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

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37 views

Why is there zero point energy at absolute zero temperature?

If we define the absolute zero as a temperature which there is no entropy in it so why should we have zero point energy?
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1answer
161 views

Dispensing with the “a priori equal probability” postulate

I find the "a priori equal probability postulate" in statistical mechanics terribly frustrating. I look at two different states of a system, and they look very different. The particles are moving in ...
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1answer
83 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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1answer
35 views

Cosmological Boltzmann equation

Consider the Boltzmann equation: $$\frac{d \ln{n^c(T)}}{d \ln{T}} = \frac{\Gamma}{H}(1 - \frac{n^c_{eq}(T)}{n^c(T)})$$ We know that the ratio $\Gamma/H$ can be considered constant, let us put it ...
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1answer
105 views

Spin drift velocity?

I am currently reading this Phys Rev paper by H C Torrey. In this paper, he derives the Bloch equations with an additional diffusion term. He says that the current density is given by $$\mathbf ...
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1answer
130 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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2answers
53 views

Probablistic interpretation of entropy

After taking a statistical mechanics course, I'm somewhat surprised that my intuitive highschool understanding of entropy doesn't match my current understanding. When I was introduced to entropy, I ...
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1answer
20 views

Monoatomic fluids and free space around atoms

In monoatomic fluids the atoms can move quite freely around each other. Is there any thermodynamic/statistical mechanic equation how much free space there is between the atoms? This has to be ...
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1answer
91 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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18 views

What is the justification for the minimum image convention in periodic boundary condition?

As the distance between first particle-second particle and first particle-image of the second particle are not same. How is it justified to use the distance from the nearest image to compute ...
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22 views

How to derive entropy transport equation from heat equation?

Suppose I have heat equation: $$ \rho (\partial_{t} + (u \cdot \nabla)) T = -\nabla \cdot \mathbf R, $$ where $\mathbf R$ - some vector and $T$ - temperature. How to get the equation for entropy $S$ ...
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106 views

Could Navier-Stokes equation be derived directly from Boltzmann equation?

I know how to derive Navier-Stokes equations from Boltzmann equation in case where bulk and viscosity coefficients are set to zero. I need only multiply it on momentum and to integrate it over ...
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62 views
+150

Thermalization of coupled classical oscillators

I would like to understand if it is possible to perform an experiment, where a bunch of classical harmonic oscillators (e.g., LC circuits or mechanical pendula) coupled in a simple manner (e.g., one ...
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5answers
4k views

Why don't things get destroyed by gas molecules flying around?

Gas molecules go at an insane velocity, and though they are miniscule, yet there is a LOT of them. Of course, because of all these molecules hurtling around, there is air pressure; yet if you envision ...
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1answer
33 views

Derive the Sackur-Tetrode equation

How do you derive the Sackur-Tetrode equation? I know that you must start off with the multiplicity of a mono-atomic ideal gas: ...
5
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2answers
229 views

Why energy at room temperature $= kT$ and not $(3/2)kT$ [duplicate]

I always see that a room temperature of $T=300\,\text{K}$ corresponds to an energy of $k_BT \approx \frac{1}{40}\,\text{eV}$. But shouldn't it be $\frac{3}{2}k_BT$ since the molecules in the air have ...
3
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2answers
205 views

$E=kT$ or $\frac32kT$?

Basically, which is the correct formula for thermal energy, and is this the same as kinetic energy? My notes are pretty conflicting on this topic, and I'm getting pretty confused.
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124 views

What is the proper time used in relativistic non-equilibrium statistical physics?

In the literature one often finds covariant relativistic generalizations of classical non equilibrium statistical equations (Boltzmann, Vlasov, Landau, Fokker-Planck, etc...) but I wonder what is the ...
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1answer
76 views

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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28 views

Number of particles reaching a spherical cap [on hold]

If I have a spherical cap positioned exactly so the centre of the sphere is at a hole of a container containing an ideal gas that satisfies the Maxwell-Boltzmann distribution, what number of particles ...
2
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1answer
43 views

Why does the superconductivity hamiltonian have a µ term, while the superfluid does not?

In every discussion of SC and SF that I read (e.g. Simons), the SC Hamiltonian (BCS) has a $\epsilon_k - \mu$ in the kinetic part of the Hamiltonian, while the SF Hamiltonian has just a $\epsilon_k + ...
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2answers
61 views

Why $\epsilon > \mu$ for Bose-Einstein distribution (but not for Fermi-Dirac)?

For fermions $$\bar{n}_{FD}=\frac{1}{e^{(\epsilon -\mu)/kT}+1}$$ and $\epsilon$ can be bigger or small than $\mu$. However, for bosons: $$\bar{n}_{BE}=\frac{1}{e^{(\epsilon -\mu)/kT}-1}$$ which ...
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19 views

Internal energy of ideal gas in the grand canonical ensemble

I am reading through Pathria/Beale StatMech and I have a problem to understand the calculation of the internal energy of an ideal gas in the grand canonical ensemble, i.e. the derivation of the ...
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1answer
78 views

What is the cause for mechanical equilibrium in statistical mechanics?

In classical thermodynamics, mechanical equilibrium is defined as the state of a system in which there is no net flow of volume as there should be no net pressure within the system. Ok. ...
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29 views

Is the Landau Free Energy U-TS or βH?

I'm having a hard time figuring out the physical meaning of the Landau Free Energy density: $$f(\phi,\nabla\phi,T) = \frac{1}{2}|\nabla\phi |^2 + \frac{a(T-T_c)}{2}|\phi |^2 + \frac{b}{4}|\phi |^4$$ ...
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3answers
93 views

Does the second law of thermodynamics take into consideration of attractive interactions between particles?

