, the heat $Q$ is defined as a type of energy in transfer, and is not a state function, which function denotes the energy of thermal motion within a system? 1) $TS$, (there is a ...
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39 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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881 views

Motivation for the use of Tsallis entropy

Every now and again I hear something about Tsallis entropy, $$ S_q(\{p_i\}) = \frac{1}{q-1}\left( 1- \sum_i p_i^q \right), \tag{1} $$ and I decided to finally get around to investigating it. I haven't ...
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19 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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1answer
105 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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3answers
112 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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1answer
89 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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22 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
153 views

Thermodynamics, chaperones : How to model polymer fragmentation

Living polymers are well described by equilibrium statistical physics. Now I would like to consider a case were living polymers undergo fragmentation due to chaperones. I can think of a kinetic ...
2
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1answer
148 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} ...
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40 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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254 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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25 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
243 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 ...
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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 ...
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1answer
941 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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10 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 ...
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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
41 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
17 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
47 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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28 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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22 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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2answers
99 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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296 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 ...
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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, ...
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957 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 ...
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1answer
105 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 ...
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1answer
391 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 ...
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59 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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23 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 ...
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2answers
407 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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30 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
308 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 ...
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1answer
28 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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70 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
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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 ...
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2answers
421 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
80 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
58 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
34 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 ...
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1answer
34 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.) ...
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48 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})= ...
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1answer
42 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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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 ...
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1answer
27 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
35 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
49 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}=-{ ...