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

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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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11 views

Mean value of number of molecules of a gas inside a volume

I'm trying to solve this problem in Statistical Mechanics but I'm not sure if my reasoning is right. The problem is as follows: we have a certain region with volume $V_0$ containing $N_0$ molecules of ...
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1answer
33 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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41 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
17 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
23 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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37 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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15 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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24 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
163 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
46 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
26 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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30 views

Particles with spin and probability theory

I'm studying Statistical Mechanics but I'm not being able to understand some points on how probability theory ideas are being applied so I'm going to ask on the context of a particular problem that is ...
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1answer
48 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
28 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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71 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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31 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
31 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
34 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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11 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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7 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
39 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
65 views

Historical Survey of Statistical Mechanics

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
28 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
92 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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25 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 ...
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1answer
111 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
45 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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2answers
68 views

Does the second law of thermodynamics take into consideration 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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18 views

Interesting: Estimate range of Coulomb interactions. [closed]

I am new to Statistical Mechanics and I just came across this question. Can anyone help me with how to solve/approach it? All the atoms in a hot plasma are completely ionized. Despite the presence ...
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39 views

Interesting question in Statistical Mechanics involving Lennard-Jones potential [closed]

Atoms in a molecule interact according to the Lennard-Jones potential: $U(r) = A/r^9 − B/r^6 , (A, B > 0)$. The coefficient of linear expansion of the molecule is defined as $αl ≡ ...
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30 views

Identity and indistinguishability in quantum and statistical mechanics [closed]

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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32 views

Is there an analytic solution for this Fokker-Planck equation? [migrated]

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial ...
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2answers
61 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
28 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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16 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
39 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
57 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
55 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 ...
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1answer
56 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
99 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
44 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
38 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
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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
67 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
18 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
62 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?
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2answers
51 views

Violently shaking object

Would violently shaking something cause a temperature change? For example, if a container of water was shook violently enough; would it be possible to make it boil?
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46 views

Thermodynamics and Axioms and the like

Can thermodynamics and any important related information be expressed as a set of axioms with various 'rules of manipulation'?
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1answer
37 views

Entropy $S$ for canonical (NVT) and isobaric (NPT) ensemble

In case of non-isolates system (NVT or NPT ensemble), I learned I can calculate the entropy, $$S=-k_B\sum_jp_j\ln(p_j)$$ where $p_j$=probability at $j$ state. but I saw that the entropy is also ...