If one searches Google or textbooks on 2nd Law of Thermodnamics, one usually finds a statement that is either equivalent or implies the following. The entropy of the universe always increases. But ...
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30 views

Canonical or microcanonical ensemble?

What of this ensembles is more honest with natural thermal equilibrium? In microcanonical ensemble the sample is isolated, and we don't now the precise value of energy. By this considerations we have ...
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1answer
64 views

Resources for introductory quantum statistical mechanics

I am currently struggling to understand my basic introductory course on quantum statistical mechanics and I have done a basic course on single particle quantum mechanics. I was wondering whether ...
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2answers
34 views

Counting classical microstates

In my notes it states that the convention for summing over the classical states is $$\sum_{\Gamma} \longrightarrow \frac{1}{N!}\int \prod_{i=1}^N \frac{d^3q_id^3p_i}{h_0^3} \tag1$$ Now I know that ...
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1answer
40 views

assuming $kT=1$ in $Z=\sum e^{-H}$ and $F=-lnZ$?

Some statistical physics book use: $Z=\sum e^{-H}$ and $F=-lnZ$ as defination for partition function and free energy. I think they should be $Z=\sum e^{-\frac{H}{kT}}$ and $F=-kT lnZ$ Are they ...
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0answers
31 views

Why 'free energy' can contain different amount of information in different settings, and what's their connection to phase transition?

I have seen 'free energy' arising from several contexts in very different forms, and each contains different amount of information (as a number, 1D function, 2D surface, etc). For example free ...
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52 views

How is partition function related to ordinary generating function?

Ordinary generating function can be used to solve combinatorial enumeration problems. Now if the energy levels are discrete, say $g_i$, and if one want to count how many ways one can add up $g_i$ ...
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1k views

The unreasonable effectiveness of the partition function

In a first course on statistical mechanics the partition function is normally introduced as the normalisation for the probability of a particle being in a particular energy level. ...
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1answer
21 views

How to understand Density of States with dispersion relation

I am having trouble understanding the Density of states concept. As I currently understand it, for the density of states $g(k)$ it is the number of microstates with wave number in the range ...
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0answers
49 views

Proving the Virial theorem

Consider the expectation in the canonical ensemble defined by $$\left\langle x_i\frac{\partial \mathcal{H}}{\partial x_j} \right\rangle=\frac{1}{Z}\int d\Gamma x_i\frac{\partial ...
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19 views

Formula for computing macrostates

I'm trying to figure out how to arrange 3 particles across 5 energy level from 0E to 4E and obtained 5 macrostates (this could be wrong). While it is possible to do so for small number of n particles, ...
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2answers
52 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
170 views

Number of microstates compatible with two boxes

From my notes I have: From one point of view there are many more microstates compatible with the LHS than the RHS, in fact the relation between the number of microstates is ...
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0answers
30 views

Construction of free energy based on Landau theory

Consider an Ising model system where the total energy is $E = −J \sum_{<ij>} S_iS_j $, $S_i = \pm 1$ and $< ij >$ implies sum over nearest neighbours. For $J < 0$ the ground state of ...
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1answer
227 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
29 views

Counting the number of microstates that there are for a given configuration. How to prove this result?

I'm doing some statistical physics and I came across a result which I'm not sure how to derive. Any help? The answer turns out to be: Can anyone help with this derivation? Thank you :D
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1answer
31 views

“Definition” of internal energy

Conversation of energy implies that if we have a thermally insulated system which goes from state 1 to state 2: $$\Delta E_{12}=E(2)-E(1)=\Delta W_{12}$$ and the 1st law of thermodynamics ...
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1answer
38 views

System of two harmonic oscillators and its quantum partition function

Consider a system of two harmonic oscillators with different frequencies $\omega_1,\omega_2$ and masses $m_1,m_2$ so the hamiltonian is $$\mathcal{H}(p_1,q_1;p_2,q_2)=\sum_{i=1}^2 ...
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0answers
68 views

Historical Survey of Statistical Mechanics [migrated]

Statistical mechanics is a subject with a particularly rich history. I think of the early debates of Boltzmann and Loschmidt, the rather confusing differences between the approaches of Gibbs and ...
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1answer
34 views

Sufficient conditions for Equipartition Theorem to hold

I was wondering what are the sufficient conditions for the Equipartition Theorem. I know there is another question (For which systems is the equipartition theorem valid?) that somewhats answers this ...
2
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1answer
503 views

Pauli paramagnetism for electrons with external magnetic field

Apparently it is to be shown that for electrons under an external magnetic field, in the limit as $B\to 0 $ $$ \chi = \frac{dM}{dB} \approx \frac{n\,\mu^{*^2}}{k\,T}\,\frac{f_{1/2}(z)}{f_{3/2}(z)} $$ ...
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1answer
147 views

How to derive the critical temperature for Bose-Einstein condensation of photon?

I found in Nature magazine that photon can have Bose-Einstein condensation. But I have a question how to derive the critical temperature for photon? Because the chemical potential of photon is zero ...
2
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1answer
142 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 ...
3
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1answer
41 views

When would the Gross-Pitaevskii equation break down as $a\rightarrow \infty$?

It is now common to use Feshbach resonance to tune the s-wave scattering length of a Bose-Einstein condensate. Apparently as $a\rightarrow \infty$, the GPE would break down. The reason is that it ...
3
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1answer
87 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 ...