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

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21 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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1answer
25 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
45 views

Cosmological Boltzmann equation [closed]

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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0answers
27 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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30 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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35 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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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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129 views

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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40 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 ...
3
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1answer
101 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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0answers
50 views

Why the non-analyticity of free energy function implies phase transition? And what's its connection with other 'higher level' free energies?

I have seen 'free energy' arising from several contexts in very different forms, and each contains different amount of information. For example free energy is defined as the logarithm of the ...
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1answer
43 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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62 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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1answer
26 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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2answers
42 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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0answers
52 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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23 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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43 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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2answers
181 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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2answers
100 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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1answer
35 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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2answers
65 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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1answer
37 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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116 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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1answer
50 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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1answer
41 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 ...
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1answer
45 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
15 views

why can noise induce multistability, particularly in (bio)chemical systems

There are several instances that people claim that a system is monostable in a deterministic model, but when they consider stochastic models, from either master equations or Fokker-Planck equations, ...
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0answers
11 views

When obtaining the thermodynamic entropy (e.g. by differentiating F) the average entropy is being found. In what sense is this an average?

If I have some expression of the entropy (or another thermodynamic quantity of a system (e.g. pressure) obtained from the Helmholtz free energy, F. Is this the mean (average) or the modal (most ...
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1answer
52 views

Showing existence of negative temperature for a quantum system

It may be shown that the partition function for a quantum system containing N distinguishable particles each of which has energy state $\epsilon_1$ and $\epsilon_2$ is given by ...
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1answer
38 views

Temperature and Renormalization Scale in QFT

A particle physicist told me that everything in Peskin & Schroder is at zero temperature, and once you consider finite-$T$ QFT, things become more complicated. Meanwhile, I sometimes see people ...
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2answers
103 views

Calculating quantum partition functions

...By quantizing we the get the following Hamiltonian operator $$\hat{H}=\sum_{\mathbf{k}}\hbar \omega(\mathbf{k})\left(\hat{n}(\mathbf{k})+\frac{1}{2} \right)$$ where ...
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1answer
31 views

How do i mathematically represent reflection in a (diffusion) Problem?

I am trying to formulate boundary conditions and it occurred to me that I never had to implement a reflective boundary before. The example is a one dimensional diffusion, where at $x=0$ the ...
2
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1answer
133 views

How does Metropolis algorithm work in the Ising model?

I was reading the proof of Metropolis algorithm. The transition probability of going from a state $i$ to a state $j$ is $\pi_{ij}$. If I understand correctly, this is the product $\pi_{i ...
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1answer
50 views

Boltzmann Distribution - why maximum number of microstates?

I've recently started to learn statistical mechanics and I've run into Boltzmann Distribution. I wanted to see how it is derived and found some articles on web, but no one of them explain why the idea ...
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3answers
97 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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2answers
67 views

Calculating temperature from molecular dynamics simulation

My understanding is that temperature is an inherently macroscopic quantity, but I've seen a number of people talk about calculating the temperature of ideal-gas simulations like this one. To take one ...
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1answer
35 views

Bose-Einstein Grand Canonical partition function derivation step

The total grand canonical partition function is $$\mathcal{Z} = \sum_{all\ states}{e^{-\beta(E-N\mu)}} = \sum_{N=0}^\infty\sum_{\{E\}}{e^{-\beta(E-N\mu)}}$$ For Bose-Einstein or Fermi-Dirac, the ...
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0answers
18 views

Singularities across the critical isotherm in Landau's phenomenological theory of phase transition

Why don't we encounter any singularities when crossing the critical isotherm when $h \neq 0$ or $m\neq0$, where $h$ is the ordering field and $m$ is the order parameter.
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1answer
43 views

Simplifying a Vector Integral

This question has (long) remained unanswered on MSE. While reading the book - Theory and Applications of Boltzmann Transport Equation by Cercignani, I found this integral which I am unable to ...
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1answer
70 views

Meaning of the symmetrisation postulate in absence of a proper model

My question is on the use of the concept of indistinguishable particles (in quantum mechanics) in a very general context and in particular in statistical mechanics. I have made clear some of my ...
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1answer
73 views

Can one apply the Hubbard-Stratonovich transformation to the exponential of the Laplacian?

Is there a generalization of the Hubbard-Stratonovich transformation that transforms the exponential of the Laplacian into a Gaussian integral? Or can anyone suggest me how I can find the ...
3
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1answer
71 views

Is Kinetic Theory part of Statistical Mechanics?

Some years ago from now I've seem some basic details about what was then called "kinetic theory of gases" where the study of property of gases was made by statistical considerations about the momentum ...
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2answers
121 views

Susceptibilities and response functions

It is often confusing whether a susceptibility is the same as a response function, specially that often they are used interchangeably, in the context of statistical mechanics and thermodynamics. Very ...
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1answer
52 views

Statistical Mechanics deals with the same systems that Thermodynamics does?

Thermodynamics deals with "equilibrium states of macroscopic matter", that is, considering macroscopic systems there are states which can be characterized fully by a few number of measured degrees of ...
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1answer
47 views

Integration over angles (volume element change)

I'm trying to change from one volume element to another, as suggested in a problem 13.2 of Reif's Statistical and Thermal Physics. My volume element is currently: $d^3$$\nu$ And I'd like to change ...
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4answers
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
81 views

Drifting Maxwellian distribution for energy

Assume I have a drifting Maxwellian distribution with velocity $\vec{a}$, say, in the x-direction, so something like $$ f(\vec{v}) = n\left(\frac{m}{2\pi ...
2
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0answers
27 views

Degrees of freedom in a diatomic gas in 2-dimensions

Question: What is the specific heat capacity at constant volume of a two-dimensional diatomic ideal gas of N particles at room temperature? My answer: A diatomic gas can move in both directions, can ...
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2answers
64 views

$N>2$ gravitating masses can never reach equilibrium [closed]

If you have $N>2$ point masses, each attracted to each other by the force of gravity, how could you go about showing that they will never reach equilibrium?